The Complexity of Cake Cutting with Unequal Shares

Abstract

An unceasing problem of our prevailing society is the fair division of goods. The problem of proportional cake cutting focuses on dividing a heterogeneous and divisible resource, the cake, among n players who value pieces according to their own measure function. The goal is to assign each player a not necessarily connected part of the cake that the player evaluates at least as much as her proportional share. In this article, we investigate the problem of proportional division with unequal shares, where each player is entitled to receive a predetermined portion of the cake. Our main contribution is threefold. First we present a protocol for integer demands, which delivers a proportional solution in fewer queries than all known protocols. By giving a matching lower bound, we then show that our protocol is asymptotically the fastest possible. Finally, we turn to irrational demands and solve the proportional cake cutting problem by reducing it to the same problem with integer demands only. All results remain valid in a highly general cake cutting model, which can be of independent interest.

Disproportionate division

“…In the computational complexity literature, cake-cutting algorithms are often represented by a sequence of queries of the form 'evaluate a given interval in a given measure' or 'mark off an interval of a given value in a given measure', and query-complexity is the object of interest; see [5], for example. We mention in closing that the algorithm described here may also be represented in this form, and this representation is easily seen to have bounded query-complexity (though we have made no attempt to optimise the query-complexity of our algorithm, since this is not our primary motivation).…”

Cutting a cake for infinitely many guests

“…An algorithm was given in Section 7 of [5] that reduces this problem to two sub-problems in one of which the number of players is smaller by one while in the other all the entitlements are rational and the number of players remains the same. We introduce two algorithms both of which solves the problem in finitely many steps and based on 'Last diminisher'-type of ideas.…”

“…The first one reduces the problem to another one in which either the number of players is smaller by one or all the entitlements are rationals and the number of players is the same. In this algorithm we need to pick rational numbers from non-degenerate intervals (which was used in the algorithms given in [5,13]). In the second algorithm no such rational approximation is needed.…”

“…Several finite procedures were developed to find a (strongly) fair division allowing irrational entitlements (see [2,5,11]). Our first contribution (Section 2) is two such proceeders which we believe are simpler than the known methods.…”

Weighted Envy-Freeness in Indivisible Item Allocation

“…Recently, Aziz et al [2019b] propose a polynomial-time algorithm for computing an allocation of a pool of goods and chores that satisfies both Pareto optimality and weighted proportionality up to one item (PROP1) for agents with asymmetric weights. Unequal entitlements have also been considered in the context of divisible goods [Zeng, 2000, Cseh andFleiner, 2018]. Unlike all of these previous works, we focus on fairness concepts based on weighted envy for the indivisible goods scenario.…”