Continuity of the value and optimal strategies when common priors change

We show that the value of a zero-sum Bayesian game is a Lipschitz continuous function of the players?? common prior belief with respect to the total variation metric on beliefs. This is unlike the case of general Bayesian games where lower semi-continuity of Bayesian equilibrium (BE) payoffs rests on the ??almost uniform?? convergence of conditional beliefs. We also show upper semi-continuity (USC) and approximate lower semi-continuity (ALSC) of the optimal strategy correspondence, and discuss ALSC of the BE correspondence in the context of zero-sum games. In particular, the interim BE correspondence is shown to be ALSC for some classes of information structures with highly non-uniform convergence of beliefs, that would not give rise to ALSC of BE in non-zero-sum games.     

Values on regular games under Kirchhoff’s laws

The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff’s laws in the circuit associated to the Hasse diagram of feasible coalitions.     

Suboptimality of M-step back strategies in Bayesian games

We consider the Bayes optimal strategy for repeated two player games where moves are made simultaneously. In these games we look at models where one player assumes that the other player is employing a strategy depending only on the previousm-move pairs (as discussed in Wilson, 1986). We show that, under very unrestrictive conditions, such an assumption is not consistent with the assumption of rationality of one's opponent. Indeed, we show that by employing such a model a player is implicitly assuming that his opponent is not playing rationally,with probability one. We argue that, in the context of experimental games, thesem-step back models must be inferior to models which are consistent with the assumption that an opponent can be rational.     

Dynamic consistency in incomplete information games under ambiguity

We develop a general framework of incomplete information games under ambiguity which extends the traditional framework of Bayesian games to the context of Ellsberg-type ambiguity. We then propose new solution concepts called ex ante and interim Γ-maximin equilibrium for solving such games. We show that, unlike the standard notion of Bayesian Nash equilibrium, these concepts may lead to rather different recommendations for the same game under ambiguity. This phenomenon is often referred to as dynamic inconsistency. Moreover, we characterize the sufficient condition under which dynamic consistency is assured in this generalized framework.     

Informational aspects of a class of subjective games of incomplete information: Static case

Subjective games of incomplete information are formulated where some of the key assumptions of Bayesian games of incomplete information are relaxed. The issues arising because of the new formulation are studied in the context of a class of nonzero-sum, two-person games, where each player has a different model of the game. The static game is investigated in this note. It is shown that the properties of the static subjective game are different from those of the corresponding Bayesian game. Counterintuitive outcomes of the game can occur because of the different beliefs of the players. These outcomes may lead the players to realize the differences in their models.This work was sponsored by the Office of Naval Research under Contract No. N00014-84-C-0485.     

A smuggling game with asymmetrical information of players

This paper deals with a smuggling game with multiple stages. Customs is allowed to patrol within the limited number of chances and obtain reward by the capture of a smuggler. The smuggler gets a reward depending on the amount of contraband he succeeds to ship in smuggling at each stage. The pay-off of the game is zero-sum. In almost all past studies, they adopt the alternative of smuggling or non-smuggling as the smuggler's strategy. From the point of view of information, some researchers assumed that both players could observe their opponent's behaviour at the past stage or a few assumed that both players had no information about their opponent. Other than these types of smuggling games with the symmetric information, we introduce the asymmetrical acquisition of information or the concept of perfect Bayesian equilibrium in the smuggling game for the first time.     

Robust game theory

We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our ``robust game' model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call ``robust-optimization equilibrium,' to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results. The research of the author was partially supported by a National Science Foundation Graduate Research Fellowship and by the Singapore-MIT Alliance. The research of the author was partially supported by the Singapore-MIT Alliance.     

Zero-sum Markov games and worst-case optimal control of queueing systems

Zero-sum stochastic games model situations where two persons, called players, control some dynamic system, and both have opposite objectives. One player wishes typically to minimize a cost which has to be paid to the other player. Such a game may also be used to model problems with a single controller who has only partial information on the system: the dynamic of the system may depend on some parameter that is unknown to the controller, and may vary in time in an unpredictable way. A worst-case criterion may be considered, where the unknown parameter is assumed to be chosen by nature (called player 1), and the objective of the controller (player 2) is then to design a policy that guarantees the best performance under worst-case behaviour of nature. The purpose of this paper is to present a survey of stochastic games in queues, where both tools and applications are considered. The first part is devoted to the tools. We present some existing tools for solving finite horizon and infinite horizon discounted Markov games with unbounded cost, and develop new ones that are typically applicable in queueing problems. We then present some new tools and theory of expected average cost stochastic games with unbounded cost. In the second part of the paper we present a survey on existing results on worst-case control of queues, and illustrate the structural properties of best policies of the controller, worst-case policies of nature, and of the value function. Using the theory developed in the first part of the paper, we extend some of the above results, which were known to hold for finite horizon costs or for the discounted cost, to the expected average cost.     

The conventionally stable sets in noncooperative games with limited observations I: Definitions and introductory arguments

This paper attempts to define a new solution concept for n-person noncooperative games. The idea of it is close to that of the von Neumann-Morgenstern stable set, more precisely, is based on their ‘standards of behavior’ interpretation of the stable set. This new approach enables us to consider new interesting problems of information. Further this approach gives us a plausible interpretation of the Nash equilibrium concept. This paper provides the definition and considers certain general properties of our solution concept. Further we consider our solution concept for zero-sum two-person games, the prisoner's dilemma, the battle of the sexes and several simple games with a continuum of players.     

