XIV IMGTA

Ischia 11 - 14 July 2001

PLENARY LECTURES SPEAKERS and SUMMARIES

PLENARY LECTURES

Jean-Pierre AUBIN, Tamer BASAR, Carmen HERRERO, Ehud KALAI, George LEITMANN,

Guillermo OWEN, Ariel RUBINSTEIN, Sylvain SORIN, Stef TIJS

DYNAMIC CORE OF FUZZY DYNAMICAL COOPERATIVE GAMES

Jean-Pierre Aubin

Centre de Recherche Viabilité, Jeux, Contrôle, Universite Paris-Dauphine

F-75775 Paris cx (16),France, e-mail: aubin@viab.ufrmd.dauphine.fr

We use in this talk the viability/capturability approach for studying the problem of characterizing the dynamic core of a dynamic cooperative game defined in a characteristic function form. In order to allow coalitions to evolve, we embed them in the set of fuzzy coalitions. Hence, we define the dynamic core as a set-valued map associating with each fuzzy coalition and each time the set of allotments such that their payoffs at that time to the fuzzy coalition are larger than or equal to the one assigned by the characteristic function of the game. We shall characterize this core through the (generalized) derivatives of a valuation function associated with the game. We shall provide its explicit formula, characterize its epigraph as a viable-capture basin of the epigraph of the characteristic function of the fuzzy dynamical cooperative game, use the tangential properties of such basins for proving that the valuation function is a solution to a Hamilton-Jacobi-Isaacs partial differential equation and use this function and its derivatives for characterizing the dynamic core.

NONCOOPERATIVE NETWORK GAMES WITH A LARGE NUMBER OF PLAYERS: ASYMPTOTIC BEHAVIOR OF NASH EQUILIBRIA

Tamer Basar

Coordinated Science Laboratory and Department of Electrical and Computer Engineering

University of Illinois, 1308 West Main Street, Urbana, IL 61801-2307, USA

ph.: (+1) 217 333-3607, fax: (+1) 217 244-1653, e-mail: tbasar@decision.csl.uiuc.edu

A current important application area for noncooperative game theory is traffic networks, particularly those that arise in communication systems, where a large number of users (players) seek to ship traffic (such as packets) from specific sources to targeted destinations while experiencing minimum loss and minimum delay. The decisions that each user is faced with, which are generally guided by an appropriate performance index, are (i) at what (flow) rate to send her traffic from a particular source to a particular destination, and (ii) how to distribute this rate over the available links joining the particular source-destination pair. An appropriate paradigm for such problems is that of a noncooperative game with a large number of players. Some important questions that arise in this context are the existence, possible uniqueness, and characterization of Nash equilibria, and their asymptotic behavior as the number of players becomes arbitrarily large.

This talk will formulate and discuss such network games, both for the case when the flows are fixed and the decision faced by each player is one of routing, and the case when the decision involves combined routing and flow control. For the former, it is possible to establish existence and uniqueness of Nash equilibrium for general topology networks, which we will discuss briefly. For the latter, we will focus on networks having the topology of parallel links, and take the utility function of each user to be the ratio of some positive power of the total throughput of that user to the average delay seen by the user. This utility function is nonconcave in the flow rates of the user, for which I will show that there exists an appropriate scaling that makes the asymptotic limit (as the number of users, N, becomes arbitrarily large) well defined . In spite of the lack of concavity, it is possible to obtain explicit expressions for the flow rates of the users and their associated routing decisions, which are in O(1/N) Nash equilibrium. This O(1/N) equilibrium solution, which is symmetric across different users and could be multiple in some cases, exhibits an appealing delay-equalizing feature among the links which carry positive flow. The talk will conclude with presentation of several numerical examples that illustrate different features of the asymptotic Nash equilibria, and with a discussion of some open problems in this area.

Plenary talk at the XIVth Italian Meeting on Game Theory and Applications, Ischia, Italy, July 11-14, 2001. Partly based on joint work with Eitan Altman (INRIA, Sophia-Antipolis, France) and R. Srikant (University of Illinois at Urbana-Champaign, Illinois, USA).

THE THREE MUSKEETERS: FOUR CLASSICAL SOLUTIONS TO BANKRUPTCY PROBLEMS

Carmen Herrero¹ and Antonio Villar²

1 Departamento de Fundamentos del Analisis Economico

Universidad de Alicante, 03071 Alicante, Spain

ph: 34965903618, fax: 34965903685, e-mail: carmen.herrero@ua.es

This paper provides a comparative analysis of some classical solutions to bankruptcy problems from an axiomatic viewpoint. These rules are the constrained equal-awards rule, the constrained equal-losses rule, the proportional rule and the Talmud rule. The purpose of this study is to facilitate the understanding of their differences and to clarify the type of situations in which each of these rules is better. Noncooperative procedures to support some of those rules are also introduced.

INFORMATION-PROOF EQUILIBRIA IN LARGE GAMES

Ehud Kalai

Kellogg School of Management, Northwestern University, Evanston, IL 60208 USA

e-mail: kalai@kellogg.northwestern.edu

An equilibrium of a one-shot simultaneous-move Bayesian-game is information proof, if it is immune to changes in the order of play, information leakage and the possibility of revisions. Such equilibrium is also immune to changes in the prior probabilities by which types are drawn.

All the equilibria of semi-anonymous Bayesian games, that satisfy continuity and independence conditions, become information proof at a uniform exponential rate as the number of players becomes large.

Information-proof equilibrium, like rational-expectations equilibrium, may be used to model ongoing rational interaction, as opposed to just instantaneous one-shot play. In one shot normal form games they may be used as a purification device.

A DIRECT METHOD OF OPTIMIZATION AND ITS APPLICATION TO A CLASS OF DIFFERENTIAL GAMES

George Leitmann

University of California, Berkeley, California, USA

e-mail: gleit@uclink4.berkeley.edu

Many problems in economics and engineering can be posed as dynamic optimization problems involving the extremization of an integral over a given class of functions subject to prescribed end conditions. These problems are usually solved via the Calculus of Variations or the Maximum Principle of optimal control theory, applying necessary conditions to obtain candidate optimal solutions and then assuring optimality via sufficient conditions, if available. These methods are variational in that they employ comparison of solutions in a neighborhood of an optimal one.

A different approach, first proposed in Leitmann, Jou. of Nonlinear Mechanics 1967 , and more recently expanded in Leitmann, Jou. of Optim. Th. and Appl. vol.108, 2001 and Jou. of Computational Technologies (in print), permits the derivation of global extrema for some classes of dynamic optimization problems directly, that is, without the use of comparison methods. It uses, in their place, coordinate transformations and the imposition of a functional identity. Problems with differential constraints as well as infinite horizon problems are treated in in Leitmann, Jou. of Optim. Th. and Appl. (in print). In Dockner and Leitmann, Jou. of Optim. Th. and Appl. vol.110, 2001, this direct optimization approach is applied to a class of open-loop differential games.

Here we present the salient features of the direct optimization approach and illustrate its use via some examples.

GAME THEORY AND CENTRALITY IN SOCIAL NETWORKS

Gulliermo Owen

NPS Code MA/NO 93953 Monterrey, USA, e-mail: gowen@nps.navy.mil

The sociological concept of centrality is frequently used in the study of social networks, but has not been given a precise definition. In this article, we show that Myerson's modification of the Shapley value, when applied to a symmetric superadditive (and preferably convex) game, can be used to define a measure of centrality which satisfies several reasonable desiderata. We also discuss weaknesses in the existing measures. In particular, we consider three possible symmetric games and study the three measures of centrality derived from these. Methods of computation, especially when the underlying graph is a tree, are discussed. Several simple networks are treated as examples.

COMMENTS ON RATIONALIZING CHOICE FUNCTIONS WHICH VIOLATE RATIONALITY

Gil Kalai¹, Ariel Rubinstein² and Ran Spiegler³

¹ Department of Mathematics, The Hebrew University, Jeruslaem

kalai@math.huji.ac.il http://www.math.huji.ac.il/ kalai

² Coffee Places, Tel Aviv and Department of Economics, Princeton University

rariel@post.tau.ac.il http://www.princeton.edu/ ariel

³ Nuffield College, Oxford and The Institute for Advanced Study, Princeton

rani@ias.edu http://www.nuff.ox.ac.uk/economics/spiegler.html

The paper presents a notion of rationalizing choice functions that violate the independence of irrelevant alternatives axiom. We argue that a choice function is rationalized by a partition of the collection of all choice sets if the choices from all sets in each cell in the partition are consistent with maximization with respect to a preference ordering assigned to the cell. We study one criterion for evaluating the rationality of a choice function: the minimal number of cells that is required for a partition rationalization.

ZERO SUM TWO-PERSON REPEATED GAMES WITH PUBLIC UNCERTAIN DURATION PROCESS

Abraham Neyman¹ and Sylvain Sorin²

¹ Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram

Jerusalem 91904, Israel, and Department of Economics, State University of

New York at Stony Brook, Stony Brook, New York 11794, USA

e-mail:aneyman@math.huji.ac.il

² Laboratoire d'Econométrie, Ecole Polytechnique,1 rue Descartes

75005 Paris, France

e-mail: sorin@poly.polytechnique.fr

We consider repeated two-person zero-sum games in which the number of repetitions q is unknown. The information about the duration is identical to both players and can change during the play of the game. We establish a recursive formula for the value in this class of games. We study asymptotic properties of the normalized value as the expected duration E (q) goes to infinity. This analysis applies in particular to stochastic games and repeated games of incomplete information.

ON ADDITIVITY AND MONOTONICITY IN COOPERATIVE GAME THEORY

S. H. Tijs

CentER and Department of Econometrics and Operations Research

Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands

e-mail: S.H.Tijs@kub.nl

In the seminal work of Lloyd Shapley of 1953 additivity appears as one of the characterizing properties of his introduced value. Young (1984) characterized the Shapley value by replacing the additivity property by strong monotonicity. What about the additivity of other solution concepts? In the first part of the lecture I will describe different cones where the nucleolus is additive and where the t-value is additive. Also cones are given where the core is an additive correspondence. Other interesting additive solutions appear in information collecting situations where the information collectors pay to the informed agents a fraction of their marginal contribution. In the second part of the lecture I deal with properties of solutions where the player set varies such as: population monotonicity, bi-monotonicity, type monotonicity and leave-monotonicity. We discuss the existence of such solutions for different classes of games which arose from economic sitations as holding situations, information sharing situations, etc.

SPEAKERS and SUMMARIES

Abdou, Agliari, Alkan, Anwar, Asheim, Ball, Battigalli, Battinelli, Belo_Carmona, Benthem, Bilbao, Bischi, Board, Bonanno, Branzei, Breton, Brink, Calleja, Calvo, Champarnaud and Cartigny, Castellanos, Cavaliere, Cavazzuti, Celentani, Cleuren, Das, De_Francesco, De_Sinipoli, Deloche and Oguer, Di_Liddo, Domansky, Dragan, Driessen, Ekes, Espinoza, Fazzino, Filipovich, Flam, Fragnelli, Frankowska, Friedenberg, Frigau, Gambarelli, Ganuza, Garcia-Jurado, Gavious, Hamiache, Hendrickx, Hofbauer, Holler, Holubiec, Horner, Inarra, Izquierdo, Josephson, Keiding, Khmelnitskaya, King, Klaus, Klijn, Kopel, Kreps, Kultti, Laffond, Lambertini, Laruelle, Leech, Lehrer, Luporini, Manzini, Mariotti, Martin-Herran, Martìnez de Albeniz, Matsuhisa, Meca Martinez, Megiddo, Meinhardt, Molina, Montero, Naumova, Norde, Novikova, Nunez-Oliva, Nurmi, Okugushi, Olcina, Osterdal, Parrondo, Patrone, Perez_de_la_Cruiz, Possajennikov, Pusillo, Puu, Ramsza, Ricon-Zapatero, Ritzberger, Rosenmuller, Rusinowska, Ryan, Saint-Pierre, Salonen, Scheffran, Schipper, Schmidtchen, Schummer, Simon, Skorin-Kapov, Slikker, Sovik, Tan-no, Taub, Timmer, Toledo Casado, Toral, Torre, Ueng, Urbano, Uriarte, Valenciano, Vannucci, Vardy, Widgren, Xifré-Oliva, Zaccour.

ON NECESSARY AND SUFFICIENT CONDITIONS FOR STRONG SOLVABILITY OF GAME FORMS

Joseph Abdou¹ and Hans Keiding²

¹ CERMSEM, Maison des Sciences Economiques, Universite Paris 1

106-112 boulevard de l'Hopital, 75647 Paris Cedex 13, France

ph.:33-144078303, fax:33-144078301, e-mail: abdou@univ-paris1.fr

²Institute of Economics, University of Copenhagen

Studiestraede 6, K-1455 Copenhagen K., e-mail: Hans.Keiding@pop.oko.ku.dk

A game form is strongly solvable if for each assigment of individual preferences over outcomes, the resulting game possesses a strong Nash equilibrium. Several necessary conditions for strong solvability can be found in the literature; in this paper we give conditions which are both necesssary and sufficient. These conditions use an extension of the concept of the effectivity function associated with the game form: this is the effectivity sheaf. A property called acyclicity of the effectivity. isdefined is proved to be a necessary and sufficient condition for strong solvability.

GLOBAL BIFURCATIONS IN A THREE-DIMENSIONAL COURNOT
OLIGOPOLY

Anna Agliari¹, Laura Gardini² and Tonu Puu³

¹ Istituto di Econometria e Matematica per le Applicazioni Economiche, Finanziarie e Attuariali

Università Cattolica di Milano, largo Gemelli 1, 20123 Milano, Italy, e-mail: agliari@unipr.it

² Università di Urbino, Italy, e-mail: gardini@econ.uniurb.it

³ University of Umea, Sweden, e-mail: tonu.puu@econ.umu.se

We study a Cournot model with three producers obtained under the hypothesis of constant though different marginal costs and ``bounded iso-elastic'' inverse demand function, i.e. the price is obtained as the reciprocal of the total quantity supplied plus a positive constant W, whose reciprocal then establishes a maximum price. Assuming, with Cournot, that each competitor assumes the others to retain their last moves even though observing the contrary, the adjustment process is described by a three dimensional map T, defined in a subset of R³, which admits the Cournot equilibrium point E^* as unique fixed point. Restricting the analysis to the region of stability of E^*, we study the existence of a region F Õ R³, whose point give raise to economic meaningful trajectories, i.e. if p ‘ F then Tⁿ(p) exists and belongs to R₊³. Also the geometrical properties of F, are investigate and we show that F undergoes some global bifurcations, which cause qualitative changes in its structure. Transitions from convex volume of the space to connected not convex and disconnected ones will be explained by the theory of critical surfaces and contact bifurcations between basin boundaries and critical surfaces.

STABLE SCHEDULE MATCHING UNDER REVEALED PREFERENCE

A. Alkan* and D. Gale

^* e – mail: alkan@sabanciuniv.edu

In a recent study Baiou and Balinski (2000) have generalized the notion of two-sided matching to that of a schedule matching in which one determines not only which partnerships will form but also how much time the partners will spend together. In their model it is assumed that each agent has a ranking of the agents on the other side of the market. In this paper we treat the scheduling problem using the more general preference structure introduced by Blair (1988) and recently refined by Alkan (2000) which allows among other things for diversity to be a motivating factor in the choice of partners, a subject of considerable current interest. The set of stable matchings for this model turns out to be a lattice with other interesting structural properties.

AFFILIATION IN MULTI-UNIT AUCTIONS

Ahmed W.Anwar

Department of Economics, Edinburgh University

William Robertson Building 50 Georgr Square

EH89JY, Edinburgh, UK, e-mail: Ahmed.Anwar@ed.ac.uk

We extend Milgrom and Weber's affiliated valuations model to the multi-unit case and restrict attention to a comparison of the discriminatory and uniform mechanisms. We find that it is possible to rank the two auctions in the pure common value case which is the case in Treasury auctions as the bonds are traded in a secondary market after the auction. The common value is simply the secondary market price. We show that the discriminatory auction has a unique equilibrium in the 2 bidder common value case. The equilibrium is equivalent to the single unit first-price equilibrium. The uniform auction on the other hand generally possesses equilibria that are extremely bad for the seller. The analysis suggests that a discriminatory auction should be used for the sale of Treasury bonds. We also look at the more general affiliated valuations case. However, a comparison in the more general case is complicated by the existence of multiple equilibria.

LEXICOGRAPHIC PROBABILITIES AND RATIONALIZABILITY IN EXTENSIVE GAMES

Geir B. Asheim¹ and Andrés Perea²

¹Department of Economics, University of Oslo, P.O. Box 1095 Blindern, N-0317 Oslo, Norway

ph.: (+47) 2285 5498, fax: (+47) 2285 5035, e-mail: g.b.asheim@econ.uio.no

The concepts of sequential and quasi-perfect rationalizability are defined in an epistemic model by means of lexicographic probabilities. These are non-equilibrium analogs to sequential and quasi-perfect equilibrium, for which epistemic characterizations are provided. The defined rationalizability concepts are shown to imply backward induction in generic perfect information games, but they do not yield forward induction. The relationship between various concepts are shown and illustrated.

A NEW PICTURE OF COALITIONS AND CONSEQUENT SOLUTION FOR n-PERSON COOPERATIVE GAMES

Mike A.Ball

Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, U.K.

e-mail: ballm@ liverpool.ac.uk

We present a new picture, superplayers, of coalitions. The coalitional-structure of superplayers can be represented by a rooted tree. The Aumann-Drèze³⁾ picture is a special case. A behavioural principle, the preference rule, makes such a tree full-binary. A c-tree is a tree and an associated imputation. X is a set of c-trees, one for each full binary tree. For each X, an equilibrium class, z_e(X), of c-trees exist. A bargaining equilibrium state means that players choose a c-tree from a set X^*, containing all possible c-trees; no superplayers want to move, in any allowed way, from such a c-tree. Parameters allocate the gains at the root. Examples include 'split the difference' (STD) and "fair" allocations. The solution has a distribution of possible imputations, which can be calculated by a formula. With the STD allocation, the average over all c-trees in X^* is the Shapley value. The STD solution sometimes violates the preference rule at level 1. Superadditive allocation parameters do not allow such violations. Requirements that dummy players get their characteristic value, that there is almost equal payoff as v(N) gets large suggest root allocations based on the Shapley values of the players. Examples of results are given and the advantages of the solution, including aggregate- and coalitional- monotonicity, are discussed.

CONTRACTING OVER TIME WHEN WRITING IS COSTLY

Pierpaolo Battigalli¹ and Giovanni Maggi²

¹ Bocconi University, Milano, Italia

e-mail: pierpaolo.battigalli@uni-bocconi.it

² Princeton University

e-mail: maggi@princeton.edu

In this paper we examine a model of contracting where parties interact repeatedly and can contract at any point in time, but writing formal enforceable contracts is costly. Enforceable contracts may contain contingent and non-contingent clauses. In order to save on writing costs, parties may modify only a subset of clauses as circumstances change. Informal, self-enforcing obligations may also arise as part of a subgame perfect equilibrium regulating the long-term relationship. We argue that the costs of writing contracts can provide a theoretical explanation for two common observations: the fact that long-term contracts are often preferred to spot contracts, and the fact that relationships are often managed by a combination of formal (externally enforced) and informal (self-enforcing) contracts.

