A consistent value for games with n players and r alternatives

In Bolger [1993], an efficient value was obtained for a class of games called games with n players and r alternatives. In these games, each of the n players must choose one and only one of the r alternatives. This value can be used to determine a player’s “a priori” value in such a game. In this paper, we show that the value has a consistency property similar to the “consistency” for TU games in Hart/Mas-Colell [1989] and we present a set of axioms (including consistency) which characterizes this value.  The games considered in this paper differ from the multi-choice games considered by Hsiao and Raghavan [1993]. They consider games in which the actions of the players are ordered in the sense that, if i >j, then action i carries more “weight” than action j.  These games also differ from partition function games in that the worth of a coalition depends not only on the partitioning of the players but also on the action chosen by each subset of the partition. Received: April 1994/final version: June 1999     

Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers

The purpose of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of trapezoidal fuzzy numbers (TrFNs), which are a type of two-person non-cooperative games with payoffs expressed by TrFNs and players’ strategies being constrained. In this methodology, it is proven that any Alfa-constrained matrix game has an interval-type value and hereby any constrained matrix game with payoffs of TrFNs has a TrFN-type value. The auxiliary linear programming models are derived to compute the interval-type value of any Alfa-constrained matrix game and players’ optimal strategies. Thereby the TrFN-type value of any constrained matrix game with payoffs of TrFNs can be directly obtained through solving the derived four linear programming models with data taken from only 1-cut and 0-cut of TrFN-type payoffs. Validity and applicability of the models and method proposed in this paper are demonstrated with a numerical example of the market share game problem.     

Total Reward Stochastic Games and Sensitive Average Reward Strategies

In this paper, total reward stochastic games are surveyed. Total reward games are motivated as a refinement of average reward games. The total reward is defined as the limiting average of the partial sums of the stream of payoffs. It is shown that total reward games with finite state space are strategically equivalent to a class of average reward games with an infinite countable state space. The role of stationary strategies in total reward games is investigated in detail. Further, it is outlined that, for total reward games with average reward value 0 and where additionally both players possess average reward optimal stationary strategies, it holds that the total reward value exists.     

On a noisy-silent-versus-silent duel with equal accuracy functions

This paper deals with the noisy-silent-versus-silent duel with equal accuracy functions. Player I has a gun with two bullets and player II has a gun with one bullet. The first bullet of player I is noisy, the second bullet of player I is silent, and the bullet of player II is silent. Each player can fire their bullets at any time in [0, 1] aiming at his opponent. The accuracy function ist for both players. If player I hits player II, not being hit himself before, the payoff of the duel is +1; if player I is hit by player II, not hitting player II before, the payoff is –1. The optimal strategies and the value of the game are obtained. Although optimal strategies in past works concerning games of timing does not depend on the firing moments of the players, the optimal strategy obtained for player II depends explicitly on the firing moment of player I's noisy bullet.     

Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games

Bimatrix and polymatrix games are expressed as parametric linear 0–1 programs. This leads to an algorithm for the complete enumeration of their extreme equilibria, which is the first one proposed for polymatrix games. The algorithm computational experience is reported for two and three players on randomly generated games for sizes up to 14 × 14 and 13 × 13 × 13.Communicated by P. M. PardalosThe authors thank Bernhard von Stengel for constructive comments on the contents of this paper.     

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.     

Games with permission structures: The conjunctive approach

This paper is devoted to the game theoretic analysis of decision situations, in which the players have veto power over the actions undertaken by certain other players. We give a full characterization of the dividends in these games with a permission structure. We find that the collection of these games forms a subspace of the vector space of all games with side payments on a specified player set.Two applications of these results are provided. The first one deals with the projection of additive games on a permission structure. It is shown that the Shapley value of these projected games can be interpreted as an index that measures the power of the players in the permission structure. The second application applies the derived results on games, where the organization structure can be analysed separately from the production capacities of the participating players.     

Perturbation theory for games in normal form and stochastic games

In this paper, the effect on values and optimal strategies of perturbations of game parameters (payoff function, transition probability function, and discount factor) is studied for the class of zero-sum games in normal form and for the class of stationary, discounted, two-person, zero-sum stochastic games.A main result is that, under certain conditions, the value depends on these parameters in a pointwise Lipschitz continuous way and that the sets of -optimal strategies for both players are upper semicontinuous multifunctions of the game parameters.Extensions to general-sum games and nonstationary stochastic games are also indicated.     

A set of axioms for a value for partition function games

Myerson (1977) derived an efficient value for games in partition function form. In this paper, we present a set of axioms which characterize a different efficient value for such games. This latter value assigns value 0 to dummies and assigns nonnegative values to players in monotone simple games.     

