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

Chuang-Chieh Lin, Chi-Jen Lu, Po-An Chen, Chih-Chieh Hung

TL;DR

The paper studies a two-party policy competition where each party selects a real-valued policy vector within a compact set and voters’ utilities are given by inner products with policy vectors. Winning probability is modeled as an affine isotonic function of total utility, enabling tractable PSNE analysis; the authors prove PSNE existence in both one- and multi-dimensional settings, provide a closed-form equilibrium in 1D, and show non-monotonicity can occur in general. They explore gradient-based learning dynamics, finding that monotonicity and cocoercivity do not hold and standard guarantees fail, yet decentralized gradient ascent often achieves approximate PSNE in practice. To guarantee computable equilibria, they introduce a polynomial-time grid-based ε-PSNE algorithm (GBA-PSNE) that discretizes the policy space and leverages Lipschitz bounds. Overall, the work extends spatial voting models to continuous multi-dimensional policies, offering both theoretical equilibrium guarantees and practical algorithms, while outlining directions for future work on alternative isotonic forms, multi-party settings, and information limitations.

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 $1/ε$.

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

TL;DR

The paper studies a two-party policy competition where each party selects a real-valued policy vector within a compact set and voters’ utilities are given by inner products with policy vectors. Winning probability is modeled as an affine isotonic function of total utility, enabling tractable PSNE analysis; the authors prove PSNE existence in both one- and multi-dimensional settings, provide a closed-form equilibrium in 1D, and show non-monotonicity can occur in general. They explore gradient-based learning dynamics, finding that monotonicity and cocoercivity do not hold and standard guarantees fail, yet decentralized gradient ascent often achieves approximate PSNE in practice. To guarantee computable equilibria, they introduce a polynomial-time grid-based ε-PSNE algorithm (GBA-PSNE) that discretizes the policy space and leverages Lipschitz bounds. Overall, the work extends spatial voting models to continuous multi-dimensional policies, offering both theoretical equilibrium guarantees and practical algorithms, while outlining directions for future work on alternative isotonic forms, multi-party settings, and information limitations.

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 -approximate PSNE of the game in time polynomial in the input size and .
Paper Structure (33 sections, 7 theorems, 9 equations, 4 figures, 2 algorithms)

This paper contains 33 sections, 7 theorems, 9 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction and Motivation
  2. Our Contributions
  3. Organization of this paper.
  4. Related work
  5. On Candidate Nominations.
  6. On spatial theory of voting.
  7. On the Hotelling-Downs Model.
  8. On Continuous Game Convergence toward PSNE
  9. Variational Inequality Formulation.
  10. Gradient and Extragradient Methods.
  11. Monotonicity vs. cocoercivity.
  12. The Model and Preliminaries
  13. Justification of the Isotonicity of the Winning Probability
  14. A dueling-bandit perspective.
  15. The Payoff Functions and Their Derivatives
  16. ...and 18 more sections

Key Result

Theorem 1

The two-party policy competition game in the one-dimensional setting has a PSNE in a closed form.

Figures (4)

  • Figure 1: Utility difference vs. probability that party $\bm{A}$ wins.
  • Figure 2: The reasonable range of a policy's angle.
  • Figure 3: Convergence analysis of decentralized projected gradient ascent. (a) The average number of iterations until convergence for 100 random starting policy profiles for each instance. (b) The frequency of the decentralized projected gradient ascent not converging to an approximate PSNE.
  • Figure 4: Multiple convergence of decentralized projected gradient ascent. Upper figure: under consensus-reachable condition; lower figure: under the non-consensus-reachable condition. Through starting from a random policy profile under both conditions, the decentralized projected gradient descent does not converge to a unique profile. Arrows represents $\bm{Q_A}$, $\bm{Q_B}$ and $\bm{Q}$ for the voters' preference vectors of $V_A$, $V_B$ and $V$ respectively. Scatter plots denote the policy profiles in the convergence.

Theorems & Definitions (10)

  • Definition 1: egoistic property
  • Theorem 1
  • Definition 2
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Theorem 3
  • Definition 3: Pseudo‐Gradient Rosen1965
  • Lemma 3
  • Theorem 4