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

In many industries a supplier's total demand from the retailers she supplie s frequently exceeds her capacity. In these situations, the supplier must a llocate her capacity in some manner. We consider three allocation schemes: proportional, linear and uniform. With either proportional or linear alloca tion a retailer receives less than his order whenever capacity binds. Hence , each retailer has the incentive to order strategically; retailers order m ore 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 mus t form expectations on how much other retailers actually desire (which is u ncertain) and how much each will actually order, knowing that all retailers face the same problem. We present methods to find Nash equilibria in the c apacity allocation game with either proportional or linear allocation. We f ind that behavior in this game with either of those allocation rules can be quite unpredictable, primarily because there may not exist a Nash equilibr ium. In those situations any order above one's desired quantity can be just ified, no matter how large. Consequently, a retailer with a high need may b e allocated less than a retailer with a low need; clearly an ex post ineffi cient allocation. However, we demonstrate that with uniform allocation ther e 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.