Set Of Pure Strategies Research Articles

On the existence of Nash equilibria in large games

We study the existence of Nash equilibria in games with an infinite number of players. We show that there exists a Nash equilibrium in mixed strategies in all normal form games such that pure strategy sets are compact metric spaces and utility functions are continuous. The player set can be any nonempty set.

Imitation games and computation

An imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. We explain how, in two different settings, observations obtained from imitation games complete a circle of ideas, showing that phenomena that had for many years seemed to be distinct are actually superficially different manifestations of a single structure. First, one can pass from a given two person finite game to an imitation game whose Nash equilibria are in one-to-one correspondence with the Nash equilibria of the given game. Second, each of the paths of the procedure described in Lemke (1965) for solving a linear complementarity problem is the projection of the path of the Lemke–Howson algorithm applied to an imitation game.

The cutting power of preparation

In a strategic game, a curb set (Basu and Weibull, Econ Lett 36:141–146, 1991) is a product set of pure strategies containing all best responses to every possible belief restricted to this set. Prep sets (Voorneveld, Games Econ Behav 48:403–414, 2004) relax this condition by only requiring the presence of at least one best response to such a belief. The purpose of this paper is to provide sufficient conditions under which minimal prep sets give sharp predictions. These conditions are satisfied in many economically relevant classes of games, including supermodular games, potential games, and congestion games with player-specific payoffs. In these classes, minimal curb sets generically have a large cutting power as well, although it is shown that there are relevant subclasses of coordination games and congestion games where minimal curb sets have no cutting power at all and simply consist of the entire strategy space.

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 “e-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.

ON THE ASYMPTOTIC BEHAVIOR OF A DISCRETE TIME INSPECTION GAME

In many material processing and storing plants an inspector performs during some reference time interval, e.g. one year, a number of inspections because it can not be excluded that the plant operator acts illegally by violating agreed rules, e.g., diverts valuable material. The inspections guarantee that any illegal action is detected at the earliest inspection following the beginning of that illegal action. We assume that the inspector wants to choose the time points for his inspections such that the time which elapses between the beginning of the illegal action and its detection is minimized whereas the operator wants to start his illegal action such that the elapsed time is maximized. Therefore, this inspection problem is modelled as a zero‐sum game with strategies and payoffs as described. Depending on the concrete situation the start of the illegal action and the inspections can take place either at a finite number of time points or at every time point of a reference period. The first case can be modelled as a zero‐sum game with finite pure strategy sets while the latter one leads to a zero‐sum game with infinite pure strategy sets and discontinuous payoff kernel.

Pure self-financing trading strategies under transaction costs

We consider a multi-asset discrete-time model of a financial market with proportional transaction costs as described by Schachermayer [7]. In this model, the set of all self-financing trading strategies contains for example trading strategies which consist of buying an asset and selling it at the same time. In the presence of transaction costs this only generates a net loss of investor’s money. We introduce the notion of naive pure self-financing trading strategy. Agents using a naive pure self-financing trading strategy do not waste money the way described above. It is well known that the only strategies which do not generate a net loss of investor’s money due to transaction costs are the boundary points of the cone of portfolios available at price zero. We call these strategies pure self-financing trading strategies or just pure self-financing strategies. Next, we give a characterization of the set of pure self-financing strategies and of the set of naive pure self-financing strategies. Furthermore, we analyze the relationship between them. Finally, we establish a hedging theorem concerning these strategies.

The Cutting Power of Preparation

In a strategic game, a curb set [Basu and Weibull, Econ.Letters 36 (1991) 141-146] is a product set of pure strategies containing all best responses to every possible belief restricted to this set.Prep sets [Voorneveld, Games Econ. Behav. 48 (2004) 403-414] relax this condition by only requiring the presence of at least one best response to such a belief.The purpose of this paper is to provide sufficient conditions under which minimal prep sets give sharp predictions.These conditions are satisfied in many economically relevant classes of games, including supermodular games, potential games, and congestion games with player-specific payoffs.In these classes, minimal curb sets generically have a large cutting power as well, although it is shown that there are relevant subclasses of coordination games and congestion games where minimal curb sets have no cutting power at all and simply consist of the entire strategy space.

