边际效用博弈
The Game of Marginal Utilities
AI总结：
该研究提出一种结合收益递减与拥挤效应的非合作资源分配博弈，证明其存在唯一纳什均衡并给出嵌套结构的均衡特征，还提出两种可求解该均衡的算法。
AI中文摘要：
我们研究了一个非合作资源分配博弈，其中m个参与者将固定资源分配给n个项目，参与者j的收益为$F^j(x) = \sum_{i=1}^n \frac{a_i x_i^j}{b_i+\sum_{\ell=1}^m x_i^\ell}$，其中$a_i$和$b_i$是项目参数，$x_i^j$是参与者j分配给项目i的资源量。该模型结合了收益递减与竞争产生的拥挤效应。我们证明该博弈存在唯一的纳什均衡，并通过等边际原理对其进行刻画。我们表明，在按$a_i/b_i$对项目排序后，每个参与者都会投资于项目的初始段，且这些段在参与者之间是嵌套的，因此均衡可分解为连续的活动区域；资源更多的参与者在每个项目上的投资弱更多。在全活跃 regime（即每个参与者都投资于所有项目）中，我们将均衡简化为关于总边际效用率的单个标量非线性方程；所有个体边际率、投资和收益均可通过显式公式推导得出。我们还提出了一种投影边际效用算法，该算法在显式步长条件下具有全局线性收敛性，以及一种利用结构的Block Pandora算法，该算法在给定嵌套截断结构的条件下重构并验证均衡。
英文摘要：
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.
