The Shapley value for airport and irrigation games

In this paper, we consider cost-sharing problems defined on rooted trees. We refer to these problems as cost-tree problems and to the induced transferable utility cooperative games as irrigation games. We introduce a formal notion of irrigation games and provide a characterization of this class of games. The well-known class of airport games (Littlechild and Thompson 1977) is shown to be a subclass of irrigation games. The Shapley value (Shapley 1953) is arguably the most widely used solution concept for transferable utility cooperative games. Dubey (1982) and Moulin and Shenker (1992) show, respectively, that the axiomatizations of the Shapley value due to Shapley (1953) and Young (1985) remain valid for the class of airport games. In this paper, we extend the results of Dubey (1982) and Moulin and Shenker (1992) to the broader class of irrigation games. Specifically, we provide two characterizations of the Shapley value for cost-sharing problems defined on rooted trees. In these characterization results, we relate the terminology of transferable utility games to that of cost-sharing problems, thereby establishing a bridge between the two fields.