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Waiter-Client and Client-Waiter games are two-player, perfect informationgames, with no chance moves, played on a finite set (board) with specialsubsets known as the winning sets. Each round of the biased $(1:q)$ game beginswith Waiter offering $q+1$ previously unclaimed elements of the board toClient, 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 beenoffered, 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 andClient-Waiter versions of two different games: the non-2-colourability game,played on the complete $k$-uniform hypergraph, and the $k$-SAT game. Inparticular, we show that the unique value of $q$ at which the winner of theClient-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 andClient-Waiter versions of the $k$-SAT game is $\frac{1}{n}\binom{n}{k}$ up to afactor that is exponential and polynomial in $k$ respectively. This shows thatthese games exhibit the "probabilistic intuition".

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