Periodic stopping games

Stopping games (without simultaneous stopping) are sequential games in which at every stage one of the players is chosen, who decides whether to continue the interaction or stop it, whereby a terminal payoff vector is obtained. Periodic stopping games are stopping games in which both of the processes that define it, the payoff process as well as the process by which players are chosen, are periodic and do not depend on the past choices. We prove that every periodic stopping game without simultaneous stopping, has either periodic subgame perfect ϵ-equilibrium or a subgame perfect 0-equilibrium in pure strategies. This work is part of the master thesis of the author done under the supervision of Prof. Eilon Solan. I am thankful to Prof. Solan for his inspiring guidance. I also thank two anonymous referees of the International Journal of Game Theory for their comments.     

N-person differential games governed by infinite-dimensional systems

This paper discussesN-person differential games governed by infinite-dimensional systems. The minimax principle, which is a necessary condition for the existence of open-loop equilibrium strategies, is proved. For linear-quadraticN-person differential games, global necessary and sufficient conditions for the existence of open-loop and closed-loop equilibrium strategies are derived.This work was supported by the Science Fund of the Chinese Academy of Sciences and the Research Foundation of Purdue University.The problems discussed in this paper were proposed by Professor G. Chen, during the author's visit to Pensylvania State University, and were completed at Purdue University. The author would like to thank Professors L. D. Berkovitz and G. Chen for their hospitality.     

Stochastic games with two non-absorbing states

In the present paper we consider recursive games that satisfy an absorbing property defined by Vieille. We give two sufficient conditions for existence of an equilibrium payoff in such games, and prove that if the game has at most two non-absorbing states, then at least one of the conditions is satisfied. Using a reduction of Vieille, we conclude that every stochastic game which has at most two non-absorbing states admits an equilibrium payoff. This paper is part of the Ph.D. thesis of the author completed under the supervision of Prof. Abraham Neyman at the Hebrew University of Jerusalem. I would like to thank Prof. Neyman for many discussions and ideas and for the continuous help he offered. I also thank Nicolas Vieille for his comments on earlier versions of the paper.     

On pure conjectural equilibrium with non-manipulable information

An information structure in a non-cooperative game determines the signal that each player observes as a function of the strategy profile. Such information structure is called non-manipulable if no player can gain new information by changing his strategy. A Conjectural Equilibrium (CE) (Battigalli in Unpublished undergraduate dissertation, 1987; Battigalli and Guaitoli 1988; Decisions, games and markets, 1997) with respect to a given information structure is a strategy profile in which each player plays a best response to his conjecture about his opponents’ play and his conjecture is not contradicted by the signal he observes. We provide a sufficient condition for the existence of pure CE in games with a non-manipulable information structure. We then apply this condition to prove existence of pure CE in two classes of games when the information that players have is the distribution of strategies in the population. This work is based on a chapter from my Ph.D. dissertation written at the School of Mathematical Sciences of Tel-Aviv University under the supervision of Prof. Ehud Lehrer. I am grateful to Ehud Lehrer as well as to Pierpaolo Battigalli, Yuval Heller, two anonymous referees, an Associate Editor and the Editor for very helpful comments and references.     

Stochastic differential games: Occupation measure based approach

A new approach based on occupation measures is introduced for studying stochastic differential games. For two-person zero-sum games, the existence of values and optimal strategies for both players is established for various payoff criteria. ForN-person games, the existence of equilibria in Markov strategies is established for various cases.     

Stochastic Games with Average Payoff Criterion

We study two-person stochastic games on a Polish state and compact action spaces and with average payoff criterion under a certain ergodicity condition. For the zero-sum game we establish the existence of a value and stationary optimal strategies for both players. For the nonzero-sum case the existence of Nash equilibrium in stationary strategies is established under certain separability conditions. Accepted 9 January 1997     

Zero-sum risk-sensitive stochastic games on a countable state space

Infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled Markov chains with countably many states are analyzed. Upper and lower values for these games are established. The existence of value and saddle-point equilibria in the class of Markov strategies is proved for the discounted-cost game. The existence of value and saddle-point equilibria in the class of stationary strategies is proved under the uniform ergodicity condition for the ergodic-cost game. The value of the ergodic-cost game happens to be the product of the inverse of the risk-sensitivity factor and the logarithm of the common Perron–Frobenius eigenvalue of the associated controlled nonlinear kernels.     

Equivalences between differential games and optimal controls

It is shown that linear differential pursuit games with linear targets, if player controls are required to be piecewise constant or if player controls areL ₁-functions (but pursuer control is bang-bang whenever quarry control is), are equivalent to a linear, autonomous control problem. As a byproduct, a sufficient condition for terminating the game, in Pontryagin's sense, is obtained.The present paper has been influenced by Prof. O. Hajek's work in differential games; the converse parts of the proofs presented here are very similar to those in Ref. 5. The author wishes to thank Dr. Hajek for his suggestions, comments, and critique. The Brasilian Government BNDE provided partial financial support.     

Convex semi-infinite games

This paper introduces a generalization of semi-infinite games. The pure strategies for player I involve choosing one function from an infinite family of convex functions, while the set of mixed strategies for player II is a closed convex setC inR ⁿ. The minimax theorem applies under a condition which limits the directions of recession ofC. Player II always has optimal strategies. These are shown to exist for player I also if a certain infinite system verifies the property of Farkas-Minkowski. The paper also studies certain conditions that guarantee the finiteness of the value of the game and the existence of optimal pure strategies for player I.Many thanks are due to the referees for their detailed comments.     