A CANONICAL FORM FOR 2x2 GAMES

Andrea Battinelli

Dipartimento di Matematica ``A.Magari''- Università di Siena

via del Capitano 15 - 53100 Siena, Italia

ph.: +39-0577-233769/02, fax: /01/30, e-mail: battinelli@unisi.it

In memory of Gianni Decima

Here is a brief report of some findings about the issue, which are discussed at greater length in Battinelli, Quaderno del dip. di mat. ''A.Magari'' n. 396, univ. di Siena, 2000 and constitute an outgrowth of some previous work Battinelli, 16th I.C.G.T.A., Genova, 1998.

In Battinelli 1998, I achieved full characterization of the boundary of the attainable set for 2×2 games under a strict independence hypothesis, thus pinpointing remarkable differences with the full correlation case. I relied on a set of tools, namely: a concept of regularity, which is a weakening of a concept introduced in Rosenmüller 1981;.the determinant signs s_A,s_B of the two appropriately bordered game matrices; four scalars, the ``game equalizers'', already known and extensively used, but in an occasional and unsystematic fashion.

Here I put forth the following claim: the game equalizers, together with s_A,s_B (in short: the tools) completely characterize the class of regular games. To this I show that: 1) the sets of maxmin strategies and a complete algebraic characterization of the Nash equilibrium set are obtained from the tools alone; 2) they are invariant with respect to increasing affine transformations of the players' utility functions; 3) an appropriate choice of the representative element in each equivalence class of game bimatrices (w.r. to such transformations) brings them in the foreground (they are directly read off in the entries). This leads me to propose a canonical form.

SUBGAME PERFECT EQUILIBRIA IN INFINITE-ACTION GAMES OF PERFECT INFORMATION

Guilherme Belo Carmona

University of Minnesota, 1035 Heller Hall 271 19th Avenue South,

Minneapolis 554545 USA, e-mail: gcarmona@econ.umn.edu

We consider games of perfect information and we seek generalizations of the existing results concerning existence of subgame perfect equilibria for those games. Our existence result extends the existing ones by weakening the assumptions on the choice sets, on the choice correspondences and on the payoff functions.

We consider infinite-action games of perfect information, either with finitely or with denumerably many players. Informally, those games are described as follows: players move in sequence, each one choosing an action from a given set (his choice set), which may depend on the actions taken by the players that have moved before (via a choice correspondence); to each players corresponds a real valued function defined over the set of all possible sequences of actions (his payoff function), which is the criteria by which players compare different strategies. We will study the question of existence of subgame perfect equilibria for those games.

Regarding the case of denumerably many players, the most general existence result is due to Harris (``Existence and Characterization of Perfect Equilibrium in Games of Perfect Information,'' Econometrica, 53, 613-628). He shows that an equilibrium exist for games of perfect information in which, for all players, the payoff functions are continuous, the choice sets are compact Hausdorff spaces and the correspondences mapping past actions into choice sets are continuous.

We note that a game of perfect information with finitely many players can be regarded as a game of perfect information with denumerably many players, simply by assigning to the ``fictitious'' players a constant payoff function; therefore, Harris' result also applies in the case of games of perfect information with finitely many players. Another known result that is not implied nor does it implies the one of Harris is due to Reny and Robson (``A Simple Proof of the Existence of Subgame Perfect Equilibria in Infinite-Action Games of Perfect Information,'' Mimeo, University of Western Ontario). Reny and Robson show that a game of perfect information with finitely many players has a subgame perfect equilibrium provided that for all players their payoff function is continuous and their choice sets are independent of past actions, sequentially compact, first countable and separable.

Our existence result is intended to generalize both the theorem of Reny and Robson and also the theorem of Harris. Specifically, we will show that a subgame perfect equilibrium exist provided that the choice sets are sequentially compact; the choice correspondence is sequentially closed; and that the payoff function for each player satisfies some continuity assumptions. Specificaly, we assume that the payoff function for each player is (jointly) upper semicontinuos in his actions and in the actions of the players that play after him; is continuous in the actions of the players that play after him; and continuous at infinity (for games with finitely many players, this last condition is trivially satisfied). Therefore, we extended the applicability of the theory of games of perfect information by weakening the assumptions on the choice sets, on the choice correspondences and on the payoff functions.

GAMES IN DYNAMIC AND EPISTEMIC LOGICS

Johan van Benthem

Amsterdam & Stanford, Stanford University, Stanford, CA 94305, USA

ph. 1 650 723-3021, e-mail: johan@csli.stanford.edu

This talk is an introduction to some recent interfaces between modern logic and game theory, showing where both mathematical structures and conceptual concerns can meet.

Indeed, logic and games relate in various ways. Logical activities such as argumentation, consistency management, or model checking can be cast as games involving inter- action between players trying to achieve their rational purposes through optimal strategies. Viewed properly, the resulting modern 'logic games' provide attractive connections between logic and game theory, including calculi of game equivalences, algebras of game-forming operations, and calculi for manipulating strategies.

In this talk, however, our starting point will be different. We discuss how general games can be viewed as models for an ascending sequence of well-known logical languages: dynamic logics of action, epistemic logics of knowledge, and prefe- rence-based logics of various sorts. In the perhaps somewhat idiosyncratic eyes of a logician, games are many-agent pro- cesses, of a particularly interesting informational sort.

COOPERATIVE GAMES UNDER AUGMENTING SYSTEMS

Jesus M. Bilbao

Matemática Aplicada II, Escuela Superior de Ingenieros

Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

e-mail: mbilbao@cica.es, http://www.esi2.us.es/126mbilbao/

Cooperative games under augmenting systems are cooperative games restricted by a combinatorial structure which generalize the family of connected subgraphs of a communication graph (Myerson, 1977; Owen, 1986) and the permission structures (Gilles, Owen, van den Brink, 1992). So, these type of cooperative games group several well-known families of games which have important applications in economics and politics. Therefore, the study of the restricted games by augmenting systems allows to unify results of various lines of research.

Let N be a finite set. A set system over N is a pair where F Õ 2^N is a family of subsets. We call a set system ( N, F) normal if N = »_{S ‘ F}S. An augmenting system is a normal set system ( N, F) with the following properties:

-

‘ F,

-

For S,T ‘ F with S«T ¼ , we have S»T ‘ F,

-

For S,T ‘ F with S Ã T, there exists i ‘ T\S such that S»i ‘ F.

Using the structural properties from the augmenting systems we will be able to express the dividends in terms of the original game. This result will be essential to provide some formulas to compute the Shapley and Banzhaf values for games under augmenting systems restrictions. In these formulas these values are computed by means of the original game without having to calculate the restricted game and taking into account only the coalitions in the augmenting system.

COMPUTING POWER INDICES IN MULTIPLE MAJORITY GAMES

J. M. Bilbao¹ , J. R. Fernández² , J. J. López³

¹ Matemática Aplicada II, Escuela Superior de Ingenieros

Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

e-mail: mbilbao@cica.es, http://www.esi2.us.es/126mbilbao/

The Shapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If the input size of the problem is n, then the function which measures the worst case running time for computing these indices is in O( n2ⁿ). We present a method based on generating functions to compute these power indices efficiently for weighted multiple majority games and we study the temporal complexity of the algorithms. Finally, we obtain the following conclusions about the triple majority systems which will be used in the European Union enlargement:

-

The difference between the Banzhaf power indices of Germany and United Kingdom, France and Italy are smaller than 1.4×10^-7 (first rule) and 1.6×10^-7 (second rule).

-

The two rules of triple majority, adopted in the Nice summit meeting, are almost equivalent to a game of simple majority (the first) or double (the second). With both rules, the required population quota to adopt a decision does not change in practice the power of the countries.

AN EVOLUTIONARY GAME WITH IMITATION DYNAMICS FOR THE DESCRIPTION OF FIRM CLUSTERING AND SPILLOVER EFFECTS

Gian-Italo Bischi¹, Herbert Dawid² and Michael Kopel³

¹ Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italia

e-mail: bischi@econ.uniurb.it

² Department of Economics, University of Southern California, USA

e-mail: dherbert@usc.edu

³ Department of Managerial Economics and Industrial Organization

University of Technology, Vienna, Austria

e-mail: kopel@ebwnov.tuwien.ac.at

We study an evolutionary game with imitation dynamics which models the evolution of two firm clusters competing on a market. Firms exit and entry a cluster on the basis of an ``imitate the better'' mechanism, based on the perceived chances for profit inside and outside the cluster. Information about profits are diffused by direct communication between firms. Internal and external spillover effects reduce production costs of firms in the clusters depending on the number of firms in the own and the competing cluster. A discrete time deterministic dynamical system describing the evolution of cluster sizes is derived. An analysis of the attractors of the system and their basins of attraction is used to compare the effects of advantages of a cluster with respect to the size of internal respectively external spillover effects. Furthermore, the implications of slow respectively fast exit and entry behavior of firms for the long run survival and size of the clusters are studied.

DYNAMIC REASONING ABOUT KNOWLEDGE AND BELIEF

Oliver Board

Department of Economics, Oxford University OX1 4AJ, Oxford, UK

e-mail: oliver.board@economics.ox.ac.uk

Strategic reasoning requires agents to form not just conjectures about each other's actions, but also about each other's knowledge and beliefs, which can then be used to infer what actions might be taken. Formal models of interactive reasoning have been developed to provide epistemic foundations for various game theoretic solution concepts. Much of this work is based around the (static) Kripke structure model of interactive epistemology, but more recently dynamic models of interactive reasoning have been developed, most notably by Stalnaker and Battigalli & Siniscalchi, and used to analyze rational play in extensive form games. But while the properties of Kripke structures are well understood, without a formal language in which belief and belief revision statements can be expressed, it is unclear exactly what are the properties of these dynamic models. Here we investigate this question, by defining such a language. A semantics (similar to that used by Stalnaker) and syntax are presented, with a soundness and completeness theorem linking the two.

THE LOGIC OF TIME, UNCERTAINTY AND ACTIONS

Giacomo Bonanno

Department of Economics, University of California, Davis, CA 95616-8578, USA

ph.: (530)-752 1574 , fax: (530)-752 9382 , email: gfbonanno@ucdavis.edu

The connections between logic (in particular, modal logic) and games have been extensively explored in the literature. So far the focus has been mainly on the epistemic foundations of solution concepts (for recent surveys see Battigalli and Bonanno, 1999, and Dekel and Gul, 1997). Only a few papers have dealt with the descriptive side of game theory, in particular the logical aspects of the sequential and informational structure of extensive games (Battigalli and Bonanno, 1999, Bonanno, 1992, 1993, van Benthem, 2001). On the other hand, game theorists have recently begun exploring the implications of imperfect memory in dynamic decision problems and in extensive games (see the special issue of Games and Economic Behavior, 1997, Vol. 20, devoted to this topic). This literature raises important questions concerning the precise and complete description of dynamic models (see, for example, Halpern, 1997). In a recent paper, van Benthem (2001) shows that dynamic epistemic logic can help shed some light on these issues.

In this paper we provide a very simple framework for the logical analysis of dynamic decision problems and extensive games. Since actions unfold over time, we take as a basic building block standard temporal logic, with its past and future operators. To the temporal precedence relation we add a belief relation in order to capture uncertainty about past events. In the first part of the paper we use the interaction of the temporal and belief operators to capture, both semantically and syntactically, the notions of memory and forgetting in a single-person setting. In the second part of the paper we add a new set of binary relations to capture the actions available to the agent over time. The interaction of the temporal, belief and action operators allows us to express formally a number of properties, such as perfect recall. In the third part of the paper we extend the framework to multiperson settings, thereby encompassing extensive games. In the last part we add one more relation for each player, capturing the notion of plan or strategy as well as beliefs about how other players will play. This allows us to talk about the solution of an extensive game along the lines of Bonanno (2001).

PEER GROUP GAMES

Rodica Brânzei¹, Vito Fragnelli² and Stef Tijs³

¹ Faculty of Computer Science, Äl.I. Cuza", University, 11 CAROL L Bd.

6600 Iasi, Romania, e-mail: branzeir@infoiasi.ro

² University of Eastern Piedmont, Italy

e-mail: fragnell@mfn.unipmn.it

³ CentER and Department of Econometrics and Operations Research

Tilburg University, The Netherlands, e-mail: S.H.Tijs@kub.nl

A class of cooperative games is introduced which arise from economic situations where agents with potential individual possibilities are connected via a hierarchy within an organization. In such a situation the important group for an agent is that consisting of the leader, the agent himself and all the intermediate agents in the given hierarchy because only by this cooperation the agent's potential economic possibilities can become effective. We call such a group of agents a peer group. It is shown that peer group games can be expressed as non-negative combinations of unanimity games based on peer groups of agents. They form a cone that lies in the intersection of convex games and monotonic veto-rich games with the organization's leader as veto-player. For peer group games classical solution concepts have nice properties. We focus on the nucleolus of peer group games from the algorithmic point of view. Different economic situations like second price sealed bid auctions, graph-restricted (binary) communication situations, flow situations, airport problems, some sequential and information collecting situations are related to peer group games.

PRICING OPTIONS EMBEDDED IN BONDS

Hatem Ben Ameur¹, Michèle Breton² and Pierre l'Écuyer³

¹ École des H.E.C. Montréal, e-mail: Hatem.Ben-ameur@hec.ca

² GERAD and École des H.E.C. Montréal, 300 Cote S. Catherine

H3T 2A7 Montreal, Canada, e-mail: Michele.Breton@hec.ca

³ GERAD and Université de Montréal, e-mail: Lecuyer@IRO.UMontreal.CA

A bond can be interpreted as a loan with a principal equal to the face value and interest payments equal to the coupons (if any). The borrower is the issuer of the bond and the lender is the investor. Several bonds contain options: for instance, the issuer may have the right to purchase back its debt for a known amount during a specified period within the bond's life. This is called the call option. The investor may also have the right to return the bond to the issuer for a known amount during a specified period within the bond's life. This is called a put option. These options are an integral part of a bond, and cannot be traded alone; they are said to be embedded in the bond. When either the issuer or the investor exercises her options, this effectively ``kills'' the option of the other player. In this paper, we address the problem of pricing options embedded in bonds as a stochastic DP formulation, the focus being on the solution of the DP equation. It is well known that the call option is more likely exercised by the bond issuer when the interest rates are low, while the put option is more likely exercised by the investor when the interest rates are high. We use the DP formulation to establish these basic intuitions, to evaluate the impact of these options on the bond price, and to determine the optimal exercise strategies of both players.

A CLASS OF CONSISTENT SHARE FUNCTIONS FOR GAMES IN COALITION STRUCTURE

Rene van den Brink¹ and Gerard van der Laan²

¹ Department of Econometrics and Operations Research

Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands

ph.: +31 13 4663425, fax: +31 13 4663280, e-mail: jbrink@kub.nl

² Department of Econometrics and Tinbergen Institute, Free University

De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands

e-mail: glaan@econ.vu.nl

A cooperative game with transferable utility -or simply a TU-game- describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games is a function which assigns to every such a game a distribution of the payoffs over the players in the game. An alternative type of solutions are share functions which assign to every player in a TU-game its share in the payoffs to be distributed.

In this paper we consider cooperative games in which the players are organized into an a priori coalition structure being a finite partition of the set of players. We introduce a general method for defining a class of share functions for such games in coalition structure using a multiplication property that states that the share of player i in the total payoff is equal to the share of player i in some internal game within i's a priori coalition, multiplied by the share of this coalition in an external game between the a priori given coalitions. We show that these coalition structure share functions satisfy certain consistency properties. We provide axiomatizations of this class of coalition structure share functions using these consistency and multiplication properties.

MULTI-ISSUE ALLOCATION GAMES

Pedro Calleja¹, Peter Borm² and Ruud Hendrickx³

¹Department of Economic, Financial and Actuarial Mathematics

University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain

e-mail: calleja@eco.ub.es

², ³CentER and Department of Econometrics and Operations Research

Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands

e-mail: {P.E.M.Borm, R.L.P.Hendrickx}@kub.nl

This paper introduces a new class of transferable-utility games, called multi-issue allocation games. These games arise from various allocation situations and are based on the concepts underlying the bankruptcy model, as introduced by O'Neill (1982). In this model, a perfectly divisible good (estate) has to be divided amongst a given set of agents, each of whom has some claim on the estate. Contrary to the standard bankruptcy model, the current model deals with situations in which the agent's claims are multi-dimensional, where the dimensions correspond to various issues.

It is shown that the class of multi-issue allocation games coincides with the class of (nonnegative) exact games. The run-to-the-bank rule is introduced as a solution for multi-issue allocation situations and turns out to be the Shapley value of the corresponding game. Finally, this run-to-the-bank-rule is characterized by means of a consistency property.

CARDINAL-ORDINAL INVARIANT SOLUTIONS

Emilio Calvo¹ and Hans Peters²

¹ Departament d'Analisi Economica, Universitat de Valencia

Campus d'els Tarongers s/n, Edificio Departamental Oriental, 46022 Valencia, Spain

ph. +34-96-3 82 82 32, fax +34-96-3 82 82 49, e-mail: Emilio.Calvo@uv.es

² University of Maastricht, The Netherlands

We consider Pure Bargaining problems in which there are two type of players: Cardinal and Ordinal players. Cardinal are players that can represent preferences over their payoffs by a unique utility function and all its positive affine transformations. Ordinal players have the freedom to represent their utility function by any differentiable order-preserving transformation. We assume that the set of cardinal players is always nonempty and that suitable smoothness regularity conditions on the Pareto boundary of the problem are satisfied. In this setting we are able to build a bargaining solution through their disagreement point sets that is invariant under the allowed utility transformations for each corresponding type of player.

DYNAMIC GAME OF SUBSIDIES BETWEEN TWO CULTURAL DEPARTMENTS

Luc Champarnaud¹ and Pierre Cartigny²

¹ Gremars, Université Lille 3 UFR Mathématiques et Sciences Sociales

BP 149, 59653 Villeneuve d'Ascq cedex, France

ph.: 33 (0)320416834, fax: 33(0)320416171, e-mail: champarnaud@univ-lille3.fr

² Université de la Méditerrannée, Centre de la Vieille Charité

2, rue de la Charité 13002, Marseille, France

ph.: 33(0)491140742, fax: 33(0)491900227, e-mail: cartigny@ehess.cnrs-mrs.fr

Once the principle of subsidising the Art admitted, the question of '' how to subsidise the Art ? " can arise to political scientists and economists. This is an old question for economists who have already emphasised that cultural interventions could lead to a non-optimal level of quality because of collusion between artists and experts in charge of subsidising the art from inside the government cultural institutions.

As a matter of fact, internal organisation of government is ruled by the separation of powers between specialised departments. Cultural administration does not escape to that rule and the bias of subsidising toward quality often informed in literature can be seen as a result of the non-cooperative attitude between cultural departments regardless to the common welfare objective. The question stressed in this paper concerns the reaction to adopt from the benevolent planificator point of view facing the non-optimality of the game between departments belonging to government cultural institutions. Is it necessary (if thinkable) to suppress these departments that are a source of non-optimality ? or is it possible (and more practicableS) to conceive an administration design leading to optimality without such an extreme process, that is : preserving natural specialisation between cultural departments inside government ?

