Parameterized Two-Player Nash Equilibrium

We study the problem of computing Nash equilibria in a two-player normal form (bimatrix) game from the perspective of parameterized complexity. Recent results proved hardness for a number of variants, when parameterized by the support size. We complement those results, by identifying three cases in which the problem becomes fixed-parameter tractable. Our results are based on a graph-theoretic representation of a bimatrix game, and on applying graph-theoretic tools on this representation.     

基于分层路径计算单元与双矩阵博弈的多域光网络静态组播专用保护算法

在预先配置组播业务的多域光网络中,如何保证静态组播业务的生存性已成为广泛关注的问题。针对此问题,通过采用基于分层路径计算单元（PCE）架构的全局拓扑信息调度计算模型,利用双矩阵博弈来生成链路不相交的组播树和组播保护树,最后提出了一种基于分层PCE与双矩阵博弈的多域光网络静态组播专用保护算法,并给出了算法的具体示例。理论分析和实验结果表明,在一定的冗余资源配置下,该算法具有较低的时间复杂度,在优化组播工作树和组播保护树的资源分配结构的同时,可明显提高多域光网络静态组播业务的生存能力。     

Well Supported Approximate Equilibria in Bimatrix Games

In view of the apparent intractability of constructing Nash Equilibria (NE in short) in polynomial time, even for bimatrix games, understanding the limitations of the approximability of the problem is an important challenge.     

Probability and possibility models of matrix games of two subjects

Probability models and their possibility counterparts of one-matrix and bimatrix games of two subjects (A and B) were defined and analyzed. For the one-matrix game possibility model, a theorem was proven saying that maximin and minimax fuzzy strategies exist and that possibilities of A winning or losing (B) in relation to these strategies are equal. The concepts of fuzzy and randomized game strategies were defined and analyzed. The problem of statistic modeling of A and B fuzzy strategies was resolved. For possible models of bimatrix games, the existence of equilibrium points was examined. For the problem of maximization of the winning A and B possibility, it was proven that equilibrium points exist. For the problem of minimizing the possibility of losses, it was shown that if equilibrium points exist, some of them are related to clear strategies, A and B.     

基于粒子群优化的混合模糊双矩阵对策求解

针对具有模糊策略集与模糊支付值的不确定性冲突环境,建立了混合模糊双矩阵对策模型。在假定模糊支付值为三角模糊数的情形下,采用了基于单个截集的模糊数线性排序函数,将模型清晰化后转化为双矩阵对策,并应用粒子群优化算法求解。最后,给出一个军事例子说明了模型的实用有效性和粒子群优化算法的高效性。     

Bayesian equilibria for uncertain bimatrix game with asymmetric information

In an uncertain bimatrix game, there are two solution concepts of \((\alpha ,\beta )\)-optimistic equilibrium strategy and \((u,v)\)-maximum chance equilibrium strategy. This paper goes further by assuming that the confidence levels \(\alpha , \beta \) and payoff levels \(u, v\) are private information. Then, the so-called uncertain bimatrix game with asymmetric information is investigated. Two solution concepts of Bayesian optimistic equilibrium strategy and Bayesian maximum chance equilibrium strategy as well as their existence theorems are presented. Moreover, sufficient and necessary conditions are given for finding the Bayesian equilibrium strategies. Finally, a two-firm advertising problem is analyzed for illustrating our modelling idea.     

Uniqueness and efficiency of Nash equilibrium in a family of randomly generated repeated games

This paper describes the results of an analysis of the Nash equilibrium in randomly generated repeated games. We study two families of games: symmetric bimatrix games G(A, B) with B = A^⊤ and nonsymmetric bimatrix games (the first includes the classical games of prisoner dilemma, battle of the sexes, and chickens). We use pure strategies, implemented by automata of size two, and different strategy domination criteria. We observe that, in this environment, the uniqueness and efficiency of equilibria outcomes is the typical result.     

The Computational Complexity of Weak Saddles

We study the computational aspects of weak saddles, an ordinal set-valued solution concept proposed by Shapley. F. Brandt et al. recently gave a polynomial-time algorithm for computing weak saddles in a subclass of matrix games, and showed that certain problems associated with weak saddles of bimatrix games are NP-hard. The important question of whether weak saddles can be found efficiently was left open. We answer this question in the negative by showing that finding weak saddles of bimatrix games is NP-hard, under polynomial-time Turing reductions. We moreover prove that recognizing weak saddles is coNP-complete, and that deciding whether a given action is contained in some weak saddle is hard for parallel access to NP and thus not even in NP unless the polynomial hierarchy collapses. Most of our hardness results are shown to carry over to a natural weakening of weak saddles.     

