Dueling over Multiple Pieces of Dessert

TL;DR

This paper investigates repeated fair division between two players, revealing the limitations of arbitrary partitions and proposing efficient strategies when using a limited number of cuts, with implications for online learning and query complexity.

Contribution

It introduces a framework analyzing the regret in repeated cake-cutting games with limited cuts and characterizes the complexity of finding approximate Stackelberg allocations.

Findings

Strongly sublinear regret is impossible with arbitrary partitions.

Limited cuts enable tractable learning and regret bounds.

Private learning rates allow universal guarantees with some vulnerability.

Abstract

We study the dynamics of repeated fair division between two players, Alice and Bob, where Alice partitions a cake into two subsets and Bob chooses his preferred one over $T$ rounds. Alice aims to minimize her regret relative to the Stackelberg value -- the maximum utility she could achieve if she knew Bob's private valuation. We show that if Alice uses arbitrary measurable partitions, achieving strongly sublinear regret is impossible; she suffers a regret of $\Omega\Bigl(\frac{T}{\log^2 T}\Bigr)$ regret even against a myopic Bob. However, when Alice uses at most $k$ cuts, the learning landscape becomes tractable. We analyze Alice's performance based on her knowledge of Bob's strategic sophistication (his regret budget). When Bob's learning rate is public, we establish a hierarchy of polynomial regret bounds determined by $k$ and Bob's regret budget. In contrast, when this learning rate is private, Alice can universally guarantee $O\Bigl(\frac{T}{\log T}\Bigr)$ regret, but any attempt to secure a polynomial rate $O(T^\beta)$ (for $\beta < 1$) leaves her vulnerable to incurring strictly linear regret against some Bob. Finally, as a corollary of our online learning dynamics, we characterize the randomized query complexity of finding approximate Stackelberg allocations with a constant number of cuts in the Robertson-Webb model.