Asymptotic Efficiency of the Proportional Compensation Scheme for a Large Number of Producers

Dmitry B. Rokhlin, Anatoly Usov

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 normative compensation scheme where the manager enforces the agents to follow an optimal cooperative strategy; (ii) the linear piece rates compensation scheme where the manager announces an optimal reward per unit good; (iii) the proportional compensation scheme where agent’s reward is proportional to his contribution to the total output. Denoting the correspondent total production levels by $s^*, \^s$ and $\=s$ respectively, where the last one is related to the unique Nash equilibrium, we examine the limits of the prices of anarchy $A_N = s^* / \=s, A'_N = \^s / \=s$ as ${N\to\infty}$. These limits are calculated for the cases of identical convex costs with power asymptotic 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 $A'_N$, and in terms of $A_N$ the loss does not exceed $31\%$.