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

2025-12-27

AI Summary

This paper investigates the existence and efficient computation of pure-strategy Nash equilibria (PSNE) in two-party policy competition. We formulate a non-cooperative game where parties select real-valued policy vectors, voters derive utility as the inner product between their preferences and the implemented policy, winning probabilities are determined via an affine monotonic transformation of total voter utility, and party payoffs equal the expected utility of their supporters. We establish, for the first time, rigorous existence proofs of PSNE in both one-dimensional and higher-dimensional policy spaces. We propose a polynomial-time grid-search algorithm that computes an ε-approximate PSNE in time poly(n, 1/ε). Furthermore, we provide theoretical convergence analysis and empirical validation demonstrating that distributed gradient methods rapidly converge in practice. Our results deliver foundational guarantees—existence, computability, and efficient solvability—for modeling policy competition as a strategic game.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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

Problem

• Proves existence of pure-strategy Nash equilibria in two-party policy competition.
• Develops algorithms to compute approximate Nash equilibria efficiently.
• Validates the isotonicity hypothesis of policy winning probabilities through simulations.

Innovation

• Validates isotonicity hypothesis through voting simulations
• Proves pure-strategy Nash equilibrium existence in multi-dimensional settings
• Introduces polynomial-time grid-based search for approximate equilibrium