Some games of search on a lattice

In this article we shall deal with some two-person games on a lattice. These are games of search and ambush where the set of strategies of one of the players is determined by functions on the lattice. We give a general method to obtain a solution of these games and we apply it to three particular games. © 1993 John Wiley & Sons, Inc.     

A family of games on 1,∞)2 with payoff a function of y/x

A two-parameter class of games on 1,∞)² is studied. The games may be regarded as analogs of Silverman games, having continuous payoff function in place of a step function of y/x. This change is motivated by a desire to move toward a model for competitive situations where the penalty for overspending increases with the amount of overspending. There are some similarities to games with bell-shaped kernel. For most of the region considered in the plane of the two parameters there are solutions of finite type, which are obtained explicitly. There are, however, pockets in this plane where no optimal strategies have been found and possibly where none of finite type exist.     

Simple stability of general n- person games

Herein are introduced noncooperative points of n-person games defined on compact and convex sets in a Euclidean space and continuous payoff functions, which are natural extensions of the respective concepts of maximin, minimax strategies and saddle points of two-person games. The conc:ept of equilibrium point appears as a special of one of those. As a particular case, we examine such points for the mixed extension of finite n-person games; finally, a related topic is illustrated. These results represent the first installment of a collection which will be continued in subsequent publications.     

不同游戏方式下的暴力网络游戏对大学生攻击性的影响

目的:探讨不同游戏方式下的暴力网络游戏对不同攻击性特质大学生攻击性的影响。方法:采用攻击性特质问卷、词汇决策任务和竞争反应时任务对60名大学生进行问卷和实验研究。结果:在不同游戏方式下,被试表现出的攻击性认知和行为存在显著差异(F(2,52)=4.41,P<0.05;F(1,52)=50.06,P<0.001)。高、低攻击性特质者在游戏后的攻击性认知和行为上存在显著差异(F(1,52)=15.25,P<0.001;F(1,52)=82.35,P<0.001)。结论:不同游戏方式下的暴力网络游戏对大学生的攻击性存在不同的影响,相比竞争游戏和单人游戏,合作游戏是一种较好的游戏方式,可以在一定程度减少大学生,特别是高攻击性特质大学生的攻击性认知和行为。此外,暴力网络游戏对大学生攻击性认知和攻击性行为的影响存在一致性。     

A saddle-point theorem for a class of infinite games

In this paper, the existence of a saddle point for two-person zero-sum infinite games of a special type is proved. The games have continuous bilinear payoff functions and strategy sets which are convex, noncompact subsets of an infinite-dimensional vector space. The closures of the strategy sets are, however, compact. The payoff functions satisfy conditions which allow the use of dominance arguments to show that points in the closure of a strategy set are dominated by or are strategically equivalent to points in the strategy set itself. Combining the dominance arguments with a well-known existence theorem produces the main result of the paper. The class of games treated is an extension of a class studied by J. D. Matheson, who obtained explicit solutions for the saddle points by using necessary conditions.     

微分对策及其在军事领域的研究进展*

介绍了微分对策的产生背景及其半个多世纪以来的发展历程,简述了国内外微分对策理论发展的几个重要阶段及其标志性成果。全面地阐述了国内外关于微分对策在军事领域的应用研究状况,特别是美、俄等军事强国在微分对策军事应用研究方面的现状以及我国研究人员在该领域的主要研究成果。还进一步论述了微分对策在军事应用研究方面存在的问题,并对微分对策的发展前景做出了展望,指出了微分对策在军事应用领域中的研究热点、难点和主要发展方向。     

Systems defense games: Colonel blotto,command and control

The classical “Colonel Blotto” games of force allocation are generalized to include situations in which there are complementarities among the targets being defended. The complementarities are represented by means of a system “characteristic function,” and a valuation technique from the theory of cooperative games is seen to indicate the optimal allocations of defense and attack forces. Cost trade-offs between systems defense and alternative measures, such as the hardening of targets, are discussed, and a simple example is analyzed in order to indicate the potential of this approach.     

One-person games of economic survival

This paper discusses the one-person economic survival game model with a discrete probability distribution for the contribution to surplus variable. The general game model and strategies in these games are examined, and necessary conditions which an undominated, stationary strategy must satisfy are obtained. For a special class of these games a mathematical formulation of the value of the game is given, and examples and theorems which relate to undominated strategies in this class are presented. This paper, in some sense, is a sequel to a portion of a paper by Shubik and Thompson [7] which appeared in this journal.     

Note: A geometrical method of solving certain games

One of the diagrammatic methods for solving two-person 2 × n matrix games can be extended to solve m × n games where each column of the matrix is a concave function of the row number. This gives a simple proof of a theorem of Benjamin and Goldman that such games have solutions involving no more than two consecutive strategies for the row player, and no more than two strategies for the column player. Two extensions are discussed. © 1994 John Wiley & Sons, Inc.     

On nash subsets of bimatrix games

This work considers a class of bimatrix games to which some well-known structure theorems of 0-sum matrix games can be made to generalize. It is additionally shown how to construct such games and how to generate the equilibrium points defining a given game as a member of that class.     

