Equilibrium analysis in cake cutting

Cake cutting is a fundamental model in fair division; it represents the problem of fairly allocating a heterogeneous divisible good among agents with different preferences. The central criteria of fairness are proportionality and envy-freeness, and many of the existing protocols are designed to guarantee proportional or envy-free allocations, when the participating agents follow the protocol. However, typically, all agents following the protocol is not guaranteed to result in a Nash equilibrium. In this paper, we initiate the study of equilibria of classical cake cutting protocols. We consider one of the simplest and most elegant continuous algorithms -- the Dubins-Spanier procedure, which guarantees a proportional allocation of the cake -- and study its equilibria when the agents use simple threshold strategies. We show that given a cake cutting instance with strictly positive value density functions, every envy-free allocation of the cake can be mapped to a pure Nash equilibrium of the corresponding moving knife game. Moreover, every pure Nash equilibrium of the moving knife game induces an envy-free allocation of the cake. In addition, the moving knife game has an epsilon-equilibrium which is epsilon-envy-free, allocates the entire cake, and is independent of the tie-breaking rule.