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

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 $s^*$, $\hat s$ and $\overline s$ respectively, where the last one is related to the unique Nash equilibrium, we examine the limits of the prices of anarchy $\mathscr A_N=s^*/\overline s$, $\mathscr A_N'=\hat s/\overline s$ 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 $\mathscr A'_N$, and in terms of $\mathscr A_N$ the loss does not exceed $31\%$.