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.     

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。     

Correlated equilibria in some classes of two-person games

A correlated equilibrium in a two-person game is “good” if for everyNash equilibrium there is a player who prefers the correlated equilibrium to theNash equilibrium. If a game is “best-response equivalent” to a two-person zero-sum game, then it has no good correlated equilibria. But games which are “almost strictly competitive” or “order equivalent” to a two-person zero-sum game may have good correlated equilibria.     

Minimax strategies for discounted “secretary problems” with interview costs

For a game-theoretical problem of optimal choice with interview costs and discounting we prove, using a saddle-point argument, the strict determinedness of the game and we give minimax strategies for both players.     

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.     

On the minimal representation of homogeneous games

It is known that the lattice-minimal representation (by natural numbers) of a weighted majority game may be not unique and may lack of equal treatment (Isbell 1959). The same is true for the total-weight minimal representation. Both concepts coincide on the class of homogeneous games. The main theorem of this article is that for homogeneous games there is a unique minimal representation. This result is given by means of a construction that depends on the natural order on the set of player types. This order coincides with one induced by the “desirability relation”. In order to compute the minimal representation inductively, while proceeding from smaller players to the greater one, we are led to distinguish two different kinds of players: some players are “replacable” by smaller ones, some not.     

An experimental study of constant-sum centipede games

In this paper, we report the results of a series of experiments on a version of the centipede game in which the total payoff to the two players is constant. Standard backward induction arguments lead to a unique Nash equilibrium outcome prediction, which is the same as the prediction made by theories of “fair” or “focal” outcomes. We find that subjects frequently fail to select the unique Nash outcome prediction. While this behavior was also observed in McKelvey and Palfrey (1992) in the “growing pie” version of the game they studied, the Nash outcome was not “fair”, and there was the possibility of Pareto improvement by deviating from Nash play. Their findings could therefore be explained by small amounts of altruistic behavior. There are no Pareto improvements available in the constant-sum games we examine. Hence, explanations based on altruism cannot account for these new data. We examine and compare two classes of models to explain these data. The first class consists of non-equilibrium modifications of the standard “Always Take” model. The other class we investigate, the Quantal Response Equilibrium model, describes an equilibrium in which subjects make mistakes in implementing their best replies and assume other players do so as well. One specification of this model fits the experimental data best, among the models we test, and is able to account for all the main features we observe in the data.     

Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian

Motivated by the “tug‐of‐war” game studied by Peres et al. in 2009, we consider a nonlocal version of the game that goes as follows: at every step two players pick, respectively, a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount ϵ > 0 (as is done in the classical case), it is an s‐stable Levy process that chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically derive a deterministic nonlocal integrodifferential equation that we call the “infinity fractional Laplacian.” We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double‐obstacle problem, both problems having a natural interpretation as tug‐of‐war games. © 2011 Wiley Periodicals, Inc.     

The Harsanyi paradox and the “right to talk” in bargaining among coalitions

We describe a coalitional value from a non-cooperative point of view, assuming coalitions are formed for the purpose of bargaining. The idea is that all the players have the same chances to make proposals. This means that players maintain their own “right to talk” when joining a coalition. The resulting value coincides with the weighted Shapley value in the game between coalitions, with weights given by the size of the coalitions. Moreover, the Harsanyi paradox (forming a coalition may be disadvantageous) disappears for convex games.     

Layers of noncooperative games

We model and analyze classes of antagonistic stochastic games of two players. The actions of the players are formalized by marked point processes recording the cumulative damage to the players at any moment of time. The processes evolve until one of the processes crosses its fixed preassigned threshold of tolerance. Once the threshold is reached or exceeded at some point of the time (exit time), the associated player is ruined. Both stochastic processes are being “observed” by a third party point stochastic process, over which the information regarding the status of both players is obtained. We succeed in these goals by arriving at closed form joint functionals of the named elements and processes. Furthermore, we also look into the game more closely by introducing an intermediate threshold (see a layer), which a losing player is to cross prior to his ruin, in order to analyze the game more scrupulously and see what makes the player lose the game.     

A single-shot game of multi-period inspection

This paper deals with an inspection game of Customs and a smuggler during some days. Customs has two options of patrolling or not. The smuggler can take two strategies of shipping its cargo of contraband or not. Two players have several opportunities to take an action during a limited number of days but they may discard some of the opportunities. When the smuggling coincides with the patrol, there occurs one of three events: the capture of the smuggler by Customs, a success of the smuggling and nothing new. If the smuggler is captured or no time remains to complete the game, the game ends. There have been many studies on the inspection game so far by the multi-stage game model, where both players at a stage know players’ strategies taken at the previous stage. In this paper, we consider a two-person zero-sum single-shot game, where the game proceeds through multiple periods but both players do not know any strategies taken by their opponents on the process of the game. We apply dynamic programming to the game to exhaust all equilibrium points on a strategy space of player. We also clarify the characteristics of optimal strategies of players by some numerical examples.     

A simple expression for the nucleolus in a special case

We present a simple algorithm for calculating the nucleolus of a game whenever (a) the characteristic function is non-positive, ie. a “cost” function, and (b) the players can be ordered by “size” in such a way that the cost of any coalition is equal to the cost of the largest player in that coalition. The cumulative nucleolus is approximately equal to the convex envelope of this cost function. A numerical and geometric illustration is given for a game based upon Birmingham airport runway costs, where there are over 13,000 players (aircraft movements) of 11 distinct (aircraft) types.     

