带策略约束的区间数双矩阵博弈的双线性规划求解方法

传统区间数双矩阵博弈理论研究局中人支付值为区间数的策略选择问题,但没有考虑局中人策略选择可能受到各种约束.创建一种求解局中人策略选择受约束且支付值为区间数的双矩阵博弈（简称带策略约束的区间数双矩阵博弈）的简单、有效的双线性规划求解方法.首先,将局中人的博弈支付看作支付值区间中数值的函数.通过证明这种函数具有单调性,据此利用支付值区间的上、下界,构造了一对辅助双线性规划模型,可分别用于显式地计算任意带策略约束的区间数双矩阵博弈中局中人区间数博弈支付的上、下界及其相应的最优策略.最后,利用考虑策略约束条件下企业和政府针对发展低碳经济策略问题的算例,通过比较其与不考虑策略约束情形下的结果,说明了提出的模型和方法的有效性、优越性及可应用性.     

Numerical construction of Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game

Numerical methods are proposed for constructing Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game with terminal cost functionals and geometric constraints on the players’ controls. The formalization of the players’ strategies and of the motions generated by them is based on the formalization and results from the theory of positional zero-sum differential games developed by N.N. Krasovskii and his school. It is assumed that the game is reduced to a planar game and the constraints on the players’ controls are given in the form of convex polygons. The problem of finding solutions of the game may be reduced to solving nonstandard optimal control problems. Several computational geometry algorithms are used to construct approximate trajectories in these problems, in particular, algorithms for constructing the convex hull as well as the union, intersection, and algebraic sum of polygons.     

A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many such allocation problems there is some hierarchical ordering of the players. In this paper we consider a class of games with a permission structure describing situations in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we provide a polynomial time algorithm for computing the nucleolus of the restricted games corresponding to a class of games with a permission structure which economic applications include auction games, dual airport games, dual polluted river games and information market games.     

Tandem antagonistic games

We study an antagonistic sequential game of two players that undergoes two phases. Each phase is modeled by multi-dimensional random walk processes. During phase 1 (or game 1), the players exchange a series of random strikes of random magnitudes. Game 1 ends whenever one of the players sustains damages in excess of some lower threshold. However, the total damage does not exceed another upper threshold which allows the game to continue. Phase 2 (game 2) is run by another combination of random walk processes. At some point of phase 2, one of the players, after sustaining damages in excess of its third threshold, is ruined and he loses the entire game. We predict that moment, along with the total casualties to both players, and other critical information; all in terms of tractable functionals. The entire game is analyzed by tools of fluctuation theory.     

Tandem antagonistic games

We study an antagonistic sequential game of two players that undergoes two phases. Each phase is modeled by multi-dimensional random walk processes. During phase 1 (or game 1), the players exchange a series of random strikes of random magnitudes. Game 1 ends whenever one of the players sustains damages in excess of some lower threshold. However, the total damage does not exceed another upper threshold which allows the game to continue. Phase 2 (game 2) is run by another combination of random walk processes. At some point of phase 2, one of the players, after sustaining damages in excess of its third threshold, is ruined and he loses the entire game. We predict that moment, along with the total casualties to both players, and other critical information; all in terms of tractable functionals. The entire game is analyzed by tools of fluctuation theory.     

Constrained core solutions for totally positive games with ordered players

In many applications of cooperative game theory to economic allocation problems, such as river-, polluted river- and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.     

On noncooperative hybrid stochastic games

The present article models and analyzes a noncooperative hybrid stochastic game of two players. The main phase (prime hybrid mode) of the game is preceded by “unprovoked” hostile actions by one of the players (during antecedent hybrid mode) that at some time transforms into a large scale conflict between two players. The game lasts until one of the players gets ruined. The latter occurs when the cumulative damage to the losing player exceeds a fixed threshold. Both hybrid modes are formalized by marked point stochastic processes and the theory of fluctuations is utilized as one of the chief techniques to arrive at a closed form functional describing the status of both players at the ruin time.     

Notions of equilibrium for differential games on intersecting game sets

We suggest new notions of conflict equilibrium and demonstrate a technique of their use for finding a solution in arbitrary game problems on a game set common for all players and especially in problems with side interests of players in the static and dynamic settings.     

有限理性条件下演化博弈行为分析

基于博弈双方有限理性的假设,运用动力系统的相关理论和方法对一般2×2非对称演化博弈过程建立了动态复制方程,并对博弈双方在演化过程中的行为进行了分析,得出博弈双方交互系统均衡点及稳定性相应的结论及其全部动力学行为.     

Rankings and values for team games

In this article we attack several problems that arise when a group of individuals is organized in several teams with equal number of players in each one (e.g., for company work, in sports leagues, etc). We define a team game as a cooperative game v that can have non-zero values only on coalitions of a given cardinality; it is further shown that, for such games, there is essentially a unique ranking among the players. We also study the way the ranking changes after one or more players retire. Also, we characterize axiomatically different ways of ranking the players that intervene in a cooperative game.     

Theories of pursuit and evasion

We give three elementary definitions of a game of pursuit and evasion in a large class of metric spaces. Our definitions are independent of the theory of differential games. We prove that the three definitions yield the same value of the game, and we study this value as a function of the initial positions of the players and their velocities. Several open problems are stated.This work was supported by an NSF Grant.     

On the foundations of game theory: The case of non-Archimedean utilities

Contrary to what appears to have become an accepted part of the folklore of game theory, a finite two-person zero-sum game with non-Archimedean utilities may have no equilibrium-point solution, and either one or both players may have no “minimax” strategy. Even when both players have “minimax” strategies, such a game may lack an equilibrium point.     

