The Computational Rise and Fall of Fairness

The fair division of indivisible goods has long been an important topic in economics and, more recently, computer science. We investigate the existence of envyfree allocations of indivisible goods, that is, allocations where each player values her own allocated set of goods at least as highly as any other player’s allocated set of goods. Under additive valuations, we show that even when the number of goods is larger than the number of agents by a linear fraction, envy-free allocations are unlikely to exist. We then show that when the number of goods is larger by a logarithmic factor, such allocations exist with high probability. We support these results experimentally and show that the asymptotic behavior of the theory holds even when the number of goods and agents is quite small. We demonstrate that there is a sharp phase transition from nonexistence to existence of envy-free allocations, and that on average the computational problem is hardest at that transition. Introduction The allocation of goods to interested agents is a central tenet of society. Some goods, like land, are divisible: a mechanism can split a single good amongst multiple agents. Others, like the houses or cars in an estate sale or divorce proceedings, are indivisible: a mechanism must allocate each good to exactly one agent. A chief concern in the assignment of divisible and indivisible goods to agents—and in the employment of divorce lawyers—concerns defining and guaranteeing the fairness of the final allocation. One formal notion of fairness is envy-freeness. An allocation of goods is envy free (EF) if each player values her own allocated set of goods at least as highly as any other player’s allocated set of goods. While EF divisions exist for any number of players in the divisible goods case (see, e.g., (Procaccia 2013), and the references therein), it is not guaranteed that such fair allocations exist when indivisible goods are considered. Indeed, consider the simple case of a single Copyright c © 2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. good and two agents, both of which have positive value for the good. Allocating the good to either agent will result in envy from its empty-handed partner. In this paper, we investigate the conditions under which EF allocations of indivisible goods exist, when agents’ values of goods are drawn at random. Under additive valuations and rather general conditions on the distributions over values of individual goods, we characterize conditions for nonexistence, showing that even when the number of goods is larger than the number of agents by a linear fraction, an EF allocation is unlikely to exist (Theorem 1). We then show that when the number of goods is larger by a logarithmic factor than the number of agents, an EF allocation exists with high probability (Theorem 2). Thus, these asymptotic existence results are almost tight. We support our theoretical results, which apply asymptotically, with an empirical exploration of the EF allocation problem on different distributions over valuations and different objectives over EF allocations using two integer programming models. The theory applies to each of these experiments even when the number of agents and goods is quite small. We also uncover a phenomenon common to many problems in artificial intelligence: that the hardest computational EF allocation problems on average occur during the (sharp) transition from nonexistence to existence. Related Work Fair division occupies an important place in AI research; see, e.g., Chevaleyre, Endriss, and Maudet (2007), Bouveret and Lang (2008), Chen et al. (2010), Cohler et al. (2011) and the survey by Chevaleyre et al. (2006). Among the many AI papers that study the EF allocation of indivisible goods, the work of Bouveret and Lang (2008) is of particular interest. They showed that determining the existence of an EF allocation is computationally hard. In contrast, we focus on typical instances, and show that EF allocations exist, or do not exist, with high probability. Similarly, the phase transition phenomenon is a staple of AI research (Cheeseman, Kanefsky, and Taylor 1991; Hogg, Huberman, and Williams 1996). In a nutshell, conProceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence