Staff Solutions to Problem Set 4

Problem 1. We’re going to characterize a large category of games as a recursive data type and then prove, by structural induction, a fundamental theorem about game strategies. The games we’ll consider are known as deterministic games of perfect information. Checkers, chess, and GO, for example, all fit this description. It’s useful to regard each game situation as a game in its own right. For example, after five moves in a chess game, we think of the players as being at the start of a new “chess” game determined by the current board position. It’s also useful to define the “payoff” of a chess game to be 1 if White wins, 1 if Black wins, and 0 if the game ends in stalement (a draw). This lets us describe White’s objective as maximizing the payoff, and Black’s objective is minimizing it. This leads to an elegant abstraction of this kind of game. We suppose there are two players, called the max-player and the min-player, whose aim is, respectively, to maximize and minimize the final score. A game will specify its set of possible first moves, each of which will simply be another game. A game with no possible moves is called a finish that determines the payoff. An unfinished game will have a label max or min indicating which player is supposed to move next. Here’s the formal definition: