Foundations of positional games

Abstract

This is the third piece of a series of papers about a new, quasiprobabilistic theory of positional games (i.e., “combinatorial games”). The strange thing is that we wrote these papers in reverse order. Chronologically the first paper, “Deterministic Graph Games and a Probabilistic Intuition,” was a highly technical one, and can be considered as part III of the series. The next paper, “Achievement Games and the Probabilistic Method,” was a survey that attempted to explain the role of the subject in discrete mathematics. We felt, however, that we did not quite succeed, and somehow the foundations were not very solid. This is why we had to write this paper, which should be considered as part I of the series. Our main object here is to explain what the basic questions of positional game theory are. The well-known algebraic theory of Nim-like games (called “combinatorial game theory”) and the quasiprobabilistic theory represent two entirely different viewpoints, and they in some sense complement each other. Indeed, combinatorial game theory (i.e., Nim-like games) is an exact local theory in the sense how seemingly complicated games start out as composites, or quickly develop into composites of several simple local games. On the other hand, the quasi-probabilistic theory attempts to solve “hopelessly complicated” Tic-Tac-Toe-like games which usually remain as single coherent entities throughout play. It is an efficient global approach which, roughly speaking, evaluates via loss probabilities. Because of the intractable complexity of the exhausting search through the game-tree, an efficient evaluation method has to approximate. So one cannot really expect from the quasi-probabilistic theory to solve evenly balanced “head-to-head games,” where a single mistake could be fatal, but it can effectively recognize and solve large classes of difficult “one-sided games.” Positional games are finite 2-player games of skill (i.e., no chance moves) with perfect information, and the payoff function has three values 1, 0, −1 only (“win,” “draw,” “loss”). These games, therefore, are deterministic, and because of the perfect information, the optimal strategies are deterministic. How can randomness then enter the story? To answer this, we very briefly summarize the simplest case of the quasi-probabilistic theory: the majority principle. The majority principle is in two parts. The first part is a probabilistic intuition that says in a nutshell that, in many complicated games, the outcome between two perfect players is the same as the “majority outcome” between two “random players” (random game). The point is that even relatively simple games are too hopelessly complicated to analyze in full depth, but to describe the “typical” behavior is usually a tractable problem to solve by using probability theory. However, the majority principle is more than merely predicting the outcomes of complicated games. The second part is to convert the probabilistic intuition, via potential techniques, into effective deterministic strategies, in fact, greedy algorithms. © 1996 John Wiley & Sons, Inc.