Do we detect and exploit mixed strategy play by opponents?

We conducted an experiment in which each subject repeatedly played a game with a unique Nash equilibrium in mixed strategies against some computer-implemented mixed strategy. The results indicate subjects are successful at detecting and exploiting deviations from Nash equilibrium. However, there is heterogeneity in subject behavior and performance. We present a one variable model of dynamic random belief formation which rationalizes observed heterogeneity and other features of the data.The minimax and Nash equilibrium solutions coincide in this setting, and we could proceed only referring to the minimax solution and strategies. However, we proceed using the Nash equilibrium framework because we wish to focus on the concept of best response.     

Some interesting properties of maximin strategies

Since the seminal paper of Nash (1950) game theoretic literature has focused mostly on equilibrium and not on maximin (minimax) strategies. We study the properties of these strategies in non-zero-sum strategic games that possess (completely) mixed Nash equilibria. We find that under certain conditions maximin strategies have several interesting properties, some of which extend beyond 2-person strategic games. In particular, for n-person games we specify necessary and sufficient conditions for maximin strategies to yield the same expected payoffs as Nash equilibrium strategies. We also show how maximin strategies may facilitate payoff comparison across Nash equilibria as well as refine some Nash equilibrium strategies.     

Picking the winners

We analyze the problem of choosing the w contestants who will win a competition within a group of n > w competitors when all jurors commonly observe who the w best contestants are, but they may be biased. We study conditions on the configuration of the jury so that it is possible to induce the jurors to always choose the best contestants, whoever they are. If the equilibrium concept is dominant strategies, the condition is very strong: there must be at least one juror who is totally impartial, and the planner must have some information about who this juror is. If the equilibrium concept is Nash (or subgame perfect) equilibria the condition is less demanding: for each pair of contestants, the planner must know that there is a number of jurors who are not biased in favor/against any of them and he must have some information about who these jurors are. Furthermore, the latter condition is also necessary for any other equilibrium concept.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

Mixed equilibrium problems with Z*-bifunctions and least element problems in Banach lattices

The purpose of this paper is to study the relations among a mixed equilibrium problem, a least element problem and a minimization problem in Banach lattices. We propose the concept of Z*-bifunctions as well as the concept of a feasible set for the mixed equilibrium problem. We prove that the feasible set of the mixed equilibrium problem is a sublattice provided that the associated bifunction is a strictly α-monotone Z*-bifunction. We establish the equivalence of the mixed equilibrium problem, the least element problem and the minimization problem under strict α-monotonicity and Z*-bifunction conditions.     

Symmetric equilibrium with random entry

This paper considers Cournot-Nash equilibrium with free entry among identical firms which possess large minimum efficient scale. We consider equilibrium in which all firms receive equal treatment by allowing firms to play mixed strategies. In particular, firms randomize over the decision to enter or not. It is shown that symmetric Cournot-Nash equilibrium in mixed strategies exists when there is a finite number of potential entrants. We then consider a sequence of such mixed strategy equilibria as the number of potential entrants gets large. It is shown that such a sequence always has a convergent subsequence whose limit is a symmetric equilibrium in mixed strategies when the number of potential entrants is infinite. An example is given which shows that increased competition, in the form of a larger pool of potential entrants, is socially harmful in that expected social surplus is decreasing in the number of potential entrants.     

Stationary,completely mixed and symmetric optimal and equilibrium strategies in stochastic games

In this paper, we address various types of two-person stochastic games—both zero-sum and nonzero-sum, discounted and undiscounted. In particular, we address different aspects of stochastic games, namely: (1) When is a two-person stochastic game completely mixed? (2) Can we identify classes of undiscounted zero-sum stochastic games that have stationary optimal strategies? (3) When does a two-person stochastic game possess symmetric optimal/equilibrium strategies? Firstly, we provide some necessary and some sufficient conditions under which certain classes of discounted and undiscounted stochastic games are completely mixed. In particular, we show that, if a discounted zero-sum switching control stochastic game with symmetric payoff matrices has a completely mixed stationary optimal strategy, then the stochastic game is completely mixed if and only if the matrix games restricted to states are all completely mixed. Secondly, we identify certain classes of undiscounted zero-sum stochastic games that have stationary optima under specific conditions for individual payoff matrices and transition probabilities. Thirdly, we provide sufficient conditions for discounted as well as certain classes of undiscounted stochastic games to have symmetric optimal/equilibrium strategies—namely, transitions are symmetric and the payoff matrices of one player are the transpose of those of the other. We also provide a sufficient condition for the stochastic game to have a symmetric pure strategy equilibrium. We also provide examples to show the sharpness of our results.     

