Minimax Algorithm Analysis

Abstract

This paper discusses the minimax algorithm and its application as the solution to the famous Wythoff’s game. It is an impartial two player game in which both players play optimally in order to emerge as the winner. In the game, both players pick a certain number of stones from two piles having m and n stones. The player can either pick any number of stones from one pile, or the same number of stones from both. The player who picks up the last stone is declared the winner.

I. Introduction

The game is a variation of the Nim game, in which there a number of piles of counters (stones, sticks etc.) and they are to be removed one by one such that none are left at the end of the game. The player who cannot move any further at a point in the game loses. To solve this game, minimax algorithm has been used. It is an analytical way of approaching a two player game in order to maximize the payoff and minimize loss for a player in the worst case scenario.

For the game, following assumptions have been taken:

 The defining feature of the game is the current configuration represented by (m,n), where m = number of stones in first pile and n = number of stones in the

For this purpose, the recursive minimax algorithm is the obvious choice to solve this game and to determine how each player should decide its strategy at each turn in the game. This is also a zero-sum game which means that one player’s payoff results in the negative payoff of the other. In other words, if one player gets a certain reward by winning, the other player loses the reward by that very amount, hence the utility received from the game, if the player loses, is negative. Since in this game either of the players will win and the other will lose and the utility of the losing player is exactly equal to and negative of the utility of the winning player, it is a zero-sum