The asymptotic value, introduced by Kannai in 1966, is an asymptotic approach to the notion of the Shapley value for games with infinitely many players. A vector measure game is a game v where the worth v(S) of a coalition S is a function f of u(S) where u is a vector measure. Special classes of vector measure games are the weighted majority games and the two-house weighted majority games, where a two-house weighted majority game is a game in which a coalition is winning if and only if it is winning in two given weighted majority games. All weighted majority games have an asymptotic value. However, not all two-house weighted majority games have an asymptotic value. In this paper, we prove that the existence of infinitely many atoms with sufficient variety suffice for the existence of the asymptotic value in a general class of nonsmooth vector measure games that includes in particular two-house weighted majority games.

We prove here the existence of a value (of norm 1) on the spaces ’NA and even ’AN, the closure in the variation distance of the linear space spanned by all games f°µ, where µ is a non-atomic, non-negative finitely additive measure of mass 1 and f a real-valued function on [0,1] which satisfies a much weaker continuity at zero and one.

Much of economic theory is concerned with the existence of prices. In particular, economists are interested in whether various outcomes defined by diverse postulates turn out to be actually generated by prices. Whenever this is the case, a theory of endogenous price formation is derived. In the present analysis, a well-known game-theoretic solution concept is considered: value. Nonatomic games are considered that are defined by finitely many nonnegative measures. Nonatomic vector measure games arise, for example, from production models and from finite-type markets. It is shown that the value of such a game need not be a linear combination of the nonatomic nonnegative measures. This is in contrast to all the values known to date. Moreover, this happens even for certain differentiable market games. In the economic models, this means that the value allocations are not necessarily produced by prices. All the examples presented are special cases of a new class of values.

We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of the maximizer’s strategies while the other player is unrestricted. We will show that if the bound (n), a function of the number of repetitions n, satisfies the condition (n)/n (n), then the maxmin value Wn ((n)) converges to (cavU)(), the concavification of the maxmin value of the stage game in which the maximizer’s actions are restricted to those with entropy at most . A similar result is obtained for the infinitely repeated games.

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.

It is shown that an exponentially small departure from the common knowledge assumption on the number T of repetitions of the prisoners’ dilemma already enables cooperation. More generally, with such a departure, any feasible individually rational outcome of any one-shot game can be ap

We introduce the entropy-based measure of uncertainty for mixed strategies of repeated games-strategic entropy. We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of player 1’s strategies while player 2 is unrestricted, as the bound grows to infinity. We apply the results thus obtained to study the asymptotic behavior of the value of the repeated games with finite automata and bounded recall.

Every two person repeated game of symmetric incomplete information, in which the signals sent at each stage to both players are identical and generated by a state and moves dependent probability distribution on a given finite alphabet, has an equilibrium payoff.

Every two person repeated game of symmetric incomplete information, in which the signals sent at each stage to both players are identical and generated by a state and moves dependent probability distribution on a given finite alphabet, has an equilibrium payoff.

Any correlated equilibrium of a strategic game with bounded payoffs and convex strategy sets which has a smooth concave potential, is a mixture of pure strategy profiles which maximize the potential. If moreover, the strategy sets are compact and the potential is strictly concave, then the game has a unique correlated equilibrium.

Four axioms are placed on a correspondence from smooth, non-atomic economies to their allocations. We show that the axioms categorically determine the (coincident) competitive-core-value correspondence. Thus any solution is equivalent to the above three if, and only if, it satisfies the axioms. In this sense our result is tantamount to an "equivalence principle." At the same time, our result implies that the three solutions themselves are determined by the axioms and so serves as an axiomatic characterization of the well-known competitive (or core, or value) correspondence.

It has been remarked that in rational interactions more information to one player, while all others’ information remains the same, may reduce his payoff in equilibrium. This classical observation relies on comparing equilibria of two different games. It is argued that this analysis is not tenably performed by comparing equilibria of two different games. Rather, one is compelled to perform the analysis in an interaction without complete information, and to compare equilibria of two interactions that are embedded in some compounded game. It is then shown that the player whose information is unilaterally refined cannot be worse off at equilibrium. 

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 subgames of v.