Simple centrifugal incentives in spatial competition

This paper studies the effects of introducing centrifugal incentives in an otherwise standard Downsian model of electoral competition. First, we demonstrate that a symmetric equilibrium is guaranteed to exist when centrifugal incentives are induced by any kind of partial voter participation (such as abstention due to indifference, abstention due to alienation, etc.) and, then, we argue that: (a) this symmetric equilibrium is in pure strategies, and it is hence convergent, only when centrifugal incentives are sufficiently weak on both sides; (b) when centrifugal incentives are strong on both sides (when, for example, a lot of voters abstain when they are sufficiently indifferent between the two candidates) players use mixed strategies—the stronger the centrifugal incentives, the larger the probability weight that players assign to locations near the extremes; and (c) when centrifugal incentives are strong on one side only—say for example only on the right—the support of players’ mixed strategies contain all policies except from those that are sufficiently close to the left extreme.     

混合策略纳什均衡的新教法

理解博弈论中的最优混合策略对本科生而言具有一定困难,而目前教材中对此内容的讲述又过于抽象.提出一个简单而有效地讲授混合策略纳什均衡的方法.首先利用猜硬币游戏引入并介绍混合策略的基本该念.再通过将混合策略加入到支付矩阵中构造拓展支付矩阵,使学生可以清晰地看到采用混合策略的结果,实现从纯策略到混合策略的自然过渡.然后引导学生思考博弈参与者采用混合策略的各种动机,并在拓展支付矩阵中检验其是否达成均衡.最后介绍最优混合策略计算的一般方法,并分析其与参与者行为动机之间的一致性.课堂实践证明,方法可以有效提高学生对混合策略纳什均衡的综合理解,学生不仅能够更好地掌握求解技术,而且能更深入地理解其经济学含义.     

Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points

Equilibrium points in mixed strategies seem to be unstable, because any player can deviate without penalty from his equilibrium strategy even if he expects all other players to stick to theirs. This paper proposes a model under which most mixed-strategy equilibrium points have full stability. It is argued that for any gameΓ the players' uncertainty about the other players' exact payoffs can be modeled as a disturbed gameΓ ^*, i.e., as a game with small random fluctuations in the payoffs. Any equilibrium point inΓ, whether it is in pure or in mixed strategies, can “almost always” be obtained as a limit of a pure-strategy equilibrium point in the corresponding disturbed gameΓ ^* when all disturbances go to zero. Accordingly, mixed-strategy equilibrium points are stable — even though the players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies — because the random fluctuations in their payoffs willmake them use their pure strategies approximately with the prescribed probabilities.     

Nash fuzzy equilibrium: Theory and application to a spatial duopoly

Consider a non-cooperative n-persons game. Each gambler has a set of mixed strategies at his disposal. The payoffs are some physical or immaterial objects. The game is a fuzzy game because (1) gamblers have more or less precise preferences for the payoffs and (2) the outcoming of payoffs is uncertain. The uncertainty can be expressed either by a distribution of possibility or by a distribution of probability. The product set of a gambler's mixed strategies is convex and compact and the payoff functions are continuous. Then a closed and convex fuzzy point-to-set mapping is defined on the product set of strategies and, by using a Butnariu theorem, the existence of a fixed point for this fuzzy point-to-set mapping is proved. The issue allows us to generalize a famous Nash result: a n-persons non-cooperative fuzzy game with mixed strategies has at least one equilibrium point. In the second part of the paper an economic application is devoted to the statement of the equilibrium existence conditions in a spatial duopoly. The model is not only more general than the classical ones, but also more relevant because new results are obtained.     

Finitely additive and epsilon Nash equilibria

We prove the existence of a mixed strategy Nash equilibrium in normal form games when the space of mixed strategies consists of finitely additive probability measures. It is then proved that from this result an existence result for epsilon equilibria with countably additive mixed strategies can be obtained. These results are applied to the classic Cournot game.     

