The Role Of Simulation In Developing Game Playing Strategies

Abstract

This paper presents results concerning the use of simulation in the analysis of finite games and the development of strategies for their solution. Programs were developed in the Scheme programming language [Rees and Clinger 1986] to play large numbers of games pitting various strategies against each other. The games discussed here range from the well-known Tic-tac-toe, through versions of ConnectThree for various sized boards, to the commercial ConnectFour game. The ConnectThree game is based on ConnectFour, but was developed as a simplification amenable to play within reasonable time by perfect strategies that look ahead to the end of the game. The game is played on a vertical array with players alternately dropping disks of their color into the array. The first player to create a row, column or diagonal of three disks of their own color wins. The ConnectFour game is similar, but requires completion of a line of four disks of one's own color to win. Conducting tournaments involving optimal strategies enabled us to predict for which board sizes the ConnectThree game is a first-player win and for which it is a draw. Armed with the results from our simulation, we have been able to prove these predictions in almost all cases. Research is continuing on the ConnectFour game in an effort to determine if and when it is a forced draw or whether the first or second player has the advantage. For this game, it is not computationally feasible to play perfect strategies that look ahead right to the end of the game. In order to develop even good strategies, it is necessary to find good heuristics for evaluating non terminal board positions. At this stage of our investigations, the role of simulation is essentially to evaluate various such heuristics by testing them as predictors of the outcome of games.