The complexity of cake cutting with unequal shares

TL;DR

This paper advances the theory of fair division by providing a more efficient protocol for proportional cake cutting with unequal shares, including both integer and irrational demands, and establishes its optimality.

Contribution

It introduces a faster protocol for integer demands, proves its asymptotic optimality, and extends solutions to irrational demands within a general cake cutting model.

Findings

Fewer queries needed for proportional division with integer demands.

Matching lower bound established for protocol efficiency.

Extension of solutions to irrational demands in a general model.

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 paper, 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 that delivers a proportional solution in fewer queries than all known algorithms. Then we show that our protocol is asymptotically the fastest possible by giving a matching lower bound. 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.