Asymptotic efficiency of the proportional compensation scheme for a large number of producers

Abstract

We consider a manager, who allocates some fixed total payment amount between $N$ rational agents in order to maximize the aggregate production. The profit of $i$-th agent is the difference between the compensation (reward) obtained from the manager and the production cost. We compare (i) the \emph{normative} compensation scheme, where the manager enforces the agents to follow an optimal cooperative strategy; (ii) the \emph{linear piece rates} compensation scheme, where the manager announces an optimal reward per unit good; (iii) the \emph{proportional} compensation scheme, where agent's reward is proportional to his contribution to the total output. Denoting the correspondent total production levels by $s^{\ast }$, $\hat{s}$ and $\overline{s}$ respectively, where the last one is related to the unique Nash equilibrium, we examine the limits of the prices of anarchy ${\mathcal{A}}_N=s^{\ast }/\overline{s}$, ${\mathcal{A}}_N^{\prime }=\hat{s}/\overline{s}$ as $N\to \infty$. These limits are calculated for the cases of identical convex costs with power asymptotics at the origin, and for power costs, corresponding to the Coob-Douglas and generalized CES production functions with decreasing returns to scale. Our results show that asymptotically no performance is lost in terms of ${\mathcal{A}}_N^{\prime }$, and in terms of ${\mathcal{A}}_N$ the loss does not exceed $31\%$.

arXiv comment17 pages