Abstract
This paper proves an optimal strategy for Ebert's hat game with three players and more than two hat colors. In general, for $n$ players and $k$ hat colours, we construct a strategy that is asymptotically optimal as $k\rightarrow \infty$. Computer calculation for particular values of $n$ and $k$ suggests that, as long as $n$ is linear with $k$, the strategy is asymptotically optimal. We conclude by comparing our strategy with the strategy of Lenstra and Seroussi and with the bound of Alon, and suggest our strategy is better when $2k \geq n \geq 7$.
Highlights
There have been several different versions of hat games and various generalizations of the games’ rules
This paper proves an optimal strategy for Ebert’s hat game with three players and more than two hat colors
We conclude by comparing our strategy with the strategy of Lenstra and Seroussi and with the bound of Alon, and suggest our strategy is better when 2k n 7
Summary
There have been several different versions of hat games and various generalizations of the games’ rules. We will focus on Ebert’s hat game, and the generalization on the number of possible hat colors. In this paper we will use n as the number of players and k as the number of colors in Ebert’s hat game. We will show, with a combinatorial optimization proof, the optimal strategy for Ebert’s hat game with n = 3 and k > 2. It is easy to show that the strategy is optimal with the winning probability 2(k − 1)/k2.) There has been a construction of strategy for Ebert’s hat game with three players and k colors [3]. We discuss that Lenstra and Seroussi’s strategy and Alon’s bound are not preferable in the case k being linear with n. Computer calculation leads to a conjecture that in the case k being linear with n, and we let n → ∞, the probability of winning still approaches this same upper bound
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