Showing papers on "Combinatorial game theory published in 1977"

Game-Theoretic Approaches to Urban Planning and Design:

Abstract: In this paper the urban-planning process is explored and modelled using a variety of concepts and techniques drawn from the theory of games, The rationale for using game theory as a basis for simulating the design process is presented first, and this serves to highlight the major features of such processes in terms of bargaining and the implied power positions of the players involved. In the second section these ideas are given substance through a description of a case study based on the choice of location of a town to accept overspill population from a large conurbation, and a number of conceptual game-theoretic models of parts of this process are presented. By developing game theory nonalgebraically in terms of this case study, it is then possible to generate a set of formal models based on stochastic game thoery, as first suggested by Shapley (1953). These models are presented theoretically in the third section, and in terms of their algorithms and application in section four. These models include seve...

Game equipment and method having simultaneously played, balanced, multiple game theories

Abstract: Game equipment and a method are disclosed in which the equipment is formed for the simultaneous play of a game by at least two differing game theories. The game equipment includes probability determining means formed to substantially balance the probability of winning by either of the differing game theories. In a preferred form, a parlor type board game is disclosed in which business development or real estate trading is simulated with the game including simultaneous play by at least two players, with one playing under a monopolistic game theory and the other playing under a competitive game theory and each having an equal chance of winning.

Blindfold Games are Harder than Games with Perfect Information

Abstract: Recently several researchers have shown an interest in the complexity of determining the existence of winning strategies in various games. The purpose of this note is to show that this problem is (probably) much more difficult for games in which the players lack perfect information about the state of the game. Familiar examples of games of this type include Kriegspiel (blindfold chess) and Battleship. In particular, we show that a simple one token game on graphs requires polynomial space to analyze in its blindfold version, although polynomial time is known to be sufficient for the version with perfect information.

Two general properties of the saddle-point solutions of dynamic games

Abstract: In deterministic team problems every closed-loop representation of an optimal open-loop solution is also optimal. This property, however, no longer holds true when the optimization problem is a zero-sum or a nonzero-sum game. In zero-sum games, two weaker (but still general enough) versions of this statement are valid, which still fail to hold in the case of nonzero-sum games. In this correspondence we state and prove these two general properties of the saddle-point solution in dynamic games.

Differential games with no information

Abstract: This paper considers a broad class of differential games of prescribed duration in which the players obtain no information about the state variables. It is shown that such games always have a value when the players can choose their controls by means of a mixed strategy.It is further shown that a player can always implement a mixed strategy by drawing a random number from the uniform distribution on the unit interval and then using a control which is determined by the number so chosen.

Embedded game analysis and international conflict control

Abstract: The use of game theoretic models for understanding international relations has been widely criticized. In this paper we suggest that the theory of embedded differential games with optimal control is a productive methodology for responding to the critics. Several noncooperative differential game models with embedded objective functions are formulated. Concepts for obtaining undominated solutions based on the game structure, the quadratic objective functions and linear differential kinematic equations are discussed. The paper concludes that models with rich empirical reference can be formulated and productively used in understanding international conflict.

Games of Pure Skill and Competitive Computers

Abstract: This chapter discusses games of pure skill and competitive computers. Games of pure skill can be devoid of all probabilistic elements. Such games are not often associated with monetary wagers, although formal admonishments against so profiting from a skill are not proclaimed in most cultures. Take away type of games are exemplified by games such as Tic-Tac-Toe and Mill—sometimes referred to as Nine Men's Morris or NIM. Furthermore, numerous games have been proposed whose solutions become apparent once their Nim-like structures are revealed. Two such examples are Northcott's Nim and Stepine's game. In graphical representation of a countdown game, each vertex represents a state of the game, and as one conventionally defines the winner as that player who leaves the zero state for his opponent, the zero Grundy function is associated with a winning vertex. From all other vertices, there always exists a path to a vertex with a zero Grundy function and from a zero Grundy function vertex there are connections only to vertices with nonzero Grundy functions; Grundy functions are suggested by the Guy–Smith classification system.

First passage game

Abstract: Appealing to the theory of stochastic games, a two-person, zero-sum first passage game, which may be viewed as a generalization of the first passage decision problem, is developed. In the first passage game, the players have stationary optional strategies and the values are unique and these can be computed using an algorithm for terminating stochastic games. It is also shown that the solution of a recurrence game is closely related to that of the first passage game. Finally, it is shown that a finite step stochastic game with nonstationary transition probabilities and payoffs can be converted to a first passage game whose solution yields a solution of the original finite step game. The first passage game so obtained has stationary transition probabilities and payoffs. Because of its special structure, the solution method reduces to a dynamic programming recursion in the context of games.

An application of non-zero-sum games to competitive decision making†

Abstract: A fairly general class of decision-making problems has been investigated in which two or more players compete to satisfy the demand for a particular commodity in a manner that will minimize their own individual costs. A situation of imperfect competition without negotiation has been assumed, which leads to a series of nonzero-sum games that must be solved. A short discussion of some of the important properties of the minhnax and Nash solution strategies for non-zero-sum games has been included. Finally, the game solution strategy is incorporated into a dynamic equation to form a solution technique for decision-making problems involving competition.

Existence of a value in semidynamical games

Abstract: The games (generalizing differential games) in which the dynamics of players is described by k-semidynamical systems are called semidynamical games. For such games two theorems on the existence of a value in the class of piecewise program strategies are proved. Examples are given to show that the conditions of these theorems impose very weak restrictions on the set of admissible controls of the players and, in the games with a fixed duration, on the set of trajectories of semidynamical systems.