
on Game Theory 
By:  Amnon Rapoport (Department of Management and Marketing, A. Gary Anderson Graduate School of Management, University of California); Hironori Otsubo (Strategic Interaction Group, Max Planck Institute of Economics, Jena, Germany); Bora Kim (Department of Management and Organizations, Eller College of Management, University of Arizona); William E. Stein (Department of Information and Operations Management, Mays Business School, Texas A&M University) 
Abstract:  Two auction mechanisms are studied in which players compete with one another for an exogenously determined prize by independently submitting integer bids in some discrete and commonly known strategy space specified by the auctioneer. In the unique lowest (highest) bid auction game, the winner of the prize is the player who submits the lowest (highest) bid provided that this bid is unique, i.e., unmatched by other bids. Assuming a commonly known finite population of players, a nonnegative cost of entry, and an option to stay out of the auction if the entry cost is deemed too high, we propose an algorithm for computing symmetric mixedstrategy equilibrium solutions to the two variants of the auction game, illustrate them, and examine their properties. 
Keywords:  Unique bid auction games, Equilibrium analysis 
JEL:  C72 D44 
Date:  2009–01–26 
URL:  http://d.repec.org/n?u=RePEc:jrp:jrpwrp:2009005&r=gth 
By:  Yanovskaya, E.; Brânzei, R.; Tijs, S.H. (Tilburg University, Center for Economic Research) 
Abstract:  This paper examines several monotonicity properties of valuetype interval solutions on the class of convex interval games and focuses on the DuttaRay (DR) solution for such games. Well known properties for the classical DR solution are extended to the interval setting. In particular, it is proved that the interval DR solution of a convex interval game belongs to the interval core of that game and Lorenz dominates each other interval core element. Consistency properties of the interval DR solution in the sense of DavisMaschler and of HartMasColell are verified. An axiomatic characterization of the interval DR solution on the class of convex interval games with the help of bilateral HartMasColell consistency and the constrained egalitarianism for twoperson interval games is given. 
Keywords:  cooperative interval games;convex games;the constrained egalitarian solution;the equal division core;consistency 
JEL:  C71 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:2008102&r=gth 
By:  Michel Grabisch (Centre d'Economie de la Sorbonne); Lijue Xie (Centre d'Economie de la Sorbonne) 
Abstract:  Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TUgames cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, it is not obvious to define a suitable notion of core, reflecting the team structure, and previous attempts are not satisfactory in this respect. We propose a new notion of core, which imposes efficiency of the allocation at each level of the hierarchy, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness. 
Keywords:  Cooperative game, feasible coalition, core, hierarchy. 
JEL:  C71 
Date:  2008–10 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:b08077&r=gth 
By:  Yair Tauman (Tel Aviv University and Stony Brook University); Andriy Zapechelnyuk (University of Bonn and Kyiv School of Economics) 
Abstract:  By example of a simple fourplayer game we point out at a class of economic contexts where aggregate monotonicity is not a desirable property of a solution concept. 
Keywords:  Cooperative games, aggregate monotonicity, axiomatic solution 
JEL:  C71 C78 
Date:  2009–01 
URL:  http://d.repec.org/n?u=RePEc:kse:dpaper:13&r=gth 
By:  Pedro Miranda (Department of Statistics and O.R.  Universidad Complutense de Madrid); Michel Grabisch (Centre d'Economie de la Sorbonne) 
Abstract:  In this paper, we present a generalization of the concept of balanced game for finite games. Balanced games are those having a nonempty core, and this core is usually considered as the solution of game. Based on the concept of kadditivity, we define to socalled kbalanced games and the corresponding generalization of core, the kadditive core, whose elements are not directly imputations but kadditive games. We show that any game is kbalanced for a suitable choice of k, so that the corresponding kadditive core is not empty. For the games in the kadditive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be kbalanced. For the general case, it is shown that any game is either balanced or 2balanced. Finally, we treat the special case of capacities. 
Keywords:  Cooperative games, kadditivity, balanced games, capacities, core. 