The Owen value values friendship

This paper presents two new axiomatizations of the Owen value for games with coalition structures. Two associated games are defined and a consistency axiom is required. The construction of the associated games presupposes that coalitions behave in an aggressive manner towards players who are not members of the same unions and in a friendly manner towards players that do belong to their unions. The consistency axiom necessitates the definition of only one associated game which is not a reduced game. Received: February 1999/Revised version: January 2000     

Totally balanced combinatorial optimization games

Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it.?We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games. Received: November 5, 1998 / Accepted: September 8, 1999?Published online February 23, 2000     

The nucleolus and kernel for simple games or special valid inequalities for 0–1 linear integer programs

An equivalence between simplen-person cooperative games and linear integer programs in 0–1 variables is presented and in particular the nucleolus and kernel are shown to be special valid inequalities of the corresponding 0–1 program. In the special case of weighted majority games, corresponding to knapsack inequalities, we show a further class of games for which the nucleolus is a representation of the game, and develop a single test to show when payoff vectors giving identical amounts or zero to each player are in the kernel. Finally we give an algorithm for computing the nucleolus which has been used successfully on weighted majority games with over twenty players.     

Stochastic games with non-observable actions

We examine n-player stochastic games. These are dynamic games where a play evolves in stages along a finite set of states; at each stage players independently have to choose actions in the present state and these choices determine a stage payoff to each player as well as a transition to a new state where actions have to be chosen at the next stage. For each player the infinite sequence of his stage payoffs is evaluated by taking the limiting average. Normally stochastic games are examined under the condition of full monitoring, i.e. at any stage each player observes the present state and the actions chosen by all players. This paper is a first attempt towards understanding under what circumstances equilibria could exist in n-player stochastic games without full monitoring. We demonstrate the non-existence of -equilibria in n-player stochastic games, with respect to the average reward, when at each stage each player is able to observe the present state, his own action, his own payoff, and the payoffs of the other players, but is unable to observe the actions of them. For this purpose, we present and examine a counterexample with 3 players. If we further drop the assumption that the players can observe the payoffs of the others, then counterexamples already exist in games with only 2 players.     

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.     

Existence of value in pursuit-evasion games with restricted phase coordinates

Almost all results referring to the problem of the existence of a value in differential games concern games without restricted phase coordinates. In this paper, we introduce a concept of value for differential games of pursuit and evasion and prove, under some general assumption, the existence of it. The players are required to satisfy some general phase constraints. The arguments employed in this paper are based on some extent on Krasovskii's method of extremal construction. We also show that the lower value in the Friedman sense is a generalization of our value. In a special linear case, the equivalence between pursuit differential games and time-optimal control problems is established.     

Values of mixed games

Aumann andShapley [1973] have investigated values of games in which all players are individually insignificant, i.e. form a non-atomic continuum, or “ocean”. In this paper we treat games in which, in addition to such an ocean, there are also some “atoms”, i.e. players who are individually significant. We define spaces of such games that are analogous to those investigated byAumann andShapley, and prove the existence of values on some of them. Unlike in the non-atomic case, we find that in general there are infinitely many values, corresponding to various ways in which the atoms can be imbedded in the ocean. The results generalize those ofMilnor andShapley [1961]. Precise statements will be found in Section 2.     

Qualitative analysis of secured outcome regions for two-target games

Qualitative (game of kind) outcomes of two-target games are analyzed in this paper, under both the zero-sum and nonzero-sum preference ordering of outcomes by the players. The outcome regions of each player are defined from a security standpoint. The secured draw and mutual-kill regions of a player depend explicitly on his preference ordering of outcomes and should be constructed separately for each player, especially in a nonzero-sum game. General guidelines are presented for identifying the secured outcome regions of players in a class of two-target games that satisfy an Isaacs-like condition, in terms of the qualitative solutions of the two underlying single-target pursuit-evasion games. A construction has been proposed for obtaining the qualitative solution of a large class of two-target games. Illustrative examples are included.This work was done while the first author was a Research Associate in the Department of Electrical Engineering at the Indian Institute of Science, Bangalore, and was financially supported by the Council of Scientific and Industrial Research, Delhi, India.     

On the role of chance moves and information in two-person games

The value of information has been the subject of many studies in a strategic context. The central question in these studies is how valuable the information hidden in the chance moves of a game is for one or more of the players. Generally speaking, only the extra possibilities that are beneficial for the players have been considered so far. In this note we study the value of information for a special class of two-person games. For these games we also investigate how “badly” the players can do, both with and without knowing the result of the chance move. In this way one can determine to what extent the players are restricted in their possibilities by the fact that some information is hidden in the chance moves of the games. This allows for a comparison of the influence of the chance move to the control that the players have over the game result.     

基于个人超出值的模糊联盟合作博弈最小二乘预核仁

本文给出了基于个人超出值的无限模糊联盟合作博弈最小二乘预核仁的求解模型,得到该模型的显式解析解,并研究该解的若干重要性质。证明了:本文给出的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的相等解(The equalizer solution),基于个人超出值的字典序解三者相等。进一步证明了:基于Owen线性多维扩展的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的经典合作博弈最小二乘预核仁相等。最后,通过数值实例说明本文提出的无限模糊联盟合作博弈求解模型的实用性与有效性。     

连续对策的公平性和刺激性

本文用刺激性(感)来描述游戏一个零和对策的两个局中人的风险性和侥幸取胜性,游戏不同的零和对策可能有不同的刺激感,刺激性越大,对策结果的公平性越小;反之亦然,本文解决了如下问题;(1)刺激性和公平性的数学描述是什么?(2)局中人如何保证他们的一局对策的对策结果是最公平的或最有刺激感的?(3)如果两个局中人希望对策结果尽量公平或尽量有刺激感,他们最好从给定的连续对策中选择哪个?