
on Game Theory 
By:  Carmona, Guilherme 
Abstract:  We show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an "nequilibria, with "n converging to zero. In our characterization, the sequence of finite games approaches the continuum game in the sense that the set of players and the distribution of characteristics and actions in the finite games converge to those of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance. Also, they suggest defining a refinement of Nash equilibria for games with a continuum of agents as limit points of equilibria of finite games. This allows us to discard those Nash equilibria that are artifacts of the continuum model, making limit equilibrium a natural equilibrium concept for games with a continuum of players. 
Keywords:  Nash equilibrium, limit equilibrium, noncooperative games, continuum of players 
Date:  2004 
URL:  http://d.repec.org/n?u=RePEc:unl:unlfep:wp442&r=gth 
By:  Burkhard C. Schipper 
Abstract:  We provide an evolutionary foundation to evidence that in some situations humans maintain optimistic or pessimistic attitudes towards uncertainty and are ignorant to relevant aspects of the environment. Players in strategic games face Knightian uncertainty about opponents' actions and maximize individually their Choquet expected utility. Our Choquet expected utility model allows for both an optimistic or pessimistic attitude towards uncertainty as well as ignorance to strategic dependencies. An optimist (resp. pessimist) overweights good (resp. bad) outcomes. A complete ignorant never reacts to opponents' change of actions. With qualifications we show that optimistic (resp. pessimistic) complete ignorance is evolutionary stable / yields a strategic advantage in submodular (resp. supermodular) games with aggregate externalities. Moreover, this evolutionary stable preference leads to Walrasian behavior in those classes of games. 
Keywords:  ambiguity, Knightian uncertainty, Choquet expected utility, neoadditive capacity, Hurwicz criterion, Maximin, Minimax, Ellsberg paradox, overconfidence, supermodularity, aggregative games, monotone comparative statics, playing the field, evolution of preferences 
JEL:  C72 C73 D43 D81 L13 
Date:  2005–11 
URL:  http://d.repec.org/n?u=RePEc:bon:bonedp:bgse35_2005&r=gth 
By:  Carmona, Guilherme 
Abstract:  We characterize Nash equilibria of games with a continuum of players (MasColell (1984)) in terms of approximate equilibria of large finite games. For the concept of ("; ") equilibrium in which the fraction of players not " optimizing is less than " we show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an ("n; "n) equilibria, with "n converging to zero. The same holds for " equilibrium in which almost all players are " optimizing provided that either players payoff functions are equicontinuous or players action space is finite. Furthermore, we give conditions under which the above results hold for all approximating sequences of games. In our characterizations, a sequence of finite games approaches the continuum game in the sense that the number of players converges to infinity and the distribution of characteristics and actions in the finite games converges to that of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance. 
Date:  2004 
URL:  http://d.repec.org/n?u=RePEc:unl:unlfep:wp466&r=gth 
By:  Carmona, Guilherme 
Abstract:  We consider an asymptotic version of MasColells theorem on the existence of pure strategy Nash equilibria in large games. Our result states that, if players payoff functions are selected from an equicontinuous family, then all sufficiently large games have an " pure, " equilibrium for all " > 0. We also show that our result is equivalent to MasColells existence theorem, implying that it can properly be considered as its asymptotic version. 
Date:  2004 
URL:  http://d.repec.org/n?u=RePEc:unl:unlfep:wp465&r=gth 
By:  Fabrizio Germano; Gábor Lugosi 
Abstract:  We show that every finite Nplayer game possesses a correlated equilibrium with a precise lower bound on the number of outcomes to which it assigns zero probability. In particular, the largest games with a unique fully supported correlated equilibrium are threeplayer 2x2x2 games; moreover, the lower bound grows exponentially in the number of players N. 
Keywords:  Correlated equilibrium, finite games 
JEL:  C72 
Date:  2005–10 
URL:  http://d.repec.org/n?u=RePEc:upf:upfgen:907&r=gth 
By:  Josef Hofbauer (Dept. of Mathematics, University College London); Joerg Oechssler (Dept. of Economics, University of Heidelberg); Frank Riedel (Dept. of Economics, University of Bonn) 
Abstract:  In John Nash’s proofs for the existence of (Nash) equilibria based on Brouwer’s theorem, an iteration mapping is used. A continuous— time analogue of the same mapping has been studied even earlier by Brown and von Neumann. This differential equation has recently been suggested as a plausible boundedly rational learning process in games. In the current paper we study this Brown—von Neumann—Nash dynamics for the case of continuous strategy spaces. We show that for continuous payoff functions, the set of rest points of the dynamics coincides with the set of Nash equilibria of the underlying game. We also study the asymptotic stability properties of rest points. While strict Nash equilibria may be unstable, we identify sufficient conditions for local and global asymptotic stability which use concepts developed in evolutionary game theory. 
Keywords:  Learning in games; evolutionary stability; BNN 
JEL:  C70 C72 
Date:  2005–12–15 
URL:  http://d.repec.org/n?u=RePEc:wpa:wuwpga:0512003&r=gth 
By:  Ron Holzman; Bezalel Peleg; Peter Sudholter 
Abstract:  Let A be a finite set of m alternatives, let N be a finite set of n players and let R<sup>N</sup> be a profile of linear preference orderings on A of the players. Let u<sup>N</sup> be a profile of utility functions for R<sup>N</sup>. We define the NTU game V<sub>u<sup>N</sup></sub> that corresponds to simple majority voting, and investigate its AumannDavisMaschler and MasColell bargaining sets. The first bargaining set is nonempty for m <FONT FACE="Symbol">£</FONT> 3 and it may be empty for m <FONT FACE="Symbol">³</FONT> 4. However, in a simple probabilistic model, for fixed m, the probability that the AumannDavisMaschler bargaining set is nonempty tends to one if n tends to infinity. The MasColell bargaining set is nonempty for m <FONT FACE="Symbol">£</FONT> 5 and it may be empty for m <FONT FACE="Symbol">³</FONT> 6. Furthermore, it may be empty even if we insist that n be odd, provided that m is sufficiently large. Nevertheless, we show that the MasColell bargaining set of any simple majority voting game derived from the kth replication of R<sup>N</sup> is nonempty, provided that k <FONT FACE="Symbol">³</FONT> n + 2. 
