Role of update dynamics in the collective cooperation on the spatial snowdrift games: Beyond unconditional imitation and replicator dynamics

In this paper, we investigate the role of update or imitation rules in the spatial snowdrift game on regular lattices. Three different update rules, including unconditional imitation (UI), replicator dynamics (RD) and the Moran process, are utilized to update the strategies of focal players during the game process in the spatial snowdrift on the lattice. We observe that the aggregate cooperation level between players is largely elevated by using the Moran process in the spatial snowdrift game, when compared to the UI or replicator dynamics. Meanwhile, we carefully explore the dynamical evolution of frequency of cooperators and the cluster formation pattern for these three update rules. Moreover, it is also shown that the evolutionary behavior under the Moran update is independent of and insensitive to the randomly initial configurations of cooperators and defectors. The current results clearly indicate that the introduction of moderate randomness in the strategy update will highly promote the maintenance and persistence of cooperation among selfish individuals, which will be greatly instrumental to deeply understand the evolution of cooperation within many natural, biological and social systems.     

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.     

A game-theoretic formulation of joint implementation of environmental projects

The aim of this paper is to provide a game-theoretic interpretation of joint implementation in environmental projects and to assess the merit of such a strategy. More specifically, we consider a two-player game and solve it under three different cases. In the first case, countries play a non-cooperative game and optimize their welfare under an environmental constraint without having access to joint implementation. In the second case, we assume countries do have access to JI, which allows us to assess its merits by comparing the players’ welfare levels achieved with and without JI. In the last case, the players jointly optimize their welfare under a collective environmental constraint. Comparing welfare levels in this case to those in the second case allows us to assess the merits of cooperation.     

Mutual punishment promotes cooperation in the spatial public goods game

Punishment has been proved to be an effective mechanism to sustain cooperation among selfish individuals. In previous studies, punishment is unidirectional: an individual i can punish j but j cannot punish i. In this paper, we propose a mechanism of mutual punishment, in which the two individuals will punish each other if their strategies are different. Because of the symmetry in imposing the punishment, one might expect intuitively the strategy to have little effect on cooperation. Surprisingly, we find that the mutual punishment can promote cooperation in the spatial public goods game. Other pertinent quantities such as the time evolution of cooperator density and the spatial distribution of cooperators and defectors are also investigated.     

Inverse Stackelberg Public Goods Game with Multiple Hierarchies Under Global and Local Information Structures

This paper studies the inverse Stackelberg game with multiple hierarchies under global and local information structures, where the players have discrete strategy spaces. For the classic public goods game, we solve the pure-strategy inverse Stackelberg equilibria under three typical hierarchical structures. The results reveal some counterintuitive characteristics within the systems with hierarchies, such as that the cooperation does not increase with the return rate at the equilibria. Furthermore, by defining a local information structure, we give an estimate of the fewest hierarchies required for full cooperation, which can be a constant multiple of the logarithm or square root of the population size or of the population size itself, according to different information structures and return rates. This paper proposes a novel mechanism to play the game and promote cooperation. Both the formulation and analysis method are different from existing works, and the results can find their ample implications in practice, which might help decision making in hierarchical systems.     

The not-quite non-atomic game: Normal approximation

Measure games with a large number of players are frequently approximated by nonatomic games. In fact, however, while it is true that values of large measure games will, under certain reasonable circumstances, converge to the value of a non-atomic game, it is also true that this convergence is quite slow. Using the multi-linear extension and the central limit theorem, we obtain an approximation which (because it is based on the normal distribution) we call the normal approximation. We show that, for two examples with several hundred and several thousand players respectively, the normal approximation is much better than the non-atomic approximation.     

Cooperation in dividing the cake

This paper defines models of cooperation among players partitioning a completely divisible good (such as a cake or a piece of land). The novelty of our approach lies in the players’ ability to form coalitions before the actual division of the good with the aim to maximize the average utility of the coalition. A social welfare function which takes into account coalitions drives the division. In addition, we derive a cooperative game which measures the performance of each coalition. This game is compared with the game in which players start cooperating only after the good has been portioned and has been allocated among the players. We show that a modified version of the game played before the division outperforms the game played after the division.     

