A three-person game theory model arising in stochastic inventory control theory

In this paper, the authors build a game theory model for a single-period inventory problem where each of three retailers tries to determine his optimal order quantity. Their products are substitutable and have random demands. Therefore, multiple direction demand transfers occur when one or more retailers are sold out. It is shown that, by using Nash equilibrium, optimal decision exists for each retailer when they work independently. The optimal order quantity in this case is larger than that under the newsboy problem model. This is because substitution can be a source of sales and hence a justification for inventory. In situations where some retailer may act irrationally to inflict a damage on the others, the defensive strategy for the latter is to totally ignore the existence of the irrational player. The authors also study the cooperation of retailers by switching excess inventory to those who have excess demand both when side payments are and are not allowed. If side payments are not allowed, secure (Nash) strategies always exist for each retailer and cooperation will reduce inventory. The latter is especially true when the depreciation of value of the item and/or inventory cost are large. The authors give conditions for cooperation in both cases and show that although all-player cooperation is often feasible it may not always be obtainable even if side payments are allowed.