On Optimal Strategies for a Hat Game on Graphs

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 occupies 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. AMS Subject Classification: 05C35, 05C57, 05C69.