An Equilibrium Analysis of Linear, Proportional and Uniform Allocation of Scarce Capacity, IIE Transactions
In many industries, a supplier's total demand from the retailers supplied frequently exceeds the supplier's capacity. In these situations, the supplier must allocate capacity in some manner. Three allocation schemes are considered: proportional, linear and uniform. With either proportional or linear allocation a retailer receives less than his order whenever capacity binds. Retailers then order more than they desire in an attempt to ensure that their ultimate allocation is close to what they truly want. If capacity does not bind, they will receive too much. In the capacity allocation game, each retailer must form expectations on how much other retailers actually desire and how much each will actually order, knowing that all retailers face the same problem. Methods to find Nash equilibria in the capacity allocation game with either proportional or linear allocation are presented. It is found 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. It is demonstrated that with uniform allocation there always exists a unique Nash equilibrium. In that equilibrium the retailers order their desired amounts. Supply chain profits across the 3 allocation schemes are compared.
Martin Lariviere, G Cachon
Lariviere, Martin, and G Cachon. 1999. An Equilibrium Analysis of Linear, Proportional and Uniform Allocation of Scarce Capacity. IIE Transactions. 31(9): 835-849.