Computing Pure-Strategy Nash Equilibria in a Two-Party Policy Competition: Existence and Algorithmic Approaches

TL;DR

This paper models two-party policy competition as a game, proves the existence of pure-strategy Nash equilibria, and introduces algorithms for finding approximate equilibria efficiently.

Contribution

It generalizes previous models, proves equilibrium existence in multi-dimensional settings, and develops algorithms for computing approximate equilibria.

Findings

Pure-strategy Nash equilibria exist in the formulated game.

Gradient-based algorithms typically converge rapidly to approximate equilibria.

A grid-based search algorithm finds ε-approximate equilibria in polynomial time.

Abstract

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 $\epsilon$-approximate PSNE of the game in time polynomial in the input size and $1/\epsilon$.