Price strategies in dynamic duopolistic markets with deregulated electricity supplies using mixed strategies

While effective competition can force service providers to seek economically efficient methods to reduce costs, the deregulated electricity supply industry still allows some generators to exercise market power at particular locations, thereby preventing the deregulated power market to be perfectly competitive. In this paper, we investigate the interdependence of pricing mechanisms and strategy behaviors of the suppliers. A multiperiod dynamic profit-maximizing problem is converted to a bimatrix game that is solved in the framework of mixed strategies. By this procedure, we have at least one Nash solution. Instead of considering only perfectly competitive price and monopoly price, we introduce other prices between these two to simulate the real market better. Numerical examples show that the new entrant that maximizes its profit will not choose the perfectly competitive price even as an entry price.     

GLOBAL ROBUST SERVOMECHANISM OF A CLASS OF NONLINEAR SYSTEMS USING OUTPUT FEEDBACK

The robust servomechanism problem (alternatively, output regulation problem) of the class of nonlinear systems in lower triangular form has been extensively studied in recent years. The semi‐global solution was first given by either state feedback or output feedback. The global solution by state feedback was given very recently. However, the global solution by output feedback has long been an open problem. In this paper, we present a set of solvability conditions of the global robust servomechanism problem for this class of nonlinear systems by output feedback.     

Random Bimatrix Games Are Asymptotically Easy to Solve (A Simple Proof)

We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player’s payoffs are bounded and independent random variables, not necessarily identically distributed, but with almost common expectations. We show that the completely mixed uniform strategy profile, i.e., the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is with high probability a $\sqrt{\frac{\ln n}{n}}$ -Nash equilibrium and a $\sqrt{\frac{3\ln n}{n}}$ -well supported Nash equilibrium, where n is the number of pure strategies available to each player. This asserts that the completely mixed, uniform strategy profile is an almost Nash equilibrium for random bimatrix games, since it is, with high probability, an ?-well-supported Nash equilibrium where ? tends to zero as n tends to infinity.     

Bargaining property of nucleolus and τ-value in a class of TU-games

First, we introduce pairwise-bargained consistency with a reference point, and use as reference points the maxmin and the minmax value within pure strategies of a certain constant-sum bimatrix game, and also the game value within mixed strategies of it. Second, we show that the pairwise-bargained consistency with reference point being the maxmin or the minmax value determines the nucleolus in some class of transferable utility games. (This result is known in the bankruptcy games and the pseudoconcave games with respect to supersets of the managers.) This class of games whose element we call a pseudoconcave game with respect to essential coalitions, of course, includes the bankruptcy games and the pseudoconcave games with respect to supersets of the managers. It is proved that this class of games is exactly the same as the class of games which have a nonempty core that is determined only by one-person and (n − 1)-person coalition constraints. And we give a sufficient condition which guarantees that the bargaining set coincides with the core in this class of games. Third, we interpret the τ-value of a quasibalanced transferable utility game by the pairwise-bargained consistency with reference point being the game value. Finally, by combining the second and the third results, if a transferable utility game in this class is also semiconvex, then the nucleolus and the τ-value are characterized by the pairwise-bargained consistency with different reference points which are given by the associated bimatrix game.     

A Game-theoretic approach to analyze interacting actors in GRL goal models

Goal-oriented requirements engineering aims to capture desired goals and strategies of relevant stakeholders during early requirements engineering stages, using goal models. Goal-oriented modeling techniques support the analysis of system requirements (especially non-functional ones) from an operationalization perspective, through the evaluation of alternative design options. However, conflicts and undesirable interactions between requirements produced from goals are inevitable, especially as stakeholders often aim for different objectives. In this paper, we propose an approach based on game theory and the Goal-oriented Requirement Language (GRL) to reconcile interacting stakeholders (captured as GRL actors), leading to reasonable trade-offs. This approach consists in building a payoff bimatrix that considers all actor’s valid GRL strategies, and computing its Nash equilibrium. Furthermore, we use two optimization techniques to reduce the size of the payoff bimatrix, hence reducing the computational cost of the Nash equilibrium. The approach goes beyond existing work by supporting nonzero-sum games, multiple alternatives, and inter-actor dependencies. We demonstrate the applicability of our game-theoretic modeling and analysis approach using a running example and two GRL models from the literature, with positive results on feasibility and applicability, including performance results.

