Meta-envy-free Cake-cutting and Pie-cutting Protocols

Abstract

This paper discusses cake-cutting protocols when the cake is a heterogeneous good, represented by an interval on the real line. We propose a new desirable property, the meta-envy-freeness of cake-cutting, which has not been formally considered before. Meta-envy-free means there is no envy on role assignments, that is, no party wants to exchange his/her role in the protocol with the one of any other party. If there is an envy on role assignments, the protocol cannot be actually executed because there is no settlement on which party plays which role in the protocol. A similar definition, envy-freeness, is widely discussed. Envy-free means that no player wants to exchange his/her part of the cake with that of any other player's. Though envy-freeness was considered to be one of the most important desirable properties, envy-freeness does not prevent envy about role assignment in the protocols. We define meta-envy-freeness to formalize this kind of envy. We propose that simultaneously achieving meta-envy-free and envy-free is desirable in cake-cutting. We show that current envy-free cake-cutting protocols do not satisfy meta-envy-freeness. Formerly proposed properties such as strong envy-free, exact, and equitable do not directly consider this type of envy and these properties are very difficult to realize. This paper then shows cake-cutting protocols for two and three party cases that simultaneously achieves envy-free and meta-envy-free. Last, we show meta-envy-free pie-cutting protocols.