The profit of i-th agent is the difference between the compensation obtained from the manager and the production cost. We compare: The normative compensation scheme where the manager enforces the agents to follow an optimal cooperative strategy, the linear piece rates compensation scheme where the manager announces an optimal reward per unit good, the proportional compensation scheme where agent’s reward is proportional to his contribution to the total output. | Yugoslav Journal of Operations Research 28 (2018), Number 4, 501–520 DOI: ASYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME FOR A LARGE NUMBER OF PRODUCERS Dmitry B. ROKHLIN Institute of Mathematics, Mechanics and Computer Sciences of the Southern Federal University, Mil’chakova str., 8a, 344090, Rostov-on-Don, Russia rokhlin@ Anatoly USOV Institute of Mathematics, Mechanics and Computer Sciences of the Southern Federal University, Mil’chakova str., 8a, 344090, Rostov-on-Don, Russia usov@ Received: September 2017 / Accepted: October 2018 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 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∗ , sˆ and s respectively, where the last one is related to the unique Nash equilibrium, we examine the limits of the prices of anarchy AN = s∗ /s, AN0 = sˆ/s as N → ∞. 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 AN0 , and in terms of AN the loss does not exceed 31%. Keywords: Proportional Compensation Scheme, Total Production, Price of Anarchy, Asymptotic Efficiency, Tullock Contest. MSC: 91B32, 91B40, 91B38. 502 .