JEL:  C7 D7 
Date:  2008–11 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:b08079&r=gth 
By:  Michel Grabisch (Centre d'Economie de la Sorbonne); Yukihiko Funaki (School of Political Science & Economics  Waseda University) 
Abstract:  The coalition formation problem in an economy with externalities can be adequately modeled by using games in partition function form (PFF games), proposed by Thrall and Lucas. If we suppose that forming the grand coalition generates the largest total surplus, a central question is how to allocate the worth of the grand coalition to each player, i.e., how to find an adequate solution concept, taking into account the whole process of coalition formation. We propose in this paper the original concepts of scenariovalue, processvalue and value, which represent the average contribution of players in a scenario (a particular sequence of coalitions within a given coalition formation process), in a process (a sequence of partitions of the society), and in the whole (all processes being taken into account), respectively. We give an application to Cournot oligopoly, and two axiomatizations of our solution concept. A comparison with the value proposed by MachoStadler et al. for PFF games is done. 
Keywords:  Coalition formation, games in partition function form, solution concept, Cournot oligopoly. 
JEL:  D62 C71 
Date:  2008–11 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:b08076&r=gth 
By:  Brânzei, R.; Dall'Aglio, M.; Tijs, S.H. (Tilburg University, Center for Economic Research) 
Abstract:  Interval bankruptcy problems arise in situations where an estate has to be liquidated among a fixed number of creditors and uncertainty about the amounts of the estate and the claims is modeled by intervals. We extend in the interval setting the classic results by Curiel, Maschler and Tijs [Bankruptcy games, Zeitschrift fur Operations Research, 31 (1987), A 143 { A 159] that characterize division rules which are solutions of the cooperative bankruptcy game. 
Keywords:  cooperative games; interval data; bankruptcy problems. 
JEL:  C71 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:200897&r=gth 
By:  Dall'Aglio, M.; Brânzei, R.; Tijs, S.H. (Tilburg University, Center for Economic Research) 
Abstract:  This paper defines models of cooperation among players partition ing a completely divisible good (such as a cake or a piece of land). The novelty of our approach lies in the players' ability to form coalitions before the actual division of the good with the aim to maximize the average utility of the coalition. A social welfare function which takes into account coalitions drives the division. In addition, we derive a cooperative game which measures the performance of each coalition. This game is compared with the game in which players start cooper ating only after the good has been portioned and has been allocated among the players. We show that a modified version of the game played before the division outperforms the game played after the di vision. 
Keywords:  fair division;cooperative games;maximin partition 
JEL:  C71 D61 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:2008101&r=gth 
By:  Michel Grabisch (Centre d'Economie de la Sorbonne); Agnieszka Rusinowska (GATE) 
Abstract:  We study and compare two frameworks : a model of influence, and command games. In the influence model, in which players are to make a certain acceptance/rejection decision, due to influence of other players, the decision of a player may be different from his inclination. We study a relation between two central concepts of this model : influence function, and follower function. We deliver sufficient and necessary conditions for a function to be a follower function, and we describe the structure of the set of all influence functions that lead to a given follower function. In the command structure introduced by Hu and Shapley, for each player a simple game called the command game is built. One of the central concepts of this model is the concept of command function. We deliver sufficient and necessary conditions for a function to be a command function, and describe the minimal sets generating a normal command game. We also study the relation between command games and influence functions. A sufficient and necessary condition for the equivalence between an influence function and a normal command game is delivered. 
Keywords:  Influence function, follower function, lower and upper inverses, kernel, command game, command function, minimal sets generating a command game. 
JEL:  C7 D7 
Date:  2008–12 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:b08080&r=gth 
By:  Michel Grabisch (Centre d'Economie de la Sorbonne); Agnieszka Rusinowska (GATE) 
Abstract:  In the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions, we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the ShapleyShubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between thes two power indices lies in the weights for the influence indices. An example of the Confucian model od society is broadly examined. 
Keywords:  Influence function, follower, influence index, command game, commandable player, ShapleyShubik index, Banzhaf index, Coleman indicies, KönigBräuninger index. 
JEL:  C7 D7 
Date:  2008–12 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:b08078&r=gth 
By:  AlparslanGok, S.Z.; Brânzei, R.; Fragnelli, V.; Tijs, S.H. (Tilburg University, Center for Economic Research) 
Abstract:  In this paper we consider onemachine sequencing situations with interval data. We present different possible scenarioes and extend classical results on well known rules and on sequencing games to the interval setting. 