     

Game and Information Theory Analysis of Electronic Countermeasures in Pursuit-Evasion Games

Two-player pursuit-evasion games in the literature typically either assume both players have perfect knowledge of the opponent's positions or use primitive sensing models. This unrealistically skews the problem in favor of the pursuer who needs only maintain a faster velocity at all turning radii. In real life, an evader usually escapes when the pursuer no longer knows the evader's position. In our previous work, we modeled pursuit evasion without perfect information as a two-player bimatrix game by using a realistic sensor model and information theory to compute game-theoretic payoff matrices. That game has a saddle point when the evader uses strategies that exploit sensor limitations, whereas the pursuer relies on strategies that ignore the sensing limitations. In this paper, we consider, for the first time, the effect of many types of electronic countermeasures (ECM) on pursuit-evasion games. The evader's decision to initiate its ECM is modeled as a function of the distance between the players. Simulations show how to find optimal strategies for ECM use when initial conditions are known. We also discuss the effectiveness of different ECM technologies in pursuit-evasion games.      

A NEW AGENT MATCHING SCHEME USING AN ORDERED FUZZY SIMILARITY MEASURE AND GAME THEORY

In this paper, an agent matching method for bilateral contracts in a multi-agent market is proposed. Each agent has a hierarchical representation of its trading commodity attributes by a tree structure of fuzzy attributes. Using this structure, the similarity between the trees of each pair of buyer and seller is computed using a new ordered fuzzy similarity algorithm. Then, using the concept of Stackelberg equilibrium in a leader–follower game, matchmaking is performed among the sellers and buyers. The fuzzy similarities of each agent with others in its personal viewpoint have been used as its payoffs in a bimatrix game. Through a case study for bilateral contracts of energy, the capabilities of the proposed agent-based system are illustrated.     

Cooperative global output regulation of heterogeneous second-order nonlinear uncertain multi-agent systems

In this paper, we study the cooperative global output regulation problem for a class of heterogeneous second order nonlinear uncertain multi-agent systems. We first introduce a type of distributed internal model that converts the cooperative global output regulation problem into the global robust stabilization problem of the so-called augmented multi-agent system. Then we further globally stabilize this augmented multi-agent system via a distributed state feedback control law, thus leading to the solution of the original problem. A special case of our result leads to the solution of the global leader-following consensus problem for the second order nonlinear multi-agent systems without satisfying the global Lipschitz condition.     

Local search of orthogonal packings using the lower bounds

Consideration was given to the problem of packing the rectangular half-infinite strip. It was suggested to solve it using a one-point evolutionary algorithm with search of the best solution in the neighborhood of the local lower bound. Proposed was an algorithm to construct this neighborhood including the packings obtained by solving a special problem of one-dimensional cutting. Its solution was shown to be the local lower bound in the considered neighborhood. An improved global lower bound was proposed. The results of numerical modeling were presented. The record value obtained was compared with the global bounds.     

最小化全局完成时间的成像卫星任务规划算法

成像侦察卫星任务规划问题是一类典型多约束组合优化问题.最小化全局完成时间是任务规划领域时效性要求较高情况下的一种优化目标.提出一种整合整数规划与约束规划方法,在最小化任务规划方案全局完成时间的目标下,求解成像侦察卫星任务规划问题的组合算法.该算法通过应用Benders分解将原约束整数规划模型划分为主问题与子问题两部分,采用软件MOSEK与GECODE对主、子问题分别求解.根据子问题求解结果生成剪枝约束,返回主问题迭代,直到获得优化解.算法有效性通过仿真实验进行了检验并取得预期效果.     

一种求解非线性约束优化全局最优的新方法

本文提出了一种求解非线性约束优化的全局最优的新方法—它是基于利用非线性互补函数和不断增加新的约束来重复解库恩-塔克条件的非线性方程组的新方法。因为库恩-塔克条件是非线性约束优化的必要条件,得到的解未必是非线性约束优化的全局最优解,为此,本文首次给出了通过利用该优化问题的先验知识,不断地增加约束来限制全局最优解范围的方法,一些仿真例子表明提出的方法和理论有效的,并且可行的。