A differential game of incomplete information

In this paper, we study a differential game of incomplete information. In such a game, the cost function depends on parameter . At the start of the game, only one of the players knows the value of this parameter, while the other player has only a (subjective) probability distribution for the parameter. We obtain explicit expressions for both the value of the game and the two players' optimal strategies.     

Crowding games are sequentially solvable

A sequential-move version of a given normal-form game Γ is an extensive-form game of perfect information in which each player chooses his action after observing the actions of all players who precede him and the payoffs are determined according to the payoff functions in Γ. A normal-form game Γ is sequentially solvable if each of its sequential-move versions has a subgame-perfect equilibrium in pure strategies such that the players' actions on the equilibrium path constitute an equilibrium of Γ.  A crowding game is a normal-form game in which the players share a common set of actions and the payoff a particular player receives for choosing a particular action is a nonincreasing function of the total number of players choosing that action. It is shown that every crowding game is sequentially solvable. However, not every pure-strategy equilibrium of a crowding game can be obtained in the manner described above. A sufficient, but not necessary, condition for the existence of a sequential-move version of the game that yields a given equilibrium is that there is no other equilibrium that Pareto dominates it. Received July 1997/Final version May 1998     

Isaacs' princess and monster game on the circle

The solution is given of a multistage pursuit-evasion game in which ablind pursuer searches for ablind evader within a finite set of locations arranged in a circle. The players traverse this set by transfering their positions, at each of a succession of instants, from the points which they occupy to ones which are adjacent to them. Some standard results of measure theory are used to construct the players' optimal strategies when the payoff to the evader is the time taken for the pursuer to find him or, more generally, an increasing function of this time.This work was carried out with the support of a CSIRO postgraduate studentship.     

Independent mistakes in large games

Economic models usually assume that agents play precise best responses to others' actions. It is sometimes argued that this is a good approximation when there are many agents in the game, because if their mistakes are independent, aggregate uncertainty is small. We study a class of games in which players' payoffs depend solely on their individual actions and on the aggregate of all players' actions. We investigate whether their equilibria are affected by mistakes when the number of players becomes large. Indeed, in generic games with continuous payoff functions, independent mistakes wash out in the limit. This may not be the case if payoffs are discontinuous. As a counter-example we present the n players Nash bargaining game, as well as a large class of “free-rider games.” Received: November 1997/Final version: December 1999     

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 Approach to Fuzzy Noncooperative Nash Games

Systems that involve more than one decision maker are often optimized using the theory of games. In the traditional game theory, it is assumed that each player has a well-defined quantitative utility function over a set of the player decision space. Each player attempts to maximize/minimize his/her own expected utility and each is assumed to know the extensive game in full. At present, it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player in these types of problems. On the other hand, in many of such complex problems, each player has a heuristic knowledge of the desires of the other players and a heuristic knowledge of the control choices that they will make in order to meet their ends.In this paper, we utilize fuzzy set theory in order to incorporate the players' heuristic knowledge of decision making into the framework of conventional game theory or ordinal game theory. We define a new approach to N-person static fuzzy noncooperative games and develop a solution concept such as Nash for these types of games. We show that this general formulation of fuzzy noncooperative games can be applied to solve multidecision-making problems where no objective function is specified. The computational procedure is illustrated via application to a multiagent optimization problem dealing with the design and operation of future military operations.     

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。     

The card game 24 and its application to math education

The card game 24 is a mathematical game that traditionally engages elementary students to practice their mental computational skills. In this paper, we use probability to formulate and explore the game. We create score of difficulty level for solvable card combination set under various setup of game rules. Based on our findings, we create new playing rules and provide suggestions for different levels of players. Our results may serve as guidelines on introducing the game 24 into the practice of math education.     

连续对策的最大熵策略密度对策解

对于正方形[0,2]×[0,2]上的连续对策,将局中人的非纯策略(概率分布函数)的导数称为这个局中人的策略密度(概率密度函数).建立了这种连续对策的最大熵理论.主要证明了当每个局中人都没有最优纯策略时,具有最大熵的最优策略密度集合的非空紧凸性,研究了最优策略密度的最大熵,给出一类带有最大熵的连续对策.     

Avoider-Enforcer: The rules of the game

An Avoider-Enforcer game is played by two players, called Avoider and Enforcer, on a hypergraph F⊆X². The players claim previously unoccupied elements of the board X in turns. Enforcer wins if Avoider claims all vertices of some element of F, otherwise Avoider wins. In a more general version of the game a bias b is introduced to level up the players' chances of winning; Avoider claims one element of the board in each of his moves, while Enforcer responds by claiming b elements. This traditional set of rules for Avoider-Enforcer games is known to have a shortcoming: it is not bias monotone.We relax the traditional rules in a rather natural way to obtain bias monotonicity. We analyze this new set of rules and compare it with the traditional ones to conclude some surprising results. In particular, we show that under the new rules the threshold bias for both the connectivity and Hamiltonicity games, played on the edge set of the complete graph Kn, is asymptotically equal to n/logn. This coincides with the asymptotic threshold bias of the same game played by two “random” players.     

