How to Cut a Cake Before the Party Ends

Abstract

For decades researchers have struggled with the problem of envy-free cake cutting: how to divide a divisible good between multiple agents so that each agent likes his own allocation best. Although an envy-free cake cutting protocol was ultimately devised, it is unbounded, in the sense that the number of operations can be arbitrarily large, depending on the preferences of the agents. We ask whether bounded protocols exist when the agents' preferences are restricted. Our main result is an envy-free cake cutting protocol for agents with piecewise linear valuations, which requires a number of operations that is polynomial in natural parameters of the given instance.

Obvious manipulations in cake-cutting

“…In cake-cutting with three or more agents, every strategy-proof Robertson-Webb mechanism assigns no cake to at least one agent, whereas with two agents every Robertson-Webb strategy-proof mechanism is a dictatorship (Kurokawa et al 2013;Brânzei and Miltersen 2015). These results assume that a Robertson-Webb protocol may cut the cake only in cut-points determined by agents' replies to queries.…”

Simultaneous Cake Cutting

“…In recent years, cake cutting has emerged as a major research topic in artificial intelligence (Procaccia 2009;Caragiannis, Lai, and Procaccia 2011;Cohler et al 2011;Brams et al 2012;Bei et al 2012;Aumann, Dombb, and Hassidim 2013;Kurokawa, Lai, and Procaccia 2013;Brânzei, Procaccia, and Zhang 2013;Brânzei and Miltersen 2013;Chen et al 2013). The growing interest in cake cutting, and fair division more broadly, is partly motivated by potential applications in AI, such as industrial procurement, manufacturing and scheduling, and airport traffic management (Chevaleyre et al 2006).…”

“…The standard model of communication between the center and agents in cake cutting was proposed by Robertson and Webb (1998), and employed in a body of work studying the complexity of cake cutting (Edmonds and Pruhs 2006b;2006a;Woeginger and Sgall 2007;Procaccia 2009;Kurokawa, Lai, and Procaccia 2013). The model restricts the interaction between the protocol and the agents to two types of queries:…”

“…This model views the cake cutting process as an interaction between the protocol and the agents, where the protocol elicits information about the agents' valuations by iteratively asking two types of queries: either cutting a piece of certain value, or evaluating a given piece. The Robertson-Webb model allows computer scientists to formally reason about the complexity of achieving fair cake divisions (Edmonds and Pruhs 2006b;2006a;Woeginger and Sgall 2007;Procaccia 2009;Deng, Qi, and Saberi 2012;Aumann, Dombb, and Hassidim 2013;Kurokawa, Lai, and Procaccia 2013;Brânzei, Procaccia, and Zhang 2013).…”

Envy-Free Mechanisms with Minimum Number of Cuts

“…One of the important properties of these valuation functions is that they can be described concisely. (Kurokawa, Lai, and Procaccia 2013) proved that finding an envy-free allocation (in Robertson-Webb model) when the valuation functions are piecewise-uniform is as hard as solving the problem without any restriction on the valuation functions.…”