Abraham Neyman

Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma

Abstract Cooperation in the finitely repeated prisoners dilemma is justified, without departure from strict utility maximization or complete information, but under the assumption that there are bounds (possibly very large) to the complexity of the strategies that the players may use.

Asymptotic Behavior of Nonexpansive Mappings in Normed Linear Spaces

LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, limn→xf(Tnx/n)=limn→x‖Tnx/n‖=α, where α≡infy∈c‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichTnx/n converges weakly for allx (infz∈fg(Tnx/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT.

Two-person repeated games with finite automata

Abstract. We study two-person repeated games in which a player with a restricted set of strategies plays against an unrestricted player. An exogenously given bound on the complexity of strategies, which is measured by the size of the smallest automata that implement them, gives rise to a restriction on strategies available to a player. We examine the asymptotic behavior of the set of equilibrium payoffs as the bound on the strategic complexity of the restricted player tends to infinity, but sufficiently slowly. Results from the study of zero sum case provide the individually rational payoff levels.

Uniqueness of the Shapley value

Abstract It is shown that the Shapley value of any given game v is characterized by applying the value axioms—efficiency, symmetry, the null player axiom, and either additivity or strong positivity—to the additive group generated by the game ν itself and its subgames.

Stochastic games and nonexpansive maps

This chapter studies asymptotic properties of the orbits of non-expansive maps defined on a normed space, and relates these properties to properties of the value of two-person zero-sum games and to properties of the minmax of n-person stochastic games.

Real Algebraic Tools in Stochastic Games

The present chapter brings together parts of the theory of polynomial equalities and inequalities used in the theory of stochastic games. The theory can be considered as a theory of polynomial equalities and inequalities over the field of real numbers or the field of real algebraic numbers or more generally over an arbitrary real closed field.

Existence of optimal strategies in Markov games with incomplete information

The existence of a value and optimal strategies is proved for the class of two-person repeated games where the state follows a Markov chain independently of players’ actions and at the beginning of each stage only Player 1 is informed about the state. The results apply to the case of standard signaling where players’ stage actions are observable, as well as to the model with general signals provided that Player 1 has a nonrevealing repeated game strategy. The proofs reduce the analysis of these repeated games to that of classical repeated games with incomplete information on one side.

Stochastic games: Existence of the MinMax

The existence of the value for stochastic games with finitely many states and actions, as well as for a class of stochastic games with infinitely many states and actions, is proved in [2]. Here we use essentially the same tools to derive the existence of the minmax and maxmin for n-player stochastic games with finitely many states and actions, as well as for a corresponding class of n-person stochastic games with infinitely many states and actions.

Voting for Public Goods

It is shown that when resources are privately owned, the institution of voting is irrelevant to the choice of non-exclusive public goods: the total bundle of such goods produced by Society is the same whether or not minority coalitions are permitted to produce them. This is in sharp contrast to the cases of redistribution and of exclusive public goods, where public decisions depend strongly on the vote. The analytic tool used is the Harsanyi-Shapley non-transferable utility value.

Continuous-time stochastic games ☆

Every continuous-time stochastic game with finitely many states and actions has a uniform and limiting-average equilibrium payoff.