We explore the fairness of a redistricting game introduced by Mixon and Villar, which provides a two-party protocol for dividing a state into electoral districts, without the participation of an impartial independent authority. We analyze the game in an abstract setting that ignores the geographic distribution of voters and assumes that voter preferences are fixed and known. We first show that the minority player can always win at least $p-1$ districts, where $p$ is proportional to the percentage of minority voters, and that when the minority is large they can win more than $p$ districts. We also show that a "cracking" strategy by the majority party limits the number of districts the minority player can win as a function of the size of the minority.

翻译：我们探讨了Mixon和Villar提出的选区划分博弈的公平性，该博弈提供了一种无需公正独立机构参与、由两党协议划分州选举区的协议。我们在忽略选民地理分布并假设选民偏好固定且已知的抽象背景下分析该博弈。首先证明少数派玩家总能赢得至少$p-1$个选区（其中$p$与少数派选民比例成正比），且当少数派规模较大时可以赢得超过$p$个选区。同时表明，多数党采取的“分裂”策略会限制少数派玩家所能赢得的选区数量，该数量是少数派规模的函数。