A construction for the hat problem on a directed graph

TL;DR

This paper investigates the hat guessing game on directed graphs, disproving a conjecture relating the hat number to maximum clique size and providing a detailed characterization of possible hat numbers for various graph classes.

Contribution

It demonstrates that the conjecture does not hold for directed graphs and characterizes the range of hat numbers for each clique size, introducing constructions with asymptotically optimal values.

Findings

Disproved Feige's conjecture for directed graphs.

Characterized the range of hat numbers for each maximum clique size.

Determined the hat number of tournaments as 0.5.

Abstract

A team of players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his or her own hat color by looking at the hat colors of other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom she or he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph. Previous works focused on the problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique. We show that the conjecture does not hold for directed graphs.Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.