Mechanism design for a multicommodity flow game in service network alliances

We study a collaborative multicommodity flow game where individual players own capacity on the edges of the network and share this capacity to deliver commodities. We present membership mechanisms, by adopting a rationality based approach using notions from game theory and inverse optimization, to allocate benefits among the players in such a game.     

合作博弈的粒子群算法求解

随着局中人人数的增加,利用传统的“占优”方法和“估值”方法进行合作博弈求解无论从逻辑上还是计算上都变得非常困难。针对此问题,将合作博弈的求解看作是局中人遵照有效性和个体理性提出分配方案,并按照一定规则不断迭代调整直至所有方案趋向一致的过程。依据该思路,对合作博弈粒子群算法模型进行构建,确定适应度函数,设置速度公式中的参数。通过算例分析,利用粒子群算法收敛快、精度高、容易实现的特点,可以迅速得到合作博弈的唯一分配值,这为求解合作博弈提供了新的方法和工具。     

From hierarchies to levels: new solutions for games with hierarchical structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many of these problems, players are organized according to either a hierarchical structure or a levels structure that restrict the players’ possibilities to cooperate. In this paper, we propose three new solutions for games with hierarchical structure and characterize them by properties that relate a player’s payoff to the payoffs of other players located in specific positions in the hierarchical structure relative to that player. To define each solution, we consider a certain mapping that transforms the hierarchical structure into a levels structure, and then we apply the standard generalization of the Shapley value to the class of games with levels structure. Such transformation mappings are studied by means of properties that relate a player’s position in both types of structure.     

Preferred coalitions in cooperative differential games

There are many interesting situations which can be described by anN-person general-sum differential game. Such games are characterized by the fact that the strategy of each player depends upon reasonable assumptions about the strategies of the remaining players; and, thus, these games cannot be considered asN uncoupled optimal control problems. In such cases, we say that the game is not strictly competitive, but involves a mutual interest which makes it possible for all of the players to reduce their costs by cooperating with one another, provided the resulting agreement can be enforced. When cooperation is allowed and there are more than two players, there is always the question of whether all possible subcoalitions will be formed with equal ease. This work considers the situation in which a particular subcoalition is preferred. A theory of general-sum games with preferred coalitions is presented, together with constructive examples of alternative approaches which are unsatisfactory.     

DYNAMIC SUPPLY FROM COMMON PROPERTY RESOURCE: WATER DIVERSIONS FROM THE GREAT LAKES

For many purposes, the five Great Lakes can be classified as an open access resource since there is no well-defined system of property rights governing these water uses in the lakes. Yet the Great Lakes Charter and the Water Resources Act of 1966 establish regulations concerning the diversion of water out of the basin. Thus, for interbasin water diversions the Great Lakes is more like a common property resource with rules governing such transfers. Moreover, water diversion decisions are interconnected since withdrawing water from one point may affect water levels in the entire lake system. This, in turn, can adversely affect hydropower production and commercial navigation. Contributing to the complexity of the problem are the eight U.S. states, two Canadian provinces and two federal governments which are involved in Great Lakes management. Game theory is used to describe this situation. Several games are constructed to describe different market structures. Of particular interest is the number of players who participate in the game, as well as the expectations they hold. Open-loop (where players commit themselves to future actions) and closed-loop (where players do not commit themselves to future actions) are compared for the ten players game (eight states and two provinces), two players game (U.S. and Canada) and one player game (a social planner's solution). The open-loop game is shown to ignore part of the externalities involved, and thus can underestimate the social loss caused by diversions from the lakes.     

Two Game Models for Cooperation with Implicit Noncooperation

We propose two flexible game models to represent and analyze cases that cannot be modeled by current game models. One is called sharing creditability game (SCG) and the other is called bottomline game (BLG). The new models transform cooperative games into new games that incorporate auxiliary information (noncooperative in nature) usually neglected in previous theories. The new games will be solved only by traditional noncooperative game theory. When the new solutions are applied to the original games, the solutions can reflect the auxiliary information in addition to the original objectives of the decision makers or players. Generally, the new solutions are different from the cooperative and the noncooperative solutions of the original games. Existing transferable utility (TU) games and noncooperative games will coincide with special cases of the two new game models. Using SCG and BLG, the prisoner’s dilemma can be reformulated and a richer set of decisions can be considered for the players. The two new game models have potential applications in military and socioeconomic situations.This research was partly funded by the College Engineering, Ohio State University.     

Generalized solution concepts in games with possibly unaware players

Most work in game theory assumes that players are perfect reasoners and have common knowledge of all significant aspects of the game. In earlier work (Halpern and Rêgo 2006, arxiv.org/abs/0704.2014), we proposed a framework for representing and analyzing games with possibly unaware players, and suggested a generalization of Nash equilibrium appropriate for games with unaware players that we called generalized Nash equilibrium. Here, we use this framework to analyze other solution concepts that have been considered in the game-theory literature, with a focus on sequential equilibrium.     

Application of Atanassov’s I-fuzzy set theory to matrix games with fuzzy goals and fuzzy payoffs

We aim to extend some results in [6, 7, 8, 2] on two person zero sum matrix games (TPZSMG) with fuzzy goals and fuzzy payoffs to I-fuzzy scenario. Because the payoffs of the matrix game are fuzzy numbers, the aspiration levels of the players are fuzzy as well. It is reasonable to believe that there is some indeterminacy in estimating the aspiration levels of both players from their respective expected pay offs. This situation is modeled in the game using Atanassov??s I-fuzzy set theory. A new solution concept is proposed for such games and a procedure is outlined to obtain the degrees of suitability of the aspiration levels for each of the two players.