The Game of Marginal Utilities

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.