General three person two color Hat Game

Three distinguishable players are randomly fitted with a white or black hat, where the probabilities of getting a white or black hat may be different for each player, but known to all the players. All players guess simultaneously the color of their own hat observing only the hat colors of the other two players. It is also allowed for each player to pass: no color is guessed. The team wins if at least one player guesses his hat color correctly and none of the players has an incorrect guess. No communication of any sort is allowed, except for an initial strategy session before the game begins. Our goal is to maximize the probability of winning the game and to describe winning strategies, using the concept of an adequate set.Comment: 7 pages. v1 is about three and four players and is incorrect; v2 is a modified version only about three players. arXiv admin note: substantial text overlap with arXiv:1612.00276, arXiv:1612.05924; v3: modifications in 2.2 and 2.3; v4: modifications in 2.2 and 2.3, new section 2.

On 1-factorizations of Bipartite Kneser Graphs

It is a challenging open problem to construct an explicit 1-factorization of the bipartite Kneser graph $H(v,t)$, which contains as vertices all $t$-element and $(v-t)$-element subsets of $[v]:=\{1,\ldots,v\}$ and an edge between any two vertices when one is a subset of the other. In this paper, we propose a new framework for designing such 1-factorizations, by which we solve a nontrivial case where $t=2$ and $v$ is an odd prime power. We also revisit two classic constructions for the case $v=2t+1$ --- the \emph{lexical factorization} and \emph{modular factorization}. We provide their simplified definitions and study their inner structures. As a result, an optimal algorithm is designed for computing the lexical factorizations. (An analogous algorithm for the modular factorization is trivial.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a odd prime powe

Generalized four person hat game

This paper studies Ebert's hat problem with four players and two colors, where the probabilities of the colors may be different for each player. Our goal is to maximize the probability of winning the game and to describe winning strategies We use the new concept of an adequate set. The construction of adequate sets is independent of underlying probabilities and we can use this fact in the analysis of our general case.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:2211.09819, arXiv:1612.00276, arXiv:1704.0424

Ebert's asymmetric Hat Game

The Hat Game (Ebert's Hat Problem) got much attention in the beginning of this century; not in the last place by its connections to coding theory and computer science. All players guess simultaneously the color of their own head observing only the hat colors of the other players. It is also allowed for each player to pass: no color is guessed. The team wins if at least one player guesses his or her own hat color correct and none of the players has an incorrect guess. This paper studies Ebert's hat problem, where the probabilities of the colors may be different (asymmetric case). Our goal is to maximize the probability of winning the game and to describe winning strategies. In this paper we introduce the notion of an adequate set. The construction of adequate sets is independent of underlying probabilities and we use this fact in the analysis of the asymmetric case. Another point of interest is the fact that computational complexity using adequate sets is much less than using standard methods.Comment: 33 page

Derandomization of auctions

We study the role of randomization in seller optimal (i.e., profit maximization) auctions. Bayesian optimal auctions (e.g., Myerson, 1981) assume that the valuations of the agents are random draws from a distribution and prior-free optimal auctions either are randomized (e.g., Goldberg et al., 2006) or assume the valuations are randomized (e.g., Segal, 2003). Is randomization fundamental to profit maximization in auctions? Our main result is a general approach to derandomize single-item multi-unit unit-demand auctions while approximately preserving their performance (i.e., revenue). Our general technique is constructive but not computationally tractable. We complement the general result with the explicit and computationally-simple derandomization of a particular auction. Our results are obtained through analogy to hat puzzles that are interesting in their own right