Communication Complexity of Cake Cutting

Abstract

We study classic cake-cutting problems, but in discrete models rather than using infiniteprecision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and movingknife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocation for any number of players) which we prove are not in the "easy" class. Our main open problem is to separate the "hard" from the "medium" classes.Moving beyond Cut-and-Choose, protocols for more players can get very complex and, in fact, a central question in the cake cutting literature is that of computing fair allocations despite the informational challenge of private preferences. The goal of a mediator (or center) is to help the parties reach a fair solution, which in turn requires it to learn enough valuation information. The existing cake cutting protocols are broadly classified in three categories as follows. The largest class contains so called "discrete" protocols, which operate in a query model (due to Robertson and Webb [RW98, WS07]), where the center can repeatedly ask the players to make a cut or evaluate an existing piece. Despite the fact that the Robertson-Webb (RW) query model is viewed as discrete in the cake cutting literature, it is an infinite precision model, which is unavoidable for the purpose of computing exact solutions (or even approximate ones for arbitrarily complex instances.) Continuous (moving knife) procedures generally involve multiple knives sliding over the cake, with stopping conditions that signal the presence of a fair allocation. In a companion paper [BN17] we formalize a general class of moving knife procedures and prove that they can be approximately simulated by logarithmically many queries in the RW model. Finally, direct revelation protocols were studied also in the context of mechanism design [MT10, MN12, CLPP13, BM15], where the players directly submit their entire valuation to the center, whose goal is to compute a fair allocation despite the strategic behavior of the players. The positive results in the direct revelation model are either only for very restricted preferences, such as piecewise uniform functions [CLPP13], or rely again on infinite precision for general preferences [MT10].In this paper we study the communication complexity [KN96, LS09] of cake cutting: we assume that each player knows its own valuation function, and study the amount of communication that the players need to exchange (between themselves or with a mediator) in order to find an ǫ-fair allocation, for various notions of fairness.There are several motivations for studying the communication complexity of fair allocation problems. First, it is natural to use discrete protocols for studying notions of comp...

Resource-monotonicity and population-monotonicity in connected cake-cutting

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Communication complexity of approximate Nash equilibria

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