Optimal strategic reasoning with McNaughton functions

The aim of the paper is to explore strategic reasoning in strategic games of two players with an uncountably infinite space of strategies the payoff of which is given by McNaughton functions—functions on the unit interval which are piecewise linear with integer coefficients. McNaughton functions are of a special interest for approximate reasoning as they correspond to formulas of infinitely valued Lukasiewicz logic. The paper is focused on existence and structure of Nash equilibria and algorithms for their computation. Although the existence of mixed strategy equilibria follows from a general theorem (Glicksberg, 1952) [5], nothing is known about their structure neither the theorem provides any method for computing them. The central problem of the article is to characterize the class of strategic games with McNaughton payoffs which have a finitely supported Nash equilibrium. We give a sufficient condition for finite equilibria and we propose an algorithm for recovering the corresponding equilibrium strategies. Our result easily generalizes to n-player strategic games which don't need to be strictly competitive with a payoff functions represented by piecewise linear functions with real coefficients. Our conjecture is that every game with McNaughton payoff allows for finitely supported equilibrium strategies, however we leave proving/disproving of this conjecture for future investigations.     

正则对策的N-M稳定集及其唯一存在定理

研究了有非对称性和负传递性偏好的无限策略对策,提出了N-M稳定集和正则对策的概念,其中N-M稳定集是将合作对策中由Von Neumann 和Morgenstern给出的相应概念引入到策略对策中的.所谓正则对策是指其Nash均衡集中每条链关于一致偏好总有上界的无限策略对策.证明了每个正则对策都有唯一N-M稳定集. 此结果及其应用例子说明正则对策N-M稳定集的概念对于策略对策的纯Nash均衡的精炼起着重要作用.     

Nash equilibrium in compact-continuous games with a potential

If the preferences of the players in a strategic game satisfy certain continuity conditions, then the acyclicity of individual improvements implies the existence of a (pure strategy) Nash equilibrium. Moreover, starting from any strategy profile, an arbitrary neighborhood of the set of Nash equilibria can be reached after a finite number of individual improvements.     

A two-population game in observable double-ended queuing systems

In this study, the existing game theoretical framework is extended to strategic queuing in search of solutions for a two-population game in observable double-ended queuing systems with zero matching times. We show that multiple Nash equilibria and one unique subgame perfect Nash equilibrium exist in this game.     

Markov stopping games with random priority

In the paper a construction of Nash equilibria for a random priority finite horizon two-person non-zero sum game with stopping of Markov process is given. The method is used to solve the two-person non-zero-sum game version of the secretary problem. Each player can choose only one applicant. If both players would like to select the same one, then the lottery chooses the player. The aim of the players is to choose the best candidate. An analysis of the solutions for different lotteries is given. Some lotteries admit equilibria with equal Nash values for the players.The research was supported in part by Committee of Scientific Research under Grant KBN 211639101.     

Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions

We consider the generalized Nash equilibrium problem which, in contrast to the standard Nash equilibrium problem, allows joint constraints of all players involved in the game. Using a regularized Nikaido-Isoda-function, we then present three optimization problems related to the generalized Nash equilibrium problem. The first optimization problem is a complete reformulation of the generalized Nash game in the sense that the global minima are precisely the solutions of the game. However, this reformulation is nonsmooth. We then modify this approach and obtain a smooth constrained optimization problem whose global minima correspond to so-called normalized Nash equilibria. The third approach uses the difference of two regularized Nikaido-Isoda-functions in order to get a smooth unconstrained optimization problem whose global minima are, once again, precisely the normalized Nash equilibria. Conditions for stationary points to be global minima of the two smooth optimization problems are also given. Some numerical results illustrate the behaviour of our approaches.     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

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.     

n人可信策略博弈

本文研究模糊支付n人策略博弈,由于支付信息的不完全性,模糊支付可视为模糊变量。基于可信性理论,在不同的决策环境中引入模糊排序方法表征支付者行为。因此,本文定义四种可信均衡,等价于纳什均衡。其次证明了可信均衡存在性定理。除此之外,在现实策略博弈中用算例说明四种可信均衡的现实意义。最后讨论四种可信均衡之间可能存在的关系。     

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.     

A non-cooperative approach to the cost spanning tree problem

We associate to each cost spanning tree problem a non-cooperative game, which is inspired by a real-life problem. We study the Nash equilibria and subgame perfect Nash equilibria of this game. We prove that these equilibria are closely related with situations where agents connect sequentially to the source.Finicial support from the Ministerio de Ciencia y Tecnologia and FEDER, and Xunta de Galicia through grants BEC2002-04102-C02-01 and PGIDIT03PXIC30002PN is gratefully acknowledged.     