Games of stopping with infinite horizon

We consider two-person zero-sum games of stopping: two players sequentially observe a stochastic process with infinite time horizon. Player I selects a stopping time and player II picks the distribution of the process. The pay-off is given by the expected value of the stopped process. Results of Irle (1990) on existence of value and equivalence of randomization for such games with finite time horizon, where the set of strategies for player II is dominated in the measure-theoretical sense, are extended to the infinite time case. Furthermore we treat such games when the set of strategies for player II is not dominated. A counterexample shows that even in the finite time case such games may not have a value. Then a sufficient condition for the existence of value is given which applies to prophet-type games.     

Dynamic competition over social networks

We propose an analytical approach to the problem of influence maximization in a social network where two players compete by means of dynamic targeting strategies. We formulate the problem as a two-player zero-sum stochastic game. We prove the existence of the uniform value: if the players are sufficiently patient, both can guarantee the same mean-average opinion without knowing the exact length of the game. Furthermore, we put forward some elements for the characterization of equilibrium strategies. In general, players must implement a trade-off between a forward-looking perspective, according to which they aim to maximize the future spread of their opinion in the network, and a backward-looking perspective, according to which they aim to counteract their opponent’s previous actions. When the influence potential of players is small, we describe an equilibrium through a one-shot game based on eigenvector centrality.     

Nash equilibria for non-binary choice rules

We prove the existence of equilibria in games with players who employ abstract (non-binary) choice rules. This framework goes beyond the standard, transitive model and encompasses games where players have non-transitive preferences (e.g., skew-symmetric bilinear preferences).      

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.     

Heights of simple games

We consider simple games which are constructed as iterated weighted majority games. It turns out that every proper simple game can be obtained in this way. The minimal number of iterations necessary to obtain a given game is called the height of this game. We investigate the behaviour ofh (n), the maximal height of a simple game withn players.     

Fundamentals of simple games from a viewpoint of blockability relations

This paper shows some elementary facts on simple games with respect to blockability relations. It is verified in this paper that fundamental concepts on simple games as null players, dictators, veto players, and so on can be expressed in terms of blockability relations. More, some new concepts as “conflict-free” and so on, are introduced from the viewpoint of blockability relations into the framework of simple games.     

Optimal cost sharing for capacitated facility location games

We consider cost sharing for a class of facility location games, where the strategy space of each player consists of the bases of a player-specific matroid defined on the set of resources. We assume that resources have nondecreasing load-dependent costs and player-specific delays. Our model includes the important special case of capacitated facility location problems, where players have to jointly pay for opened facilities. The goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability. We investigate two classes of protocols: basic protocols guarantee the existence of at least one pure Nash equilibrium and separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource. We find optimal basic and separable protocols that guarantee the price of stability/price of anarchy to grow logarithmically/linearly in the number of players. These results extend our previous results (cf. von Falkenhausen & Harks, 2013), where optimal basic and separable protocols were given for the case of symmetric matroid games without delays.     

Weighted and roughly weighted simple games

In this paper we give necessary and sufficient conditions for a simple game to have rough weights. We define two functions f(n) and g(n) that measure the deviation of a simple game from a weighted majority game and roughly weighted majority game, respectively. We formulate known results in terms of lower and upper bounds for these functions and improve those bounds. We also investigate rough weightedness of simple games with a small number of players.     

Maximum principle via Malliavin calculus for regular-singular stochastic differential games

We consider non-zero-sum regular-singular stochastic differential games, where the informations available to the two players are asymmetry partial informations. The control strategy of each player consists of two components: regular control and singular control. Applying the Malliavin calculus approach, we establish a necessary maximum principle for the games, where the adjoint processes are explicitly represented by the parameters and the states of the system.     

Sequence of opponents and reduced strategies

We consider the framework of repeated two-person zero-sum games with lack of information on one side. We compare the equilibrium payoffs of the informed player in two cases: where he is facing either a) a single long-lived uninformed player, or b) a sequence of short-lived uninformed players. We show: 1) that situation b) is always (weakly) better than a), 2) that it can be strictly better in some cases, 3) that the two cases are equivalent if the long uninformed player has an optimal strategy independent of his own moves. Received: March 1999/Revised: May 2000     

Unconditionally secure key assignment schemes

In this paper we propose an information-theoretic approach to the access control problem in a scenario where a group of users is divided into a number of disjoint classes. The set of rules that specify the information flow between different user classes in the system defines an access control policy. An access control policy can be implemented by using a key assignment scheme, where a trusted central authority (CA) assigns an encryption key and some private information to each class.We consider key assignment schemes where the key assigned to each class is unconditionally secure with respect to an adversary controlling a coalition of classes of a limited size. Our schemes are characterized by a security parameter r, the size of the adversary coalition. We show lower bounds on the size of the private information that each class has to store and on the amount of randomness needed by the CA to set up any key assignment scheme. Finally, we propose some optimal constructions for unconditionally secure key assignment schemes.