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 $O(1/\sqrt{T})$.

多次公平分割研究中，考虑了两个玩家——爱丽丝和鲍勃——对蛋糕的私人评估。通过使用类似二分搜索的策略，爱丽丝可以逐渐准确地判断鲍勃的喜好，从而在资源分配中获得了不成比例的份额。通过与Blackwell可达性的连接，证明了在每一轮的游戏中，通过保证对手的平均效用近似为1/2，并保证自己的平均效用至少近似为1/2，在极限情况下玩家可以实现公平的效用分配。最后，通过分析称为虚构博弈的自然动态，证明了虚构博弈以O(1/√T)的速率收敛到公平的效用分配。

决斗甜点，掌握反复蛋糕切割的艺术