On the existence of pure strategy equilibria in large generalized games with atomic players

We consider a game with a continuum of players where only a finite number of them are atomic. Objective functions and admissible strategies may depend on the actions chosen by atomic players and on aggregate information about the actions chosen by non-atomic players. Only atomic players are required to have convex sets of admissible strategies and quasi-concave objective functions. We prove the existence of a pure strategy Nash equilibria. Thus, we extend to large generalized games with atomic players the results of equilibrium existence for non-atomic games of Schemeidler (1973) and Rath (1992). We do not obtain a pure strategy equilibrium by purification of mixed strategy equilibria. Thus, we have a direct proof of both Balder (1999, Theorem 2.1) and Balder (2002, Theorem 2.2.1), for the case where non-atomic players have a common non-empty set of strategies and integrable bounded codification of action profiles. Our main result is readily applicable to many interesting problems in general equilibrium. As an application, we extend Aumann (1966) result on the existence of equilibrium with a continuum of traders to a standard general equilibrium model with incomplete asset markets.Generalized games; Non-convexities; Pure-strategy Nash equilibrium

Evolutionarily stable strategies, preferences and moral values, in n-player Interactions

We provide a generalized definition of evolutionary stability of heritable types in arbitrarily large symmetric interactions under random matching that may be assortative. We establish stability results when these types are strategies in games, and when they are preferences or moral values in games under incomplete information. We show that a class of moral preferences, with degree of morality equal to the index of assortativity are evolutionarily stable. In particular, selfishness is evolutionarily unstable when there is positive assortativity in the matching process. We establish that evolutionarily stable strategies are the same as those played in equilibrium by rational but partly morally motivated individuals, individuals with evolutionarily stable preferences. We provide simple and operational criteria for evolutionary stability and apply these to canonical examples

Evolution leads to Kantian morality

We provide a generalized definition of evolutionary stability of heritable types in arbitrarily large symmetric interactions under random matching that may be assortative. We establish stability results when these types are strategies in games, and when they are preferences or moral values in games under incomplete information. We show that a class of moral preferences, with degree of morality equal to the index of assortativity are evolutionarily stable. In particular, selfishness is evolutionarily unstable when there is positive assortativity in the matching process. We establish that evolutionarily stable strategies are the same as those played in equilibrium by rational but partly morally motivated individuals, individuals with evolutionarily stable preferences. We provide simple and operational criteria for evolutionary stability and apply these to canonical examples

Evolution leads to Kantian morality

We provide a generalized definition of evolutionary stability of heritable types in arbitrarily large symmetric interactions under random matching that may be assortative. We establish stability results when these types are strategies in games, and when they are preferences or moral values in games under incomplete information. We show that a class of moral preferences, with degree of morality equal to the index of assortativity are evolutionarily stable. In particular, selfishness is evolutionarily unstable when there is positive assortativity in the matching process. We establish that evolutionarily stable strategies are the same as those played in equilibrium by rational but partly morally motivated individuals, individuals with evolutionarily stable preferences. We provide simple and operational criteria for evolutionary stability and apply these to canonical examples

A probabilistic approach to quantum Bayesian games of incomplete information

A Bayesian game is a game of incomplete information in which the rules of the game are not fully known to all players. We consider the Bayesian game of Battle of Sexes that has several Bayesian Nash equilibria and investigate its outcome when the underlying probability set is obtained from generalized Einstein-Podolsky-Rosen experiments. We find that this probability set, which may become non-factorizable, results in a unique Bayesian Nash equilibrium of the game.Comment: 18 pages, 2 figures, accepted for publication in Quantum Information Processin

Perfect Prediction in Minkowski Spacetime: Perfectly Transparent Equilibrium for Dynamic Games with Imperfect Information

