An equilibrium analysis of linear, proportional and uniform allocation of scarce capacity

Abstract

In many industries a supplier's total demand from the retailers she supplies frequently exceeds her capacity. In these situations, the supplier must allocate her capacity in some manner. We consider three allocation schemes: proportional, linear and uniform. With either proportional or linear allocation a retailer receives less than his order whenever capacity binds. Hence, each retailer has the incentive to order strategically; retailers order more than they desire in an attempt to ensure that their ultimate allocation is close to what they truly want. Of course, they will receive too much if capacity does not bind. In the capacity allocation game, each retailer must form expectations on how much other retailers actually desire (which is uncertain) and how much each will actually order, knowing that all retailers face the same problem. We present methods to find Nash equilibria in the capacity allocation game with either proportional or linear allocation. We find that behavior in this game with either of those allocation rules can be quite unpredictable, primarily because there may not exist a Nash equilibrium. In those situations any order above one's desired quantity can be justified, no matter how large. Consequently, a retailer with a high need may be allocated less than a retailer with a low need; clearly an ex post inefficient allocation. However, we demonstrate that with uniform allocation there always exists a unique Nash equilibrium. Further, in that equilibrium the retailers order their desired amounts, i.e., there is no order inflation. We compare supply chain profits across the three allocation schemes.