Games on fuzzy communication structures with Choquet players

Myerson (1977) used graph-theoretic ideas to analyze cooperation structures in games. In his model, he considered the players in a cooperative game as vertices of a graph, which undirected edges defined their communication possibilities. He modified the initial games taking into account the graph and he established a fair allocation rule based on applying the Shapley value to the modified game. Now, we consider a fuzzy graph to introduce leveled communications. In this paper players play in a particular cooperative way: they are always interested first in the biggest feasible coalition and second in the greatest level (Choquet players). We propose a modified game for this situation and a rule of the Myerson kind.     

Link formation in cooperative situations

In this paper we study the endogenous formation of cooperation structures or communication graphs between players in a superadditive TU game. For each cooperation structure that is formed, the payoffs to the players are determined by an exogenously given solution. We model the process of cooperation structure formation as a game in strategic form. It is shown that several equilibrium refinements predict the formation of the complete cooperation structure or some structure which is payoff-equivalent to the complete structure. These results are obtained for a large class of solutions for cooperative games with cooperation structures. Received September 1995/Revised version I October 1996/Revised version II April 1997/Final version September 1997     

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.     

Multilayers in a modulated stochastic game

We are concerned with an antagonistic stochastic game between two players A and B which finds applications in economics and warfare. The actions of the players are manifested by a series of strikes of random magnitudes at random times exerted by each player against his opponent. Each of the assaults inflicts a random damage to enemy's vital areas. In contrast with traditional games, in our setting, each player can endure multiple strikes before perishing. Predicting the ruin time (exit) of player A, along with the total amount of casualties to both players at the exit is a main objective of this work. In contrast to the time sensitive analysis (earlier developed to refine the information on the game) we insert auxiliary control levels, which both players will cross in due game before the ruin of A. This gives A (and also B) an additional opportunity to reevaluate his strategy and change the course of the game. We formalize such a game and also allow the real time information about the game to be randomly delayed. The delayed exit time, cumulative casualties to both players, and prior crossings are all obtained in a closed-form joint functional.     

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.     

Cooperative Markov decision processes: time consistency, greedy players satisfaction, and cooperation maintenance

We deal with multi-agent Markov decision processes (MDPs) in which cooperation among players is allowed. We find a cooperative payoff distribution procedure (MDP-CPDP) that distributes in the course of the game the payoff that players would earn in the long run game. We show under which conditions such a MDP-CPDP fulfills a time consistency property, contents greedy players, and strengthen the coalition cohesiveness throughout the game. Finally we refine the concept of Core for Cooperative MDPs.     

Characterization of the core in games with restricted cooperation

Games with restricted cooperation describe situations in which the players are not completely free in forming coalitions. The restrictions in coalition formation can be attributed to economic, hierarchical, political or ethical reasons. In order to manage these situations, the model includes a collection of coalitions which determines the feasible agreements among the agents. The purpose of this paper is to extend the characterization of the core of a cooperative game, made by Peleg [International Journal of Game Theory 15 (1986) 187–200; Handbook of Game Theory with Economic Applications, vol. I. Elsevier Science Publishers B.V., pp. 397–412] to the context of games with restricted cooperation. In order to make the approach as general as possible, we will consider classes of games with restricted cooperation in which the collection of feasible coalitions has a determined structure, and we will impose conditions on that structure to generalize the Peleg’s axiomatization.     

Enhanced cooperation in prisoner’s dilemma with aspiration

We develop an aspiration-based dynamic model which leads to enhanced cooperation in the Prisoner’s Dilemma game played by the continuous population of agents. The main idea is to limit the aggregate information available to the agents. The model–a system of three nonlinear differential equations–describes the evolution of the aspiration levels of players who use different strategies, and the evolution of the mean frequency of the cooperative strategy in the system of players. The stationary (partial) cooperation level is calculated explicitly. We demonstrate that, contrary to the similar model with only one global aspiration level, the stationary cooperation level can be greater than half.     

