Waiter-Client and Client-Waiter colourability games on a $k$-uniform hypergraph and the $k$-SAT game

TL;DR

This paper analyzes the threshold biases in Waiter-Client and Client-Waiter games on hypergraphs and the k-SAT problem, providing estimates that align with probabilistic intuition for these combinatorial games.

Contribution

It provides the first estimates for threshold biases in these games on hypergraphs and k-SAT, revealing their alignment with probabilistic intuition.

Findings

Threshold bias for non-2-colourability game in Client-Waiter is approximately (1/n) * C(n,k) * 2^{-k}

Threshold bias for non-2-colourability game in Waiter-Client is approximately (1/n) * C(n,k) * 2^{Θ_k(k)}

Threshold bias for the k-SAT game is approximately (1/n) * C(n,k) within exponential and polynomial factors

Abstract

Waiter-Client and Client-Waiter games are two-player, perfect information games, with no chance moves, played on a finite set (board) with special subsets known as the winning sets. Each round of the biased $(1:q)$ game begins with Waiter offering $q+1$ previously unclaimed elements of the board to Client, who claims one. The $q$ elements remaining are then claimed by Waiter. If Client fully claims a winning set by the time all board elements have been offered, he wins in the Client-Waiter game and loses in the Waiter-Client game. We give an estimate for the threshold bias of the $(1:q)$ Waiter-Client and Client-Waiter versions of two different games: the non-2-colourability game, played on the complete $k$-uniform hypergraph, and the $k$-SAT game. In particular, we show that the unique value of $q$ at which the winner of the Client-Waiter version of the non-2-colourability game changes is $\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$ and, for the Waiter-Client version, the corresponding value of $q$ is $\frac{1}{n}\binom{n}{k}2^{\Theta_k(k)}$. Additionally, we show that the threshold bias for the Waiter-Client and Client-Waiter versions of the $k$-SAT game is $\frac{1}{n}\binom{n}{k}$ up to a factor that is exponential and polynomial in $k$ respectively. This shows that these games exhibit the "probabilistic intuition".