Random-Restart Best-Response Dynamics for Large-Scale Integer Programming Games and Their Applications

This paper presents scalable algorithms for computing pure Nash equilibria (PNEs) in large-scale integer programming games (IPGs), where existing exact methods typically handle only small numbers of players. Motivated by a county-level aquatic invasive species (AIS) prevention problem with 84 decision makers, we develop and analyze random-restart best-response dynamics (RR-BRD), a randomized search framework for PNEs. For IPGs with finite action sets, we model RR-BRD as a Markov chain on the best-response state graph and show that, whenever a PNE exists and the restart law has positive probability of reaching a PNE within the round cap, RR-BRD finds a PNE almost surely. We also propose a Monte Carlo sampling-and-simulation procedure to estimate success behavior under a fixed round cap, which informs our instance-dependent performance characterization. We then embed RR-BRD as a randomized local-search subroutine within the zero-regret (ZR) framework, yielding BRD-incorporated zero-regret (BZR). Using solver callbacks, RR-BRD searches for and supplies PNEs, while ZR separates and adds equilibrium inequalities to tighten the formulation. We introduce edge-weighted budgeted maximum coverage (EBMC) games to model AIS prevention and establish PNE existence results for both selfish and locally altruistic utilities. Computational experiments on synthetic EBMC and knapsack problem game instances show that RR-BRD and BZR scale equilibrium computation up to $n \le 30$ players. We further solve a real-world EBMC game derived from the Minnesota AIS dataset with $n = 84$ county players.

💡 Research Summary

This paper tackles the scalability bottleneck in computing pure Nash equilibria (PNE) for integer programming games (IPGs), where each player solves an individual integer program and utilities are interdependent. Existing exact methods, such as branch‑and‑prune or zero‑regret (ZR), are limited to very small numbers of players (typically n ≤ 3) or a tiny total number of decision variables. Motivated by a real‑world application—county‑level aquatic invasive species (AIS) prevention in Minnesota involving 84 counties—the authors develop a randomized algorithm called Random‑Restart Best‑Response Dynamics (RR‑BRD) and embed it within the ZR framework, yielding a hybrid method they call BZR.

Algorithmic contribution.
RR‑BRD augments classic best‑response dynamics (BRD) with two sources of randomness: (1) random initialization of the joint strategy profile, and (2) a fixed “round cap” after which the current trajectory is abandoned and a new random start is launched. The process is modeled as a finite Markov chain on the best‑response state graph, where vertices are pure strategy profiles and directed edges correspond to a single player’s best‑response update. The authors prove two key results: (i) if a PNE exists and the restart distribution assigns positive probability to the basin of attraction of at least one PNE, then RR‑BRD converges to a PNE with probability one; (ii) the expected number of best‑response solves before success is bounded by a function of the shortest distance from the initial state to a PNE and the minimal restart probability. Importantly, these guarantees hold without assuming potential‑game structure or BR‑weakly‑acyclicity.

Monte‑Carlo performance estimator.
Because the round cap is fixed in practice, the probability that a single attempt succeeds cannot be derived analytically for arbitrary instances. The authors therefore propose a Monte‑Carlo sampling‑and‑simulation procedure: repeatedly draw random initial profiles, run RR‑BRD up to the cap, and record success. The empirical success rate informs an instance‑specific estimate of expected attempts and total runtime, and also highlights worst‑case versus best‑case graph topologies.

Integration with Zero‑Regret (BZR).
Zero‑Regret is a cutting‑plane method that iteratively adds equilibrium inequalities to a master mixed‑integer program (MIP). BZR uses solver callbacks to feed integer‑feasible incumbents from the MIP as random starting points for RR‑BRD. Whenever RR‑BRD discovers a new best‑response trajectory, the corresponding equilibrium inequality is generated and added to the master problem, tightening the feasible region. This synergy dramatically improves scalability: while vanilla ZR solves only a handful of instances with n ≤ 3, BZR solves many with n up to 30 and even the 84‑player real instance.

Edge‑Weighted Budgeted Maximum Coverage (EBMC) games.
To demonstrate applicability, the paper introduces EBMC games, a new class of IPGs. Players control subsets of vertices in a weighted graph, each subject to a budget constraint, and aim to maximize the total weight of edges covered by their selected vertices. Two utility specifications are studied: (a) selfish utilities, where a player counts only the edges it covers; (b) locally altruistic utilities, where a player also receives a fraction of the edge weight covered by neighboring players. The authors prove that locally altruistic EBMC games are potential games and thus always admit a PNE; for selfish EBMC games they provide sufficient conditions (e.g., tree graphs or large budgets) guaranteeing existence.

Computational experiments.
Synthetic experiments cover three families of games: two EBMC variants and knapsack‑problem games (KPGs), each with 10, 20, and 30 players, yielding 135 test cases. RR‑BRD finds a PNE in 128 cases; BZR finds at least one PNE in all 128 cases and discovers a total of 179 EBMC PNEs and 3,160 KPG PNEs, far surpassing the 33 successes of plain ZR. The real‑world Minnesota AIS dataset (84 counties) is modeled as an EBMC game. For the selfish version, RR‑BRD identifies a socially optimal PNE that would be invisible to standard MIP formulations. For the locally altruistic version, the algorithm finds a PNE that coincides with the welfare‑maximizing solution, and the authors show that altruistic equilibria yield higher individual utilities for most counties.

Implications and future work.
The paper establishes that a simple random‑restart scheme, when analyzed through the lens of Markov chains on best‑response graphs, provides strong probabilistic convergence guarantees without restrictive game‑class assumptions. Embedding this scheme into a cutting‑plane framework yields a practical tool (BZR) capable of handling IPGs with dozens to hundreds of players—orders of magnitude beyond prior exact methods. Future directions include extending the approach to mixed integer‑continuous games, dynamic or stochastic settings, and parallel/distributed implementations of RR‑BRD for even larger networks.