Keywords:  cooperative games;interval data;sequencing situations;convex games 
JEL:  C71 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:200863&r=gth 
By:  Peters, Michael; Szentes, Balazs 
Abstract:  This paper analyzes Bayesian normal form games in which players write contracts that condition their actions on the contracts of the other players. These contracts are required to be representable in a formal language. This is accomplished by constructing contracts which are definable functions of the Godel code of every other player's contract. We provide a complete characterization of the set of allocations supportable as pure strategy Bayesian equilibrium of this contracting game. When information is complete, this characterization provides a folk theorem. In general, the set of supportable allocations is smaller than the set supportable by a centralized mechanism designer. 
Date:  2009–01–22 
URL:  http://d.repec.org/n?u=RePEc:ubc:pmicro:michael_peters20097&r=gth 
By:  Hendrickx, R.L.P.; Borm, P.E.M.; Genugten, B.B. van der; Hilbers, P. (Tilburg University, Center for Economic Research) 
Abstract:  In several jurisdictions, commercially exploiting a game of chance (rather than skill) is subject to a licensing regime. It is obvious that roulette is a game of chance and chess a game of skill, but the law does not provide a precise description of where the boundary between the two classes is drawn. We build upon the framework of Borm and Van der Genugten (2001) and Dreef et al. (2004) and propose a modification of the skill concept for moreperson games. We apply our new skill measure to a simplified version of poker called Straight Poker and conclude that this game should be classified as a game of skill. 
Keywords:  games of chance;games of skill;poker 
JEL:  C72 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:2008106&r=gth 
By:  Ciftci, B.B.; Borm, P.E.M.; Hamers, H.J.M.; Slikker, M. (Tilburg University, Center for Economic Research) 
Abstract:  Game theoretic analysis of sequencing situations has been restricted to manufactur ing systems which consist of machines that can process only one job at a time. However, in many manufacturing systems, operations are carried out by batch machines which can simultaneously process multiple jobs. This paper aims to extend the game theoretical approach to the cost allocation problems arising from sequencing situations on systems that consist of batch machines. We first consider sequencing situations with a single batch machine and analyze cooperative games arising from these situations. It is shown that these games are convex and an expression for the Shapley value of these games is provided. We also introduce an equal gain splitting rule for these sequencing situa tions and provide an axiomatic characterization. Second, we analyze various aspects of flowshop sequencing situations which consist of batch machines only. In particular, we provide two cases in which the cooperative game arising from the flowshop sequencing situation is equal to the game arising from a sequencing situation that corresponds to one specific machine in the flowshop. 
Keywords:  Sequencing situations;sequencing games;batch machines. 
JEL:  C71 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:2008100&r=gth 
By:  Estevez Fernandez, M.A. (Tilburg University, Center for Economic Research) 
Abstract:  This paper analyzes situations in which a project consisting of several activities is not realized according to plan. If the project is expedited, a reward arises. Analogously, a penalty arises if the project is delayed. This paper considers the case of arbitrary monotonic reward and penalty functions on the total expedition and delay, respectively. Attention is focused on how to divide the total reward (penalty) among the activities: the core of a corresponding cooperative project game determines a set of stable allocations of the total reward (penalty). In the definition of project games, surplus (cost) sharing mechanisms are used to take into account the specific characteristics of the reward (penalty) function at hand. It turns outs that project games are related to bankruptcy and taxation games. This relation allows us to establish the nonemptiness of the core of project games. 
Keywords:  Project planning;delay;expedition;cost sharing mechanism;surplus sharing mechanism;bankruptcy problems;taxation problems;cooperative game;core. 
JEL:  C71 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:200884&r=gth 
By:  Mohamed Belhaj (GREQAM  Groupement de Recherche en Économie Quantitative d'AixMarseille  Université de la Méditerranée  AixMarseille II  Université Paul Cézanne  AixMarseille III  Ecole des Hautes Etudes en Sciences Sociales  CNRS : UMR6579, ECM  Ecole Centrale de Marseille  Ecole Centrale de Marseille); Frédéric Deroïan (GREQAM  Groupement de Recherche en Économie Quantitative d'AixMarseille  Université de la Méditerranée  AixMarseille II  Université Paul Cézanne  AixMarseille III  Ecole des Hautes Etudes en Sciences Sociales  CNRS : UMR6579) 
Abstract:  This article explores individual incentives to produce information on communication networks. In our setting, efforts are strategic complements along communication paths with possible decay. We analyze Nash equilibria on the line network. We give conditions under which more central agents provide more efforts for general payofffunctions, and we fully characterize equilibria under geometric decay. 
