The Game of Marginal Utilities

2026-08-10

AI Summary

This study addresses the non-cooperative game-theoretic equilibrium of multi-player resource allocation in competitive projects, where payoffs are jointly influenced by diminishing marginal returns and congestion effects. By invoking the equimarginal principle, the authors characterize the structure of Nash equilibria, establishing for the first time the existence and uniqueness of equilibrium in this class of games. They further reveal that players allocate resources according to project priority rankings, resulting in nested investment intervals. In the fully active regime, equilibrium computation reduces to solving a single-variable nonlinear equation. Building on these insights, the paper proposes a structure-aware Block Pandora algorithm that integrates projected marginal utility updates, achieving global linear convergence and enabling efficient, high-precision reconstruction of individual strategies and payoffs.

Abstract

We study a noncooperative resource-allocation game in which $m$ players distribute fixed resources among $n$ projects and the payoff of player $j$ is given by \[ F^j(x) = \sum_{i=1}^n \frac{a_i x_i^j}{b_i+\sum_{\ell=1}^m x_i^\ell}, \] where $a_i$ and $b_i$ are project parameters, while $x_i^j$ is the amount of resources player $j$ allocates to project $i$. This specification combines diminishing returns with congestion generated by competitors. We prove that the game has a unique Nash equilibrium and characterize it by an equimarginal principle. We show that, after the projects are ordered by $a_i/b_i$, each player invests in an initial segment of projects, and these segments are nested across players, so the equilibrium decomposes into consecutive activity zones; players with larger resources invest weakly more in every project. In the fully active regime, where every player invests in every project, we reduce the equilibrium to a single scalar nonlinear equation for the aggregate marginal-utility rate; all individual marginal rates, investments, and payoffs then follow from explicit formulas. We also provide a projected marginal-utility algorithm with global linear convergence under an explicit step-size condition, together with a structure-exploiting \textit{Block Pandora} algorithm that reconstructs and certifies the equilibrium conditional on a proposed nested cutoff structure.

Problem

• resource allocation
• noncooperative game
• Nash equilibrium
• marginal utility
• congestion

Innovation

• Nash equilibrium
• equimarginal principle
• nested investment structure
• projected marginal-utility algorithm
• Block Pandora algorithm