We formulate two-party policy competition as a two-player non-cooperative game, generalizing Lin et al.'s work (2021). Each party selects a real-valued policy vector as its strategy from a compact subset of Euclidean space, and a voter's utility for a policy is given by the inner product with their preference vector. To capture the uncertainty in the competition, we assume that a policy's winning probability increases monotonically with its total utility across all voters, and we formalize this via an affine isotonic function. A player's payoff is defined as the expected utility received by its supporters. In this work, we first test and validate the isotonicity hypothesis through voting simulations. Next, we prove the existence of a pure-strategy Nash equilibrium (PSNE) in both one- and multi-dimensional settings. Although we construct a counterexample demonstrating the game's non-monotonicity, our experiments show that a decentralized gradient-based algorithm typically converges rapidly to an approximate PSNE. Finally, we present a grid-based search algorithm that finds an $ε$-approximate PSNE of the game in time polynomial in the input size and $1/ε$.

翻译：我们将两党政策竞争建模为双人非合作博弈，推广了Lin等人（2021）的研究。每个政党从欧几里得空间的紧致子集中选择一个实值政策向量作为其策略，选民对政策的效用由该向量与其偏好向量的内积给出。为刻画竞争中的不确定性，我们假设政策的获胜概率随其在所有选民中的总效用单调递增，并通过仿射保序函数对此进行形式化定义。参与者的收益定义为其支持者获得的期望效用。本文首先通过投票模拟检验并验证了保序性假设。随后，我们证明了一维与多维情形下纯策略纳什均衡的存在性。尽管我们构造了反例证明该博弈的非单调性，但实验表明基于梯度的分散式算法通常能快速收敛至近似纯策略纳什均衡。最后，我们提出一种基于网格的搜索算法，该算法可在输入规模与$1/ε$的多项式时间内找到博弈的$ε$近似纯策略纳什均衡。