On the subject of non optimal play in zero sum extensive games: “The trap phenomenon”

This paper discusses the best reply to be played after a mistake is committed in a zero sum extensive game. First it is shown that the minimax strategies do not, in general, provide specific answers. Two possible criteria are proposed to narrow the set of optimal strategies. Then, a surprising feature of the selected optimal strategy may be observed: it may set the stage in advance so that, if the mistake is committed, the penalty will be maximal. This is called: the “trap phenomenon”.     

The solution of the soccer “dream league” game

On January 5, 1996, Maariv, one of the two leading daily newspapers in Israel, announced “The Dream League” game. Every participant in this game was required to “purchase” from a pool of all the soccer players in the Israeli National League, a team which according to his judgment would be chosen as the best team at the end of the season. Purchasing the players was subject to a given budget and to several other constraints. After the soccer season was over, we were requested by Maariv to find the optimal “Dream Team”.The problem of finding the optimal team is shown to be a generalized version of the well-known knapsack problem. It is formulated as an integer program and solved to optimality by the software NAG. Evidently, the optimal Dream Team is much better (in terms of the total cumulative grade) than the actual winning team chosen by the readers of Maariv. A possible heuristic procedure for solving the game in larger settings is also discussed.     

One-way flow network formation under constraints

We study the effects of institutional constraints on stability and efficiency in the “one-way flow” model of network formation. In this model the information that flows through a link between two players runs only towards the player that initiates and supports the link, so in order for it to flow in both directions, both players must pay whatever the unit cost of a directional link is. We assume that an exogenous “societal cover” consisting of a collection of possibly overlapping subsets covering the set of players specifies the social organization in different groups or “societies,” so that a player may initiate links only with players that belong to at least one society that he/she also belongs to, thus restricting the feasible strategies and networks. In this setting, we examine the impact of such societal constraints on stable/efficient architectures and on dynamics.     

Intervening duality and bargaining with a farmer–landowner example

In a 1995 paper in this journal I introduced the intervening duality idea and showed how it could be used to analyse a simple matching pennies game in which each of two players anticipated a gain from a single toss of a “fair” coin. This paper generalises that earlier analysis and results. Firstly it introduces more general production related classes of cases in which individuals may seek to gain by making themselves dual to the duals of an intervening specification of production plans, strategies and payoffs. Secondly, through this explicitly economic context, the paper introduces and analyses issues pertaining to framing (e.g. with reference to choices of production of crops vs. “setaside”), bargaining and learning.     

具有超图合作结构的赋权Position值

具有超图交流结构的可转移效用合作对策,也称为超图对策,它由一个三元组(N,v,H)所组成,其中(N,H)是一个可转移效用对策(简称TU-对策),而(N,H)是一个超图(超网络)。在超图对策中,除Myerson值(Myerson)外,Position值(Meessen)是另一个重要的分配规则。该模型要求把超图结构中每条超边Shapley的值平均分配给它所包含的点,而不考虑每个点的交流能力或合作水平。本文引入超图结构中点的度值来度量每条超边中每个点的交流能力或合作水平,并结合Haeringer提出用于推广Shapley值的权重系统,并由此定义了具有超图合作结构的赋权Position值。我们证明了具有超图合作结构的赋权Position值可以由“分支有效性”、“冗余超边性”、“超边可分解性”、“拟可加性”、“弱积极性”和“弱能转换”六个性质所唯一确定,并且发现参与者获得的支付随其度值的增加而增加,参与者分摊的成本随其度值的增加而降低。     

On the Structural Stability of Values for Cooperative Games

It is generally assumed that any set of players can form a feasible coalition for classical cooperative games. But, in fact, some players may withdraw from the current game and form a union, if this makes them better paid than proposed. Based on the principle of coalition split, this paper presents an endogenous procedure of coalition formation by levels and bargaining for payoffs simultaneously, where the unions formed in the previous step continue to negotiate with others in the next step as “individuals,” looking for maximum share of surplus by organizing themselves as a partition. The structural stability of the induced payoff configuration is discussed, using two stability criteria of core notion for cooperative games and strong equilibrium notion for noncooperative games.

     

Optimal strategies for node selection games on oriented graphs: Skew matrices and symmetric games

In the node selection game Γ_D each of the two players simultaneously selects a node from the oriented graph D. If there is an arc between the selected nodes, then there is a payoff from the “dominated” player to the “dominating” player. We investigate the set of optimal strategies for the players in the node selection game Γ_D. We point out that a classical theorem from game theory relates the dimension of the polytope of optimal strategies for Γ_D to the nullity of certain skew submatrix of the payoff matrix for Γ_D. We show that if D is bipartite (with at least two nodes in each partite set), then an optimal strategy for the node selection game Γ_D is never unique. Our work also implies that if D is a tournament, then there is a unique optimal strategy for each player, a result obtained by Fisher and Ryan [Optimal strategies for a generalized “scissors, paper, and stone” game, Amer. Math. Monthly 99 (1992) 935–942] and independently by Laffond, Laslier, and Le Breton [The bipartisan set of a tournament game, Games Econom. Behav. 5 (1993) 182–201].     

Nim games with arbitrary periodic moving orders

A well known two person game called Nim consists in two players, alternately taking matches from a single heap of matches. In our paper we extend this game to arbitrary periodic moving orders, for exampleAABAB, whereA andB are the players. The main result states, that the player moving more frequently always can force a win, provided there are “enough” matches in the heap initially. Thus in the exampleA will win.