Evolutionary games

This paper deals with a mathematical game. As the name implies, the game concept is formulated with biological evolution in mind. An evolutionary game differs from the usual game concepts in that the players cannot choose their strategies. Rather, the strategies used by the players are handed down from generation to generation. It is the survival characteristics of a strategy that determine the outcome of the evolutionary game. Players interact and receive payoffs according to the strategies they are using. These interactions, in turn, determine the fitness of players using a given strategy. The survival characteristics of strategy are determined directly from the fitness functions. Necessary conditions for determining an evolutionarily stable strategy are developed here for a continuous game. Results are illustrated with an example.Dedicated to G. LeitmannThis work was supported by NSF Grant No. INT-82-10803 and The University of Western Australia (Visiting Fellowship, Department of Mathematics, 1983).     

模糊支付合作博弈的可信性准核仁

合作博弈是处理局中人之间协同行为的数学理论。有诸如核心、稳定集、沙普利值、准核仁和核仁等不同的解概念。在很多情形,除了借助专家经验和主观直觉,没有恰当的方式来确定支付函数,由此产生了具有模糊支付的合作博弈模型。准核仁是一种重要的解概念,在模糊支付合作博弈中如何恰当定义准核仁是个重要的问题。本文在可信性理论的框架下研究了这个问题,定义了两类可信性准核仁概念并证明了它们的存在性和唯一性,同时研究了可信性核心、可信性核仁与它们之间的关系。     

Prisoner's dilemma game on adaptive networks under limited foresight

We study the emergence of cooperation in an environment where players in prisoner's dilemma game (PDG) not only update their strategies but also change their interaction relations. Different from previous studies in which players update their strategies according to the imitation rule, in this article, the strategies are updated with limited foresight. We find that two absorbing states—full cooperation and full defection—can be reached, assuming that players can delete interaction relations unilaterally, but new relations can only be created with the mutual consent of both partners. Simulation experiments show that high levels of cooperation in large populations can be achieved when the temptation to defect in PDG is low. Moreover, we explore the factors which influence the level of cooperation. These results provide new insights into the cooperation in social dilemma and into corresponding control strategies. © 2012Wiley Periodicals, Inc. Complexity, 2012     

Limiting distributions of the number of pure strategy Nash equilibria in n-person games

We study the number of pure strategy Nash equilibria in a “random” n-person non-cooperative game in which all players have a countable number of strategies. We consider both the cases where all players have strictly and weakly ordinal preferences over their outcomes. For both cases, we show that the distribution of the number of pure strategy Nash equilibria approaches the Poisson distribution with mean 1 as the numbers of strategies of two or more players go to infinity. We also find, for each case, the distribution of the number of pure strategy Nash equilibria when the number of strategies of one player goes to infinity, while those of the other players remain finite.     

乡村振兴战略下农村人居环境整治PPP模式合作行为演化博弈分析

农村人居环境整治是乡村振兴战略的重要内容,基于有限理性的演化博弈理论,构建了以地方政府、社会资本和农村居民为博弈主体的农村人居环境整治PPP模式合作行为演化博弈模型,运用Matlab软件分析了三方主体的初始意愿和政府规制对三方主体行为策略演化的影响。研究发现:(1)政府规制对农村人居环境整治PPP模式合作博弈系统演化的影响从大到小依次是:政府对社会资本的补贴资助、投机罚金、合作奖励和政府对农村居民的参与奖励。(2)与提高地方政府对社会资本的补贴资助力度和惩罚力度相比,提高地方政府的激励规制初始意愿更能促进农村人居环境整治PPP项目的顺利落地实施。(3)农村居民参与PPP项目的初始意愿的提升关键在于提高农村居民参与收益。最后,为促进PPP模式在农村环境治理领域的应用与发展和推动农村人居环境整治的可持续改善提出相应建议。