Multi-period inventory games with information update

Abstract

This paper addresses the multi-period inventory game of two substitutable products with learning, in which the demand distribution form in each period is known but one or more parameters are unknown in advance. The retailers update the estimates of the unknown parameters using the historical demand information as the transaction proceeds. At each period, the retailers check the initial inventories and update the forecasts about their demands first. Then they place the orders, which are delivered immediately. Finally the demands are realized and satisfied by the available inventories. The unsatisfied demand of one retailer switches to another and is satisfied by the other׳s left inventory. Unsatisfied demand is lost and the excess inventories incur a holding cost. All the information is common knowledge. The objectives of the two retailers are to make proper ordering decisions to maximize their total expected discount profits respectively. We model the problem as a dynamic Bayesian game with multi-dimension state-space. We give the conditions under which the state-space can be reduced. We prove the existence and uniqueness of Nash equilibrium and show that the equilibrium ordering strategy is the base-stock policy, the computation of the base-stock levels can be reduced to a game with one-dimension state-space. Moreover, for the case with perishable product, the computation of the base-stock levels can be reduced to the Newsboy game.