Dueling over Multiple Pieces of Dessert

TL;DR

This work studies repeated two-player cake-cutting where Alice, as the Stackelberg leader, partitions the cake and Bob chooses a piece each round. It reveals sharp learnability boundaries: with fully measurable cuts, Alice cannot achieve strongly sublinear regret against even a myopic Bob; but when restricting to at most $k$ cuts, the learning landscape becomes tractable and yields distinct regret regimes depending on Bob’s suspected learning rate (public vs private) and his strategic sophistication (myopic vs non-myopic). The results provide tight (up to polylog factors) upper and lower bounds for $k=2$ and $k\ge3$ cuts, including regimes with public $\alpha$-regret budgets and adaptive strategies, and extend to a Robertson-Webb query-model corollary detailing $\varepsilon$-Stackelberg allocations via $O(k/\varepsilon)$ queries. Collectively, the findings illuminate the fundamental trade-offs between partitioning flexibility and learnability in repeated Stackelberg-style cake cutting, with implications for online learning in strategic division problems and for query-complexity in classical cake-cutting models.

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 $Ω\Bigl(\frac{T}{\log^2 T}\Bigr)$ 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 $O(T^β)$ (for $β< 1$) 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.

Figures (6)

• Figure 1: A Stackelberg cut for $k=2$. Bob's value density $v_B$ is plotted in blue. Alice's value density (not pictured) is assumed to be uniform: $v_A(x)=1$$\forall x \in [0,1]$. The region between the red dashed cuts has value $1/2$ to Bob and maximizes Alice's value among all such regions; thus, $u_A^*(2) = 5/8$.
• Figure 2: An example outcome of the first two rounds of a $2$-cut game. In each round, Alice partitions $[0, 1]$ into two pieces by making at most $2$ cuts. The pieces Bob chose are in blue. In this case, they are the pieces he would prefer under the value density from Figure \ref{['fig:two-cut-Stackelberg']}.
• Figure 3: Illustration of an alternating partition for $k=6$. The positive-length intervals of $Z_1$ and $Z_2$ are colored blue and red, respectively. Setting some cut points equal to each other, as with $x_2=x_3$ and $x_6=x_7$, allows Alice to effectively make fewer than $k$ cuts if desired.
• Figure 4: An illustration of the transformations in Theorem \ref{['thm:measurable-cut-myopic-lower-bound']}, focusing on the mass moved in and out of a single $(L_i, R_i)$ pair. Blue indicates the elements of each of the three sets. The starting $B^j_1$ is an arbitrary piece Alice's strategy could make. From $B^j_1$ to $B^j_2$, an equal amount of mass is removed from $L_i$ and $R_i$ until one of them is empty, in this case $L_i$. Twice this much mass is added to $[1/2, 1]$. From $B^j_2$ to $B^j_3$, mass is moved back into the partially filled $R_i$ to complete it. The resulting $B^j_3$ then contains exactly one set from each $(L_i, R_i)$ pair while still having similar value to $B^j_1$ for both players.
• Figure 5: Superimposed plots of the unspiked value density function $\sigma_0^k$ and an example spiked value density $\sigma_{w;z}^k$ for $k=5$. The segments in purple are common to $\sigma_0^k$ and $\sigma_{w;z}^k$. The functions differ only in a small interval centered at $z$ called the spike, where $\sigma_0^k$ maintains its value in red and $\sigma_{w;z}^k$ diverges in blue.

Theorems & Definitions (91)

• Theorem 1
• Theorem 2: $k \geq 2$ cuts; Myopic Bob
• Theorem 3: 2 cuts; Non-Myopic Bob with Public Learning Rate
• Theorem 4: $k \geq 3$ cuts; Non-Myopic Bob with Public Learning Rate
• Theorem 5: $k \geq 2$ cuts; Non-Myopic Bob with Private Learning Rate
• Definition 1
• Theorem 6
• Lemma 1
• Theorem 6
• proof : Proof sketch of Theorem \ref{['thm:measurable-cut-myopic-lower-bound']}