József Beck

Remarks on positional games. I

1. Introduction and results We start with some terminology. A hypergraph is a collection of sets. The sets in the hypergraph are called edges, and the elements of these edges are called vertices. We deal with finite hypergraphs only. IA [ denotes the number of elements of the set A. Let p and q be positive integers and ~ be a hypergraph. A (p, q, ,r is a game in which two players select previously unselected vertices of ~. The first player selects p vertices per move and the second player selects q vertices per move. The first player wins whenever he selects all the vertices of some eddge of aye, other wise the second player wins. In the case p=q=l, ERD6S and SELFRIDGE [3] have found a sufficient condition for the second players win: If 2-lal < 1/2, AE~

Random Graphs and Positional Games on the Complete Graph

There seems to be a mysterious and exciting analogy between the evolution of random graphs and biased positional games on the complete graphs. The purpose of this paper is to point out two known instances of this analogy, and then to prove a theorem providing a third instance. The theorem concerns a game that involves hamiltonian graphs (see Theorem 3.B below). This paper can be considered as Part II of [2].

Van der waerden and ramsey type games

Let us consider the following 2-player game, calledvan der Waerden game. The players alternately pick previously unpicked integers of the interval {1, 2, ...,N}. The first player wins if he has selected all members of ann-term arithmetic progression. LetW*(n) be the least integerN so that the first player has a winning strategy.By theRamsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the completek-uniform hypergraph ofN verticesKNk, and the first player wins if he has selected allk-tuples of ann-set. LetRk*(n) be the least integerN so that the first player has a winning strategy.We prove (W* (n))1/n → 2,R2*(n)<(2+ε)n andRk*n<2nk/k! fork ≧3.

On positional games

Let {Ai} be a family of sets and let S = ∩iAi. By a positional game we shall mean a game played by two players on {Ai}. The players alternately pick elements of S and that player wins who fist has all the elements of one of the Ai. This paper deals with almost disjoint hypergraphs only, i.e., |Ai∪Aj| ⩽ 1 if i ≠ j. Let M∗(n) be the smallest integer for which there is an almost disjoint n-uniform hypergraph |T| = M∗(n), so that the first player has a winning strategy. It is shown that limn [M∗(n)]1n = 4, which was conjectured by Erdos. The same method is applied to prove a conjecture of Hales and Jewett on r-dimensional tick-tack-toe if r is large enough. Finally we prove that for an arbitrary almost disjoint n-uniform hypergraph the second player has such a strategy that the first player unable to win in his mth move if m < (2 − ϵ)n.

Foundations of positional games

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.

Inevitable randomness in discrete mathematics

Mathematics has been called the science of order. The subject is remarkably good for generalizing specific cases to create abstract theories. However, mathematics has little to say when faced with highly complex systems, where disorder reigns. This disorder can be found in pure mathematical arenas, such as the distribution of primes, the 3n 1 conjecture, and class field theory. The purpose of this book is to provide examples - and rigorous proofs - of the complexity law: discrete systems are either simple or they exhibit advanced pseudorandomness; a priori probabilities often exist even when there is no intrinsic symmetry. Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics. This book considers randomness and complexity. The traditional approach to complexity - computational complexity theory - is to study very general complexity classes, such as P, NP and PSPACE. What Beck does is very different: he studies interesting concrete systems, which can give new insights into the mystery of complexity. The book is divided into three parts. Part A is mostly an essay on the big picture. Part B is partly new results and partly a survey of real game theory. Part C contains new results about graph games, supporting the main conjecture. To make it accessible to a wide audience, the book is mostly self-contained.

Games, Randomness and Algorithms

The object of this 50 % survey and 50 % “theorem-proof” paper is to demonstrate recent developments of some of the ideas initiated by Erdős [17, 18], Erdős and Selfridge [201], Erdős and Lovasz [19] and Erdős and Chvatal [15].