Stability in Single-Peaked Strategic Resource Selection Games

TL;DR

This work introduces a Resource Selection Game with heterogeneous (red/blue) agents and single-peaked utilities centered at $\Lambda\in(0,1)$, and studies stability under impact-blind and impact-aware dynamics across cycles, binary trees, and arbitrary graphs. By deriving exact thresholds $L_\Delta(\mathbf{Q})$ and $U_\Delta(\mathbf{Q})$ and constructing targeted counterexamples, the authors establish tight conditions for the existence of equilibria on arbitrary graphs (with linear $p$) and provide complete stability results on cycles and binary trees. The analysis relies on monotonicity properties of the single-peaked utility, two notions of improvement moves, and a potential function framework to prove finite improvement properties in key cases. The results illuminate how heterogeneity and peak-directed preferences shape equilibria and demonstrate the limits of standard methods in broader parameter regimes, offering precise guidance for stability in such resource sharing environments.

Abstract

The strategic selection of resources by selfish agents has long been a key area of research, with Resource Selection Games and Congestion Games serving as prominent examples. In these traditional frameworks, agents choose from a set of resources, and their utility depends solely on the number of other agents utilizing the same respective resource, treating all agents as indistinguishable or anonymous. Only recently, the study of the Resource Selection Game with heterogeneous agents has begun, meaning agents have a type and the fraction of agents of their type at their resource is the basis of their decision-making. In this work, we initiate the study of the Resource Selection Game with heterogeneous agents in combination with single-peaked utility functions, as some research suggests that this may represent human decision-making in certain cases. We conduct a comprehensive analysis of the game's stability within this framework. We provide tight bounds that specify for which peak values equilibria exist across different dynamics on cycles and binary trees. On arbitrary graphs, in a setting where agents lack information about the selection of other agents, we provide tight bounds for the existence of equilibria, given that the utility function is linear on both sides of the peak. Agents possessing this information on arbitrary graphs creates the sole case where our bounds are not tight, instead, we narrow down the cases in which the game may admit equilibria and present how several conventional approaches fall short in proving stability.

Figures (5)

• Figure 1: Example of a linear and single-peaked function $p$.
• Figure 2: Red dots are red agents, blue dots are blue agents and black dots are resources. An agent $a$ has access to a resource $q$ if and only if $a$ and $q$ are adjacent in the above graph.
• Figure 3: Overview of transitions between resource states with $3$$0$, $0$$3$ and $0$$0$ omitted, as the first two states can never be entered and the last one cannot be left.
• Figure 4: Red dots are red agents, blue dots are blue agents and black dots are resources. An agent $a$ has access to a resource $q$ if and only if $a$ and $q$ are adjacent in the above graph.
• Figure 5: Red dots are red agents, blue dots are blue agents and black dots are resources. An agent $a$ has access to a resource $q$ if and only if $a$ and $q$ are adjacent in the above graph.

Theorems & Definitions (44)

• Definition 1: The function $p$
• Definition 2: Utility Function
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof
• proof