Abstract not found

We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players ’ strategies. We show that this definition spans a spectrum of eliminability criteria from strict dominance (when the sets are as small as possible) to Nash equilibrium (when the sets are as large as possible). We show that checking whether a strategy is eliminable according to this criterion is coNP-complete (both when all the sets are as large as possible and when the dominator sets each have size 1). We then give an alternative definition of the eliminability criterion and show that it is equivalent using the Minimax Theorem. We show how this alternative definition can be translated into a mixed integer program of polynomial size with a number of (binary) integer variables equal to the sum of the sizes of the eliminee sets, implying that checking whether a strategy is eliminable according to the criterion can be done in polynomial time, given that the eliminee sets are small. Finally, we study using the criterion for iterated elimination of strategies. Categories and Subject Descriptors

Abstract. A recent sequence of results established that computing Nash equilibria in normal form games is a PPAD-complete problem even in the case of two players [11,6,4]. By extending these techniques we prove a general theorem, showing that, for a far more general class of families of succinctly representable multiplayer games, the Nash equilibrium problem can also be reduced to the twoplayer case. In view of empirically successful algorithms available for this problem, this is in essence a positive result — even though, due to the complexity of the reductions, it is of no immediate practical significance. We further extend this conclusion to extensive form games and network congestion games, two classes which do not fall into the same succinct representation framework, and for which no positive algorithmic result had been known. 1

A central question in game theory, learning, and other fields is how a rational intelligent agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational or otherwise) for each strategy. While a zero-sum game with strategy costs is no longer zerosum, we show that its Nash equilibria have an interesting structure and the game has a new type of “value. ” We also show that potential games with strategy costs remain potential games. Both zero-sum and potential games with strategy costs maintain a very appealing property: simple learning dynamics converge to Nash equilibrium. 1 The Approach and Basic Model How should an intelligent agent play a complicated game like chess, given that it does not have unlimited time to think? This question reflects one fundamental aspect of “bounded rationality, ” a term coined by Herbert Simon [1]. However, bounded rationality has proven to be a slippery concept to formalize (prior work has focused largely on finite automata playing simple repeated games such

Recently, algorithms for computing game-theoretic solutions have been deployed in real-world security applications, such as the placement of checkpoints and canine units at Los Angeles International Airport. These algorithms assume that the defender (security personnel) can commit to a mixed strategy, a so-called Stackelberg model. As pointed out by Kiekintveld et al. (Kiekintveld et al. 2009), in these applications, generally, multiple resources need to be assigned to multiple targets, resulting in an exponential number of pure strategies for the defender. In this paper, we study how to compute optimal Stackelberg strategies in such games, showing that this can be done in polynomial time in some cases, and is NP-hard in others.

We study the problem of finding approximate Nash equilibria of two player games. We show that for any 0 < ǫ < 1, there is no 1-approximate equilibrium with strategies of 1+ǫ log n support O( ǫ2). Categories and Subject Descriptors

We present a technique for reducing a normal-form (aka. (bi)matrix) game, O, to a smaller normal-form game, R, for the purpose of computing a Nash equilibrium. This is done by computing a Nash equilibrium for a subcomponent, G, of O for which a certain condition holds. We also show that such a subcomponent G on which to apply the technique can be found in polynomial time (if it exists), by showing that the condition that G needs to satisfy can be modeled as a Horn satisfiability problem. We show that the technique does not extend to computing Pareto-optimal or welfaremaximizing equilibria. We present a class of games, which we call ALAGIU (Any Lower Action Gives Identical Utility) games, for which recursive application of (special cases of) the technique is su#cient for finding a Nash equilibrium in linear time. Finally, we discuss using the technique to compute approximate Nash equilibria.

Abstract. This is the 26th edition of a column that covers new developments in the theory of NP-completeness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NP-Completeness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at

We study the price of malice in linear congestion games using the technique of no-regret analysis in the presence of Byzantine players. Our assumptions about the behavior both of rational players, and of malicious players are strictly weaker than have been previously used to study the price of malice. Rather than assuming that rational players route their flow according to a Nash equilibrium, we assume only that the play so as to have no regret. Rather than assuming that malicious players myopically seek to maximize the social cost of the game, we study Byzantine players about whom we make no assumptions, who may be seeking to optimize any utility function, and who may engage in an arbitrary degree of counterspeculation. Because our assumptions are strictly weaker than in previous work, the bounds we prove on two measures of the price of malice hold also for the quantities studied by Babaioff et al. [2] and Moscibroda et al. [17] We prove tight bounds both for the special case of parallel link routing games, and for general congestion games.

Game-theoretic solution concepts, such as Nash equilibrium, are playing an ever increasing role in the study of systems of autonomous computational agents. A common criticism of Nash equilibrium is that its existence relies on the possibility of randomizing over actions, which in many cases is deemed unsuitable, impractical, or even infeasible. In work dating back to the early 1950s Lloyd Shapley proposed ordinal setvalued solution concepts for zero-sum games that he refers to as strict and weak saddles. These concepts are intuitively appealing, they always exist, and are unique in important subclasses of games. We initiate the study of computational aspects of Shapley’s saddles and provide polynomial-time algorithms for computing strict saddles in normal-form games and weak saddles in a subclass of symmetric zero-sum games. On the other hand, we show that certain problems associated with weak saddles in bimatrix games are NP-hard. Finally, we extend our results to mixed refinements of Shapley’s saddles introduced by Duggan and Le Breton.