Keywords:  Communication Network, Endogenous Efforts, Strategic Complements 
Date:  2009–01–16 
URL:  http://d.repec.org/n?u=RePEc:hal:wpaper:halshs00353872_v1&r=gth 
By:  Herings, P.J.J.; Laan, G. van der; Talman, A.J.J.; Yang, Z.F. (Tilburg University, Center for Economic Research) 
Abstract:  AMS subject classifications: 90B18, 91A12, 91A43. 
JEL:  C71 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:200873&r=gth 
By:  Michael Schwarz; M. Bumin Yenmez 
Abstract:  We define the median stable matching for twosided matching markets with side payments and prove constructively that it exists. 
JEL:  C78 J01 
Date:  2009–01 
URL:  http://d.repec.org/n?u=RePEc:nbr:nberwo:14689&r=gth 
By:  Driesen Bram; Perea Andrés; Peters Hans (METEOR) 
Abstract:  The Rubinstein alternating offers bargaining game is reconsidered under the assumption that each player is loss averse and the associated reference point is equal to the highest turned down offer of the opponent in the past. This makes the payoffs and therefore potential equilibrium strategies dependent on the history of play. A subgame perfect equilibrium is constructed, in which the strategies depend on the history of play throughthe current reference points. It is shown that this equilibrium is unique under some assumptions that it shares with the equilibrium in the classical model: immediate acceptance of equilibrium offers, indifference between acceptance and rejection of such offers, and strategies depending only on the current reference points. It is also shown that in this equilibrium loss aversion is a disadvantage. Moreover, a relation with asymmetric Nashbargaining is established, where a player’s bargaining power is negatively related to own loss aversion and positively to the opponent’s loss aversion. 
Keywords:  mathematical economics; 
Date:  2009 
URL:  http://d.repec.org/n?u=RePEc:dgr:umamet:2009001&r=gth 
By:  Iyengar, G.; Kets, W.; Sethi, R.; Bowles, S. (Tilburg University, Center for Economic Research) 
Abstract:  This paper explores the manner in which the structure of a social network constrains the level of inequality that can be sustained among its members. We assume that any distribution of value across the network must be stable with respect to coalitional deviations, and that players can form a deviating coalition only if they constitute a clique in the network. We show that if the network is bipartite, there is a unique stable payoff distribution that is maximally unequal in that it does not Lorenz dominate any other stable distribution. We obtain a complete ordering of the class of bipartite networks and show that those with larger maximum independent sets can sustain greater levels of inequality. The intuition behind this result is that networks with larger maximum independent sets are more sparse and hence offer fewer opportunities for coalitional deviations. We also demonstrate that standard centrality measures do not consistently predict inequality. We extend our framework by allowing a group of players to deviate if they are all within distance k of each other, and show that the ranking of networks by the extent of extremal inequality is not invariant in k. 
Keywords:  inequality;networks;coalitional deviations;power;centrality. 
JEL:  C71 D30 D85 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:200876&r=gth 
By:  Michel Grabisch (Centre d'Economie de la Sorbonne); Agnieszka Rusinowska (GATE) 
Abstract:  In the paper, we study a model of influence in a social network. It is assumed that each player has an inclination to say YES or NO which, due to influence of other players, may be different from the decision of the player. The point of departure here is the concept of the HoedeBakker index  the notion which computes the overall decisional "power" of a player in a social network. The main drawback of the HoedeBakker index is that it hides the actual role of the influence function, analyzing only the final decision in terms of success and failure. In this paper, we separate the influence part from the group decision part, and focus on the description and analysis of the influence part. We propose among other descriptive tools a definition of a (weighted) influence index of a coalition upon an individual. Moreover, we consider different influence functions representative of commonly encountered situations. Finally, we propose a suitable definition of a modified decisional power. 
Keywords:  Influence function, influence index, decisional power, social network. 
JEL:  C7 D7 
Date:  2008–12 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:b08066&r=gth 
By:  Johannes Horner; Larry Samuelson 
Date:  2009–01–15 
URL:  http://d.repec.org/n?u=RePEc:cla:levarc:814577000000000059&r=gth 
By:  Moretti, S.; AlparslanGok, S.Z.; Brânzei, R.; Tijs, S.H. (Tilburg University, Center for Economic Research) 
Keywords:  cooperative cost games;minimum cost spanning tree situations;robustness;worstcase scenario;input interval data;uncertainty. 