We show that a very simple and practicable rule of redistribution between departments can lead to optimality and correct the non-cooperative game inefficiency.

A NEW EMPIRICAL STUDY OF THE MEXICAN TREASURY SECURITIES PRIMARY AUCTIONS: IS THERE MORE UNDERPRICING?

Sara Castellanos

Economic Studies Division, Banco de México, Avenida Cinco de mayo 18, Mexico City 06059, Mexico

ph: (52) 5237-2643, fax: (52) 5237-2687, e-mail: sgcastel@banxico.org.mx

Examination of new data of the Mexican government securities primary auctions shows, not only that there exists underpricing with respect to the secondary market in CETES of 28, 91, 182, and 365 days maturity, but also that such underpricing tends to increase over time, especially during the period of 1996-2000. This study also finds that changes to the auction format employed by the Mexican Treasury do not fully account for this pattern because in spite that since 1995 the discriminatory format is in place, some modifications have been adopted with the purpose of improving sales conditions in the primary auctions, like setting common maximum bidding limits for all participants or reopening securities issues with high observed interest rates. Usual factors assumed to affect auction prices by existing models, like competition level, bidder participation and market uncertainty seem to play a role. However, evidence is also found that some factors related to secondary market features suggested by the finance microstructure literature, like liquidity, inventory and adverse selection costs may be playing an even more important role. Overall, these findings provide confirmation of the relevance of resale markets and suggests that: 1) it may be fruitful to add some of these latter features in future theoretical models about government securities markets and 2) to improve secondary market conditions may be important to enhance efficiency in the primary market.

COORDINATION AND PROVISION OF A DISCRETE PUBLIC GOOD BY CORRELATED EQUILIBRIA

Alberto Cavaliere

Dipartimento di Economia Pubblica-Università degli Studi di Pavia

Corso Strada Nuova 65, 27100 Pavia, Italia

ph. +39 0382 504358, e-mail cavalier@unipv.it

The strategic analysis of the private provision of a discrete public goods has shown the existence of multiple Nash Equilibria with the efficient number of players voluntarily contributing. However the coordination issue is left unexplained by the past literature. The experimental evidence shows very clearly that communication among players is used by them to achieve coordination and that cooperation increases significantly when communication is allowed. Moreover players seem to use random devices to reach the distribution decision. We claim that from the strategic point of view coordination can be achieved by playing a correlated equilibrium in the general public good game modelled as Chicken that was originally presented by Palfrey and Rosenthal. We show the existence of such equilibria as feasible cordination mechanism to achieve public goods provision in the general n-people game and in some examples. On the basis of the existence of such equilibria we derive a condition relating the contribution amount, the total number of players and the minimal number of players that is necessary to provide the public good. In order to discuss efficiency we further distinguish between the ``classic''case of a discrete public goods and the case of a discrete public goods with spillover effects that increase with the number of contributors once the minimal set of contributions is achieved (an example could be fundraising for medical research). In the first case Pareto efficiency can be achieved by economic agents that play any correlated equilibrium that results in a convex combination of Nash equilibrium pay-offs. In the second case this kind of correlated equilibria are no more Pareto efficient. However one can find correlated equilibria that Pareto dominate the convex combinations of Nash Equilibria and search for constrained Pareto efficient allocations that can be implemented by correlated equilibria. These are incentive effcient mechanism (Myerson, 1985)for the private provision of the public good, once strategic free-riding is seen as a moral hazard problem. One example concerning the opportunity of coalition formation among some players is also presented. This example can be useful to discuss the incentives for people to start any institution devoted to the private provision of a public good and the social benefits that result from this decision.

NASH EQUILIBRIUM, VARIATIONAL INEQUALITY AND DYNAMIC SYSTEMS

E.Cavazzuti¹, M.Pappalardo² and M.Passacantando³

¹ Dipartimento di Matematica, Università di Modena, Via Campi 213/b,

41100 Modena, Italia, ph. +390103536835, e-mail: pacchiarotti@unimo.it

^2,3 Università di Pisa

We only consider games in normal form and Nash equilibrium solutions and we expose some necessary, sufficient or necessary and sufficient conditions for Nash equilibrium; these conditions are expressed in terms of quasi-variational inequalities of Minty or Stampacchia type. To semplify the exposition, we split the family of non cooperative games accordingly with the properties of payoffs, considering three cases: the general, the quasi-convex and the pseudo-convex case.

We also consider dinamical systems related to games and analyze convergence properties of the generated trajectories. Special emphasis will be given to the ones which are generated by "gradient" or Newton method.Moreover we consider an heuristic method.

EXECUTIVE COMPENSATION AND INFORMATION

Marco Celentani¹ and Rosa Loveira²

¹ Department of Economics, Universidad Carlos III de Madrid, Getafe (Madrid) 28903, Spain

fax: +34-91-624 9875, e-mail: celentan@eco.uc3m.es

²Department of Economics, Universidad Carlos III de Madrid, Getafe (Madrid) 28903, Spain

fax: +34-91-624 9329, e-mail: rloveira@eco.uc3m.es

This paper considers a market for managerial services in which managers are assumed to have superior information on their own ability to make an appropriate decision on a given investment project. Managers' investment decisions are affected by their concern for their future careers. We propose a simple two-period model in which firms compete for managers offering short period contingent contracts. We study the impact on equilibrium contracts and investment decisions of exogenous noisy information on the manager's ability that becomes costlessly available after the latter has accepted an employment contract. We consider three different possibilities: 1) The exogenous information is privately observed by the firm and the manager and is contractible; 2) The exogenous information is privately observed by the firm and the manager and is noncontractible; 3) The exogenous information is publicly observed and is contractible.

We find that, in contrast to the informativeness principle of Holmström (1979), in cases 1 and 2 it is possible that equilibrium contracts do not make payments contingent on the available information. We use case 2 to characterize equilibrium contracts which are made dependent on internal and noncontractible managerial evaluations. We use case 3 to study relative performance compensation in a case in which the public outcome of the industry provides an indirect, noisy signal of a manager's ability whose informativeness is increasing in the competitiveness of the industry.

BROWNIAN DONKEYS AND DONKEY GAMES

C. Van den Broeck and B. Cleuren^*

Limburgs Universitair Centrum, B-3590, Diepenbeek, Belgium

^* e-mail: bart.cleuren@luc.ac.be

A Brownian donkey is a Brownian motor that possesses negative mobility, i.e., it moves always opposite to the direction of an applied (small) force. A similar property can be observed in a new, simple game. It consists of tossing a biased coin, the capital of the player being increased or decreased by one, depending on the outcome. The additional, history-dependent rule is that whenever the player won (or lost) N times successively, these last N results are not taken into account. The analytic expression of the average gain is derived and it is shown that the history dependence results in an unexpected behavior, namely a "negative mobility" with respect to the coin bias.

HIGH-PERFORMANCE BIDDING AGENTS FOR THE CONTINUOUS DOUBLE AUCTION

Gerald Tesauro and Rajarshi Das

Institute for Advanced Commerce, IBM T. J. Watson Research Center

30 Saw Mill River Road, Hawthorne, NY 10532, USA

e-mail: {gtesauro, rajarshi}@us.ibm.com

The Continuous Double Auction institution (CDA), which is known for high market efficiency, and which is pervasive in real-world markets, also provides a well-defined economic environment for testing economic theory-especially game theory with incomplete information. An increasingly important focus of research in agent-based electronic commerce is the design of robust heuristic bidding algorithms for a variety of auctions, including the CDA. In this paper we develop two CDA bidding algorithms, that offer what we believe to be the strongest known performance of any published CDA bidding strategy. Our algorithms, which are based on extensions of two published strategies (Cliff, 1997) and (Gjerstad & Dickhaut, 1998), can handle trading multiple units in real-time markets. We have tested these strategies against each other and against the sniping strategy of (Rust et al., 1992) and the baseline ``Zero Intelligence'' strategy of (Gode and Sunder, 1992). Our results show that, under a variety of market rules and limit price distributions, our modified Gjerstad-Dickhaut strategy utperforms and generally dominates the other strategies. We discuss insights from these simulations into when bidders do well (or do poorly), either as an individual or as a group, and we suggest mechanisms by which the strategies can be improved.

ON BERTRAND COMPETITION UNDER NOT SO LARGE AN EXCESS OF TOTAL CAPACITY

Massimo De Francesco

Dipartimento di Economia Politica, Università degli Studi di Siena

P.za S.Francesco 7, 53100 Siena, Italia, e-mail: defrancesco@unisi.it

We consider a homogeneous product market where, given their capacities, existing firms compete in prices. First, pricing at the constant short-run average-marginal cost - i. e., Bertrand outcome - is shown to be a Nash equilibrium of the static price game provided total capacity is sufficiently higher than the quantity demanded at a price equal to marginal cost; most importantly, the minimum amount of excess capacity that is required is quite modest when the one-firm concentration ratio is sufficiently small. Then a study of repeated price decisions is carried out in a context where less than "fully rational" firms in each period aim at maximising current profits. The convergence result hinted at by Bertrand for a duopoly is extended straightforwardly to the n-firm case: prices converge to short-run average cost under iterated best responses as well as under milder restrictions on the learning process. These results suggest that in an unconcentrated homogenous-product industry self-interested behaviour can easily be "destructive" to the firms - making them not covering anything towards their fixed costs under even a modest excess of total capacity.

A SPATIAL VOTING MODEL WHERE PROPORTIONAL RULE LEADS TO TWO-PARTY EQUILIBRIA

Francesco De Sinopoli^* and Giovanna Iannantuoni

^* Francesco De Sinopoli, CORE-UCL, 34 Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium

ph: ++32-10-474352, fax: ++32-10-474301, e-mail: desinopoli@core.ucl.ac.be

In this paper we show that in a simple spatial model where the government is chosen under strict proportional rule, if the outcome function is a linear combination of parties' positions, with coefficients equal to their share of seats, only a two-party voting equilibrium basically exists. The two parties taking a positive number of votes are the two extremist ones. Applications of this result include an extension of the well-known Alesina and Rosenthal's model of divided government as well as a modified version of Besley and Coate's model of representative democracy. This result cannot be extended to a general outcome function but, however, when the policy is determined by the two leading parties, in pure strategies, only two-party equilibria can emerge. Analogous result holds for coalitions of parties.

STACKELBERG SOLUTIONS OF A DIFFERENTIAL GAME MODELING AN INNOVATION DIFFUSION PROBLEM INCLUDING SUBSIDIES POLICIES

Luigi De Cesare¹, Andrea Di Liddo² and Stefania Ragni³

¹ Facoltà di Economia, Università di Lecce, via per Monteroni-Ecotekne, Lecce, Italia

IRMA-CNR, Via Amendola 122/I, Bari, Italia

² Facoltà di Economia, Università di Foggia, Via IV novembre, Foggia, Italia

IRMA-CNR, Via Amendola 122/I, Bari, Italia, e-mail: a.diliddo@unifg.it

³ Facoltà di Economia, Università di Bari, Via Camillo Rosalba,

IRMA-CNR, Via Amendola 122/I, Bari, Italia

Let us consider a new, non renewable, technology which is produced by a monopolist firm which sells it in a market. We assume that the firm permits the technology by a suitable advertising policy and we take into account the diffusion of information about the new product by interpersonal contacts. We consider a government that wishes to accelerate the adoption of the new technology by using an assigned budget to give price subsidies directly to the consumers. The problem we are dealing with is a dynamic game which is played in the Stackelberg leader-follower fashion. The two players are the government and the firm. The government takes the leader's role and announces its strategy before the firm makes its decisions. If the government's decision is irrevocable then the announcement of a subsidy scheme is credible and open-loop Stackelberg equilibrium strategies make sense. On the other hand if binding commitments on the entire time horizon are not put into effect, then the government has an incentive to reoptimize at any instant of time so that feedback strategies seem to be more appropriate. The problem of time consistency of open-loop equilibrium strategies is also addressed. Numerical experiments are performed in order to gain an insight into dependence of equilibribium on the various parameters.

WHAT'S NEW ABOUT TRICKERY? BACCARAT AND (PSYCHOLOGICAL) GAMES

Regis Deloche¹ and Fabienne Oguer²

¹ Università de Franche-Comte, U.F.R. des sciences Economiques © et juridiques

Avenue de l'Observatoire, F-25030 Besaneon Cedex, France

e-mail: regis.deloche@univ-fcomte.fr

² Università de Franche-Comte, U.F.R. des sciences Economiques et juridiques

Avenue de l'Observatoire, F-25030 Besaneon Cedex, France

fabienne.oguer@univ-fcomte.fr

Baccarat is nowadays a casino game built on a card game with rigid rules. In the beginning, it was a parlor card game where players had full discretion in their actions according to the cards dealt during the play. We may look at baccarat parlor game as a non-cooperative dynamic game of incomplete information with uncertainty. Its solution is perfect Bayesian equilibrium with behavioral strategies. Using the game-theoretic concept of kernel prompts one to obtain the same strategies. Moreover, baccarat parlor game rules allow a tricking behavior, such that experienced players choose an action while letting their opponent believe they are used to choose another action. Hence, we may also look at baccarat parlor game as a psychological game. Its solution is psychological Nash equilibrium. Analyses of baccarat of the late 19th century and the early 20th century enable searching after such solution of baccarat.

RANDOMIZED OPTIMAL STOPPING RULES FOR A CLASS OF STOPPING GAMES

Victor Domansky

St.Petersburg Institute for Economics and Mathematics Russian Academy of

Sciences, Tchaikovskogo 1,191187 St.Petersburg, Russia

e-mail: victor@agd.stud.pu.ru

We consider games of stopping for Markov chain in the formulation introduced by Dynkin (1969). Two players observe a Markov sequence and may stop it at any stage. When the chain is stopped the game ends and Player 1 receives from Player 2 the sum depending on the player who stopped the chain and on its current state.

We describe solutions for a class of stopping games with a countable state space. Payoffs are nonnegative and payoff is zero if Player 1 stops the chain but not Player 2. Estimations of value are obtained with use of randomized stopping times. The qualitative characteristics of solutions are determined with the "limiting" behavior of payoffs.

The explicit solutions are given for the case when  p(i,i+1) = 1.

ON SEMIVALUES AND TU GAMES ON MATRIODS

Irinel Dragan

University of Texas at Arlington, Arlington 76019-0408, USA

e-mail: dragan@uta.edu

The concept of TU game on a matroid and the Shapley value of a game on a matroid have been recently introduced by M.Bilbao, T.Driessen, J.Losada and E.Lebron in TR Univ. Seville 1999. A system of four axioms defines uniquely the Shapley value, precisely the linearity, symmetry, a modified dummy player and a modified efficiency. In our paper, we are looking for an alternative concept of Shapley value, by starting from a Semivalue and imposing a matroid-efficiency axiom, via a formula for the Power Game of a game relative to a Semivalue. We show the formula for the new value and discuss some other properties connected to the potential of this value.

A MULTIPLICATIVE POTENTIAL APPROACH TO SOLUTIONS FOR COOPERATIVE TU-GAMES: THE MULTIPLICATIVILY PROPORTIONAL VALUE

Theo S.H. Driessen

Faculty of Mathematical Sciences, University of Twente

P.O. Box 217, 7500 AE Enschede, The Netherlands

e-mail: t.s.h.driessen@math.utwente.nl

Concerning the solution theory for cooperative games with transferable utility, it is well-known that the Shapley value is the most appealing representative of the family of (not necessarily efficient) game-theoretic solutions with an additive potential representation. This paper introduces a new solution concept, called Multiplicativily Proportional (MP) value, that can be regarded as the counterpart of the Shapley value if the additive potential approach to the solution theory is replaced by a multiplicative potential approach in that the difference of two potential evaluations is replaced by its quotient. One out of two main equivalence theorems states that every solution with a multiplicative potential representation is equivalent to this specifically chosen efficient value in that the solution of the initial game coincides with the MP value of an auxiliary game. The associated potential function turns out to be of a multiplicative form (instead of an additive form) with reference to the worth of all the coalitions. The second equivalence theorem presents four additional characterizations of solutions that admit a multiplicative potential representation, e.g., preservation of discrete ratios or path independence.

CORE AND EQUILIBRIA IN LARGE MODELS OF HOUSEHOLD ECONOMY

Maria Ekes

Institute of Econometrics, Warsaw School of Economics

Al. Niepodleglosci 162, 02-554 Warszawa, Poland

ph: (48 22) 646 61 04, fax: (48 22) 48 68 01, e-mail: mmroman@sgh.waw.pl

The paper presents a discussion of relations between the core and equilibria of a model of household economy with infinitely many agents classified into a finite number of types. We describe the model presenting two different approaches. In the first approach we focus on the notion of competitive equilibrium. In the second one we give the definitions of core and semi-core. Then we formulate the theorem on the relations between the market equilibrium and core and semi-core in the model, analogous to the results of Aumann, Vind and others.

ENDOGENOUS FORMATION OF COMPETING PARTNERSHIPS WITH MORAL HAZARD

María Paz Espinosa¹ and Inés Macho-Stadler²

¹Universidad del País Vasco, Av. Lehendakari Aguirre 83,

48015 Bilbao, Spain, e-mail: jepesalp@bs.ehu.es

²Universidad Autónoma de Barcelona

e-mail: Ines.Macho@uab.es

In a linear Cournot model, we analyze the formation of competing partnerships as a sequential game with moral hazard within coalitions. When moral hazard within coalitions is very severe, no partnership will form. However, we show that when moral hazard is not too severe, so that some partnership forms, the coalition structure will be either similar to or more concentrated than it is without moral hazard. Concerning industry profits, without moral hazard too many coalitions are formed in equilibrium as compared to the efficient outcome, but moral hazard may be responsible for an inefficiency of opposite sign.

NON LINEAR DYNAMICS AND THE PARRONDO PARADOX

P.Arena, S.Fazzino, L.Fortuna^* and P.Maniscalco

^* Dipartimento Elettrico Elettronico e Sistemistico, Università

degli Studi di Catania,Viale A.Doria 6,Catania,95125 Italia

ph.: +39 095 7382307, fax: +39 095 339535, e-mail: lfortuna@dees.unict.it

In this paper we deal with a new research topic regarding the role of chaos in a particular game problem: the Parrondo Paradox. In this paradox, it has been proved how two separate losing games can be combined following a random or periodic strategy in order to have a resulting winning game. In particular, we have studied and discussed, through simulations, how the introduction of a strategy based on various chaotic time series could improve the gain in the following cases: the classic two Parrondo games, a three game problem obtained from the classical one with the introduction of a third game and a more generalized N game problem.