The Expected Number of Nash Equilibria of a Normal Form Game

Fix finite pure strategy sets S1,..., S,,, and let S = S, x... x S-n. In our model of a random game the agents' payoffs are statistically independent, with each agent's payoff uniformly distributed on the unit sphere in R-A. For given nonempty T-1 subset of S-1,..., T-n subset of S-n we give a computationally implementable formula for the mean number of Nash equilibria in which each agent i's mixed strategy has support T-i. The formula is the product of two expressions. The first is the expected number of totally mixed equilibria for the truncated game obtained by eliminating pure strategies outside the sets Ti. The second may be construed as the probability that such an equilibrium remains an equilibrium when the strategies in the sets S-i\T-i become available.

Preparation

A product set of pure strategies is a prep set (‘prep’ is short for ‘preparation’) if it contains at least one best reply to any consistent belief that a player may have about the strategic behavior of his opponents. Minimal prep sets are shown to exist in a class of strategic games satisfying minor topological conditions. The concept of minimal prep sets is compared with (pure and mixed) Nash equilibria, rationalizability, minimal curb sets, and persistent retracts.

Individually rational pure strategies in large games

In “randomly selected” large finite normal form games, pure strategy “strongly individually rational” outcomes (all payoffs exceeding pure strategy minimax payoffs) exist with high probability. As a corollary, pure strategy Nash equilibria also exist with high probability in certain infinitely repeated games. Specifically, the pure strategy Folk Theorem, which can be true in a vacuous sense, has nontrivial substance with probability approaching one as pure strategy sets increase in cardinality, provided strategy set growth rates are not too different. To prove this, and noting a possible “bounded rationality” interpretation, we begin with a study of “approximate equilibria” in finite normal form games. Given a positive integer l, these are pure strategy outcomes where each player's payoff is among the l largest available given strategy choices of opponents. When pure strategy sets are large, such outcomes are highly probable and usually have payoffs dominating pure minimax payoffs.

On finite strategy sets for finitely repeated zero-sum games

We study finitely repeated two-person zero-sum games in which Player 1 is restricted to mixing over a fixed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 along with an optimal mixed strategy. We show that the entropy of this mixed strategy appears as a factor in an exact formula for the value of the game and thus is seen to have a direct numerical effect on the game's value. We develop upper and lower bounds on the value of these games that are within an additive constant and discuss how our results are related to the work of Neyman and Okada on strategic entropy (Neyman and Okada, 1999, Games Econ. Behav. 29, 191–223). Finally, we use these results to bound the value of repeated games in which one of the players uses a computer with a bounded memory and is further restricted to using a constant amount of time at each stage.

Which four-part partition games are isomorphic to Silverman's game?

A four-part partition game on a rectangle is a two-person zero-sum game where the pure strategy sets are intervals and the resulting rectangle is partitioned by three curves into four regions, on each of which the payoff is constant. Such a game is essentially a generalization of Silverman's game, which has been studied extensively by the author and others. It has been shown previously that if two of the partitioning curves of suitable type are given, and the four payoffs are suitably related, there is a unique choice of the third curve for which the game is isomorphic to Silverman's game. In the present note this third curve is expressed explicitly in terms of the first two, provided that one of the two given curves is the middle one of the three.

The Lagging Anchor Algorithm: Reinforcement Learning in Two-Player Zero-Sum Games with Imperfect Information

The article describes a gradient search based reinforcement learning algorithm for two-player zero-sum games with imperfect information. Simple gradient search may result in oscillation around solution points, a problem similar to the “Crawford puzzle”. To dampen oscillations, the algorithm uses lagging anchors, drawing the strategy state of the players toward a weighted average of earlier strategy states. The algorithm is applicable to games represented in extensive form. We develop methods for sampling the parameter gradient of a player's performance against an opponent, using temporal-difference learning. The algorithm is used successfully for a simplified poker game with infinite sets of pure strategies, and for the air combat game Campaign, using neural nets. We prove exponential convergence of the algorithm for a subset of matrix games.

