Non-cooperative matching games

Consider a market in which two distinct groups of agents face each other. Every agent can improve upon his status quo if he is matched with a member of the opposite group and if he agrees with his partner how to split the realized gain. The paper presents two non-cooperative games in which the agents construct the allocation without the help of an auctioneer. In the first game the set of equilibria coincides with the cooperative “stable solutions” which are well-established in this context. In the second game it is shown that a change of the order of the moves is sufficient to arrive at a unique equilibrium payoff.     

Non-cooperative monopolistic games and monopolistic market games

A class of non-cooperative games is discussed in which one player (“the monopolist”) by choosing his strategy restricts the other players to subsets of their strategy sets. Examples of such games in various fields are given. In particular it is shown that some very important economic situations fall within this class of games. A solution concept is defined and sufficient conditions for its existence are derived. The question of the advantages a player derives from being a monopolist is raised and conditions are derived for him to benefit from being a monopolist.     

Non-cooperative games with minmax objectives

We consider noncooperative games where each player minimizes the sum of a smooth function, which depends on the player, and of a possibly nonsmooth function that is the same for all players. For this class of games we consider two approaches: one based on an augmented game that is applicable only to a minmax game and another one derived by a smoothing procedure that is applicable more broadly. In both cases, centralized and, most importantly, distributed algorithms for the computation of Nash equilibria can be derived.     

Non-cooperative two-person games in biology: A classification

This article compares evolutionary equilibrium notions with solution concepts in rational game theory. Both static and dynamic evolutionary game theory are treated. The methods employed by dynamic theory, so-called “game dynamics”, could be discovered to be relevant for rational game theory also.     

On stochastic games

In this paper, we consider the stochastic games of Shapley, when the state and action spaces are all infinite. We prove that, under certain conditions, the stochastic game has a value and that both players have optimal strategies.Part of this research was supported by NSF grant. The authors are indebted to L. S. Shapley for the useful discussions on this and related topics. The authors thank the referee for pointing out an ambiguity in the formulation of Lemma 2.4 in an earlier draft of this article.     

Operational asymptotic stochastic dominance

Levy (2016) proposes asymptotic first-degree stochastic dominance as a distribution ranking criterion for all non-satiable decision makers with infinite investment horizons. Given Levy’s setting, this paper defines and offers the equivalent distributional conditions for asymptotic second-degree stochastic dominance, as well as operational asymptotic first- and second-degree stochastic dominance. Interestingly, the operational asymptotic stochastic dominance provides a full rank over assets with lognormal returns and different means. Empirical applications show that our conditions can be readily implemented in practice.     

Linear-quadratic stochastic pursuit-evasion games

A linear-quadratic differential game in which the system state is affected by disturbance and both players have access to different measurements is solved. The problem is first converted to an optimization problem in infinite-dimensional state space and then solved using standard techniques. For convenience, L ₂-white noise instead of Wiener process setup is used.     

Two-person zero-sum stochastic games

Two-person zero-sum stochastic games with finite state and action spaces are considered. The expected average payoff criterion is introduced. In the special case of single controller games it is shown that the optimal stationary policies and the value of the game can be obtained from the optimal solutions to a pair of dual programs. For multichain structures, a decomposition algorithm is given which produces such optimal stationary policies for both players. In the case of both players controlling the transitions, a generalized game is obtained, the solution of which gives the optimal policies.     

Non-differentiable transformations preserving stochastic dominance

In this paper, we solve the following problem: when does a stochastic improvement in one risk maintain itself under a non everywhere continuously differentiable transformation of this risk? Using the notion of divided differences, we show that stochastic dominance at the third (and higher) order, and sometimes at the second one, is not preserved after simple piecewise linear transformation of the initial risk. Our analysis complements the one that exists for everywhere continuously differentiable transformations.     

Instrument effects and stochastic dominance

This paper investigates how individual choice is affected by increases in risk when the choice variable (instrument) affects the distribution of the random variable as well as the objective function. The effect of increased risk on optimal choice is shown to depend on attitudes towards risk and the interaction between exogenous uncertainty and the instrument. The latter is described in terms of an extension of the notion of stochastic dominance to a comparison of changes in probability distributions (signed measures) rather than the direct comparison of distributions (probability measures). Sufficiency conditions for signing comparative statistics exercises are presented and applied to an insurance example involving moral hazard.     

