The One Dimensional Kakutanis Fixed Point Theorem: A Classroom Capsule

In problems of fair division of a given bundle of infinitely divisible commodities amongst a finite number of agents, the standard framework pioneered by Thomson [1988] has been one where a choice correspondence associates with each profile of preferences and a given aggregate initial endowment vector, a subset of the set of feasible allocations. The literature on this topic is so vast that the possibility of a single survey doing justice to all aspects of the problem is rather remote. However, a near adequate survey of the relevant literature is the monograph by Thomson [1995]. A feature of the study of such problems which is difficult to miss is that there is a clear dichotomy between the analytical methodology concerning the study of problems of fair division of just one commodity and the analytical techniques involving the study of problems of fair division of more than one commodity. However, within the study of problems concerning the fair division of more than one commodity, there does not appear to be a major difference involving the number of commodities. This observation by and large applies to the theorems, examples and counter examples pertaining to the relevant literature. To an extent, this phenomenon is not very surprising. The major difference that arises between one commodity fair division problems and multi-commodity fair division problems is the presence of the possibility of trading off the consumption of one commodity for another in the latter case and its absence in the former. This possibility, to the extent that it is invoked in the analysis of fair division problems does not depend on the number of commodities involved provided, the number of commodities is atleast two.