The assumptions of necessary rationality and necessary knowledge of strategies, also known as perfect prediction, lead to at most one surviving outcome, immune to the knowledge that the players have of them. Solutions concepts implementing this approach have been defined on both dynamic games with perfect information and no ties, the Perfect Prediction Equilibrium, and strategic games with no ties, the Perfectly Transparent Equilibrium. In this paper, we generalize the Perfectly Transparent Equilibrium to games in extensive form with imperfect information and no ties. Both the Perfect Prediction Equilibrium and the Perfectly Transparent Equilibrium for strategic games become special cases of this generalized equilibrium concept. The generalized equilibrium, if there are no ties in the payoffs, is at most unique, and is Pareto-optimal. We also contribute a special-relativistic interpretation of a subclass of the games in extensive form with imperfect information as a directed acyclic graph of decisions made by any number of agents, each decision being located at a specific position in Minkowski spacetime, and the information sets and game structure being derived from the causal structure. Strategic games correspond to a setup with only spacelike-separated decisions, and dynamic games to one with only timelike-separated decisions. The generalized Perfectly Transparent Equilibrium thus characterizes the outcome and payoffs reached in a general setup where decisions can be located in any generic positions in Minkowski spacetime, under necessary rationality and necessary knowledge of strategies. We also argue that this provides a directly usable mathematical framework for the design of extension theories of quantum physics with a weakened free choice assumption.Comment: 25 pages, updated technical repor

Extensive Games with Possibly Unaware Players

Standard game theory assumes that the structure of the game is common knowledge among players. We relax this assumption by considering extensive games where agents may be unaware of the complete structure of the game. In particular, they may not be aware of moves that they and other agents can make. We show how such games can be represented; the key idea is to describe the game from the point of view of every agent at every node of the game tree. We provide a generalization of Nash equilibrium and show that every game with awareness has a generalized Nash equilibrium. Finally, we extend these results to games with awareness of unawareness, where a player i may be aware that a player j can make moves that i is not aware of, and to subjective games, where payers may have no common knowledge regarding the actual game and their beliefs are incompatible with a common prior.Comment: 45 pages, 3 figures, a preliminary version was presented at AAMAS0

On the existence of pure strategy equilibria in large generalized games with atomic players

We consider a game with a continuum of players where only a finite number of them are atomic. Objective functions and admissible strategies may depend on the actions chosen by atomic players and on aggregate information about the actions chosen by non-atomic players. Only atomic players are required to have convex sets of admissible strategies and quasi-concave objective functions. We prove the existence of a pure strategy Nash equilibria. Thus, we extend to large generalized games with atomic players the results of equilibrium existence for non-atomic games of Schemeidler (1973) and Rath (1992). We do not obtain a pure strategy equilibrium by purification of mixed strategy equilibria. Thus, we have a direct proof of both Balder (1999, Theorem 2.1) and Balder (2002, Theorem 2.2.1), for the case where non-atomic players have a common non-empty set of strategies and integrable bounded codification of action profiles. Our main result is readily applicable to many interesting problems in general equilibrium. As an application, we extend Aumann (1966) result on the existence of equilibrium with a continuum of traders to a standard general equilibrium model with incomplete asset markets

On the existence of pure strategy equilibria in large generalized games with atomic players

We consider a game with a continuum of players where only a finite number of them are atomic. Objective functions and admissible strategies may depend on the actions chosen by atomic players and on aggregate information about the actions chosen by non-atomic players. Only atomic players are required to have convex sets of admissible strategies and quasi-concave objective functions. We prove the existence of a pure strategy Nash equilibria. Thus, we extend to large generalized games with atomic players the results of equilibrium existence for non-atomic games of Schemeidler (1973) and Rath (1992). We do not obtain a pure strategy equilibrium by purification of mixed strategy equilibria. Thus, we have a direct proof of both Balder (1999, Theorem 2.1) and Balder (2002, Theorem 2.2.1), for the case where non-atomic players have a common non-empty set of strategies and integrable bounded codification of action profiles. Our main result is readily applicable to many interesting problems in general equilibrium. As an application, we extend Aumann (1966) result on the existence of equilibrium with a continuum of traders to a standard general equilibrium model with incomplete asset markets