New Constructions and Bounds for Winkler's Hat Game

Gadouleau, Maximilien; Georgiou, Nicholas

Abstract

Hat problems have recently become a popular topic in combinatorics and discrete mathematics. These have been shown to be strongly related to coding theory, network coding, and auctions. We consider the following version of the hat game, introduced by Winkler and studied by Butler et al. A team is composed of several players; each player is assigned a hat of a given color; they do not see their own color but can see some other hats, according to a directed graph. The team wins if they have a strategy such that, for any possible assignment of colors to their hats, at least one player guesses their own hat color correctly. In this paper, we discover some new classes of graphs which allow a winning strategy, thus answering some of the open questions of Butler et al. We also derive upper bounds on the maximal number of possible hat colors that allow for a winning strategy for a given graph.

Citation

Gadouleau, M., & Georgiou, N. (2015). New Constructions and Bounds for Winkler's Hat Game. SIAM Journal on Discrete Mathematics, 29(2), 823-834. https://doi.org/10.1137/130944680

• Journal Article Type: Article
• Acceptance Date: Jan 26, 2015
• Online Publication Date: Apr 21, 2015
• Publication Date: Apr 1, 2015
• Deposit Date: Apr 28, 2015
• Publicly Available Date: Apr 29, 2015
• Journal: SIAM Journal on Discrete Mathematics
• Print ISSN: 0895-4801
• Electronic ISSN: 1095-7146
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Issue: 2
• Pages: 823-834
• DOI: https://doi.org/10.1137/130944680
• Public URL: https://durham-repository.worktribe.com/output/1406611