PRODUCT IDENTIFICATION IN THE ABSENCE OF TRADEMARKS

Dragan Filipovich

El Colegio de Mexico, Lope de Vaga 504, Drept. 9, 11570 Mexico D.F.,

Mexico, e-mail: dfilipovich@colmex.mx

This paper looks at whether it is possible for firms to differentiate their products from each other by choosing meaningless, worthless `tags' in the absence of exogenous rules prohibiting imitation. It shows that even when there is general consensus regarding the ranking of firms' products (from worse to better sell), `endogenous' differentiation might result. Moreover, it shows that for differentiation of this sort to obtain, the quality distribution must not be biased upwards.

NEWTONIAN MECHANICS AND NASH PLAY

Sjur D. Flàm¹ and Jacqueline Morgan²

¹Department of Economics, University of Bergen 5007 Norway

e-mail: Sjur.Flaam@econ.uib.no

²Dipartimento di Matematica e Statistica, Università di Napoli Federico II

Via Cinthia, 80125 Napoli, Italia

e-mail: morgan@unina.it

Repeated play among noncooperative agents is here modelled as a second-order process, motivated - and interpreted - in two ways: One presumes that players do not quite pursue directions of steepest-payoff-ascent, but rather use deviations from such directions to undertake appropriate acceleration. The other motivation, purely mechanical in nature, draws on the fact that friction, via energy loss, stabilizes systems.

No player need here much information or competence. Nonetheless, convergence to Nash equilibrium obtains under weak and natural conditions. A most important one is that marginal payoffs, when accumulated along the path of play, be bounded above.

AN ENVY-FREE PROCEDURE FOR AN INSURANCE PROBLEM

Vito Fragnelli¹ and Maria Marina²

¹Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale

C.so Borsilano 54, I-15100 Alessandria, Italia

e-mail: fragnell@mfn.unipmn.it

²Dipartimento di Economia e Metodi Quantitativi, Università di Genova

e-mail: marina@economia.unige.it

We consider a random variable R, representing a risk that has to be insured by n companies and which a premium f is assigned to. How should they share the risk and the premium in order to be better off? We suppose that every company i expresses her valuation of a random variable X as the real number H_i(X). Under suitable hypotheses on the functionals H_i, there exists an optimal decomposition in constant quotas of the risk. According to this division, we develop a modification of the fair division procedure of Haake et al. (2000), that guarantees an envy-free division of the premium f and of the risk R.

HAMILTON-JACOBI EQUATIONS WITH DISCONTINUOUS HAMILTONIAN

Helene Frankowska

Ecole Polytechnique, 1 rue Descartes,PARIS, F-75005, France

ph.:+33-(0)1-55-55-80-89, fax:+33-(0)1-43-25-29-44

e-mail: franko@poly.polytechnique.fr

It is well known that in two person zero-sum differential games, the optimal strategies of players tend to be discontinuous. On the other hand, in nonlinear optimal control theory, the optimal feedback may be derived from the Bellman equation and it is usually discontinuous. The Isaacs equation, in principle, may be used for the same purposes, but the "double" discontinuity (because of the presence of the second player) makes the regularity much worse. In this talk, I will discuss a possible approach to Hamilton-Jacobi equations with discontinuous Hamiltonians.

WHEN DOES COMMON ASSUMPTION OF RATIONALITY LEAD TO NASH EQUILIBRIUM?

Adam Brandenburger¹ and Amanda Friedenberg²

¹ Harvard Business School Boston, MA 02163, USA

e-mail: abrandenburger@hbs.edu

² Harvard University, Littauer Center, North Yard Cambridge, MA 02138, USA

e-mail: afrieden@fas.harvard.edu

It has often been written that if the players in a game are rational, then their strategies must constitute a Nash equilibrium. A common variant on this statement hypothesizes not just rationality of the players, but common assumption of rationality (CAR)-i.e. each player assumes that each other player is rational, assumes that each other player assumes this, and so on ad infinitum. However, in the modern, epistemic treatment of games, CAR - a fortiori, rationality alone - in general does not yield Nash equilibrium. This paper considers the class of perfect-information (PI) games. It is shown that in a PI game, CAR implies that the path of play is a pure-Nash-equilibrium path. Thus, we provide a formal justification, within a certain class of games at least, for the older, pre-epistemic intuition linking CAR and equilibrium. An immediate corollary of our main result is that CAR yields the backward induction outcome in any PI game where all pure Nash equilibria are outcome equivalent. The much studied Centipede game is in this class. Thus, the paper also provides a logical basis for backward induction in Centipede, a matter that has been the subject of much debate in the literature.

FINDING NASH EQUILIBRIA IN BIMATRIX GAMES VIA VERTICES ENUMERATION

P. Frigau¹ and G. Quaranta²

¹Istituto Tecnico Commerciale ''G. Peano'', Firenze, Italia

² Università degli Studi di Pisa, Pisa, Italia, e-mail: quaranta@unisi.it

The set of all Nash equilibria ofa bimatrix games is characterized by quadratic conditions which can be transformed into linear complementary conditions. For each player the set of all strategies satisfying the equilibrium conditions is a subset of a convex polyhedron whose extreme points are called extreme strategies. Checking complementarity conditions between the two different players' extreme strategies is necessary in order to determine all Nash equilibria.

Computation of equilibria can be performed in two phases:

- vertices enumeration for the two polyhedra associated to the game;

- analysis of the complementary conditions.

The aim of this work is to present an automatic procedure (written in C++) to determine N. E.. The first stage is implemented in two different ways: using the Balinski algorithm (1961), and using the Avis-Fukuda Algorithm (1992); the second phase is implemented referring to an algorithm proposed by H.M. Winkels(1979).

The work contains a short description of the characterization of the N.E. and of the main properties of convex polyhedra, and a full analysis of the algorithms used to perform vertices enumeration.

Computational experiences are listed in the paper, and it reports also small examples created to show how the algorithms works.

FROM VOTING GAMES TO GAMES IN NORMAL FORM

Gianfranco Gambarelli

Department of Mathematics, Statistics, Computer Science and Applications

University of Bergamo, P. Rosate, Bergamo, Italy

e-mail: gambarex@unibg.it

It is known that there are three principle forms of representing Games: normal, extended and characteristic. It has been proved that Games in extended form can be transfor med into normal form, to the advantage therefore of the relative theoretical results. This paper will propose a similar operation, transforming games in characteristic form (in particular, voting games) into normal form, enabling the former to exploit the existing literature on normal-form games. This transformation also enables us, using different logic, to tackle the problem of different probabilities of coalition formation. This new approach leads to different solutions from the classic ones.

COMPETITION AND COST OVERRUNS: OPTIMAL MISSPECIFICATION OF PROCUREMENT CONTRACTS

Juan-José Ganuza

Department of Economics, Universitat Pompeu Fabra, Carrer Ramón Trias Fargas, 23-27

Barcelona 08005, Spain, e-mail: juanjo.ganuza@econ.upf.es

Most cases of cost overruns in public procurement are related to important changes in the initial project design. This paper deals with the problem of design specification in public procurement and provides a rationale for design misspecification. We propose a model in which the sponsor decides how much to invest in design specification and awards competitively the project to a contractor. After the project has been awarded the sponsor engages in bilateral renegotiation with the contractor, in order to accommodate changes in the initial project's design that new information makes desirable. When procurement takes place in the presence of horizontally differentiated contractors, the design's specification level is seen to affect the resulting degree of competition. The paper highlights this interaction between market competition and design specification and shows that the sponsor's optimal strategy, when facing an imperfectly competitive market supply, is to underinvest in design specification so as to make significant cost overruns likely. Since no such misspecification occurs in a perfectly competitive market, cost overruns are seen to arise as a consequence of lack of competition in the procurement market.

COOPERATION AND COMPETITION IN INVENTORY GAMES

Ana Meca¹ , Ignacio García-Jurado² and Peter Borm³

¹ Center of Operations Research, Miguel Hernández University

e-mail: ana.meca@umh.es

² Department of Statistics and OR, Faculty of Mathematics

Santiago de Compostela University, 15782 Santiago de Compostela, Spain

ph: +34 981 563100, fax: +34 981 597054, e-mail: ignacio@zmat.usc.es

³CentER and Department of Econometrics and Operations Research

Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands

e-mail: P.E.M.Borm@kub.nl

In this paper we consider the class of basic inventory games. These games arise when several firms facing an EOQ model decide to collaborate and make their orders jointly. This collaboration produces a reduction of the total costs. The basic inventory games (from now on, IG) are those describing the cost allocation problem corresponding to this situation, in which players have to allocate the total costs when ordering together.

We first show that, when considering more complex inventory models (EOQ with shortages, EPQ, EPQ with shortages), the class of games arising from the corresponding cost allocation problems is the same IG. Then, we give a new axiomatic characterization of the SOC-rule (this rule consists of allocating the ordering costs proportionally to the squared individual optimal orders per time unit, as declared by the players, and proposing each player to pay his storage cost).

Finally, we consider that players can behave strategically when declaring their individual optimal orders per time unit. These parameters are crucial when using the SOC-rule. We define a non cooperative game modelling this situation, give necessary and sufficient conditions for the existence of a Nash equilibrium, and compute such equilibria when they exist.

AUCTIONS WITH RESERVATION PRICES

Arieh Gavious¹ and Aner Sela²

¹Ben-Gurion University, Faculty of Engineering Sciences, School of Industrial

Engineering and Management, P.O. Box 653, Beer-Sheva 84105, Israel

e-mail: ariehg@bgumail.bgu.ac.il

² Ben-Gurion University, Department of Economics, P.O. Box 653, Beer-Sheva 84105, Israel

e-mail: anersela@bgumail.bgu.ac.il

We study classical auction mechanisms with a single item and n  bidders who have private information about their valuations for the item. Each player's (bidders and seller) valuation for the item is drawn independently from the same interval according to a distribution function that is the same for all the players and that is common knowledge. All bidders bear a cost of bidding that is an increasing function of their bids. We show that in second-price auctions as well as in first-price auctions, independent of the form of each bidder's cost function, setting a reservation price is profitable for the seller. On the other hand, unlike first-price and second-price auctions, in all-pay first-price auctions, setting a reservation price is not necessarily profitable for the seller who wishes to maximize the average bid. In particular, we show that in all-pay auctions where bidders have convex cost functions (i.e., increasing marginal costs), it might not be profitable  for the seller facing a sufficiently large number of bidders to set a reservation price or, alternatively, an entry fee.

A VALUE FOR GAMES WITH COALITION STRUCTURES

Gérard Hamiache

CERESUR, Université de La Réunion, Faculté de Droit et d'Économie

15 Avenue René Cassin, 97715 Saint-Denis, B.P. 7151 MESSAG. CEDEX 9

Ile de La Réunion, France

ph.: +262 93 84 52, fax: +262 93 84 80, e-mail: gerard.hamiache@univ-reunion.fr

This paper presents an axiomatization of a value for games with coalition structures which is an alternative to the Owen Value. The motor of this new axiomatization is a consistency axiom based on an associated game, which is not a reduced game. The new value of unanimity games with coalition structures is more in keeping with the intuition that unity is strength. For example, in a unanimity game, the new value allocates to bigger coalitions a larger share of the total wealth.

ON CONVEXITY FOR NTU-GAMES

Ruud Hendrickx¹, Peter Borm² and Judith Timmer³

¹,² CentER and Department of Econometrics and Operations Research

Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands

e-mail: {ruud, P.E.M.Borm}@kub.nl

³ Faculty of Mathematical Sciences, University of Twente

P.O. Box 217, 7500 AE Enschede, The Netherlands

ph: +31 53 489 3419, fax: + 31 53 489 3069, e-mail: j.b.timmer@math.utwente.nl

For cooperative games with transferable utility, convexity has turned out to be an important and widely applicable concept. Convexity can be defined in a number of ways, each having its own specific attractions. Basically, these definitions fall into two categories, namely those based on a supermodular interpretation and those based on a marginalistic interpretation. For games with non-transferable utility, however, the literature only offers two kinds of convexity, ordinal and cardinal convexity, which both extend the supermodular interpretation. In this talk, we introduce and analyse three new types of convexity for NTU-games that generalise the marginalistic interpretation of convexity.

EVOLUTION AND LEARNING IN GAMES WITH RANDOMLY DISTURBED PAYOFFS

Josef Hofbauer¹ and William H. Sandholm²

¹ Institut für Mathematik, Universität Wien, Strudlhofgasse 4

A-1090 Vienna, Austria, e-mail: josef.hofbauer@univie.ac.at

http://mailbox.univie.ac.at/Josef.Hofbauer

² Department of Economics, University of Wisconsin,1180 Observatory

Drive Madison, WI 53706, USA, e-mail: whs@ssc.wisc.edu

http://www.ssc.wisc.edu/ whs

We consider four models of evolution and learning in games which rely on perturbations of payoffs, including stochastic fictitious play. In all cases, we establish global stability results for zero-sum games, games with an interior ESS, potential games, and supermodular games.

CONSTRAINED MONOTONICITY AND THE MEASUREMENT OF POWER

Manfred J. Holler

Institute of SocioEconomics (IAW) University of Hamburg, Von-Melle-Park 5,20146, Germany

ph. +4940/42838-4458, e-mail: holler@hermes1.econ.uni-hamburg.de

In this paper we will discuss constraints on the distribution of votes such that local monotonicity is satisfied for the Public Good Index. These results are compared to properties which are related to constraints on the redistribution of votes (such as implied by global monotonicity). The discussion shows that monotonicity is not a straightforward criterion for power measure.

ON THE ARTIFICIAL INTELLIGENCE METHODS APPROACH TO VOTING PROBLEMS

Jerzy Holubiec

Warsaw, Poland, e-mail: holubiec@ibspan.waw.pl

In the first part of the paper the application of machine learning method from examples to forecast voting behaviour is presented. Applied inductive learning method gives explicit decision rules of " if - then " type. Identification of the rule " if " one can formulate analysing positions taken by parliamentary groups during voting procedures. The next step " then " is the forecast of the voting results for determined parliamentary groups. Practical results for the Polish Parliament are given.

The second part of the paper is on the rough set approach to determination of the electorate preferences during voting procedures. Analysing the elements of the core one can receive even unexpected results. Some examples from Poland are presented.

At the end some consideration concerning the possibility of fuzzy and multiobjective games to voting problems are discussed.

DURABLE-GOODS MONOPOLISTS, SECOND-HAND MARKETS AND THE NATURE OF THE GOOD

Johannes Hörner^* and Morton Kamien

^* KGSM-MEDS, 2001 Sheridan Road, Leverone Hall, 5th Floor

Northwestern University, Evanston, IL 60208-2001

ph.: 847-491-8681, e-mail: j-horner@kellogg.nwu.edu

What is the optimal strategy of a durable goods monopolist when there is a second-hand market? This paper analyzes the subgame perfect Nash equilibria of an infinite horizon game between a monopolist and consumers who become resellers once they consumed the good. We examine the two extreme cases of information goods (freely replicable) and rival goods. In the first case, when agents are patient enough relative to the market size, a ``folk theorem'' holds and the monopolist may extract the whole surplus. Otherwise, its profits are lower and decrease in the discount factor. In particular, a patient monopolist in a very large market obtains an average payoff close to zero in every SPNE. Similarly, second-hand markets restrict the market power of a monopolist producing rival goods, although this monopolist is better off than in the first case. This difference is often substantial, and sometimes extreme.

SURPLUS SHARING OF EXCLUDABLE PUBLIC GOODS

R. Branzei¹, E. Inarra², S. Tijs³ and J.M. Zarzuelo

¹ Faculty of Computer Science, Äl.I. Cuza", University, Iasi, Romania

e-mail: branzeir@infoiasi.ro

² Universidad del Pais Vasco, Avd. Lehendakari Aguirre 83,

48015 Bilbao, Spain, e-mail: jepingae@bs.ehu.es

³ CentER and Department of Econometrics and Operations Research

Tilburg University, The Netherlands, e-mail: S.H.Tijs@kub.nl

In this paper we deal with excludable public goods without rivalry and no congestion. We consider that these goods can only be produced at finitely different levels and we allow agents to consume different amounts of the good. In particular, we assume that each potential consumer reports the quantity of the public good he wishes to consume and his willingness to pay. Only quantities reported are feasible provided the set of consumers jointly cover the cost of the largest amount. The technology of the production of the public good is given by the cost function which is increasing with quantity. Two relevant questions in this context are: Q.1 What will be the optimal quantity of the public good to be produced? Q.2 What will be the contributions of the consumers to the cost of the optimal quantity? Examples of these goods include all sorts of public facilities whenever are not congested such as highways, airports, parks, irrigation services, water treatment services, etc. The airport game discussed first by Littlechild and Thomson [1977] we believe provides the first systematic discussion of cost sharing of a non-rival and excludable good of this type. In the first part of this paper we determine the optimal level of the public good in connection to question Q.1. Question Q.2 will be tackle by constructing a cooperative game which allows us to use well known sharing rules from cooperative game theory. We give a suitable characterization of core elements which is useful in finding the extreme points of the core starting with introduced the adjusted Bird rule. It turns out that these games are convex. Then we introduce a non-cooperative game related to the described situation. Using our core representation we can prove that the strong Nash equilibria of the non-cooperative game correspond to the core elements of the cooperative game. We finally propose an interesting solution to finance the production of the public good. It happens that this proposal coincides with the nucleolus of the class of cooperative games defined. Some remarks about the use of alternative solution concepts in this setting conclude the paper.

COALITIONALLY MONOTONIC SET-SOLUTIONS FOR COOPERATIVE TU GAMES

Josep M. Izquierdo¹ and Carles Rafels²

¹ Department of Actuarial, Financial and Economic Mathematics, and CREB

University of Barcelona, Avda. Diagonal 690 E-08034 Barcelona, Spain

e-mail: izquier@eco.ub.es

² Department of Actuarial, Financial and Economic Mathematics, and CREB

University of Barcelona, Avda. Diagonal 690 E-08034 Barcelona, Spain

rafels@eco.ub.es

A static comparative study on set-solutions for cooperative TU games is carried out. The analysis focuses on studying the compatibility between two classical and sensible properties introduced by Young (1985) in the context of single valued solutions, namely core-selection and coalitional monotonicity. As a main result, it is showed that coalitional monotonicity is not only incompatible with the core-selection property but also with a more general property called bargaining-selection property. This new impossibility result strengthens the existing trade-off between these kind of interesting and intuitive economic properties.

Positive results about compatibility between desirable economic properties are given changing the core-selection requirement by the core-extension property. In particular, the Weber set, the Weber set of order k and the individual core are examples of set-solutions meeting both properties. Actually, the core itself, not being in general coalitionally monotonic, is always the intersection of all its coalitionally monotonic extensions.

Finally, we show that the core - the most important set-solution concept - still preserves some kind of weak coalitional property.

STOCHASTIC BETTER-REPLY DYNAMICS IN GAMES

Jens Josephson

Stockholm School of Economics, Department of Economics,

P.O. Box 6501, 113 83 Stockholm, Sweden, e-mail: jj74@cornell.edu

In Young (1993, 1998) agents are recurrently matched to play a finite game and almost always play a myopic best reply to a frequency distribution based on a sample from the recent history of play. He proves that in a generic class of finite n-player games, as the mutation rate tends to zero, only strategies in certain minimal sets closed under best replies will be played with positive probability. In this paper I alter Young's behavioral assumption and allow agents to choose not only best replies, but also better replies. The better-reply correspondence maps distributions over the player's own and her opponents' strategies to those pure strategies which gives the player a weakly better expected payoff. I prove that in all finite n-player games, the limiting distribution will put positive probability only on strategies in certain minimal sets closed under better replies. This result is consistent with and extends Ritzberger's and Weibull's (1995) results on the equivalence of asymptotically stable strategy-sets and closed sets under better replies in a deterministic continuous-time model with sign-preserving selection dynamics.

