Partial pooling by independent firms with allocation according to contribution to pool

We consider two firms which pool some of their inventory. The pool is created by the firms' contributions, and a firm's entitlement for an allocation from the pool (if needed) is a function of its contribution. Transshipment from the pool is costly, but the firms can benefit from reduced risk through inventory sharing using the pool. We analyze the resulting non-cooperative game. We prove existence of a Nash equilibrium and compare it to a model with centralized control. An appropriate compensation cost for using the other firms contribution to the pool can induce the retailers to achieve centralized solutions. We also compare the optimal partial pooling strategy to the special cases of no pooling and complete pooling and discuss situations where it is likely that one of the special cases will be optimal. Numerical results confirm that in the prevalent practice of partial pooling the retailers can achieve higher expected profits than under no pooling or complete pooling and that there is a significant difference between a setting with independent players and a model of central control.