Maximal Element Theorems with Applications to Generalized Games and Systems of Generalized Vector Quasi-Equilibrium Problems in <Emphasis Type="Italic">G</Emphasis>-Convex Spaces

By applying the maximal element theorems on product of G-convex spaces due to the first author, some equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in G-convex spaces. As applications, some existence theorems of solutions for the system of generalized vector quasiequilibrium problem are established in noncompact product of G-convex spaces. Our results improve and generalize some recent results in the literature to product of G-convex spaces.The authors thank the referees for valuable comments and suggestionsThe research of this author was supported by the National Science Foundation of China, Sichuan Education Department.The research of this author was supported by the National Science Council of the Republic of China.     

On finding curb sets in extensive games

We characterize strategy sets that are closed under rational behavior (curb) in extensive games of perfect information and finite horizon. It is shown that any such game possesses only one minimal curb set, which necessarily includes all its subgame perfect Nash equilibria. Applications of this result are twofold. First, it lessens computational burden while computing minimal curb sets. Second, it implies that the profile of subgame perfect equilibrium strategies is always stochastically stable in a certain class of games.I am grateful to J. Kamphorst, G. van der Laan and X. Tieman, who commented on the earlier versions of the paper. I also thank an anonymous referee and an associate editor for their helpful remarks. The usual disclaimer applies.     

Sufficient conditions for Nash equilibria inN-person games over reflexive Banach spaces

Sufficient conditions are obtained for the existence of Nash equilibrium points inN-person games when the strategy sets are closed, convex subsets of reflexive Banach spaces. These conditions require that each player's cost functional is convex in that player's strategy, weakly continuous in the strategies of the other players, weakly lower semicontinuous in all strategies, and furthermore satisfies a coercivity condition if any of the strategy sets is unbounded. The result is applied to a class of linear-quadratic differential games with no information, to prove that equilibrium points exist when the duration of these games is sufficiently small.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

Hedonic coalition formation games: A new stability notion

This paper studies hedonic coalition formation games where each player’s preferences rely only upon the members of her coalition. A new stability notion under free exit-free entry membership rights, referred to as strong Nash stability, is introduced which is stronger than both core and Nash stabilities studied earlier in the literature. Strong Nash stability has an analogue in non-cooperative games and it is the strongest stability notion appropriate to the context of hedonic coalition formation games. The weak top-choice property is introduced and shown to be sufficient for the existence of a strongly Nash stable partition. It is also shown that descending separable preferences guarantee the existence of a strongly Nash stable partition. Strong Nash stability under different membership rights is also studied.     

Approximation of Noncooperative Semi-Markov Games

An approximation of a general V-ergodic semi-Markov game with Borel state space by discrete-state space strongly-ergodic games is studied. The standard expected ratio-average criterion as well as the expected time-average criterion are considered. New theorems on the existence of ∊-equilibria are given.Communicated by D. A. CarlsonThe authors thank an anonymous referee for constructive comments. This work is supported by MEiN Grant 1P03A 01030.     

On the existence of equilibria in discontinuous games: three counterexamples

We study whether we can weaken the conditions given in Reny [4] and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy ɛ-equilibria for all ɛ>0. We show by examples that there are:1. quasiconcave, payoff secure games without pure strategy ɛ-equilibria for small enough ɛ>0 (and hence, without pure strategy Nash equilibria),2. quasiconcave, reciprocally upper semicontinuous games without pure strategy ɛ-equilibria for small enough ɛ>0, and3. payoff secure games whose mixed extension is not payoff secure.The last example, due to Sion and Wolfe [6], also shows that non-quasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.I wish to thank the editor, an associate editor and an anonymous referee for very helpful comments. I thank also John Huffstot for editorial assistance. Any remaining error is, of course, mine     

On the distance coefficient between isomorphic function spaces

IfX, Y are compact countable metric spaces such thatY contains no subset homeomorphic toX, then for any isomorphismΦ ofC(X) intoC(Y), ‖ φ ‖ ‖ φ⁻¹ ‖≧3. This result and some variants of it are established here, and prove a special case of a conjecture raised in [1]. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. J. Lindenstrauss. I wish to thank Prof. Lindenstrauss and Prof. A. Dvoretzky for their guidance and the interest they showed in this work.     

Zero-sum risk-sensitive stochastic games for continuous time Markov chains

We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game, we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game, we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.     

Linear-quadratic,two-person,zero-sum differential games: Necessary and sufficient conditions

We consider linear-quadratic, two-person, zero-sum perfect information differential games, possibly with a linear target set. We show a necessary and sufficient condition for the existence of a saddle point, within a wide class of causal strategies (including, but not restricted to, pure state feedbacks). The main result is that, when they exist, the optimal strategies are pure feedbacks, given by the classical formulas suitably extended, and that existence may be obtained even in the presence of a conjugate point within the time interval, provided it is of a special type that we calleven.The partial support of the Trieste Unit of the GNAS, Italian CNR, is gratefully acknowledged.     

Prem-mappings,triple self-intersection points of oriented surfaces,and the Rokhlin signature theorem

We find a connection between the Rokhlin theorem on the signature of a four-dimensional manifold and the notion of a prem-mapping that arises from the theory of embeddings of smooth manifolds. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 803–810, June, 1996. I thank Prof. A. Szucs for informing me about the existence of a canonical orientationO on the double points curve of a prem-mapping of an oriented surface into ℝ³.