Envy-free cake cutting: A polynomial number of queries with high probability

In this article we propose a probabilistic framework in order to study the fair division of a divisible good, e.g., a cake, between n players. Our framework follows the same idea than the ''Full independence model'' used in the study of fair division of indivisible goods. We show that, in this framework, there exists an envy-free division algorithm satisfying the following probability estimate:$$\mathbb{P}\big( C(\mu_1, \ldots,\mu_n) \geq n^{7+b}\big) = \mathcal{O}\Big(n^{-\frac{b-1}{3}+1+o(1)}\Big),$$where $\mu_1,\ldots, \mu_n$ correspond to the preferences of the $n$ players,$C(\mu_1, \ldots,\mu_n)$ is the number of queries used by the algorithm and $b>4$. In particular, this gives$$\lim_{n \rightarrow + \infty}\mathbb{P}\big( C(\mu_1, \ldots,\mu_n) \geq n^{12}\big) = 0.$$ It must be noticed that nowadays few things are known about the complexity of envy-free division algorithms. Indeed, Procaccia has given a lower bound in $\Omega(n^2)$ and Aziz and Mackenzie have given an upper bound in $n^{n^{n^{n^{n^{n}}}}}$. As our estimate means that we have $C(\mu_1, \ldots, \mu_n)<n^{12}$ with a high probability, this gives a new insight on the complexity of envy-free cake cutting algorithms. Our result follows from a study of Webb's algorithm and a theorem of Tao and Vu about the smallest singular value of a random matrix.