In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is St-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's move depends on moves of other players. We say that a game has small neighborhood if the " utility function for each player depends only on (the actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph I-I(G). Among other results, we show that if jC has small neighborhood and if I-I(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class _NC ~ of highly parallelizable problems. 1 Introduction and Overview of Results The theory of strategic games and Nash equilibria has important applications in economics and decision making [31, 2]. Determining whether Nash equilibria exist, and effectively computing

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An outcome in a noncooperative game is said to be self-enforcing, or a strategic equilibrium, if, whenever it is recommended to the players, no player has an incentive to deviate from it. This paper gives an overview of the concepts that have been proposed as formalizations of this requirement and of the properties and the applications of these concepts. In particular the paper discusses Nash equilibrium, together with its main coarsenings (correlated equilibrium, rationalizibility) and its main re…nements (sequential, perfect, proper, persistent and stable equilibria). There is also an extensive discussion on equilibrium selection.

We develop a general game-theoretic framework for reasoning about strategic agents performing possibly costly computation. In this framework, many traditional game-theoretic results (such as the existence of a Nash equilibrium) no longer hold. Nevertheless, we can use the framework to provide psychologically appealing explanations to observed behavior in well-studied games (such as finitely repeated prisoner’s dilemma and rock-paper-scissors). Furthermore, we Consider the following game. You are given a random odd n-bit number x and you are supposed to decide whether x is prime or composite. If you guess correctly you receive $2, if you guess incorrectly you instead have to pay a penalty of $1000. Additionally you have the choice of “playing safe” by giving up, in which case you receive $1. In traditional game theory, computation is considered

We propose a measure of riskiness of “gambles ” (risky assets) that is objective: it depends only on the gamble and not on the decision maker. The measure is based on identifying for every gamble the critical wealth level below which it becomes “risky ” to accept the gamble. I.

I discuss the merits and drawbacks of game theory in economics from the perspective of Austrian economics. I begin by arguing that Austrians have neglected game theory at their peril, and then suggest that game theoretic reasoning could be one way of modelling key Austrian insights. However, admittedly some aspects of game theory dont square easily with Austrian economics. Moreover, a major stumbling block for an Austrian acceptance of game theory may lie in the traditional Austrian resistance to formal methods.

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Both noncooperative and cooperative game theory have been applied to business strategy. We propose a hybrid noncooperative-cooperative game model, which we call a biform game. This is designed to formalize the notion of business strategy as making moves to try to shape the competitive environment in a favorable way. (The noncooperative component of a biform game models the strategic moves. The cooperative component models the resulting competitive environment.) We give biform models of various well-known business strategies. We prove general results on when a business strategy, modelled as a biform game, will be efficient.

Real-world games often involve both structured and unstructured interaction. For the clearly delineated moves and countermoves in such games, a non-cooperative game model can be used. But for unstructured interaction, when the moves are more fluid and free-form, a cooperative game model is more natural. This paper develops a hybrid non-cooperative/cooperative formalism, called the biform game model, designed for the study of such mixed games. As a test of the biform model, it is applied to the analyses of monopoly and perfect competition. 1