Divide-and-choose games and the Shapley value

Abstract

We establish an equivalence between two classic approaches to fairness: the Shapley value and divide-and-choose. We study the allocation of a single indivisible item among N players with heterogeneous values, where the player receiving the item compensates the others. On the cooperative side, we introduce sharing games, in which players are partitioned into syndicates and a coalition's value is determined by the highest-valued player accessible through its members' syndicates. On the procedural side, we introduce an N-player divide-and-choose game inspired by Kuhn's classic cake-cutting procedure. Our main result links the two approaches: for every ordering of proposers in the divide-and-choose game, there is a unique assortative syndicate structure for which the Shapley value of the corresponding sharing game coincides with the payoffs under maxmin-perfect play. Conversely, the Shapley value of every assortative sharing game can be implemented in a divide-and-choose game with an appropriate ordering of proposers. The equivalence provides a procedural foundation for the Shapley value and a cooperative interpretation of fairness in divide-and-choose.