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.     

0-1对策的完全混合Nash均衡的代数求解法

将求解一般0-1策略对策的完全混合Nash均衡的问题转化为求解根为正的纯小数的高次代数方程组的问题.作为一种特殊而重要的情形,利用Pascal矩阵,Newton矩阵(对角元素为Newton二项式系数的对角矩阵)和Pascal-Newton矩阵(Pascal矩阵和Newton矩阵的逆阵的乘积)将求解对称0-1对策的完全混合Nash均衡的问题转化为求解根为正的纯小数的高次代数方程的问题,并给出第二问题的反问题(由完全混合Nash均衡求解对称0-1对策族)的求解方法.同时,给出了一些算例来说明对应问题的算法.     

Some projection-like methods for the generalized Nash equilibria

A generalized Nash game is an m-person noncooperative game in which each player’s strategy depends on the rivals’ strategies. Based on a quasi-variational inequality formulation for the generalized Nash game, we present two projection-like methods for solving the generalized Nash equilibria in this paper. It is shown that under certain assumptions, these methods are globally convergent. Preliminary computational experience is also reported.     

Nash equilibria in optimal reinsurance bargaining

We introduce a strategic behavior in reinsurance bilateral transactions, where agents choose the risk preferences they will appear to have in the transaction. Within a wide class of risk measures, we identify agents’ strategic choices to a range of risk aversion coefficients. It is shown that at the strictly beneficial Nash equilibria, agents appear homogeneous with respect to their risk preferences. While the game does not cause any loss of total welfare gain, its allocation between agents is heavily affected by the agents’ strategic behavior. This allocation is reflected in the reinsurance premium, while the insurance indemnity remains the same in all strictly beneficial Nash equilibria. Furthermore, the effect of agents’ bargaining power vanishes through the game procedure and the agent who gets more welfare gain is the one who has an advantage in choosing the common risk aversion at the equilibrium.     

A Shared-Constraint Approach to Multi-Leader Multi-Follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over this shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization problem are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.     

Persistent equilibria in strategic games

A perfect equilibrium [Selten] can be viewed as a Nash equilibrium with certain properties of local stability. Simple examples show that a stronger notion of local stability is needed to eliminate unreasonable Nash equilibria. The persistent equilibrium is such a notion. Properties of this solution are studied. In particular, it is shown that in each strategic game there exists a pesistent equilibrium which is perfect and proper.     

Strategically supported cooperation in dynamic games with coalition structures

The problem of strategic stability of long-range cooperative agreements in dynamic games with coalition structures is investigated. Based on imputation distribution procedures, a general theoretical framework of the differential game with a coalition structure is proposed. A few assumptions about the deviation instant for a coalition are made concerning the behavior of a group of many individuals in certain dynamic environments.From these, the time-consistent cooperative agreement can be strategically supported by ε-Nash or strong ε-Nash equilibria. While in games in the extensive form with perfect information, it is somewhat surprising that without the assumptions of deviation instant for a coalition, Nash or strong Nash equilibria can be constructed.     

Bounded rationality, strategy simplification, and equilibrium

It is frequently suggested that predictions made by game theory could be improved by considering computational restrictions when modeling agents. Under the supposition that players in a game may desire to balance maximization of payoff with minimization of strategy complexity, Rubinstein and co-authors studied forms of Nash equilibrium where strategies are maximally simplified in that no strategy can be further simplified without sacrificing payoff. Inspired by this line of work, we introduce a notion of equilibrium whereby strategies are also maximally simplified, but with respect to a simplification procedure that is more careful in that a player will not simplify if the simplification incents other players to deviate. We study such equilibria in two-player machine games in which players choose finite automata that succinctly represent strategies for repeated games; in this context, we present techniques for establishing that an outcome is at equilibrium and present results on the structure of equilibria.     

Equilibrium existence and uniqueness in network games with additive preferences

A directed network game of imperfect strategic substitutes with heterogeneous players is analyzed. We consider concave additive separable utility functions that encompass the quasi-linear ones. It is found that pure strategy Nash equilibria verify a non-linear complementarity problem. By requiring appropriate concavity in the utility functions, the existence of an equilibrium point is shown and equilibrium uniqueness is established with a P-matrix. For this reason, it appears that previous findings on network structure and sparsity hold for many more games.