Dueling Over Dessert, Mastering the Art of Repeated Cake Cutting

TL;DR

This paper studies repeated two-player cake cutting with private valuations, analyzing both sequential and simultaneous cut-and-choose dynamics. It shows that Alice can exploit a nearly myopic Bob via a binary-search-like strategy in the sequential setting, while both players can enforce an equitable $(1/2,1/2)$ payoff profile in the long run, leveraging Blackwell approachability and robust representations of valuations. It also proves that fictitious play converges to the equitable outcome at a rate of $O(1/\sqrt{T})$, highlighting the role of learning dynamics in achieving fairness. The results distinguish the sequential and simultaneous settings, provide explicit regret bounds, and connect dynamic fair division to classical concepts in game theory and learning, with implications for repeated resource allocation under private preferences.

Abstract

We consider the setting of repeated fair division between two players, denoted Alice and Bob, with private valuations over a cake. In each round, a new cake arrives, which is identical to the ones in previous rounds. Alice cuts the cake at a point of her choice, while Bob chooses the left piece or the right piece, leaving the remainder for Alice. We consider two versions: sequential, where Bob observes Alice's cut point before choosing left/right, and simultaneous, where he only observes her cut point after making his choice. The simultaneous version was first considered by Aumann and Maschler (1995). We observe that if Bob is almost myopic and chooses his favorite piece too often, then he can be systematically exploited by Alice through a strategy akin to a binary search. This strategy allows Alice to approximate Bob's preferences with increasing precision, thereby securing a disproportionate share of the resource over time. We analyze the limits of how much a player can exploit the other one and show that fair utility profiles are in fact achievable. Specifically, the players can enforce the equitable utility profile of $(1/2, 1/2)$ in the limit on every trajectory of play, by keeping the other player's utility to approximately $1/2$ on average while guaranteeing they themselves get at least approximately $1/2$ on average. We show this theorem using a connection with Blackwell approachability. Finally, we analyze a natural dynamic known as fictitious play, where players best respond to the empirical distribution of the other player. We show that fictitious play converges to the equitable utility profile of $(1/2, 1/2)$ at a rate of $O(1/\sqrt{T})$.

Figures (11)

• Figure 1: Illustration of densities for Alice and Bob, with blue and red, respectively. Figure (a) shows Alice's midpoint at $m_A$ and Bob's midpoint at $m_B$. Figure (b) shows Alice's Stackelberg value, captured by the blue shaded area.
• Figure 2: Illustration of Alice's and Bob's average payoff in a randomly generated instance of valuations. The X axis shows the time and the Y axis shows the average payoff up to that round.
• Figure 6: Illustration of the sequences $\{\alpha_t\}_{t=1}^{\infty}$, $\{\beta_t\}_{t=1}^{\infty}$, and $\{\rho_t\}_{t=1}^{\infty}$ for the instance with trajectories shown in Figure \ref{['fig:fictitious_play']}. The X axis shows the round number $t = 1, \ldots T$ and the Y axis shows the value of the variable being plotted.
• Figure 7: Scatter plot of the sequence $(\alpha_t, \beta_t)_{t \geq 1}$, illustrating the spiral for the instance with trajectories shown in Figure \ref{['fig:fictitious_play']}, where the sequences $\alpha_t$ and $\beta_t$ are illustrated separately in Figure \ref{['fig:fictitious_play_variables']}.
• Figure 8: Alice's algorithm against myopic Bob in the exploration phase. Alice's density is shown with blue and her midpoint is $m_A$, while Bob's density is shown with red and his midpoint is $m_B$. The algorithm initialized $\ell_1 = 0$ and $r_1 =1$ and then re-computes them iteratively depending on Bob's answers. The constructed interval $[\ell_t,r_t]$ shrinks exponentially and becomes closer to $m_B$ as the time $t$ increases.

Theorems & Definitions (87)

• Proposition 1
• Theorem 1: Exploiting a nearly myopic Bob
• Theorem 2: Alice enforcing equitable payoffs; informal
• Theorem 3: Bob enforcing equitable payoffs; informal
• Theorem 4: Fictitious Play; informal
• Definition 1: Stackelberg regret
• Definition 2: Regret
• Proposition 2
• proof : Proof sketch
• Proposition 3