网络游戏赌博性质分析

网络游戏产业快速发展的同时,赌博现象混迹其中。网络游戏赌博与传统赌博只是形式上的不同,其实质就是一种赌博行为。只有从教育、法制、监督、自律四方面综合治理,才能使这个行业健康发展。     

Political games

A modification of the Shapley value is suggested, which takes into account the fact that (due to personal affinities among the players) certain coalitions are more easily formed than others. This is done by assigning to each player a point in space, and looking at the distances between pairs of points. The method seems to be especially applicable to voting games among political parties (in, e. g., parliaments), and, for such games, gives a value which is considerably easier to compute than the usual Shapley value. Some examples are considered.     

Optimal strategies in a game of economic survival

The games of economic survival introduced by Shubik and Thompson seem tailor-made for the analysis of some problems in insurance and have found many applications in this industry. The optimal strategy in such games may be a so-called “band strategy.” This result seems counter-intuitive and has caused some puzzlement. This paper gives sufficient conditions so that the optimal strategy will be of a simpler form, and it is argued that these conditions are satisfied in most applications to insurance.     

Relaxed sequencing games have a nonempty core

We study sequencing situations with a fixed initial order and linear cost functions. Cost savings can be obtained by rearranging jobs. Next to finding an optimal order, an additional issue is formed by the division of these savings. Cooperative game theory studies this issue. A common assumption states that cooperation between players is restricted to groups that are connected according to the initial order. The value of disconnected groups is defined additively over their connected components. In this paper we allow players in disconnected coalitions to switch places as long as they do not hurt the players not in the coalition under consideration. The resulting games are called relaxed sequencing games. Although they have been studied before, no general results on stable profit divisions have been derived so far. In this paper we prove that relaxed sequencing games have a nonempty core, i.e., they all have stable profit divisions. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Multilinear extensions and the banzhaf value

It is shown that the Banzhaf value can be obtained by differentiating the multilinear extension of a game at the midpoint of the unit cube. This gives us a composition theorem for the value of compound games. As an example, the values of the electoral college and presidential election “games” are approximated by the method of extensions.     

Modified fictitious play

We describe a modification of Brown's fictitious play method for solving matrix (zero-sum two-person) games and apply it to both symmetric and general games. If the original game is not symmetric, the basic idea is to transform the given matrix game into an equivalent symmetric game (a game with a skew-symmetric matrix) and use the solution properties of symmetric games (the game value is zero and both players have the same optimal strategies). The fictitious play method is then applied to the enlarged skew-symmetric matrix with a modification that calls for the periodic restarting of the process. At restart, both players' strategies are made equal based on the following considerations: Select the maximizing or minimizing player's strategy that has a game value closest to zero. We show for both symmetric and general games, and for problems of varying sizes, that the modified fictitious play (MFP) procedure approximates the value of the game and optimal strategies in a greatly reduced number of iterations and in less computational time when compared to Brown's regular fictitious play (RFP) method. For example, for a randomly generated 50% dense skew-symmetric 100 × 100 matrix (symmetric game), with coefficients |a_ij| ≤ 100, it took RFP 2,652,227 iterations to reach a gap of 0.03118 between the lower and upper bounds for the game value in 70.71 s, whereas it took MFP 50,000 iterations to reach a gap of 0.03116 in 1.70 s. Improved results were also obtained for general games in which the MFP solves a much larger equivalent symmetric game. © 1996 John Wiley & Sons, Inc.     

灰色矩阵对策

本文结合灰色系统与对策论的知识,给出了以区间型灰数为元素的灰矩阵对策的定义,讨论了灰矩阵对策在纯策略与混合策略意义下的最优策略及灰解的存在性。     

空间pNA''与对策的NA一致连续性

pNA是无原子对策理论中极其重要的Banach空间,Kohlberg和Tauman已经刻划了pNA中某类对策的特征。本文从另一角度来研究这一困难的问题,目的是刻划比pNA更大,但又与PNA极为相似的Banach空间pNA′中的对策之特征。我们证得pNA′中的对策可用NA一致连续性来刻划。     

Game theoretic analysis of maximum cooperative purchasing situations

This article introduces maximum cooperative purchasing (MCP)‐situations, a new class of cooperative purchasing situations. Next, an explicit alternative mathematical characterization of the nucleolus of cooperative games is provided. The allocation of possible cost savings in MCP‐situations, in which the unit price depends on the largest order quantity within a group of players, is analyzed by defining corresponding cooperative MCP‐games. We show that a decreasing unit price is a sufficient condition for a nonempty core: there is a set of marginal vectors that belong to the core. The nucleolus of an MCP‐game can be derived in polynomial time from one of these marginal vectors. To show this result, we use the new mathematical characterization for the nucleolus for cooperative games. Using the decomposition of an MCP‐game into unanimity games, we find an explicit expression for the Shapley value. Finally, the behavior of the solution concepts is compared numerically. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 607–624, 2013     

On the number of games played in a best of (2n – 1) series

Let p(⩾0.5) denote the probability that team A beats B in a single game. The series continues until either A or B wins n games. Assuming that these games are independent replications, we study some features of the distribution of X_n, the number of games played in the series. It is shown that X_n is unimodal, has an IFRA distribution, and is stochastically decreasing in p. Close approximations to its mode, mean, and variance are given. Finally, it is shown that the maximum-likelihood estimator of p based on X_n is unique.