A Solution for the Reward Allocation Problem in Linear Production Games with Flexible Resources

Abstract

In this paper we study an extension of the linear production game (LPG) introduced by Owen (1975). We discuss LPG with players who have flexible resources. Here, flexibility means that a player has a universal resource (e.g. working time) which he can split up (e.g. to execute different tasks). To model resource flexibility we extend the original LPG with constraints similar to the condition in Hall’s famous marriage theorem. The resources of the players can be used in several linear production processes, alone or together with other players. The resulting profit has to be divided among the players according to an allocation rule. We will show that this new variant of cooperative games – like the original linear LPG – is balanced, hence, has a nonempty core, and that we can obtain an allocation in the core by using the shadow prices of the flexible resources.