On Optimal Strategies for a Hat Game on Graphs

Abstract

The following problem was introduced by Marcin Krzywkowski as a generalization of a problem of Todd Ebert. After initially coordinating a strategy, n players each occupy a different vertex of a graph. Either blue or red hats are placed randomly and independently on their heads. Each player sees the colors of the hats of players in neighboring vertices and no other hats (and hence, in particular, the player does not see the color of his own hat). Simultaneously, each player either tries to guess the color of his own hat or passes. The players win if at least one player guesses correctly and no player guesses wrong. The value of the game is the winning probability of the strategy that maximizes this probability. Previously, the value of such games was derived for certain families of graphs, including complete graphs of carefully chosen sizes, trees, and the 4-cycle. In this manuscript we conjecture that on every graph there is an optimal strategy in which all players who do not belong to the maximum clique always pass. We provide several results that support this conjecture, and determine among other things the value of the hat game for any bipartite graph and any planar graph that contains a triangle.

A New Variation of Hat Guessing Games

“…The following lemma characterizes the relationship between the maximum winner probability P n,k and the minimum k-dominating set of Q n . The same result was showed in [5] for k = 1.…”

“…Moreover, the following proposition shows that the simple necessary condition can't be sufficient. The first counterexample is (5,3), it is not perfect while it satisfies the simple necessary condition. But (13, 3) is perfect by Theorem 1 and more generally for all s ≥ 4, (2 s − 3, 3) is perfect.…”

“…Several different hat guessing games have been studied in recent years [1,2,3,4,5,6,7]. In this paper we investigate a variation where players can either give a guess or pass.…”

A Construction for the Hat Problem on a Directed Graph

“…For an undirected graph G, it is known that if G contains a triangle, then h (G) ≥ 3/4, and it is conjectured in [9] that if G is triangle-free, then h (G) = 1/2. Do directed graphs introduce anything in between?…”

“…The hat problem was solved for trees [11], cycles [9,12,13], bipartite graphs [9], perfect graphs [9], and planar graphs containing a triangle [9]. Feige [9] conjectured that for any graph the hat number is equal to the hat number of its maximum clique. He proved this for graphs with clique number 2 k − 1.…”