A complete solution to the biased Alon-Krivelevich-Spencer-Szabó criterion problem for the discrepancy game

Let $H=(V,\mathcal{E})$ be a finite hypergraph. For positive integers $p$ and $q$, the $(p:q)$-biased discrepancy game on $H$ is played in complete rounds. In each round, Balancer first claims $p$ previously unclaimed vertices, and then Unbalancer claims $q$ previously unclaimed vertices. Let $B$ and $U$ be the final sets of vertices claimed by Balancer and Unbalancer, respectively. For an edge $e\in \mathcal{E}$, define $D_e=q\mathrm{\mid }B\cap e\mathrm{\mid }-p\mathrm{\mid }U\cap e\mathrm{\mid }=(p+q)\mathrm{\mid }B\cap e\mathrm{\mid }-p\mathrm{\mid }e\mathrm{\mid }$. Thus $D_e$ measures the deviation of Balancer’s share of $e$ from the density $p\mathrm{/}(p+q)$. In 2005, Alon, Krivelevich, Spencer and Szabó proved a Chernoff-type potential criterion for the unbiased alternating discrepancy game, corresponding to the case $p=q=1$, and asked for a biased analogue for general $p,q$. In this paper, we prove a complete biased analogue in the complete-round formulation. More precisely, for every finite hypergraph $H=(V,\mathcal{E})$ and every fixed bias $(p:q)$, we give an explicit exponential condition under which Balancer has a strategy forcing $-L_e^{-}\le D_e\le L_e^{+}$ for every $e\in \mathcal{E}$, where $L_e^{+}$ and $L_e^{-}$ are prescribed edge-dependent target values.