Keywords:  NTU game; voting game; majority rule; bargaining set 
JEL:  C71 D71 
Date:  2005–11 
URL:  http://d.repec.org/n?u=RePEc:huj:dispap:dp410&r=gth 
By:  Carmona, Guilherme 
Abstract:  We show that a distribution of a game with a continuum of players is an equilibrium distribution if and only if there exists a sequence of symmetric approximate equilibrium distributions of games with fi nite support that converges to it. Thus, although not all games have symmetric equilibrium distributions, this result shows that all equilibrium distributions can be characterized by symmetric distributions of simpler games (i.e., games with a finite number of characteristics). 
Date:  2004 
URL:  http://d.repec.org/n?u=RePEc:unl:unlfep:wp441&r=gth 
By:  Hoernig, Steffen 
Abstract:  We analyze how sharing rules affect Nash equilibria in Bertrand games, where the sharing of profits at ties is a decisive assumption. Necessary conditions for either positive or zero equilibrium profits are derived. Zero profit equilibria are shown to exist under weak conditions if the sharing rule is signpreserving. For Bertrand markets we define the class of expectation sharing rules, where profits at ties are derived from some distribution of quantities. In this class the winnertakeall sharing rule is the only one that is always signpreserving, while for each pair of demand and cost functions there may be many others. 
Keywords:  Bertrand games, Sharing rule, Tiebreaking rule, Signpreserving sharing rules, Expectation sharing rules 
JEL:  C72 D43 L13 
Date:  2005 
URL:  http://d.repec.org/n?u=RePEc:unl:unlfep:wp468&r=gth 
By:  Edward Castronova 
Abstract:  Games like EverQuest and Dark Age of Camelot occasionally produce natural experiments in social science: situations that, through no intent of the designer, offer controlled variations on a phenomenon of theoretical interest. This paper examines two examples, both of which involve the theory of coordination games: 1) the location of markets inside EverQuest, and 2) the selection of battlefields inside Dark Age of Camelot. Coordination game theory is quite important to a number of literatures in political science, economics, sociology, and anthropology, but has had very few direct empirical tests because that would require experimental participation by large numbers of people. The paper argues that games, unlike any other social science research technology, provide for both sufficient participation numbers and careful control of experimental conditions. Games are so wellsuited to the latter that, in the two cases we examine, the natural experiments that happened were, in fact, perfectly controlled on every relevant factor, without any intention of the designer. This suggests that large games should be thought of as, in effect, social science research tools on the scale of the supercolliders used by physicists: expensive, but extremely fruitful. 
JEL:  L86 
Date:  2005 
URL:  http://d.repec.org/n?u=RePEc:ces:ceswps:_1621&r=gth 
By:  Abraham Neyman; Daijiro Okada 
Abstract:  One way to express bounded rationality of a player in a game theoretic models is by specifying a set of feasible strategies for that player. In dynamic game models with finite automata and bounded recall strategies, for example, feasibility of strategies is determined via certain complexity measures: the number of states of automata and the length of recall. Typically in these models, a fixed finite bound on the complexity is imposed resulting in finite sets of feasible strategies. As a consequence, the number of distinct feasible strategies in any subgame is finite. Also, the number of distinct strategies induced in the first T stages is bounded by a constant that is independent of T. In this paper, we initiate an investigation into a notion of feasibility that reflects varying degree of bounded rationality over time. Such concept must entail properties of a strategy, or a set of strategies, that depend on time. Specifically, we associate to each subset Ψ<sub>i</sub> of the full (theoretically possible) strategy set a function <FONT FACE="Symbol">y</FONT><sub>i</sub> from the set of positive integers to itself. The value <FONT FACE="Symbol">y</FONT><sub>i</sub>(t) represents the number of strategies in Ψ<sub>i</sub> that are distinguishable in the first t stages. The set Ψ<sub>i</sub> may contain infinitely many strategies, but it can differ from the fully rational case in the way <FONT FACE="Symbol">y</FONT><sub>i</sub> grows reflecting a broad implication of bounded rationality that may be alleviated, or intensified, over time. We examine how the growth rate of <FONT FACE="Symbol">y</FONT><sub>i</sub> affects equilibrium outcomes of repeated games. In particular, we derive an upper bound on the individually rational payoff of repeated games where player 1, with a feasible strategy set Ψ<sub>1</sub>, plays against a fully rational player 2. We will show that the derived bound is tight in that a specific, and simple, set Ψ<sub>1</sub> exists that achieves the upper bound. As a special case, we study repeated games with nonstationary bounded recall strategies where the length of recall is allowed to vary in the course of the game. We will show that a player with bounded recall can guarantee the minimax payoff of the stage game even against a player with full recall so long as he can remember, at stage t, at least K log(t) stages back for some constant K >0. Thus, in order to guarantee the minimax payoff, it suffices to remember only a vanishing fraction of the past. A version of the folk theorem is provided for this class of games. 
Keywords:  bounded rationality; strategy set growth; strategic complexity; nonstationary bounded recall; repeated games; entropy 
Date:  2005–11 
URL:  http://d.repec.org/n?u=RePEc:huj:dispap:dp411&r=gth 