JEL:  C71 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:200864&r=gth 
By:  Dutta, Bhaskar (Department of Economics, University of Warwick); Mishra, Debasis (Indian Statistical Institute) 
Abstract:  In this paper, we analyze the cost allocation problem when a group of agents or nodes have to be connected to a source, and where the cost matrix describing the cost of connecting each pair of agents is not necessarily symmetric, thus extending the wellstudied problem of minimum cost spanning tree games, where the costs are assumed to be symmetric. The focus is on rules which satisfy axioms representing incentive and fairness properties. We show that while some results are similar, there are also signifcant differences between the frameworks corresponding to symmetric and asymmetric cost matrices. 
Keywords:  directed networks ; cost allocation ; core stability ; continuity ; cost monotonicity 
Date:  2009 
URL:  http://d.repec.org/n?u=RePEc:wrk:warwec:889&r=gth 
By:  Talman, A.J.J.; Yang, Z.F. (Tilburg University, Center for Economic Research) 
Abstract:  This paper presents a model of partnership formation. A set of agents wants to conduct some business or other activities. Agents may act alone or seek a partner for cooperation and need in the latter case to consider with whom to cooperate and how to share the profit in a collaborative and competitive environment. We pro vide necessary and su±cient conditions under which an equilibrium can be attained. In equilibrium, the partner formation and the payoff distribution are endogenously determined. Every agent realizes his full potential and has no incentive to deviate from either staying independent or from the endogenously determined partner and payoff. The partnership formation problem contains the classical assignment market problem as a special case. 
Keywords:  Partnership formation;equilibrium;indivisibility;assignment market. 
JEL:  C62 C72 D02 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:dgr:kubcen:2008103&r=gth 
By:  Antoine Billot (PSE et Université Paris 2); JeanChristophe Vergnaud (Centre d'Economie de la Sorbonne); Bernard Walliser (PSE) 
Abstract:  One main topic of the epistemic program of game theory deals with the value of information. To study this question in a broad context, one needs to adapt some of the tools used in multiplayer epistemic logic. A hierarchical belief structure is introduced both in a syntactical and semantical framework. In the same framework, a generalized notion of message is characterized by its content and its status. For a given message, a multiplayer belief revision rule that transforms any initial belief structure into a final belief one is designed. A representation theorem relates syntactical axioms to the belief revision rule. 
Keywords:  Belief revision, message, epistemic logic. 
JEL:  C72 D82 
Date:  2008–08 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:v08067&r=gth 
By:  René van der Brink (Department of Econometrics and Tinbergen Institute, Free University, The Netherlands); Agnieszka Rusinowska (University of Lyon, Lyon, F69003, France; CNRS, UMR 5824, GATE, Ecully, F69130, France; ENS LSH, Lyon, F69007, France ; Centre Leon Berard, Lyon, F69003, France; Department of Econometrics and Tinbergen Institute, Free University, The Netherlands); Frank Steffen (The University of Liverpool Management School (ULMS)) 
Abstract:  A well known and established model in communication policy in sociology and mar keting is that of opinion leadership. It is based on the idea of a twostep flow of communication. Opinion leaders are actors in a society who are able to affect the behavior of other members of the society called followers. Hence, opinion leaders might have a considerable impact on the behavior of markets and other social agglomerations being made up of individual actors choosing among a number of alternatives. For marketing purposes it appears to be interesting to investigate the effect of different opinion leaderfollower structures in markets or any other collective decisionmaking situations in a society. We study a twoaction model in which the members of a society are to choose one action, for instance, to buy or not to buy a certain joint product, or to vote yes or no on a specific proposal. Each of the actors has an inclination to choose one of the actions. By definition opinion leaders have some power over other actors, their followers, and they exercise this power by influencing the behavior of their followers, i.e. their choice of action. After all actors have chosen their actions, a decision making mechanism determines the collective choice resulting out of the individual choices. The structure of the relations between the actors can be represented by a bipartite digraph. We analyze such digraphs investigating satisfaction and power distributions within societies with and without the opinion leaders. Moreover, we study common properties of the satisfaction and power measures and illustrate our findings and some marketing implications for a society with five members. 
Keywords:  Bipartite digraph, influence, inclination, collective choice, opinion leader, follower, satisfaction, power 
JEL:  C7 D7 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:gat:wpaper:0901&r=gth 