STRONGLY SOLVABLE GAME FORMS: A CHARACTERIZATION

Joseph Abdou¹ and Hans Keiding²

¹ CERMSEM, Maison des Sciences Economiques, Universite Paris 1

106-112 boulevard de l'Hopital, 75647 Paris Cedex 13, France

ph.:33-144078303, fax:33-144078301, e-mail: abdou@univ-paris1.fr

²Institute of Economics, University of Copenhagen

Studiestraede 6, K-1455 Copenhagen K., Denmark

e-mail: Hans.Keiding@pop.oko.ku.dk

A game form is strongly solvable if for each assigment of individual preferences over outcomes, the resulting game possesses a strong Nash equilibrium. Several necessary conditions for strong solvability can be found in the literature; in this paper we give conditions which are both necesssary and sufficient. These conditions use an extension of the concept of the effectivity function associated with the game form and property of acyclicity of this extended effectivity function.

SEMIPROPORTIONAL VALUES FOR TU GAMES

Anna B.Khmelnitskaya¹ and Theo S.H.Driessen²

¹SPb Institute for Economics and Mathematics Russian Academy of Sciences

1 Tchaikovsky St., 191187 St.Petersburg, Russia

e-mail: anna@AK3141.spb.edu

² Faculty of Mathematical Sciences, University of Twente

P.O. Box 217, 7500 AE Enschede, The Netherlands

e-mail: t.s.h.driessen@math.utwente.nl

The goal of the paper is to introduce a family of values for cooperative TU games that are proportional for two-person games and as well satisfying some combinatorial structure composed by contributions of complementary coalitions or, to less extent, marginalistic contributions by players.

EFFICIENT FAIR DIVISION: HELP THE WORST OFF OR AVOID ENVY?

Steven J. Brams¹ and Daniel King²

¹ Dept. of Politics 715 Broadway, 4th Fl.New York University New York, NY 10003, USA

ph.: (212) 998-8510, e-mail: steven.brams@nyu.edu

² Dept. of Mathematics Sarah Lawrence College One Mead Way

Bronxville, NY 10708, USA, ph.: (914) 395-2424, e-mail: dking@slc.edu

Two or more players rank a set of indivisible items from best to worst. A fair allocation of the items is efficient and may satisfy such distributional properties as

(1) maximin: maximizes the minimum rank of items received by the players;

(2) Borda maximin: maximizes the minimum Borda score of items received by by the players;

(3) envy-avoidance: prevents, insofar as possible, any player from receiving a set of items that it considers worse than a set received by another player.

For some ranking of the items by the players, the conflict between properties (1) and (3) is inescapable; while there is not always a conflict between properties (2) and (3), there are preference profiles in which such a conflict is unavoidable. On the brighter side, if maximin and Borda maximin allocations do not expunge envy entirely, there is always an efficient allocation that does not ensure envy. Maximin and Borda maximin allocations may require allocating unequal numbers of items to the players. Thereby ïnequality" may be a virtue rather than a vice. Moreover, equal allocations do not necessarily maximize the sum of the Borda scores of all players, which may be viewed as an indicator of overall well-being. Computer calculations show that maximin allocations lead to envy quite often, but Borda maximin allocations do so only rarely. Implications of the theoretical findings for real-world fair-division problems are briefly discussed.

RESOURCE-MONOTONICITY FOR HOUSE ALLOCATION PROBLEMS

Lars Ehlers, Bettina Klaus^* and Szilvia Papai

^* Department of Economics, 368 College of Business Administration

University of Nebraska at Lincoln, Lincoln, NE 68588-0489, USA

ph: (402) 472-2129, fax: (402) 472-9700, e-mail:bklaus1@unl.edu

http://www.cba.unl.edu/faculty/bklaus.html

We study the simple model of assigning indivisible objects (e.g., houses, jobs, offices, etc.) to agents. Each agent receives at most one object and monetary compensations are not possible. We completely describe the allocation rules satisfying Pareto efficiency, independence of irrelevant objects (coalitional strategy-proofness), and resource-monotonicity. The characterized rules assign the objects by an iterative procedure in which at any time no more than two agents "trade" objects from their hierarchically specified ëndowments."

WEAK STABILITY AND A BARGAINING SET FOR THE MARRIAGE MODEL

Flip Klijn¹ and Jordi Massó²

¹ Departamento de Estatística e I.O., Universidade de Vigo, Vigo, Spain

e-mail: fklijn@uvigo.es

² Departament d'Economia i d'Història Econòmica and CODE

Universitat Autònoma de Barcelona, Barcelona, Spain

e-mail: jordi.masso@uab.es

In this note we introduce weak stability, a relaxation of the concept of stability for the marriage model by assuming that the agents are no longer myopic in choosing a blocking pair. The new concept is based on threats within blocking pairs: an individually rational matching is weakly stable if for every blocking pair one of the members can find a more attractive partner with whom he forms another blocking pair for the original matching. Our main result is that under the assumption of strict preferences, the set of weakly stable and weakly efficient matchings coincides with the bargaining set of Zhou (1994) for this context.

A DUOPOLY MODEL OF COMMERCIAL FISHING: THE ROLE OF EXPECTATIONS

Michael Kopel¹ and Gian-Italo Bischi²

¹ Department of Managerial Economics and Industrial Organization

University of Technology, Vienna, Austria

e-mail: kopel@ebwnov.tuwien.ac.at

² Istituto di Scienze Economiche, University of Urbino, Italy

e-mail: bischi@econ.uniurb.it

We consider a dynamic model of international commercial fishing under imperfect competition. Two countries harvest fish in an open access sea and sell the fish not only in their own country but also in the other country, i.e. each firm serves a domestic and a foreign market. The dynamics of the fish population in the common sea follows a logistic law for natural growth with an extra mortality term due to harvesting. The countries do not have perfect knowledge of the fish stock. Instead they have a common expectation of the fish stock available in the following period, which they use to determine their optimal harvesting amounts. This gives rise to an expectations feedback system, with expectations (on the fish stock) yielding harvesting amounts, and harvesting (together with reproduction) determining the actual fish stock. We analyze the number of equilibria of this system and determine the local and global stability for different expectation schemes (e.g. naive and adaptive). In particular, we show that the basins of attraction of these equilibria undergo important changes as parameter vary.

SOCIAL EQUILIBRIA FOR COMPETITIVE RESOURCE ALLOCATION MODELS

Victor Domansky and Victoria Kreps^*

^* St.Petersburg Institute for Economics and Mathematics Russian Academy

of Sciences, Tchaikovskogo 1, 191187 St.Petersburg, Russia

fax: 7812 273 7953, e-mail: vita@agd.stud.pu.ru

We study multistage models of resource allocation with several agents. Individual interests of agents are determined with their private objective functions, given by discounted sums of one-step utilities over the whole period of planning. Social preferences are expressed with public objective functions. We define the utility functions in such a way that aggregation of individual utilities results in the exponential public utility. Constructed private objective functions are competitive in the sense that the actions of agents affect the incomes of partners. Consequently, these models are treated as non- cooperative dynamic games. We construct the Nash Equilibria for these games satisfying the criteria of maximization of public utility. Both finite and infinite horizons of planning are examined.

VON NEUMANN-MORGENSTERN STABLE SETS IN N-PERSON BARGAINING PROBLEMS

Klaus Kultti¹ and Hannu Vartiainen²

¹Department of Economics, Helsinki School of Economics Business Administration

Pohjolnen rautatiekatu 21 b, 00100, Helsinki, Finland

e-mail: kultti@hkkk.fi

²Department of Economics, University of Helsinki

We use von Neumann-Morgenstern stable sets as a solution concept in a cake sharing problem that emanates, e.g. from Rubinsteinís alternating offers bargaining situation. It turns out that regardless of the number of agents there exists a unique stable set. The stable set has several attractive features: a) Not all elements are Pareto-optimal, b) In the case of linear utility the extreme points of the stable set correspond to the demands of the players in the corresponding alternating offers bargaining, c) When the playersí discount factors approach unity the stable set approaches the Nash-bargaining solution.

CONDORCET CHOICE, SCORING RULES AND THE OSTROGORSKI PARADOX

Gilbert Laffond¹ and Jean Laine²

¹ Laboratoire d'économétrie, Conservatoire National

des Arts et Métiers, 2 rue Conté, 75003 Paris, France

& CREST-LSM, Campus de Ker Lann, rue Blaise Pascal, 35170 Bruz, France

e-mail: laffond@vcnam.cnam.fr

²Ecole Nationale de la Statistique et de l'Analyse de l'Information (ENSAI)

and CREST-LSM, Campus de Ker Lann, rue Blaise Pascal, 35170 Bruz, France

e-mail: jlaine@ensai.fr

The Ostrogorski paradox refers to the fact that, when a society faces finitely many dichotomous issues, choosing issue-wise according to the majority rule may lead to a majority defeated overall outcome. This paper studies a strong version of the paradox, stating that the issue-wise majority winner may even be rejected by some social choice rule. Attention is paid to scoring rules and to two specific Condorcet rules, the Uncovered Set (UC) and the Top-Cycle (TC). We prove that a strong paradox may prevail for UC. Moreover, restrictions on issue-wise majority margins avoiding the strong paradox are the same as those avoiding the Ostrogorski paradox. Furthermore, there is no paradox for TC ; however, either the Ostrogorski paradox does not occur, or TC selects every possible outcome. Finally, we prove the existence of a unique paradox-free scoring rule.

ADVERTISING IN A DIFFERENTIAL GAME OF SPATIAL COMPETITION

Giorgia Bertuzzi¹ and Luca Lambertini²

¹Department of Economics, University of Bologna, Strada Maggiore 45, 40125 Bologna, Italy

e-mail: bergiot@tiscalinet.it

²Department of Economics, University of Bologna, Strada Maggiore 45, 40125 Bologna, Italia

ph.: +39-051-2092623, fax: +39-051-2092664, e-mail: lamberti@spbo.unibo.it

http://www.dse.unibo.it/lamberti/lamberti.htm

We investigate a dynamic duopoly game with horizontal product differentiation, to show that the standard approach to spatial competition fails to produce a pure strategy equilibrium in prices when treated in a differential game framework. This holds independently of the shape of the transportation cost function. Then, we introduce an endogenous costs associated with the choice of location and characterise the open-loop and closed-loop equilibria of the model, showing that in the closed-loop case firms invest more in product differentiation and less in advertising, than they do in the open-loop setting. This happens because the gains from product differentiation can be more easily internalised than those associated with advertising.

VOTING POWER MEASURES AND PARADOXES REVISITED

Annick Laruelle¹ and Federico Valenciano²

¹ Departamento de Economia Aplicada IV, Universidad del Pais Vasco

Avenida Lehendakari Aguirre, 83, 48015 Bilbao, Spain, e-mail: elxlaxxa@bsdx01.bs.ehu.es

² Departamento de Economia Aplicada IV, Universidad del Pais Vasco

Avenida Lehendakari Aguirre, 83, 48015 Bilbao, Spain

ph.: 34-946013696, fax: 34-946017028, e-mail: elpvallf@bs.ehu.es

In order to compare power indices, some authors have proposed natural properties that the power indices are supposed to satisfy. The violation of these properties by the indices are called paradoxes. In the literature there exist various paradoxes. In this paper we revisit these paradoxes when power is directly defined as the probability of being decisive in a vote.

COMPUTING POWER INDICES FOR LARGE WEIGHTED VOTING GAMES

Dennis Leech

Department of Economics, University of Warwick, Coventry CV4 7AL, U.K.

ph: (+44)(0)(24)76523047, fax: (+44)(0)(24)76523032

e-mail: ecrac@csv.warwick.ac.uk, http://www.warwick.ac.uk/fac/soc/Economics/leech/

Voting Power Indices enable the analysis of the distribution of power in a legislature or voting body in which different members have different numbers of votes. Although this approach to the measurement of power, based on co-operative game theory, has been known for a long time its empirical application has been to some extent limited, in part by the difficulty of computing the indices when there are many players. This paper presents new algorithms for computing the classical power indices, those of Shapley and Shubik and of Banzhaf, which are essentially modifications of approximation methods due to Owen, and have been shown to work well in real applications. They are of most utility in situations where both the number of players is large and their voting weights are very non-uniform, some members having considerably larger numbers of votes than others, where Owen's approximation methods are likely to be least accurate.

RELATIVE UTILITIES

Galit Ashkenazi and Ehud Lehrer^*

^* School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

home page: www.math.tau.ac.il/ lehrer

The primitive of the model is a partial order that tells which of two agent is better treated. This partial order may depend on the initial as well as of the final endowments of the agents. We give natural sufficient conditions that ensure that this partial order can be represented in a linear fashion. This implies the existence of an agent's utility function that depends on her own situation as well as on that of others. A connection to well-being indices in the context of cooperative game is also given.

BOARD EFFICIENCY AND INTERNAL CORPORATE CONTROL MECHANISM

Clara Graziano¹ and Annalisa Luporini²

¹ Dipartimento di Scienze economiche, Università di Udine

via Tomadini 30/a, 33100 UDINE, Italy

e-mail: Clara.Graziano@dse.uniud.it

² Dipartimento di Scienze Economiche e Statistiche

Università di Trieste, Piazzale europa 1, 34127 Trieste, Italy

e-mail: Annalisa.Luporini@econ.univ.trieste.it

We analyse the interactions between internal and external control mechanisms in a framework in which both the CEO's quality and the type of the board of directors are unknown. The novel aspect of our paper is that the board has two different tasks: to select the CEO and to decide whether to retain or dismiss him after observing a signal. The type of the board is defined by its ability of selecting the good CEO. Then, the quality of the CEO depends on the type of the board and the dismissal/retention decision provides information not only on the quality of the CEO but also on the board's type. Since manager dismissal may signal that the board is inefficient and this in turn may trigger a takeover, the board prefers not to dismiss the manager even if it received a very low signal on his/her quality. Our model endogenously derive a collusive behavior between board and CEO in which the board does not fire a bad CEO. This behavior emerges as an attempt of hiding mistakes in the first task, CEO selection, by distorting the second task, the CEO retention/dismissal decision. Many empirical studies have shown that board of directors are an inefficient in their monitoring task. Board of directors have been criticized for being "hostages" of the CEO that they are supposed to monitor and fire. Our model shows that such a behavior can be the board's optimal reponse to the takeover threat.

SAYING IT WITH A SMILE: CAN SECRET HANDSHAKES ENHANCE COORDINATION?

Paola Manzini¹, Abdolkarim Sadrieh², and Nicolaas J. Vriend³

¹ Department of Economics, Queen Mary, University of London

Mile End Road, London E1 4NS, UK

ph: 020-7882 5083, e-mail: p.manzini@qmw.ac.uk

² Department of Economics and CentER, Tilburg University

³Department of Economics, Queen Mary, University of London

We investigate in an experimental setting whether a non-specific signal (secret handshake) will acquire endogenously a commonly understood meaning and will be established as a means to achieve efficient coordination in a simple coordination game. Randomly matched pairs of subjects play a minimum effort game. To study the emergence of a secret handshake, we add a payoff irrelevant signaling stage to the game. Prior to play, each subject must report his/her ³state² by choosing between the states ³ready² and ³smiling². The state choices are reported to the two players before they chose their effort level. Since there is a joint interest of coordinating on the efficient high effort equilibrium, but the risk of costly coordination failure is substantial, the state signal may be turned into a secret handshake, i.e. into a coordination device. We find that subjects actually try to coordinate using the state signal. On average, chosen effort levels are significantly higher in pairs in which both players report that they are smiling than in pairs reporting the ready state. Hence, when subjects choose to report a smiling state instead of a ready state, they often intend to send a message to their counterpart suggesting a coordination on a high effort level.

ALLIANCES AND NEGOTIATIONS

Paola Manzini¹ and Marco Mariotti²

¹ Department of Economics, Queen Mary, University of London, Mile End Road, London E1 4NS, UK

ph: 020-7882 5083, e-mail: p.manzini@qmw.ac.uk

² Department of Economics, University of Exeter

Streatham Court, Rennes Drive, Exeter EX4 4PU, UK

ph: 01392-263219, e-mail: m.mariotti@exeter.ac.uk

A characteristic of many bargaining situations is that the negotiators represents the interests of a set of parties (trade unions, political parties, etc.) with composite interests, whose bargaining behaviour is regulated by some collective decision mechanism. In this paper we provide a natural model of such circumstances, and show how different preference aggregation procedures within the composite player affect the bargaining outcome. In particular we find that unanimity procedures lead to `more aggressive' behaviour than majority procedures, and that procedures which introduce minimum safeguards for the members of an alliance may result in agreements that are worse than without those safeguards.

SELF-ENFORCING AGREEMENT ON TRANSBOUNDARY POLLUTION AND INTERNATIONAL TRADE: A DIFFERENTIAL GAME

F. Cabo¹, E. Escudero² and G. Martin-Herran³

¹ Dept. Economí a Aplicada (Matemáticas). Universidad de Valladolid

Avda. Valle Esgueva, 6. 47011-Valladolid. Spain. e-mail: pcabo@eco.uva.es

²Dept. Fundamentos de Análisis Económico. Universidad de Valladolid

Avda. Valle Esgueva, 6. 47011-Valladolid. Spain. e-mail: ee@eco.uva.es

³Dpto. Economí a Aplicada (Matemáticas). Universidad de Valladolid

Avda. Valle Esgueva, 6. 47011-Valladolid. Spain. e-mail: guiomar@eco.uva.es

To achieve an international environmental agreement, it is assumed that countries negotiate on the environmental issue as well as on the international trade issue. A transboundary pollution problem is analyzed within a North-South trade framework. These two different regions are interconnected by a double relationship. On the one hand, pollution is a by-product of production in each regions. On the other hand, a good produced in the North is traded to the South. These two regions play a differential game. The South decides the demand for the intermediate good and hence total production. As a consequence the emission of pollutants is determined. The North fixes the international price for this traded good. Pollution decline due to cooperation would make both regions better-off (specially the North, which values the environment to a greater extent). Likewise, the South would benefit from a lower price under cooperation. Thus, one country gains on the environment whereas the other gains on trade. If cooperation fails because cheating is detected, players apply threat strategies under which both regions obtain less welfare. These subgame-perfect strategies are credible and consequently, a self-enforcing agreement is achieved where no commitment is necessary.

INFORMATION RESTRICTION IN COOPERATION AND THE WEBER SET

Javier Martínez-de-Albéniz and Carles Rafels

Universitat de Barcelona and CREB, Avda. Diagonal 690 E-08034 Barcelona, Spain

e-mail: {albeniz, rafels}@eco.ub.es

The standard model of cooperative games assumes that no restriction to cooperation is given, and all coalitions can be formed and evaluated. But in practice, cost allocation methods do not make use of the worth of all coalitions.

