Search Games on Hypergraphs

Abstract

The main motivation behind this thesis is a certain type of win-lose games that are played on hypergraphs and can be translated into the following puzzle. Suppose there are two persons, say Alice and Bob. There are n biscuits, where n is a positive integer, and Alice chooses s of them uniformly at random. Bob possesses h grams of poison, where h is greater than or equal to 1, and the lethal dose is 1 gram. How should Bob distribute the poison over the biscuits in order to maximize the probability of poisoning Alice? This problem is due to Ken Kikuta and William Ruckle who, driven by less devious motives, formulated it in terms of accumulation games between two players. They conjectured that the optimal distribution of poison uses dosages of 1/j grams in as many biscuits as possible, where j is a positive integer that depends on h,n,s. In Chapter 1 we introduce the poisoning problem and discuss its relation to known results from the literature. The conjecture of Kikuta and Ruckle is related to two other conjectures, one from extremal combinatorics and one from the theory of probability. The combinatorial flavor of the Kikuta-Ruckle conjecture is its relation to the matching conjecture of Paul Erdos and its fractional analogue. Its probabilistic flavor is its relation to a conjecture of Stephen Samuels on a tail probability problem. We also consider a poisoning problem on more general ground spaces. This leads to a geometric problem that generalizes the isodiametric one. In Chapter 2 we settle the Kikuta-Ruckle conjecture in case n=2s-1. This case corresponds to the, so called, Odd graph. We also settle the conjecture for a few more cases using elementary combinatorial and game-theoretic arguments. In Chapter 3 we consider the poisoning game on the cyclic graph. In this game the n biscuits are arranged cyclically and Alice chooses s consecutive of them uniformly at random. We find the value of this game along with the optimal strategies of both players. In addition, we give a characterization of the fractional covering number of uniform hypergraphs obtained from the cyclic graph. Chapter 4 deals with the analysis of a network coloring game. This is a game that is motivated by conflict resolution situations and is played on a graph. The vertices of the graph are thought of as players having a fixed set of available colors. The game is played in rounds and in each round all players simultaneously and individually choose a color with the perspective of ending up with a color that is different from the colors chosen by their neighbors. We analyze the network game by introducing a very simple search game. The optimal strategy of the searchers in this game involves tosses of fair colored coins and leads to the following combinatorial probability problem that is interesting on its own. Suppose that you can color n fair coins with n colors. It is not allowed to colors both sides of a coin with the same color, but all other combinations are allowed. Let X be the number of different colors after a toss of the coins. In what way should you color the coins such that you maximize the median of X? We solve this problem and consider its natural generalization to the case of biased coins. The later case builds on the study of Bernoulli random variables whose total number of successes is of fixed parity.