N-Person stochastic games with upper semi-continuous payoffs

Every n-person stochastic game with a countable state space, finite action sets for the players and bounded, upper semi-continuous payoffs has an ε-equilibrium for every ε>0.     

Finitely additive stochastic games with Borel measurable payoffs

We prove that a two-person, zero-sum stochastic game with arbitrary state and action spaces, a finitely additive law of motion and a bounded Borel measurable payoff has a value. Received December 1996/Final version November 1997     

Zero-sum stochastic games with average payoffs: New optimality conditions

In this paper we study zero-sum stochastic games. The optimality criterion is the long-run expected average criterion, and the payoff function may have neither upper nor lower bounds. We give a new set of conditions for the existence of a value and a pair of optimal stationary strategies. Our conditions are slightly weaker than those in the previous literature, and some new sufficient conditions for the existence of a pair of optimal stationary strategies are imposed on the primitive data of the model. Our results are illustrated with a queueing system, for which our conditions are satisfied but some of the conditions in some previous literatures fail to hold.     

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

In this paper we deal with the problem of existence of a smooth solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.     

On discounted stochastic games with incomplete information on payoffs and a security application

This paper presents a robust optimization model for n$n$-person finite state/action stochastic games with incomplete information on payoffs. For polytopic uncertainty sets, we propose an explicit mathematical programming formulation for an equilibrium calculation. It turns out that a global optimal of this mathematical program yields an equilibrium point and epsilon-equilibria can be calculated based on this result. We briefly describe an incomplete information version of a security application that can benefit from robust game theory.     

Bargaining with stochastic disagreement payoffs

We study a bargaining model where (i) players interim disagreement payoffs are stochastic and (ii) in any period, the proposer may postpone making an offer without losing the right to propose in the following period. This bargaining model has a generically unique perfect equilibrium payoff for each player, and the equilibrium outcome is inefficient in some cases, featuring a stochastically delayed agreement. We show that both the variation of players interim disagreement payoffs and the proposers ability to postpone making an offer without losing the right to propose are necessary for the existence of such a unique and inefficient perfect equilibrium outcome. Received April 2002/Final version April 2003     

Cooperative equilibria in discounted stochastic sequential games

This paper addresses the problem of computation of cooperative equilibria in discounted stochastic sequential games. The proposed approach contains as a special case the method of Green and Porter (developed originally for repeated oligopoly games), but it is more general than the latter in the sense that it generates nontrivial equilibrium solutions for a much larger class of dynamic games. This fact is demonstrated on two examples, one concerned with duopolistic economics and the other with fishery management.     

A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals

In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of doubly controlled backward stochastic differential equations. Our results extend former ones by Buckdahn et al. (2004) [3] and are based on a backward stochastic differential equation approach.     

Equilibrium payoffs of dynamic games

We give a characterization of the equilibrium payoffs of a dynamic game, which is a stochastic game where the transition function is either one or zero and players can only use pure actions in each stage. The characterization is in terms of convex combinations of connected stationary strategies; since stationary strategies are not always connected, the equilibrium set may not be convex. We show that subgame perfection may reduce the equilibrium set.     

Nonzero-sum semi-Markov games with the expected average payoffs

Nonzero-sum ergodic semi-Markov games with Borel state spaces are studied. An equilibrium theorem is proved in the class of correlated stationary strategies using public randomization. Under some additivity assumption concerning the transition probabilities stationary Nash equilibria are also shown to exist.Received: October 2004 / Revised: January 2005     

Note Equilibrium payoffs in stochastic games of incomplete information: the general symmetric case

We prove that the existence of equilibrium payoffs for stochastic games of incomplete symmetric information follows from the same result for stochastic games with complete information. Received January 1999/Revised October 2001     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

Lower equilibrium payoffs in two-player repeated games with non-observable actions

We characterize, by the one-shot game terms, the set of lower equilibrium payoffs of the undiscounted repeated game with non-observable actions.     

A reformulated Shapley-like value for cooperative games with interval payoffs

In this paper, a new value for cooperative interval games is proposed which may remedy the disadvantages of the interval Shapley-like value and of the improved interval Shapley-like value introduced by Han et al. (2012). Moreover, it is shown that the reformulated interval value uniquely satisfies the properties of efficiency, indifference null player, symmetry, and additivity.     

On three Shapley-like solutions for cooperative games with random payoffs

Three solution concepts for cooperative games with random payoffs are introduced. These are the marginal value, the dividend value and the selector value. Inspiration for their definitions comes from several equivalent formulations of the Shapley value for cooperative TU games. An example shows that the equivalence is not preserved since these solutions can all be different for cooperative games with random payoffs. Properties are studied and a characterization on a subclass of games is provided.2000 Mathematics Subject Classification Number: 91A12.The authors thank two anonymous referees and an associate editor for their helpful comments.This author acknowledges financial support from the Netherlands Organization for Scientific Research (NWO) through project 613-304-059.Received: October 2000     

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

A new solution concept for two-player zero-sum matrix games with multi-dimensional payoffs is introduced. It is based on extensions of the vector order in \(\mathbb {R}^d\) to order relations in the power set of \(\mathbb {R}^d\), so-called set relations, and strictly motivated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates for games with multi-dimensional payoffs. It is shown that–in contrast to games with one-dimensional payoffs–the corresponding strategies are different from equilibrium strategies for games with multi-dimensional payoffs. The two concepts are combined into new equilibrium notions for which existence theorems are given. Relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax and maximin solutions as well as Pareto optimal security strategies are clarified.     

Multi-player stopping games with redistribution of payoffs and BSDEs with oblique reflection

We examine the connections between a novel class of multi-person stopping games with redistribution of payoffs and multi-dimensional reflected BSDEs in discrete- and continuous-time frameworks. Our goal is to provide an essential extension of classic results for two-player stopping games (Dynkin games) to the multi-player framework. We show the link between certain multi-period m$m$-player stopping games and a new kind of m$m$-dimensional reflected BSDEs. The existence and uniqueness of a solution to continuous-time reflected BSDEs are established. Continuous-time redistribution games are constructed with the help of reflected BSDEs and a characterization of the value of such stopping games is provided.     

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.     

Cooperative games with large cores

A payoff vector in ann-person cooperative game is said to be acceptable if no coalition can improve upon it. The core of a game consists of all acceptable vectors which are feasible for the grand coalition. The core is said to be large if for every acceptable vectory there is a vectorx in the core withx?y. This paper examines the class of games with large cores.