We introduce information restrictions in a T.U. cooperative game by 'forgetting' the worth of several coalitions, because we do not want to evaluate it, or because we have only limited information on the game. Informational costs are also to be considered.

We consider, for any T.U. cooperative game (N,v), the game (N,v_k), where the worth of any coalition of size k or less is substituted by the modular game arising from the individual worth.

We study the marginal worth vectors and their convex hull, the so-called Weber set, from the original game and the transformed one, which is called the Weber set of level k.

It is proved that the core is included in the Weber set of level k, for any k, and that the Weber sets of consecutive levels form a chain if and only if the game is 0-monotonic. Even if the game is not 0-monotonic, it is proved that intersection of the Weber sets for consecutive levels is always not empty, what is not the case for non-consecutive ones. Several consequences of this fact are given.

COMMUNICATION LEADING TO EPSILON-NASH EQUILIBRIUM

Takashi Matsuhisa

Ibaraki National College of Technology, Nakne 866, Hitachinaka-shi

Ibaraki-ken 312-8508, Japan

ph.: +81292712865, fax: +81292712857, e-mail: mathisa@ge.ibaraki-ct.ac.jp

A pre-play communication-process in a knowledge system is presented which leads to an e-Nash equilibrium of a strategic form game. In the communication process each player is supposed to have the two prior distributions: the common prior and the subjective prior. The latter represents a perturbation unable to be controlled by the player when the other players play actions. He/she predicts the other players' actions, and he/she communicates privately his/her conjecture through message according to a protocol. Each player receiving the messages learns and revises his/her conjecture. We show that the profile of players' conjectures in the revision process leads to an e-Nash equilibrium of a game in the long run if the protocol contains no cycle.

INVENTORY GAMES WITH CONTINUOUS DISCOUNT

Andres Toledo Casado¹ and Ana Meca Martinez²

¹ Centro de Investigación Operativa. Universidad Miguel Hernández de Elche

Avd. Ferrocerril s/n 3202 Elche, Spain

e-mail: andres_toledo@dmr.com

² Centro de Investigación Operativa. Universidad Miguel Hernández de Elche

Avd. Ferrocerril s/n 3202 Elche, Spain

e-mail: ana.meca@umh.es

In Toledo and Meca (2001) several inventory models involving different price policies have been studied. In this work, following the philosophy of Meca et al.(1999), we analyze different cooperative games that model those situations in which several firms or shops, demanding a same product, cooperate according to different criterions. Each one of these firms has its own private demand and its own private storage possibilities for the good. There is a single supplier to whom all firms place their orders at the same ordering cost. If the purchasing cost established by the supplier can be model by certain decreasing function depending on the order size, together with several smoothness properties, these firms can reduced their average inventory cost per unit time by means of placing orders simultaneously or ordering and holding together their demand in one warehouse. To study all situations derived from these two cooperation criterions we define a class of cooperative TU games that involves that situations in which the mentioned price policy is one of the given in Toledo and Meca (2001) or the schedule discount called all-units discount (Tersine, 1994). This class of games is called Inventory Games with Discount in the Purchasing Cost. Next we define different allocation rules that, in several cases, allow us to affirm that this class is balanced. Finally, we present several particular cooperative games that the class of Inventory Game with Discount in Purchasing Cost involves.

FORMATION OF PREFERENCES AND STRATEGIC ANALYSIS OF GAMES

Nimrod Megiddo

IBM Almaden Research Center, 650 Harry Road, K53-B2

San Jose, CA 95120, U.S.A., e-mail: megiddo@almaden.ibm.com

Game-theoretic analysis is typically based on the preferences of players over the set of outcomes of the game. It is widely recognized that players may find it quite difficult to formulate their own preferences. The problem is even more severe because each player has to know the preferences of all the other players; otherwise, the game has to be reformulated according to Harsanyi's Bayesian theory of games of incomplete information, which involves beliefs of players about each other's preferences. Despite all these difficulties, very often the analysis of a game begins with a representation in which outcomes are specified as utility tuples. It is also understood that the outcome of the game in a single play is not merely what happens at the end of the play but rather consists of the entire play.

The purpose of this paper is to argue that sometimes the determination of preferences and the analysis of the game itself are inseparable. This inseparability is due in part to the phenomenon that the strategy that a player chooses may affect the preferences of other players.

Players' preferences over outcomes obviously depend not only on what they receive during the play that terminates in the outcome, but also on what other players receive during the play. For example, suppose a player has to choose between getting a dollar while another player sustains some damage, versus not getting the dollar while the other player does not sustain any damage. For most people, the main factor in the preference is what happens to the other player rather than the dollar payoff. However, a player may revise his preferences after learning something about the other players during the game.

A more severe problem is that the utility of a player from a play may depend not only on what each and every player receives during the play, but also on what the players could receive in other plays. In the game depicted above the payoffs to the players are in dollars rather than utility units. Conventional game-theoretic analysis requires fixing the preferences of the players over the outcome set A,B,C,D before the decisions can be analyzed, and if these preferences are not common knowledge, then a larger tree has to be constructed, to account for the different possible types of the players. The issue here is what would player 1 do if the play reaches the second node of player 1. The decision would depend on how seriously player 1 would take the ïnsult" involved in player 2's playing Right. Further complications arise if the node B is replaced by some subgame.

(AVERAGE-) CONVEXITY OF COMMON POOL AND OLIGOPOLY TU-GAMES

Theo Driessen¹ and Holger Meinhardt²

¹ Faculty of Mathematical Sciences, University of Twente

P.O. Box 217, 7500 AE Enschede, The Netherlands

e-mail: t.s.h.driessen@math.utwente.nl

² Institute for Statistics and Economic Theory, University of Karlsruhe

P.O. Box 69 80, Zirkel 2, D-76128 Karlsruhe, Germany

e-mail: hme@vwl3.wiwi.uni-karlsruhe.de

The paper studies both the convexity and average-convexity properties for a particular class of cooperative TU-games called common pool games. The common pool situation involves a private cost function as well as a (weakly decreasing) average joint production function. Firstly, it is shown that, if the relevant cost function is a linear function, then the common pool games are convex games. The convexity, however, fails whenever cost functions are arbitrary. We present sufficient conditions involving the cost functions (like weakly decreasing marginal costs as well as weakly decreasing average costs) and the average joint production function in order to guarantee the convexity of the common pool game. A similar approach is effective to investigate a relaxation of the convexity property known as the average-convexity property for a cooperative game. An example illustrates that oligopoly games are a special case of common pool games whenever the average joint production function represents an inverse demand function.

AN INTEGRATED TRANSPORTATION SYSTEM FOR ALACANT'S STUDENTS. UNIVERCITY

J. Sánchez-Soriano, N. Llorca, A. Meca, E. Molina, M. Pulido

Centro de Investigación Operativa. Universidad Miguel Hernández de Elche

Avd. Ferrocerril s/n 3202 Elche, Spain

e-mail: {joaquin,nllorca,anameca,e.molina,manpul}@umh.es

The university system in Alacant is very disperse; however, the social structure in Spain is focused on the family as a cellular unit and the majority of university students live with their parents while they study. Moreover, the public transportation system in this province is poor and expensive. Therefore, there is a real social need for a transportation system capable of efficiently connecting the villages and cities of Alacant to the different university campus. The Comunitat Valenciana government is aware of this problem and it has established a grant mechanism to reduce the fees and to encourage private firms to offer the transportation service which society is demanding. This transportation grants system is such that Student Unions, Town Halls and private firms can all apply for financial support. Nowadays, all these entities fail to cooperate among each other. Yet, there are many advantages to changing this situation and promoting a cooperative setting.

On this basis, we have proposed a mechanism to design a better and cheaper transportation system for university students. To favour cooperation among all Town Halls and Student Unions in the province of Alacant is the main point of our proposal. The analysis of this problem is carried out in order to make a proposal to the Diputació d'Alacant, where all Town Halls are represented. This will be the intermediary between the students (through their corresponding Town Hall) and the Generalitat Valenciana for arranging an integrated university transport system in the whole province. So as to design such a centralized transportation system we consider three different steps. The last one consists on evaluating all the costs derived from the hiring and exploitation of that system and distributing them among the agents involved in this situation: the customers, which are the students, and the governmental institutions (Generalitat Valenciana and Diputació). The last stage is precisely the one which has been developed in this paper and in which Game Theory is used. We will proceed as follows: first of all, we will divide the full network into various transportation subsystems, which we will refer to as trees of routes with common sections. Then, we distribute the cost of each subsystem among the students who use the set of routes which form the transportation subnetwork, and the Diputació. The cost allocated to the Generalitat Valenciana is determined by the rules of the grant system, and it is directly deducted from the global amount that a ``city'' (i.e., all its students) must pay.

NONCOOPERATIVE BARGAINING IN APEX GAMES AND THE KERNEL

Maria Montero

University of Dortmund, Department of Economics (Micro), D-44221

Dortmund, Germany, e-mail: maria.montero@wiso.uni-dortmund.de

This paper studies non-cooperative bargaining with random proposers in apex games. Two different protocols are considered: the egalitarian protocol, which selects each player to be the proposer with the same probability, and the proportional protocol, which selects each player with a probability proportional to his number of votes. Expected equilibrium payoffs coincide with the kernel for the grand coalition regardless of the protocol. The protocol influences the probabilities of the different minimal winning coalitions being formed and expected payoffs conditional on a coalition. Given a coalition of the apex player with a minor player, an egalitarian protocol yields a nearly equal split whereas a proportional protocol leads to a proportional split.

OVERESTIMATION AXIOMS FOR WEIGHTED ENTROPY AND LEAST SQUARE SOLUTIONS OF COOPERATIVE GAMES

Natalia I. Naumova

Dep. of Mathematics and Mechanics, St.Petersburg State University

St.Petersburg, Russia, e-mail: Natalia.Naumova@pobox.spbu.ru

We consider axiomatic justification of single-point solution for cooperative transferable utility games. Values of characteristic functions are interpreted as requests of coalitions in bargaining process.

The main proposed axiom is the following. For each partition P of the set of players, the special overestimation of the requests of coalitions belonging to P preserves the solution point. If this special overestimation is proportional, then the axiom and the continuity assumption lead us to the maximal weighted entropy solution with weights equal to values of coalitions. If we replace the proportional principle of overestimation by equal sharing principle, then the same arguments lead us to the least square solution.

MINIMUM COST SPANNING TREE GAMES AND POPULATION MONOTONIC ALLOCATION SCHEMES

Henk Norde^*, Stefano Moretti and Stef Tijs

^* CentER and Department of Econometrics and Operations Research

Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands

e-mail: H.Norde@kub.nl

In this paper we present the Subtraction Algorithm that computes for every classical minimum cost spanning tree game a population monotonic allocation scheme. As a basis for this algorithm serves a decomposition theorem that shows that every minimum cost spanning tree game can be written as a nonnegative combination of minimum cost spanning tree games corresponding to 0-1 cost functions. It turns out that the Subtraction Algorithm is closely related to the famous algorithm of Kruskal for the determination of minimum cost spanning trees. For variants of the classical minimum cost spanning tree games we show that population monotonic allocation schemes do not necessarily exist.

TWO-PERSON MULTICRITERIAL GAMES MODELING THE PROBLEM OF NETWORK SURVIVABILITY

E.M. Kreines¹, Yu.E. Malashenko², N.M. Novikova³ and I.I. Pospelova⁴

¹System Analysis dept., Faculty of Computational Mathematics and Cybernetics

Moscow State University, Vorobjovy Gory, Moscow, 119899, Russia

² Computing Center of RAS, Vavilov st. 40, 117967 Moscow GSP-1, Russia

e-mail: malashen@ccas.ru

³ Computing Center of RAS, Vavilov st. 40, 117967 Moscow GSP-1, Russia

e-mail: nnovik@ccas.ru

⁴ OR dept., Faculty of Computational Mathematics and Cybernetics, Moscow State

University, Vorobjovy Gory, Moscow, 119899, Russia, e-mail: irina@ccas.ru

In multicriterial decision making all the criteria are important, but the other model is considered in multicriterial antagonistic games (MAGs). It corresponds to the multicriterial case where an arbitrary criterion (may be the single one) is to be maximized. This nonstandard formulation is also multicriterial since we have not fixed the maximized criterion beforehand. The model appears in cooperative games without side-payments.

Opposite to MAG, a multicriterial zero-sum game (MZSG) is a two-person game with vector payoff function of Player II equal to minus vector payoff function of Player I. Thus, the both players of MZSG are multicriterial decision makers in a common sense. In order to differ MAG from MZSG, we consider two types of vector estimates: guaranteed and weak.

In the problem of survivability analysis of a multicommodity network, Player I is interested in maximizing the multiflow vector, i.e., the flows of different commodities that can be simultaneously shipped via the network. Player II reduces capacity of network edges in order to minimize the multiflow (in MZSG) or an arbitrary flow (in MAG). In the paper, we consider and compare the both games. Also two-stage formulations of the games are investigated.

THE FAIR SOLUTION FOR THE ASSIGNMENT GAME

Marina Núñez and Carles Rafels

Universitat de Barcelona, Departament de Matematica, Economica, Financiera

i Actuarial, Av. Diagonal 690, 08034 Barcelona, Spain

e-mail: {mnunez, rafels}@eco.ub.es

The fair solution defined by Thompson for assignment games as the midpoint of the buyers-optimal core allocation and the sellers-optimal core allocation can also be defined in an extended class of games: the assignment games with reservation prices, which are strategically equivalent to an assignment game. For convex assignment games, the fair solution coincides with the Shapley value. When an assignment game with reservation prices is not convex, a convex game can be defined which is strategically equivalent to an assignment game and has the same buyers-optimal core allocation and the same sellers-optimal core allocation as the initial game.

This first result will be used to give an axiomatization of the fair solution in the wider class of assignment games with reservation prices, in terms of two axioms: coincidence with the Shapley value for convex games and a restricted form of strong monotonicity.

MONOTONIC CHOICE FUNCTIONS, POWER AND SCHWARTZ'S PARADOX

Hannu Nurmi

Department of Political Science, University of Turku, FIN-20014 TURKU, Finland

ph: +358-2-3335389, fax: +358-2-3335090, e-mail: hnurmi@utu.fi

The standard requirement of monotonicity of a voting procedure states that an improvement in the ranking of the winning alternative, ceteris paribus, should not make it non-winning. This property has an obvious counterpart in MCDM contexts. A concept apparently closely linked to monotonicity is known as the participation axiom which requires that it should never be disadvantageous for a voter to abstain rather than to vote according to his/her preferences.

EXISTENCE OF NASH-COURNOT EQUILIBRIUM IN OLIGOPOLY GAME WITH POLLUTION TREATMENT COST-SHARING

Koji Okuguchi¹ and Ferenc Szidarovszky²

¹ Department of Economics and Information, Gifu Shotoku Gakuen University

Nakauzura,Gifu-shi,Gifu-ken 500-8288,Japan

e-mail: okuguchi@gifu.shotoku.ac.jp

² Systems and Industrial Engineering Department

The University of Arizona, Tucson, Arizona 85721-0020,USA

e-mail: szidar@sie.arizona.edu

A cost sharing game is formulated for oligopoly where polluting firms are required to bear the industry cost of pollution treatment in proportion to the ratios of their outputs to the total industry output. First,a unique Nash(-Cournot) equilibrium is shown to exist under two fundamental assumptions, which will then be interpreted from economic point of view. Second,the effects on the equilibrium of a change in pollution treatment technology are analyzed.An improvement in pollution treatment technology does not necessarily lead to a larger equilibrium industry output. A condition is derived which ensures a larger equilibrium industry output in the event of an improvement in pollution treatment technology. Finally,the Nash(-Cournot) equilibria are compared for oligopolies with and without pollution treatment cost-sharing.

BARGAINING WITH COMMITMENT UNDER AN UNCERTAIN DEADLINE

V.Calabuig, A.Cunyat¹ and G.Olcina²

¹ Universitat de Valencia, Valencia (Spain)

ph: +34 96 382 87 75, e-mail: Antonio.Cunat@uv.es

http://www.uv.es/ acunat

² Universitat of Valencia, Campus dels Tarongers. Av/ dels Tarongers

s/n. Edifici Departamental Oriental, 46022 Valencia (Spain)

e-mail: Gonzalo.Olcina@uv.es

We consider an infinite horizon bargaining game in which a deadline can arise with positive probability and where players possess an endogenous commitment device. We show that for any truncation of the game, the equilibrium agreement can only take place if the deadline arises within this finite horizon. Since the deadline is an uncertain event, the equilibrium exhibits agreements which are delayed with positive probability.

BARGAINING POWER IN REPEATED GAMES

Lars Peter Osterdal

Institute of Public Health, Department of Health Services Research

University of Copenhagen, Panum Institute, Blegdamsvej 3, 2200 Copenhagen N, Denmark

e-mail: L.P.Osterdal@pubhealth.ku.dk

A new equilibrium concept, denoted Bargaining Perfect Equilibrium, is suggested. The Bargaining Perfect Equilibria are renegotiation-proof refinements of Subgame Perfect Equilibria in the sense that players do not ex ante expect future bargaining power to be correlated with bygone actions.

The Bargaining Perfect Equilibria turn out to have very tractable properties. As the discounting becomes low, the equilibria follows in the limit directly from the structure of the stage game. If the bargaining power is equal among players in symmetric oligopolistic supergames, the Bargaining Perfect Equilibrium outcome is unique and have a stick-and-carrot structure regardless of the discounting factor.

PARADOXICAL GAMBLING GAMES BASED ON BROWNIAN RATCHETS

Juan M. R. Parrondo

Universidad Complutense de Madrid, Madrid, Spain

Two loosing gambling games can yield, when played alternately, a steady increase of capital. These games have been inspired by the functioning of flashing ratchets. They point out that result of the alternation of stochastic dynamics can be unexpected. In the seminar, I will present and discuss the original paradox and some of its variants.

STACKELBERG PROBLEMS: TIKHONOV REGULARIZATION AND SUBGAME PERFECT EQUILIBRIA

Jacqueline Morgan¹ and Fioravante Patrone²

¹ University of Naples, Department of Mathematics and Statistics, Via Cinthia,

80126 Naples, Italy, e-mail: morgan@unina.it

² Department of Mathematics, Via Dodecaneso 35, University of Genova, 16146

Genova, Italy, e-mail: patrone@dima.unige.it

Stackelberg equilibria, when the best reply of the follower is not uniquely determined, are quite problematic. In particular, the so-called weak Stackelberg problem, which incorporates a pessimistic attitude of the leader, may fail to have a solution even under continuity and concavity assumptions. We approach the problem via Tikhonov regurarization and prove that this method gives a sequence converging to a Subgame Perfect Equilibrium for the ``standard'' interpretation of a Stackelberg problem as a two stage game where the leader moves first.