On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms

Consider nonempty finite pure strategy sets S1,…,Sn, let S=S1×⋅⋅⋅×Sn, let Ω be a finite space of “outcomes,” let Δ(Ω) be the set of probability distributions on Ω, and let θ: S→Δ(Ω) be a function. We study the conjecture that for any utility in a generic set of n-tuples of utilities on Ω there are finitely many distributions on Ω induced by the Nash equilibria of the game given by the induced utilities on S. We give a counterexample refuting the conjecture for n≥3. Several special cases of the conjecture follow from well known theorems, and we provide some generalizations of these results.

Patterns of evolutionarily stable strategies: the maximal pattern conjecture revisited.

A conflict is defined by a set of pure strategies [1,..., n] and a payoff matrix, and may have many evolutionarily stable strategies (ESSs). A collection of subsets of the set of pure strategies is called a pattern. If there is an n x n matrix which has ESSs whose supports match those of the pattern, then that pattern is said to be attainable. Much of the work on patterns of ESSs relied upon an unproved conjecture. Subject to some relaxation of the definition of attainability, this conjecture is proved.

On comparing equilibrium and optimum payoffs in a class of discrete bimatrix games

In an m 1× m 2 bimatrix game, consider the case where payoffs to each player are randomly drawn without replacement, independently of payoffs to the other player, from the set of integers 1,2,…, m 1 m 2. Thus each player’s payoffs represent ordinal rankings without ties. In such ‘ordinal randomly selected’ games, assuming constraints on the relative sizes of m 1 and m 2 and ignoring any implications of mixed strategies, it is shown that payoffs to pure Nash equilibria (second-degree) stochastically dominate payoffs to pure Pareto optimal outcomes. Thus in such games where pure strategy sets do not differ much in size and payoffs conform with concave von Neumann-Morgenstern utility functions over ordinally ranked outcomes, players would prefer (ex ante) a ‘random pure strategy Nash equilibrium payoff’ to a ‘random pure Pareto optimal outcome payoff’.

Quantum Strategies

We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). While not every two-person zero-sum finite game has an equilibrium in the set of pure strategies, von Neumann showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player's expected payoff. We show by example, however, that a player who implements a quantum strategy can increase his expected payoff, and explain the relation to efficient quantum algorithms.

On the number of pure strategy Nash equilibria in finite common payoffs games

In a two-person “random” common payoffs game, defined as a finite game in which the players receive the same payoff at each outcome, let X represent the number of pure strategy Nash equilibria occurring. Treating both the cases where players have strictly and weakly ordinal preferences over outcomes, we observe that the expected value of X approaches infinity as the sizes of the pure strategy sets of the players increase without bound. Furthermore, we show that for any fixed positive integer k, the probability that X exceeds k approaches one as pure strategy sets increase in size without bound.

A fuzzy approach to strategic games

A game is a decision-making situation with many players, each having objectives that conflict with each other. The players involved in the game usually make their decisions under conditions of risk or uncertainty. In the paper, a fuzzy approach is proposed to solve the strategic game problem in which the pure strategy set for each player is already defined. Based on the concepts of fuzzy set theory, the approach uses a multicriteria decision-making method to obtain the optimal strategy in the game, a method which shows more advantages than the classical game methods. Moreover, with this approach, some useful conclusions are reached concerning the famous prisoner's dilemma problem in game theory.

On saddlepoints of two-person zero-sum games with applications to data verification tests

A set of sufficient conditions for the existence of saddlepoint strategies of two-person zero-sum games is given which may be described as follows: The set of pure strategies for one of the two players is a compact metric space. His opponent has a best answer to each randomized strategy. The payoff function satisfies a continuity condition concerning the weak convergence of probability measures. These conditions are neither covered by those of well known existence theorems for saddlepoints nor do they generalize them. They are applied to treat important practical problems that have not been solved before.