On a game in manufacturing

We analyse a non-zero sum two-person game introduced by Teraoka and Yamada to model the strategic aspects of production development in manufacturing. In particular we investigate how sensitive their solution concept (Nash equilibrium) is to small variations in their assumptions. It is proved that a Nash equilibrium is unique if it exists and that a Nash equilibrium exists when the capital costs of the players are zero or when the players are equal in every respect. However, when the capital costs differ, in general a Nash equilibrium exists only when the players' capital costs are high compared to their profit rates.     

Loss aversion equilibrium

The Nash equilibrium solution concept for games is based on the assumption of expected utility maximization. Reference dependent utility functions (in which utility is determined not only by an outcome, but also by the relationship of the outcome to a reference point) are a better predictor of behavior than expected utility. In particular, loss aversion is an important element of such utility functions.  We extend games to include loss aversion characteristics of the players. We define two types of loss-aversion equilibrium, a solution concept endogenizing reference points. The two types reflect different procedures of updating reference points during the game. Reference points emerge as expressions of anticipation which are fulfilled.  We show existence of myopic loss-aversion equilibrium for any extended game, and compare it to Nash equilibrium. Comparative statics show that an increase in loss aversion of one player can affect her and other players' payoffs in different directions. Received August 1998/Revised version February 2000     

Nash bargaining solution under externalities

We define a Nash bargaining solution (NBS) of partition function games. Based on a partition function game, we define an extensive game, which is a propose–respond sequential bargaining game where the rejecter of a proposal exits from the game with some positive probability. We show that the NBS is supported as the expected payoff profile of any stationary subgame perfect equilibrium (SSPE) of the extensive game such that in any subgame, a coalition of all active players forms immediately. We provide a necessary and sufficient condition for such an SSPE to exist. Moreover, we consider extensions to the cases of nontransferable utilities, time discounting and multiple-coalition formation.     

基于训练和过滤的期望均衡及其实现路径

期望均衡是博弈局中人或局外人对于博弈均衡点的一致期望,强调互利共赢,它要求博弈群体的每个成员对期望均衡点有一个共同的预期.显然,基于纳什均衡的帕累托优化组合策略是一个比纳什均衡更有效的期望均衡.要实现期望均衡,可采用局中人参与的训练与学习使得群体的预期一致,也可采用第三方过滤器来达到期望目标.在期望均衡的概率分布下,个体行为的偏离不能比均衡态取得更多收益.否则,训练就是无效的,第三方过滤器就是不公平的.     

Games of timing with resources of mixed type

The paper considers a game of timing which is closely related to the so-called duels. This is a game connected with the distribution of resources by two players. Each of the players is in possession of some amount of resource to be distributed by him in the time interval [0, 1]. In his behavior, Player 1 is restricted by the necessity of taking all of his resources at a single point, while Player 2 has no restrictions. For the payoff function, defined as for duels, the game is solved; explicit formulas on the value of the game and the optimal strategies for the players are found.     

Formulation and analysis of combat problems as zero-sum bicriterion differential games

In this paper, we present a formulation and analysis of a combat game between two players as a zero-sum bicriterion differential game. Each player's twin objectives of terminating the game on his own target set, while simultaneously avoiding his opponent's target set, are quantified in this approach. The solution in open-loop pure strategies is sought from among the Pareto-optimal security strategies of the players. A specific preference ordering on the outcomes is used to classify initial events in the assured win, draw, and mutual kill regions for the players. The method is compared with the event-constrained differential game approach, recently proposed by others. Finally, a simple example of the turret game is solved to illustrate the use of this method.     

Cores of convex games

The core of ann-person game is the set of feasible outcomes that cannot be improved upon by any coalition of players. A convex game is defined as one that is based on a convex set function. In this paper it is shown that the core of a convex game is not empty and that it has an especially regular structure. It is further shown that certain other cooperative solution concepts are related in a simple way to the core: The value of a convex game is the center of gravity of the extreme points of the core, and the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core.     

A Bidding Game with Heterogeneous Players

A one-sided limit order book is modeled as a noncooperative game for several players. Agents offer various quantities of an asset at different prices, competing to fulfill an incoming order, whose size is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random incoming order. In a previous paper, the existence of a Nash equilibrium was established by means of a fixed point argument. The main issue discussed in the present paper is whether this equilibrium can be obtained from the unique solution to a two-point boundary value problem, for a suitable system of discontinuous ordinary differential equations. Some additional assumptions are introduced, which yield a positive answer. In particular, this is true when there are exactly two players, or when all players assign the same exponential probability distribution to the incoming order. In both of these cases, we also prove that the Nash equilibrium is unique. A counterexample shows that these assumptions cannot be removed, in general.     

A two-person zero-sum Markov game with a stopped set

We consider a two-person zero-sum Markov game with continuous time up to the time that the game process goes into a fixed subset of a countable state space, this subset is called a stopped set of the game. We show that such a game with a discount factor has optimal value function and both players will have their optimal stationary strategies. The same result is proved for the case of a nondiscounted Markov game under some additional conditions, that is a reward rate function is nonnegative and the first time τ (entrance time) of the game process going to the stopped set is finite with probability one (i.e., p(τ < ∞) = 1). It is remarkable that in the case of a nondiscounted Markov game, if the expectation of the entrance time is bounded, and the reward rate function need not be nonnegative, then the same result holds.