A construction for the hat problem on a directed graph

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.Comment: 9 pages. v2: updated title and abstract to match journal versio

A New Variation of Hat Guessing Games

Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of $n$ players is assigned a hat from a given color set. Simultaneously, each player tries to guess the color of his/her own hat by looking at colors of hats worn by other players. In this paper, we consider a new variation of this game, in which we require at least $k$ correct guesses and no wrong guess for the players to win the game, but they can choose to "pass". A strategy is called {\em perfect} if it can achieve the simple upper bound $\frac{n}{n+k}$ of the winning probability. We present sufficient and necessary condition on the parameters $n$ and $k$ for the existence of perfect strategy in the hat guessing games. In fact for any fixed parameter $k$, the existence of perfect strategy can be determined for every sufficiently large $n$. In our construction we introduce a new notion: $(d_1,d_2)$-regular partition of the boolean hypercube, which is worth to study in its own right. For example, it is related to the $k$-dominating set of the hypercube. It also might be interesting in coding theory. The existence of $(d_1,d_2)$-regular partition is explored in the paper and the existence of perfect $k$-dominating set follows as a corollary.Comment: 9 pages; The main theorem was improve

The three-colour hat guessing game on the cycle graphs

We study a cooperative game in which each member of a team of $N$ players, wearing coloured hats and situated at the vertices of a cycle graph $C_N$, is guessing their own hat colour merely on the basis of observing the hats worn by their two neighbours without exchanging the information. Each hat can have one of three colours. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colours. We prove that a winning strategy exists if and only if $N$ is divisible by $3$ or $N=4$. This problem represents an example of a relational system using incomplete information about an unpredictable situation, where at least one participant has to act properly.Comment: 13 pages, 5 figures. Presented to a seminar of the Dept. of Mathematics and Informatics, Jagiellonian University, on 8 April 201

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.