Equilibria of Generalized Cut and Choose Protocols

Classic cake cutting protocols — which fairly allocate a divisible good among agents with heterogeneous preferences — are susceptible to manipulation. Do their strategic outcomes still guarantee fairness? To answer this question we adopt a novel algorithmic approach, proposing a concrete computational model and reasoning about the game-theoretic properties of algorithms that operate in this model. Specifically, we show that each protocol in the class of generalized cut and choose (GCC) protocols — which includes the most important discrete cake cutting protocols — is guaranteed to have approximate subgame perfect Nash equilibria. Moreover, we observe that the (approximate) equilibria of proportional protocols — which guarantee each of the n agents a 1/n-fraction of the cake — must be (approximately) proportional, and design a GCC protocol where all Nash equilibrium outcomes satisfy the stronger fairness notion of envy-freeness. Finally, we show that under an obliviousness restriction, which still allows the computation of approximately envy-free allocations, GCC protocols are guaranteed to have exact subgame perfect Nash equilibria.