ENVY-FREE CAKE CUTTING: A POLYNOMIAL NUMBER OF QUERIES WITH HIGH PROBABILITY
Résumé
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)
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https://hal.science/hal-02561963
Soumis le : lundi 4 mai 2020-11:37:33
Dernière modification le : lundi 20 novembre 2023-11:44:19
Citer
Guillaume Chèze. ENVY-FREE CAKE CUTTING: A POLYNOMIAL NUMBER OF QUERIES WITH HIGH PROBABILITY. 2020. ⟨hal-02561963v1⟩
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