Fair and Efficient Cake Division with Connected Pieces

Abstract

The classic cake-cutting problem provides a model for addressing fair and efficient allocation of a divisible, heterogeneous resource (metaphorically, the cake) among agents with distinct preferences. Focusing on a standard formulation of cake cutting, in which each agent must receive a contiguous piece of the cake, this work establishes algorithmic and hardness results for multiple fairness/efficiency measures.First, we consider the well-studied notion of envy-freeness and develop an efficient algorithm that finds a cake division (with connected pieces) wherein the envy is multiplicatively within a factor of 3+o(1). The same algorithm in fact achieves an approximation ratio of 3+o(1) for the problem of finding cake divisions with as large a Nash social welfare (NSW) as possible. NSW is another standard measure of fairness and this work also establishes a connection between envy-freeness and NSW: approximately envy-free cake divisions (with connected pieces) always have near-optimal Nash social welfare. Furthermore, we develop an approximation algorithm for maximizing the ρ-mean welfarethis unifying objective, with different values of ρ, interpolates between notions of fairness (NSW) and efficiency (average social welfare). Finally, we complement these algorithmic results by proving that maximizing NSW (and, in general, the ρ-mean welfare) is APX-hard in the cake-division context.

Optimal Bounds on the Price of Fairness for Indivisible Goods

“…If there is an unassigned connected bundle U ∈ U(P) that some agent envies, then the algorithm finds an inclusion-wise minimal subset of U that is envied and allocates it to an envious agent. While this preserves exact envy-freeness in the cake-cutting setting of Arunachaleswaran et al [3], in our setting we argue that inclusion-wise minimality preserves the EF1 property of P. We also argue that this iterative process terminates at a partial allocation P that satisfies the desired social welfare guarantee. Finally, we use the algorithm of Lipton et al [20] (see also [2]) to extend this partial EF1 allocation into a complete EF1 allocation without losing social welfare.…”

“…For subadditive valuations, given the absence of succinct representation, we assume access to a demandquery oracle. 3 As a consequence of this result, we also settle the price of a weaker fairness notion-proportionality up to one good (Prop1)-as Θ( √ n) for additive valuations.…”

“…This allocation trivially satisfies two properties: it is EF1, and each P i is a connected bundle in L. The algorithm then iteratively updates P to improve its social welfare while maintaining both these properties. This iterative process is inspired by a similar algorithm for (divisible) cake-cutting (Algorithm 1, Arunachaleswaran et al [3]).…”

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

“…While the approximations of 1/3 and 1/4 may not seem impressive, they represent the first non-trivial approximations for polynomial-time algorithms in the context of contiguous cake cutting (besides the very recent algorithm of Arunachaleswaran et al (2019), achieving the weaker additive approximation of 1/2 as discussed in Section 1.2). Indeed, even the classical Dubins-Spanier protocol, which guarantees every agent a value of 1/n, may allocate a piece which an agent values 1/n and another piece to a second agent which the first agent values (n − 1)/n-this leads to an envy of (n − 2)/n, which converges to 1 for large n. Improving these approximations or establishing lower bounds is an interesting and important question for future research.…”

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

“…Recently, Arunachaleswaran et al (2019) developed an efficient algorithm that computes a contiguous cake division with multiplicatively bounded envy-in particular, each agent's envy is bounded by a multiplicative factor of 3. We remark that our approximation algorithms are incomparable to their result.…”