Stochastic dominance equilibria in two-person noncooperative games

Two-person noncooperative games with finitely many pure strategies are considered, in which the players have linear orderings over sure outcomes but incomplete preferences over probability distributions resulting from mixed strategies. These probability distributions are evaluated according to t-degree stochastic dominance. A t-best reply is a strategy that induces a t-degree stochastically undominated distribution, and a t-equilibrium is a pair of t-best replies. The paper provides a characterization and an existence proof of t-equilibria in terms of representing utility functions, and shows that for large t behavior converges to a form of max–min play. Specifically, increased aversion to bad outcomes makes each player put all weight on a strategy that maximizes the worst outcome for the opponent, within the supports of the strategies in the limiting sequence of t-equilibria.The paper has benefitted from the comments of four referees and an associate editor.     

Perfect equilibria in stochastic games

We examine stochastic games with finite state and action spaces. For the -discounted case, as well as for the irreducible limiting average case, we show the existence of trembling-hand perfect equilibria and give characterizations of those equilibria. In the final section, we give an example which illustrates that the existence of stationary limiting average equilibria in a nonirreducible stochastic game does not imply the existence of a perfect limiting average equilibrium.Support was provided by the Netherlands Organization for Scientific Research NWO via the Netherlands Foundation for Mathematics SMC, Project 10-64-10.     

Fictitious play in stochastic games

In this paper we examine an extension of the fictitious play process for bimatrix games to stochastic games. We show that the fictitious play process does not necessarily converge, not even in the 2 × 2 × 2 case with a unique equilibrium in stationary strategies. Here 2 × 2 × 2 stands for 2 players, 2 states, 2 actions for each player in each state.     

Algorithms for discounted stochastic games

In this paper, a two-person zero-sum discounted stochastic game with a finite state space is considered. The movement of the game from state to state is jointly controlled by the two players with a finite number of alternatives available to each player in each of the states. We present two convergent algorithms for arriving at minimax strategies for the players and the value of the game. The two algorithms are compared with respect to computational efficiency. Finally, a possible extension to nonzero sum stochastic game is suggested.This research was supported in part by funds allocated to the Department of Operations Research, School of Management, Case Western Reserve University under Contract No. DAHC 19-68-C-0007 (Project Themis) with the U.S. Army Research Office, Durham, North Carolina. The authors thank the referees for their valuable suggestions.     

On stochastic games,II

In this paper, we consider positive stochastic games, when the state and action spaces are all infinite. We prove that, under certain conditions, the positive stochastic game has a value and that the maximizing player has an -optimal stationary strategy and the minimizing player has an optimal stationary strategy.The authors are grateful to Professor David Blackwell and the referee for some useful comments.     

On noncooperative hybrid stochastic games

The present article models and analyzes a noncooperative hybrid stochastic game of two players. The main phase (prime hybrid mode) of the game is preceded by “unprovoked” hostile actions by one of the players (during antecedent hybrid mode) that at some time transforms into a large scale conflict between two players. The game lasts until one of the players gets ruined. The latter occurs when the cumulative damage to the losing player exceeds a fixed threshold. Both hybrid modes are formalized by marked point stochastic processes and the theory of fluctuations is utilized as one of the chief techniques to arrive at a closed form functional describing the status of both players at the ruin time.     

Bayesian games as stochastic processes

This paper examines stochastic processes generated by sequential games each of whose “states” includes a distribution and whose up-dating rules for state transitions may include Bayes operators. General properties of discrete-time Markov processes are applied to specific games such as iterated matrix games, including Prisoners' Dilemma, and to a new model for the Asch experiments on conflict and conformity.     

Cooperative games with stochastic payoffs

This paper introduces a new class of cooperative games arising from cooperative decision making problems in a stochastic environment. Various examples of decision making problems that fall within this new class of games are provided. For a class of games with stochastic payoffs where the preferences are of a specific type, a balancedness concept is introduced. A variant of Farkas' lemma is used to prove that the core of a game within this class is non-empty if and only if the game is balanced. Further, other types of preferences are discussed. In particular, the effects the preferences have on the core of these games are considered.     

Semi-markov strategies in stochastic games

For a stochastic game with countable state and action spaces we prove, that solutions in the game where all players are restricted to semi-markov strategies are solutions for the unrestricted game. In addition we show, that if all players, except for one, fix a stationary strategy, then the best the remaining player can do, is solving a markov decision problem, corresponding to the fixed stationary strategies.     

On stochastic games in economics

In this paper we show that many results on equilibria in stochastic games arising from economic theory can be deduced from the theorem on the existence of a correlated equilibrium due to Nowak and Raghavan. Some new classes of nonzero-sum Borel state space discounted stochastic games having stationary Nash equilibria are also presented. Three nontrivial examples of dynamic stochastic games arising from economic theory are given closed form solutions. Research partially supported by MNSW grant 1 P03A 01030.