We propose a general class of symmetric games called position-optimization games. Given a probability distribution $Q$ over a set of targets $\mathcal{Y}$, the $n$ players each choose a position in a space $\mathcal{X}$. A player's utility is the $Q$-mass of targets they are closest to under some proximity measure, with ties broken evenly. Our model captures Hotelling games and forecasting competitions, among other applications. We show that for sufficiently large $n$, both pure and symmetric mixed Nash equilibria exist, and moreover are extreme: all players play on a finite set of pseudo-targets $\mathcal{X}^* \subseteq \mathcal{X}$. We further show that both pure and symmetric mixed equilibria converge to the distribution $P$ on $\mathcal{X}^*$ induced by $Q$, and bound the convergence rate in $n$. The generality of our model allows us to extend and strengthen previous work in Hotelling games, and prove entirely new results in forecasting competitions and other applications.

翻译：我们提出一类称为位置优化博弈的广义对称博弈模型。给定目标集合 $\mathcal{Y}$ 上的概率分布 $Q$，$n$ 位玩家各自在空间 $\mathcal{X}$ 中选择一个位置。每位玩家的效用是其在某种邻近度量下最接近的目标的 $Q$ 测度，当出现平局时均等分配。我们的模型涵盖了霍特林博弈与预测竞赛等多种应用场景。我们证明，当 $n$ 充分大时，纯策略与对称混合纳什均衡均存在，且具有极端性：所有玩家均在一个有限伪目标集 $\mathcal{X}^* \subseteq \mathcal{X}$ 上行动。我们进一步证明，纯策略均衡与对称混合均衡均收敛于由 $Q$ 在 $\mathcal{X}^* 上诱导的分布 $P$，并给出了关于 $n$ 的收敛速率界。本模型的普适性使我们能够扩展并强化霍特林博弈中的已有研究，并在预测竞赛及其他应用领域证明全新的结论。