A FEASIBLE AND STABLE SOLUTION FOR COALITION FORMATION IN A CLASS OF TASK ORIENTED DOMAINS

Maria-V. Belmonte, Ricardo Conejo, Jose-L. Perez-de-la-Cruz and Francisco Triguero

Dpto. Lenguajes y Ciencias de la Computacion, Universidad de Malaga

Campus de Teatinos, P.O.B. 4114, 29080, Malaga, Spain

e-mail: {mavi,conejo,perez}@lcc.uma.es

In this paper, we analyze a class of problems that arise from the modeling of a real world problem and give a feasible procedure for computing a payment configuration that lies inside the core. We consider a set of n agents that must perform a certain initial amount of a task each. The agents can communicate via Internet and change the amounts of tasks initially assigned to each one, in order to achieve a better global efficiency. Under certain hypotheses, we model the problem as a linear programming problem. By solving this and applying sensitivity analysis techniques, we can compute a payoff configuration that lies in the core.

COOPERATIVE PRISONERS AND AGGRESSIVE CHICKENS: EVOLUTION OF STRATEGIES ANDPREFERENCES

Alex Possajennikov

University of Dortmund, Department of Economics (Micro), D-44221 Dortmund, Germany

e-mail: A.Possajennikov@wiso.uni-dortmund.de

By means of simulations I investigate a two-speed dynamic on strategies and preferences in a prisoners' dilemma and in a chicken game. Players learn strategies according to their preferences while evolution leads to change in preferences. With complete information cooperation in the prisoners' dilemma is often achieved, with preferences that use a kind of ''secret handshake''. In the chicken game a symmetric correlated strategy profile is played that is more efficient than the equilibrium. Among preferences only pure ''hawkish'' preferences and ''selfish'' preferences survive. With incomplete information, an equilibrium of the material payoff game is played. Despite of this, all types of preferences are present in the population.

POTENTIAL GAMES AND WELL-POSEDNESS

M. Margiocco¹ and L. Pusillo²

¹Dipartimento di Matematica dell'Università di Genova, Via Dodecaneso 35, 16132 Genova, Italia

²Dipartimento di Matematica dell'Università di Genova, Via Dodecaneso 35, 16132 Genova, Italia

e-mail: pusillo@dima.unige.it

In this paper we study Tihkonov-well posedness (briefly T-wp)for potential games taking into account that a maximum point for the potential function is a Nash equilibrium for the game.

We introduce further a new concept of well-posedness: (e-k)well-posedness and we study its relation with T-wp .

COURNOT DUOPOLY WITH KINKED LINEAR DEMAND ACCORDING TO PALANDER AND WALD

Laura Gardini¹, Tonu Puu² and Irina Sushko³

¹University of Urbino, Italy, e-mail: gardini@uniurb.it

²University of Umea, Department of Economics, Umea se-90187,

Sweden, e-mail: tonu.puu@econ.umu.se

³Academy of Sciences Kiev, Ukraine, e-mail: sushko@imath.kiev.ua

In 1936 and 1939, Tord Palander focused some interesting dynamics problems in Cournot duopoly when the demand curve was kinked linear and the marginal revenue curve jumped up, producing two different local profit maxima. The Cournot reaction functions became piecewise linear, and could produce several coexistent equilibria. Palander specified the basins for these and a 2-perod oscillation. For another case, where the reaction functions did not intersect, Palander recognised the existence of a 3-period cycle. Pakander described the basins, though not completely, and missed the existence of a 6-cycle in his second case. Simultaneously with Palander, Abraham Wald considered the same type of problems though the two line segments were smoothly joined by a curved segment, meeting the lines at tan-gency. Wald did not elaborate the dynamics, but he noted the existence of a nondenumerable infinity of Cournot equilibria In fact the continuum of Cournot equilibria are attractive only for the sparse initial conditions along the main diagonal, and that otherwise the system goes to a 2-period oscillation. More interesting dynamics from these cases, including chaos, are obtained if the constant production costs are replaced by more general cost functions. This is not possible in the Wald cases, as they then cannot be solved in closed form, but for the Palander cases they provide interesting possibilities, especially in the region of decreasing marginal cost, as the slopes of the reaction functions are then raised.

FIRST-ORDER STOCHASTIC DOMINANCE AND STABILITY OF PURE STRATEGY SAMPLING EQUILIBRIA

Michael Ramsza

Institute of Econometrics, Warsaw School of Economics

Aleje Niepodleglosci 162, 02 - 554, Warszawa, Poland

e-mail: mramsz@sgh.waw.pl

The concept of procedurally rational player was introduced by Osborne and Rubinstein (1998). The resulting equilibrium was called sampling equilibrium. Sethi (1998) developed evolutionary approach to sampling equilibrium and provided condition for instability of sampling equilibria. Stability conditions, based on first-order stochastic dominance, for pure strategy sampling equilibria as well as some application to coordination games will be presented and discussed during the talk.

ON CONSERVATION LAWS AND DIFFERENTIAL GAMES

Juan Pablo Rincón-Zapatero

Department of Applied Economics (Mathematics), University of Valladolid

Avda. Valle Esgueva, 6, 47011 Valladolid, Spain

ph.: 983 423330, fax: 983 423299, e-mail: zapatero@eco.uva.es

In this paper we propose a new approach in the study of differential games, based on a quasilinear system of conservation laws, instead of the classical Hamilton-Jacobi-Bellman equation. Applying well known tools of the theory of quasilinear systems and conservation laws, we are able to establish sufficient conditions guaranteeing optimality -in the sense of Nash- of the entropy solution. Applications of the theory to some economic models are given.

FROM EVOLUTIONARY TO STRATEGIC STABILITY

Stefano Demichelis¹ and Klaus Ritzberger²

¹CORE, 34, Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium

e-mail: stefano@core.ucl.ac.be

²Department of Economics and Finance, Institute for Advanced Studies

Stumpergasse 56, A-1060 Vienna, Austria

ph.:(+43-1) 599 91-153, fax: (+43-1) 599 91-163, e-mail: ritzbe@ihs.ac.at

A component of Nash equilibria is (dynamically) potentially stable if there exits an evolutionary selection dynamics from a broad class for which the component is asymptotically stable. A necessary condition for potential stability is that the component's index agrees with its Euler characteristic. Second, if the latter is nonzero, the component contains a strategically stable set. If the Euler characteristic would be zero, the dynamics (which justifies potential stability) could be slightly perturbed so as to remove all zeros close to the component. Hence, any robustly potentially stable component contains equilibria which satisfy the strongest rationalistic refinement criteria.

LARGE GAMES AND THE MODICLUS

J. Rosenmüller^* and P. Sudhölter

^* IMW, University of Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

ph.:++49-521-106-4909/4907, fax: ++49-521-106-2997

e-mail: imw@wiwi.uni-bielefeld.de, http://www.wiwi.uni-bielefeld.de/ imw/

The modiclus or modified nucleolus is a solution concept for TU-Games that, other than the core or the Shapley value or related concepts, does not converge towards the dual solution (the Walrasian equilibrium) when the number of players in a replication becomes large.

We analyze linear production games with orthogonal initial assignments (generalized glove games) in which the modiclus shows a tendency to explain the formation of cartels in the various corners of the market. These games are totally balanced and can also be interpreted as market games. We show that the modiclus represents the external discussion ('between the cartels') as well as the internal discussion inside a cartel (which is actually performed according to the 'Talmudian' or contested garment solution (a la Aumann-Maschler)). We also show that, with slight additional conditions concerning the number of small players, this behavior of the modiclus is robust against replication: the long side of the market may receive a profit even when the game becomes large.

PARADOXES OF VOTING POWER IN DUTCH POLITICS

Agnieszka Rusinowska¹ and Ad van Deemen²

¹ Tilburg University, Department of Philosophy, P.O. Box 90153, 5000 LE Tilburg

The Netherlands, ph.: (+31-13) 466 28 22, e-mail: a.rusinowska@kub.nl

Warsaw School of Economics, Institute of Econometrics,

Al. Niepodleglosci 162, 02-554 Warsaw, Poland

² University of Nijmegen, P.O. Box 9108, 6500 HK Nijmegen

The Netherlands, e-mail: a.vandeemen@bw.kun.nl

In this paper, thirteen parliamentary elections in the Netherlands in the period 1956-1998 are studied. For each election we compute a number of power index vectors in order to analyze the results of that election. The power indices used in this paper are the Shapley-Shubik index, the normalized and non-normalized Banzhaf indices, the Deegan-Packel index and the Holler-Packel index.

Subsequently, we investigate the occurrence of a number of paradoxes of voting power indices in this period. In particular, we search for the paradox of redistribution, the paradox of large size and the paradox of new members. For each paradox its frequency for each separate party partcipating in the Dutch elections (1956-1998) is calculated. Then we compute the overall frequency of the paradoxes for the Dutch elections. For the paradox of new members all the real cases of entrances of new parties in Dutch elections are investigated. For the paradox of large size, all real cases of unions, established among Dutch parties in the course of time are examined.

REGULATORY UNCERTAINTY AND WIN:WIN BASED REGULATION

Michael J Ryan

Department of Economics, University of Hull Hull HU6 7RX, England

ph.: (0)1482 466217, fax: (0)1482 466216, e-mail: M.J.Ryan@econ.hull.ac.uk

This paper will investigate how in practice an explicitly uncertainty and learning based methodology can be used to give a more central place to win:win principles as distinct from adversarial principles in the policies and decisions of regulatory agencies. Under conditions of certainty or of risk where contingent outcomes are known, insofar as regulation restricts choice it will generally be seen as undesirable by those regulated. In practice, however, conditions of uncertainty may also obtain for example concerning nuclear safety, food safety, or concerning possible types and chances of railway or aircraft disasters. Such uncertainty may arise from incomplete knowledge of potentially forthcoming states for outputs and/or from incomplete knowledge of the probabilities with which such states may be forthcoming. Uncertainty may arise, too, on the production side from incomplete knowledge of relevant production processes. The key idea in the paper is that, given these three distinct types of uncertainty, even though regulatory prohibitions or taxes or subsidies may still be relatively undesirable to those regulated, such relatively negative effects may be partly or wholly offset by the informational value of: i) research by a regulatory agency to determine the nature of otherwise wholly uncertain events; ii) statements of the probabilities with which future states may be forthcoming and/or; iii) policies designed to alter innovation costs. These are three related yet distinct means of securing potentially mutually advantageous agreements between regulators and those they regulate.

VIABLE CAPTURE BASIN METHODS FOR STUDYING DISCRETE AND DIFFERENTIAL GAMES: THEORY, APPLICATIONS AND NUMERIACL ISSUES

Patrick Saint-Pierre

Centre de Recherche Viabilité, Jeux, Contrôle, Université Paris - Dauphine, 75116 Paris, France

ph.: 33(0)144054654, fax: 33(0)144054911, e-mail: saint-pierre@viab.dauphine.fr

The recent developments of Viability Theory in the context of either control or differential games lead to the emergence of methods for characterizing and approximating victory sets. After recalling the basic tools that are required for defining properly what we call Viable Capture Basins, we present three recent results in the field of discrete and continuous differential games. The first result deals with state constrained differential games, the second concerns discrete conditional and guaranteed viability and numerical methods, the third illustrates what uncertainty can be considered, tychastic or stochastic, for computing victory domains through a numerical example arising in finance.

ON COMPLETENESS OF KNOWLEDGE MODELS

Hannu Salonen

Department of Economics, University of Turku, 20014 Turku, Finland

e-mail: hansal@utu.fi

We study knowledge operators defined on countable Boolean algebras. Lindenbaum algebras corresponding to a propositional logic or modal logic S5 with countable atomic formulas are examples of countable Boolean algebras. The Stone space corresponding to such a Boolean algebra is interpreted as the set of states of nature. We show that that a complete knowledge model based on such a state space exists. The types in such a model may be identified with the usual infinite (of length w) hierarchies of knowledge, if the Boolean algebra is infinite. If the Boolean algebra is finite, the types may be identified with knowledge hierarchies of length w + w.

EXTENDED POWER VALUES AND COALITION FORMATION

Frank Goebeler¹ and Juergen Scheffran²

¹ Department of Mathematics, University of Hamburg, Bundesstr. 55

20146 Hamburg, Germany

ph.: 0049-40-41235183, e-mail: frank.goebeler@math.uni-hamburg.de

² e-mail: scheffran@hrzpub.tu-darmstadt.de

The classical power values (e.g. Shapley, Banzhaf) consider every possible coalition to be equally likely. In real voting systems (e.g. parliaments or in the UN Security Council) this is not a realistic assumption. Because of political, ideological, cultural, economic or military relations between the players some coalitions are more likely than others. The probability that a coalition occurs must be taken into account. This probability is not always the same in every voting. It depends on the probablility of a yes-vote or (which is equivalent) a no-vote of every player. The vote is dependent on the player's position (e.g. in O'Neill's ideological space) on the issue of the current voting. This leads to a power value matrix in which the rows show the power of a player in every issue while the columns represent the power distribution on every issue. The power value of the players over all issues can be obtained by the mean of the rows. The coalition formation can be interpreted as a bargain process in which two or more players can change the probabilities of their yes-votes by way of a cooperative optimization in order to maximize their individual power value over all issues.

CAN YOU GUESS THE GAME YOU'RE PLAYING?

Jorg Oechssler¹ and Burkhard Schipper²

¹Department of Economics, University of Bonn, Adenauerallee 24, 53113 Bonn, Germany

e-mail: oechssler@uni-bonn.de

²Department of Economics, University of Bonn, Adenauerallee 24, 53113 Bonn, Germany

e-mail: burkhard.schipper@wiwi.uni-bonn.de

Recently there has been much theoretical and experimental work on learning in games. However, learning usually means ``learning about the strategic behavior of opponents" rather than ``learning about the game" as such. In contrast, here we report on an experiment designed to test whether players learn a) the relevant features of the payoff structure of a 2x2 game (e.g. the best reply structure) and b) whether they play according to equilibrium predictions regardless of how question a) is being answered. Question a) is being tested by asking subjects to guess the payoff structure of their opponents in a repeated encounter and rewarding subjects for correct answers.

POWER RELATIONS IN CONSENSUS-ORIENTED DECISION-MAKING PROCEDURES: THE SWISS INFLUENCE IN THE BRETTON WOODS INSTITUTES

Susanne Schmidtchen

NADEL, ETH Zurich, Switzerland, e-mail: schmidtchen@nadel.ethz.ch

This paper compares power relations derived from a standard Banzhaf power analysis with a modified power measure that mirrors the de facto decision-making process in the Executive Boards of the Bretton Woods Institutes. Special focus is given to the power distribution in 1992/1993, when Switzerland joined the IMF and the World Bank. Given our survey evaluating the preference structure in the Executive Boards in 1999/2000, this latter period is examined as well. Following the consensus-oriented power analysis, even small constituencies have the potential to influence outcomes given that these constituencies are ready to cooperate with like-minded country groups. The extent of the power redistribution among the blocks and remaining single actors due to the participation of Switzerland in the Bretton Woods Institutes depends upon several factors, e.g. the number of blocks of like-minded constituencies, the size in voting shares of those blocks the Swiss-lead constituency engages in etc. Although consensus-oriented power analysis provides a more appropriate instrument to cover real world decision-making procedure in intergovernmental voting bodies, it is not able to capture qualitative aspects of power. Therefore the discussion is finally extended by characteristics that reflect a memberís creative power, e.g. excellent and proactive engagement within a block of like-minded constituencies.

ALMOST-DOMINANT STRATEGY IMPLEMENTATION

James Schummer

MEDS Department, Kellogg Graduate School of Management

Northwestern University, Evanston, IL 60208-2009, USA e-mail: schummer@nwu.edu

http://www.kellogg.nwu.edu/faculty/schummer/ftp/research/

Though some economic environments provide reasonable allocation rules that are implementable in dominant strategies (i.e., strategy-proof), a significant number of environments yield impossibility results. On the other hand, while there are quite general possibility results regarding implementation in Nash or Bayesian equilibrium, these equilibrium concepts make strong assumptions about the knowledge that players possess, or about the way they deal with uncertainty. As a compromise between these two notions, we propose a solution concept built on one premise: Players who do not have much to gain by manipulating an allocation rule will not bother to manipulate it.

We search for efficient allocation rules for 2-agent exchange economies that never provide players with large gains from cheating. Though we show that such rules are inequitable, we also show that some such rules are significantly more flexible than those that satisfy the stronger condition of strategy-proofness, even when the allowable gains from manipulation are made arbitrarily small.

A TOPOLOGICAL SOLUTION TO QUITTING GAMES

Robert Samuel Simon

Institut fuer Mathematische Stochastik (Institute of Mathematical Stochastics)

University of Goettingen, Lotze str. 13, 37083 Goettingen, Germany

e-mail: rsimon@math.huji.ac.il

This paper presents a topological conjecture and demonstrates that its confirmation would establish the existence of approximate equilibria in all quitting games. A quitting game is an un-discounted stochastic game where every player has only two moves, to end the game with certainty or to allow the game to continue. If nobody ever acts to end the game, all players receive payoffs of 0.

ON COST ALLOCATION FOR STEINER TREE BASED MULTICAST ROUTING IN NETWORKS

Darko Skorin-Kapov

School of Business, Adelphi University, Garden City, NY 11530, USA

ph.:( 516) 877-4662, fax: (516) 877-4607, e-mail: skorin@adelphi.edu

http://www.adelphi.edu/ skorind

We are concerned with the cost allocation problem associated with th e problem of broadcasting information of common interest (for example, financial news) from some source to specific network users. Multicast routing uses a tree connecting all the receivers to the source. Namely, whenever a message needs to be broadcast to a subset of receivers, multicast routing chooses a minimum cost Steiner tree that spans the source and all the receivers. The objective of this paper is to address the cost allocation associated with the practical methods used in multicasting. We assume that the best-known Steiner tree is obtained using distributed minimum spanning tree-based heuristic. We must then allocate the cost of edges to the users, who are represented as nodes. We define our multicast game in the characteristic function form as follows. For a subset of users the value of the characteristic function is the best-known cost of the Steiner tree obtained by the minimum cost spanning tree-based heuristic. Then, we construct a polynomial algorithm that finds the stable cost allocation. Moreover, we prove that our algorithm constitutes the population monotonic cost allocation scheme.

JOINT ORDERING IN MULTIPLE NEWS-VENDOR PROBLEMS:
A GAME-THEORETICAL APPROACH

Marco Slikker^*, Jan Fransoo and Marc Wouters

^* Department of Technology Management, Eindhoven University of Technology

P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

e-mail: M.Slikker@tm.tue.nl

We will study a situation with n retailers, each of them facing a news-vendor problem, i.e., selling to customers over a finite period of time (product with a short life cycle, such as fashion). Groups of retailers might improve their expected joint profit by cooperating. We will analyze these situations by defining a cooperative game, called a news-vendor game, for such a situation with n retailers. We concentrate on whether it makes sense to cooperate by studying properties of news-vendor games. We prove convexity for specific demand distributions. Our main result states that news-vendor games are balanced games.

