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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 \(n \leq 30\) players. We further solve a real-world EBMC game derived from the Minnesota AIS dataset with \(n = 84\) county players.