Game Theoretic Analysis of a Distribution System with Customer Market Search

Consider a distribution system with one supplier and two retailers. When a stockout occurs at one retailer customers may go to the other retailer. We study a single period model in which the supplier may have infinite or finite capacity. In the latter case, if the total quantity ordered (claimed) by the retailers exceeds the supplier’s capacity, an allocation policy is involved to assign the limited capacity to the retailers. We analyze the inventory control decisions for the retailers using a game theoretical approach. The necessary and sufficient conditions are derived for the existence of a unique Nash equilibrium. A computational procedure is also proposed to calculate the Nash equilibrium. In case the Nash equilibrium does not exist, we use the concept of Stackelberg game to develop optimal strategies for both the leader and the follower. The work was partially supported by the National Textile Center of the US Department of Commerce under Grant No. I01-S01. The second author is supported in part by NSF under DMI-0196084 and DMI-0200306.     

Juche versus Sadae: Game-theoretic approach

We argue that to some degree Juche is represented by the concept of Nash equilibrium, and Sadae by Thompson and Faith's truly perfect information equilibrium. We characterize the latter, and show that for a Pareto optimal Nash equilibrium, Juche is as good as, or better than Sadae. This includes the game of brinkmanship.     

Nondominated Nash points: application of biobjective mixed integer programming

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.     

Finding mixed strategies with small supports in extensive form games

The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for thesize of a mixed strategy is the number of strategies in itssupport — the set of pure strategies to which it gives positive probability. This paper investigates the existence of “small” mixed strategies in extensive form games, and how such strategies can be used to create more efficient algorithms. The basic idea is that, in an extensive form game, a mixed strategy induces a small set ofrealization weights that completely describe its observable behavior. This fact can be used to show that for any mixed strategy μ, there exists a realization-equivalent mixed strategy µ′ whose size is at most the size of the game tree. For a player with imperfect recall, the problem of finding such a strategy µ′ (given the realization weights) is NP-hard. On the other hand, if μ is a behavior strategy, µ′ can be constructed from μ in time polynomial in the size of the game tree. In either case, we can use the fact that mixed strategies need never be too large for constructing efficient algorithms that search for equilibria. In particular, we construct the first exponential-time algorithm for finding all equilibria of an arbitrary two-person game in extensive form.     

On the geometry of Nash equilibria and correlated equilibria

It is well known that the set of correlated equilibrium distributions of an n-player noncooperative game is a convex polytope that includes all the Nash equilibrium distributions. We demonstrate an elementary yet surprising result: the Nash equilibria all lie on the boundary of the polytope.We are grateful to Francoise Forges, Dan Arce, the editors, and several anonymous referees for helpful comments. This research was supported by the National Science Foundation under grant 98–09225 and by the Fuqua School of Business.The use of correlated mixed strategies in 2-player games was discussed by Raiffa (1951), who noted: it is a useful concept since it serves to convexify certain regions [of expected payoffs] in the Euclidean plane. (p. 8)Received: April 2002 / Revised: November 2003     

基于模糊数的政府与绿色制造商博弈分析

近年来环境问题已经成为人们迫切需要解决的重要问题,促使制造商采取绿色制造模式也是各国政府正在面临的一大难题。因此文章结合三角模糊数与博弈理论建立了政府与绿色制造商的模糊博弈模型,将博弈结果分为纯策略和混合策略两类,并讨论了各种情形下政府与绿色制造商的不同策略以及影响因素,分析了最优博弈结果以及相应的管理策略,为绿色制造模式的顺利实施提供了相关建议。研究结果表明,政府的补贴和惩罚等策略在绿色制造模式的采取方面起着举足轻重的作用。最后利用三角结构元法对算例分纯策略和混合策略两种情况进行分析求解,从而验证了结论的正确性和可行性。     

Nondominated Equilibrium Solutions of a Multiobjective Two-Person Nonzero-Sum Game and Corresponding Mathematical Programming Problem

With reference to a multiobjective two-person nonzero-sum game, we define nondominated equilibrium solutions and provide a necessary and sufficient condition for a pair of mixed strategies to be a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. We give a numerical example and demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.     

Nonzero-sum non-stationary discounted Markov game model

The goal of this paper is provide a theory of K-person non-stationary Markov games with unbounded rewards, for a countable state space and action spaces. We investigate both the finite and infinite horizon problems. We define the concept of strong Nash equilibrium and present conditions for both problems for which strong Nash or Nash equilibrium strategies exist for all players within the Markov strategies, and show that the rewards in equilibrium satisfy the optimality equations.