THE SEMANTICS OF PREFERENCE-BASED BELIEF OPERATORS

Geir B. Asheim and Ylva Sòvik

Department of Economics, University of Oslo, P.O.Box 1095 Blindern, N-0317 Oslo, Norway

e-mail: {g.b.asheim, ylva.sovik}@econ.uio.no

Properties of preferences will place restrictions on beliefs, and we would like to deduce the properties of belief that follow from certain properties of preferences. Morris (1997) shows how this approach gives a unified framework for studying alternative notions of decision making to that of expected utility maximization. Asheim & Dufwenberg (2000) and Asheim (1999) consider preference-based belief operators when we allow preferences that are not represented by subjective probabilities. This paper develops the semantics for these belief-operators and thereby describes the properties of the belief operators and the corresponding belief revision.

We show that the belief operators can be derived from a Kripke model with nested binary accessibility relations R₁, ..., R_L. These accessibility relations are serial, transitive and euclidean, and therefore the belief operators are KD45. Furthermore, these nested binary accessibility relations can be derived from a binary epistemic priority ordering relation Q. The properties of this priority ordering are similar to but more general than those found for instance in Stalnaker (1996, 1998) since reflexivity is not required. Provided all observable events are subjectively possible the relation can be used to define a conditional belief operator and we show that this operator fulfills the usual properties found in the belief revision literature.

THE EQUILIBRIUM CHARACTERIZATION WITH LONG-RUN AND SHORT-RUN PLAYERS

Tadanobu Tan-no

Dolce-ishihara 202, 4-19-1 Kitayama, Fuchu, Tokyo 183-0041, Japan

ph.: +81-(0)42-573-1319, e-mail: pg00306@srv.cc.hit-u.ac.jp

http://www.geocities.co.jp/WallStreet/6613/

This paper examines equilibria in the class of infinitely repeated games with long-run and short-run players when the former players' discount factor goes to one. In order to compute equilibrium payoffs in the limit, Fudenberg and Levine (1994) proposed a general algorithm maximal score method that uses many linear programming problems. To the contrary their claim about the algorithm, any equilibrium does not necessarily imply a payoff characterized by their algorithm. We show that any interior point of the payoff set characterized by all but countably many LP problems becomes a limit equilibrium. Their algorithm needed all LP problems. Equilibrium outcomes in our refined algorithm are generally larger than those in their algorithm. Two algorithms are applied to investment game. Although their characterization cannot tell whether or not all efficient outcomes satisfying short-run player's incentive are attainable, by implementing our algorithm we show that all possible outcomes satisfying the incentive can be supported in equilibria.

QUANTUM ISLANDS

Bart Taub

University of Illinois, Department of Economics, 330 Commerce West,

61820 Champaign, IL, USA, ph.: (217) 333-3467, fax: (217) 244-6678

e-mail: b-taub@equilibrium.econ.uiuc.edu

This paper constructs a model in which business cycle fluctuations are desirable: they are the manifestation of efficient, incentive-constrained communication. The model demonstrates that assets are inextricably connected to these fluctuations since assets are the expression of a private-information contract.

COLLECTING INFORMATION TO IMPROVE DECISION-MAKING: GAMES AND SOLUTIONS

Rodica Brânzei¹, Stef Tijs² and Judith Timmer³

¹ Faculty of Computer Science, ``Al.I. Cuza'' University, Iasi, Romania

² Department of Econometrics and Operations Research

Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands

e-mail: S.H.Tijs@kub.nl

³ Faculty of Mathematical Sciences, University of Twente

P.O. Box 217, 7500 AE Enschede, The Netherlands

ph: +31 53 489 3419, fax: + 31 53 489 3069, e-mail: j.b.timmer@math.utwente.nl

Imagine a single decision-maker who is faced with uncertainty. He has to take an action which generates a reward that also depends upon the uncertainty. The decision-maker can improve upon his reward by collecting information from other agents and by basing his choice of action on this extra info. One can think for example of a hot-dog seller in a football stadium who is collecting information about the number of spectators of next week's football match. Based on this number he determines how much ingredients to buy. An interesting question that arises is: ``What to pay the informants for their information?''. For this, we introduce information collecting (IC) situations as described above. Related to an IC situation with decision-maker k we define a cooperative game, the IC game. We start by analyzing this type of games. Nice properties are k-concavity and convexity. Further, we can show that a cooperative game is an IC game with decision-maker k if and only if it is a k-monotonic game in which player k has veto power. After this analysis we turn our attention to special solutions for IC games, like the marginal based allocation rule and bi-monotonic allocation schemes.

INVENTORY MODELS WITH CONTINUOUS DISCOUNT

Andres Toledo Casado¹ and Ana Meca Martinez²

¹ Centro de Investigación Operativa. Universidad Miguel Hernández de Elche

Avd. Ferrocerril s/n 3202 Elche, Spain

e-mail: andres_toledo@dmr.com

² Centro de Investigación Operativa. Universidad Miguel Hernández de Elche

Avd. Ferrocerril s/n 3202 Elche, Spain

e-mail: ana.meca@umh.es

Inventory management studies how a single firm can minimize the average cost per time unit of its inventory. Works as Hadley and Within (1963) describe and study several deterministic inventory models, and Montgomery (1974), Tersine and Toelle (1985) and Tersine (1994) present the purchasing cost depending on the order size.

In this article we consider the study of four new inventory models derived from the four classical inventory models: EOQ, EOQ with backordering, EPQ and EPQ with backordering, when the price policy applied by the supplier is defined according to two different schedules. The price schedules are defined according to a decreasing function f: R₊₊Æ R₊₊ depending on the order size, Q, such that satisfies several smoothness properties. The inventory models presented are completely original. Finally we present different economic applications of them. In these applications the supplier of the product can be interpreted as a competitive or controlled market. This shows that the inventory models here defined can be of great interest to enlarge the study done in this paper. In fact, we prove that the second of the above price schedule is a generalization of the schedule called incremental discount.

Two reasons motivate this work. The first one is to study new inventory models. This study can be extended in a more general context. We mean this situation could be modelled by a decreasing function. The second reason is to define and study different classes of Cooperative Games which arises from inventory models studied here. As it is indicated in Meca et al.(1999) and Meca (2000) a natural extension of Inventory Games studied in those works is consider the purchasing cost involved in the inventory costs, because of it depends on the order size. In the former the purchasing cost was not relevant in the inventory cost because it was constant.

INVENTORY GAMES WITH CONTINUOUS DISCOUNT

Andres Toledo Casado¹ and Ana Meca Martinez²

¹ Centro de Investigación Operativa. Universidad Miguel Hernández de Elche

Avd. Ferrocerril s/n 3202 Elche, Spain

e-mail: andres_toledo@dmr.com

² Centro de Investigación Operativa. Universidad Miguel Hernández de Elche

Avd. Ferrocerril s/n 3202 Elche, Spain

e-mail: ana.meca@umh.es

In Toledo and Meca (2001) several inventory models involving different price policies have been studied. In this work, following the philosophy of Meca et al.(1999), we analyze different cooperative games that model those situations in which several firms or shops, demanding a same product, cooperate according to different criterions. Each one of these firms has its own private demand and its own private storage possibilities for the good. There is a single supplier to whom all firms place their orders at the same ordering cost. If the purchasing cost established by the supplier can be model by certain decreasing function depending on the order size, together with several smoothness properties, these firms can reduced their average inventory cost per unit time by means of placing orders simultaneously or ordering and holding together their demand in one warehouse. To study all situations derived from these two cooperation criterions we define a class of cooperative TU games that involves that situations in which the mentioned price policy is one of the given in Toledo and Meca (2001) or the schedule discount called all-units discount (Tersine, 1994). This class of games is called Inventory Games with Discount in the Purchasing Cost. Next we define different allocation rules that, in several cases, allow us to affirm that this class is balanced. Finally, we present several particular cooperative games that the class of Inventory Game with Discount in Purchasing Cost involves.

PARRONDO'S GAMES: EFFECT OF COOPERATION

Raúl Toral

Instituto Mediterráneo de Estudios Avanzados, IMEDEA (CSIC-UIB),

Campus UIB, 07071 Palma de Mallorca, Spain, e-mail: raul@galiota.uib.es

We introduce new families of Parrondo's games of alternating losing strategies in order to get a winning result. In our version of the games we consider an ensemble of players and use ``social" rules in which the probabilities of the games are defined in terms of the actual state of the neighbors of a given player. Different types of rules will be discussed, including diffusion of capital between the players. We analyse one and two-dimensional topologies such as regular and small-world lattices. Besides the existence of the paradox itself, we focus on the capital distribution amongst the players and the clustering of losing and winning individuals. We will present numerical evidence as well as the results of a simplified mean-field type theory.

TIKHONOV WELL-POSEDNESS FOR THE NUCLEOLUS

Vito Fragnelli¹, Fioravante Patrone² and Anna Torre³

¹ Università del Piemonte Orientale - Dip. di Scienze e Tecnologie Avanzate

Corso Borsalino 54 - 15100 Alessandria, Italia, e-mail: fragnell@mfn.unipmn.it

² Università di Genova -Dip. di Matematica - via Dodecaneso 35 - 16146 Genova, Italia

e-mail: patrone@dima.unige.it

³ Università di Pavia - Dip. di Matematica - via Ferrata 1 -27100 Pavia, Italia

e-mail: atorre@dimat.unipv.it

We study Tikhonov well posedness for the nucleolus ( a kind of solution for cooperative games introduced by Schmeidler ). This kind of well-posedness is not trivial, because the nucleous is the minimum for a preorder defined using the lexicographic order, that, as it is well known, is not representable by a utility function and it is not continuous w.r.t. the euclidean topology.

EFFICIENCY OF POLICY CHOICE AND POLITICAL AUCTION

Shyh-Fang Ueng

Institute of Economics, Academia Sinica, Taipei, 115, Taiwan

e-mail: us21@gate.sinica.edu.tw

This study constructs a two-stage model to study policy choice in one round of political competition. The competition starts with election campaign and ends with the tenure of the elected government. The first-stage election competition is investigated through the perspective of posted-offer auction. The second-stage policy confrontation between the elected government and opposition parties is analyzed as a double-auction. It is shown that second-stage referendum on widely contested policy drives the campaign platforms of candidates closer to the real interest of the majority; nevertheless referendum on narrowly concerning issues only legitimizes the interest of a specific group at the cost of the majority. The correction effect of referendum on widely contested policy hinges on the information revelation in the double auction between the elected government and opposition parties.

COMPUTATIONALLY RESTRICTED UNMEDIATED TALK UNDER INCOMPLETE INFORMATION

A. Urbano^* and J. Vila

Departamento de Analisis Economico, Universitat de València,

Campus del Tarongers Avd. Del Tarongers s/n Edificio Departamento

Oriental 46022 Valencia, Spain

^* e-mail: amparo.urbano@uv.es

We show the role of computationally restricted unmediated talk as both an information transmission and a coordination device for the class of two-player games with incomplete information. We prove that any communication equilibrium payoff of such games can be reached as the Bayesian-Nash equilibrium payoff of the game extended by a two-phase universal mechanism of interim computationally restricted pre-play communication. The communication protocols are designed with the help of modern cryptographic tools. A familiar context in which our results could be applied is bilateral trading with incomplete information.

REPEATED GAMES WITH PROBABILISTIC HORIZON

I. Arribas and A. Urbano^*

Departamento de Analisis Economico, Universitat de València, Spain

^* e-mail: amparo.urbano@uv.es

Repeated games with probabilistic horizon are defined as those games where players have a common probability structure over the length of the game's repetition, T. In particular, for each t, they assign a probability p(t) to the event that "the game ends in period t". In this framework we analyze Generalized Prisoners' Dilemma games in both finite stage and differentiable stage games. Our construction shows that it is possible to reach cooperative equilibria under some conditions on the distribution of the discrete random variable T even if the expected length of the game is finite. More precisely, we completely characterize the existence of sub-game perfect cooperative equilibria in finite stage games by the (first order) convergence speed: the behavior in the limit of the ratio between the ending probabilities of two consecutive periods. Cooperation in differentiable stage games is determined by the second order convergence speed, which gives a finer analysis of the probability convergence process when the first convergence speed is zero. Leptokurtic distributions are defined as those distributions for which the (first order) convergence speed is zero and they preclude cooperation in finite stage games with probabilistic horizon. However, this negative result is obtained in differential stage games only for a subset of these distributions.

ADAPTIVE CHOICES COMPATIBLE WITH SIMILARITY RELATIONS AND EVOLUTIONARY DRIFT

José Ramón Uriarte

Universidad del Pais Vasco-Euskal Herriko Unibertsitatea

Departamento de Fundamentos del Análisis Económico

Avenida del Lehendakari Aguirre, 83,48015 Bilbao, Basque Country-Spain

e-mail: jepurayj@bsdx01.bs.ehu.es

A new approach to modelling perturbations in deterministic selection dynamics is developed. A noisy agent is a fully developed decision-maker endowed with preferences defined on the product space of expected payoffs-strategy proportions attached to his current pure strategy. His strategy revision rule is governed by the goal of minimizing the distance from the current expected payoff-strategy proportion vector to its corresponding aspiration or preferred set.

Mistake probabilities are endogenously determined by both expected payoffs and strategy proportions. This feature endows drift with a more cutting power than previous models. In particular, when drift is highly sensitive to the proportion of agents playing the strategies of a non subgame perfect equilibrium with unreached information sets then, even a Nash component with empty interior can have strong stability properties.

A plausible relation may be established between drift and the cultural traits of a country, that might explain the laboratory results obtained with the full ultimatum game.

A PROBABILISTIC REFOUNDATION OF POWER MEASURES

Annick Laruelle¹ and Federico Valenciano²

¹ Departamento de Economia Aplicada IV, Universidad del Pais Vasco

Avenida Lehendakari Aguirre, 83, 48015 Bilbao, Spain, e-mail: elxlaxxa@bsdx01.bs.ehu.es

² Departamento de Economia Aplicada IV, Universidad del Pais Vasco

Avenida Lehendakari Aguirre, 83, 48015 Bilbao, Spain

ph.: 34-946013696, fax: 34-946017028, e-mail: elpvallf@bs.ehu.es

In this paper we provide a clear and rigourous probabilistic re-foundation of the theory of the measurement of voting power. A simple and transparent redefinition of a priori voting power in voting situations as a probability is proposed. It provides a clear conceptual common basis to reinterpret coherently in terms of power (on the domain of simple games) from a unified point of view some well-known (but not so well understood) concepts to be found in the literature, as probabilistic values, weak semivalues and semivalues. It provides as well a dispassionate standpoint to reinterpret critically different power indices and much of the literature about them.

ON COALITIONAL GAME CONTEXTS AND THEIR CONCEPT LATTICES

Stefano Vannucci

Dipartimento di Economia Politica, Università di Siena, P.za S. Francesco 7, 53100, Siena, Italia

e-mail: vannucci@unisi.it

Coalitional Game Contexts -the objects of a nonfull subcategory of the category of 'Chu spaces' or 'formal contexts'- are introduced and shown to encompass virtually all coalitional game formats as currently employed in the game- and social choice-theoretic literature. Concept lattices of CGCs are also discussed, and the resulting order dimension for CGCs is defined. Some basic spectral properties of those lattices are studied. In particular, it is shown that for any positive integer k there exists a preconvex -hence strongly core-stable- CGC with a concept lattice of width k.

THE VALUE OF COMMITMENT IN STACKELBERG LEADER GAMES WITH OBSERVATION COSTS

Felix J.J. Várdy

Department of Economics, Princeton University, G8544 Princeton NJ, USA

e-mail: fjjvardy@Princeton.edu

We study a modified version of the Stackelberg leader game in which the follower faces a cost for observing the leader's action. We show that both the Bagwell (1995) and the van Damme and Hurkens (vDH, 1997) results on the value of commitment carry over to this almost-perfect-information environment. The vDH result is strictly strengthened. For our `costly leader game', the leader's value of commitment is preserved completely, in all noisy Stackelberg equilibria. Also, the probability that the follower looks at the leader's action is independent of the cost of looking. This result offers improved possibilities for testing the relevance of the Bagwell and vDH claims experimentally. (See Várdy, 2001.)

Evolutionary arguments show that noisy Stackelberg equilibria are not stable. Instead, the only evolutionary stable strategy profiles are those in which the follower never looks at the leader's action, even when the cost of looking becomes arbitrarily small.

MEASURING POWER IN SPATIAL VOTING GAMES

Mika Widgrén¹ and Stefan Napel²

¹Turku School of Economics and CEPR, Rehtorinpellonkatu 3, 20500 Turku, Finland

ph.: +358-2-3383337, fax: 358-2-3383302, e-mail: mika.widgren@tukkk.fi

² Institut für Wirtschaftstheorie und Operations Research (WIOR)

Universität Karlsruhe (TH), 76128 Karlsruhe, Germany

ph.: +49-721-608-2697, fax: +49-721-608-3082, e-mail: napel@wior.uni-karlsruhe.de

The criticism towards power indices of cooperative games usually stems from two factors. First, and which is closely related to our example above, power indices are not able to take players' strategic interaction into account. Second, their capability of modelling complicated institutional features, like agenda-setting, is limited. In this paper we extend the notion of inferior players into spatial voting games. We define the concept of spatially strategic power index using a simple agenda setting model with spatial preferences and derive an explicit index which is able to take strategic agenda-setting and preferences into accout in power analysis. Then we explicitly demonstrate how the Shapley-Shubik index of power and the new concept are related to each other.

SCIENTIFIC RESEARCH: COMMUNICATION AND COMPETITION

Ramón Xifré-Oliva

Department of Economics, Universidad Carlos III de Madrid, Getafe (Madrid) 28903, Spain

fax: +34-91-624 9875, e-mail: xifre@eco.uc3m.es

We analyse a non-cooperative, static game with incomplete information. Two researchers receive private, noisy signals about the state of nature. Researchers may have different abilities determined by the precision of their signal. Each researcher has to take an action based on available information and his payoff is determined both by his own and the opponent's action. In an attempt to study the impact of scientific interaction, we allow researchers to communicate with each other before taking their own actions. After receiving a noisy signal on the state of nature, each researcher sends a message to his opponent. Researchers are assumed to be of one of two possible types, cooperative and competitive. Other things equal, a cooperative type prefers the action of his opponent to be as close as possible to the state of nature, while a competitive type prefers it to be as far as possible from it. We characterise pure and mixed strategy equilibria of the game and we show how informative communication depends on the relative ability of the researchers and the relative likelihoods of each type of researcher.

TIME CONSISTENCY IN COOPERATIVE DIFFERENTIAL GAMES

Steffen Jorgensen¹ and Georges Zaccour²

¹ Department of Organization and Management University of Southern Denmark

Odense University, Odense, Denmark

² GERAD and École des H.E.C. Montréal, 300 Cote S. Catherine

H3T 2A7 Montreal, Canada, e-mail: georges.zaccour@hec.ca

This paper addresses the problem whether a cooperative agreement, made at the start of a game, can be sustained over time. The players can reopen negotiations or reconsider their strategies at any instant of time during the play of the game. Research in differential games has addressed the question of individual rationality over time under headings such as time consistency, dynamic stability, agreeability, or acceptability, and often in an applied context. The question is whether a bargained solution, satisfying individual rationality at the start of the game, will remain individually rational as the state vector evolves over time. The paper collects various research works on intertemporal individual rationality.

File translated from T_EX by T_TH, version 2.25.
On 2 May 2001, 11:44.