Algorithms for Finding the Best Pure Nash Equilibrium in Edge-weighted Budgeted Maximum Coverage Games

TL;DR

The paper introduces the Edge-weighted Budgeted Maximum Coverage ($EBMC$) game, a large-scale non-cooperative integer programming framework for AIS prevention, and the Best Response Plus ($BR\text{-}plus$) heuristic for finding the best Pure Nash Equilibrium ($PNE$). It connects BR dynamics with IPG theory through a $BRS$-free subroutine and a $PNE$-bounded model to iteratively improve equilibria, demonstrating superior performance to the Zero-Regret ($ZR$) approach on both synthetic and real Minnesota data. The authors prove existence of $PNE$ in locally altruistic EBMC games via exact potential game structure and analyze existence (and nonexistence) in selfish variants, supported by extensive computational results across varying numbers of players ($N$) and AIS scenarios ($k=1$ and $k\geq 2$). The work offers a scalable, practical equilibrium-search framework for distributed AIS prevention planning, with implications for stakeholding agencies and network-inspired resource allocation, and suggests future directions combining BR-based methods with $ZR$ and exploring best mixed equilibria.

Abstract

This paper introduces a new integer programming game (IPG) named the Edge-weighted Budgeted Maximum Coverage (EBMC) game and proposes a new algorithm, the Best Response Plus (BR-plus) algorithm, for finding the best Pure Nash Equilibrium (PNE). We demonstrate this methodology by optimizing county-level decisions to prevent aquatic invasive species (AIS) in Minnesota lakes, where each county-level decision makers has self-serving objectives while AIS is an interconnected issue that crosses county borders. Specifically, we develop EBMC games to model the strategic interactions among county-level decision-makers with two variations in utility functions. We also study and prove the existence of a PNE in these models under specified conditions. We advance the current state-of-the-art, which is limited to only a few players, by presenting the BR-plus algorithm that can handle a large set of players via utilizing the best response dynamics for finding PNE in normal-form games. Experimental results show that our BR-plus algorithm offers computational advantages over the ZR algorithm, especially in larger games, on both random and real-world networks.

Figures (4)

• Figure 1: An example featuring three counties (A, B, and C) and a single AIS. The distinctions among edges are crucial from County A's perspective, highlighting which edges are considered in various optimization problems that a county might undertake. The network for a single AIS scenario can be viewed as a bipartite graph due to the distinct categorization of infested and uninfested lakes.
• Figure 2: The interactions in $\mathcal{G}^{\text{Self}}_{k=2}$ for two counties A and B. The bold circles and the dashed lines represent the selected lakes and the edges covered by the choices of counties, respectively. For example: in Fig. (\ref{['fig:subgraph_selfish_k=2_0']}), AIS-carrying boats from $B_1$ to $A_1$ are inspected at lake $A_1$, while those from $B_1$ to $B_2$ are not inspected.
• Figure 3: No PNE in single AIS case
• Figure 4: No PNE in two AIS cases

Theorems & Definitions (13)

• Definition 1: Induced Arc Sets
• Lemma 1
• Corollary 1
• proof
• Theorem 1
• proof
• Corollary 2
• proof
• Theorem 2
• Lemma 1