The Random Assignment Problem with Submodular Constraints on Goods

Abstract

Problems of allocating indivisible goods to agents in an efficient and fair manner without money have long been investigated in the literature. The random assignment problem is one of them, where we are given a fixed feasible (available) set of indivisible goods and a profile of ordinal preferences over the goods, one for each agent. Then, using lotteries, we determine an assignment of goods to agents in a randomized way. A seminal paper of Bogomolnaia and Moulin (2001) shows a probabilistic serial (PS) mechanism to give an ordinally efficient and envy-free solution to the assignment problem.In this paper we consider an extension of the random assignment problem to submodular constraints on goods. We show that the approach of the PS mechanism by Bogomolnaia and Moulin is powerful enough to solve the random assignment problem with submodular (matroidal and polymatroidal) constraints. Under the agents' ordinal preferences over goods we show the following.(1) The obtained PS solution for the problem with unit demands and matroidal constraints is ordinally efficient, envy-free, and weakly strategy-proof with respect to the associated stochastic dominance relation. (2) For the multi-unit demand and polymatroidal constraint problem the PS solution is ordinally efficient and envy-free but is not strategy-proof in general. However, we show that under a mild condition (that is likely to be satisfied in practice) the PS solution is a weak Nash equilibrium.

A simple construction of complete single-peaked domains by recursive tiling

“…As mentioned in the Introduction, our work is motivated by random assignment problems (Fujishige et al 2018; Moulin 2017) related to strategy-proofness. The recent paper given by Puppe (2018) also hints a relation to the structure of the base polyhedra of submodular system (Fujishige 2005;Zhan 2005), which our extended assignment problems are based on Fujishige et al (2018).…”

“…Our motivations for this research began with the random assignment problem (Fujishige et al 2018). Single-peakedness is important to keep required properties for further generalization (Achuthankutty and Roy 2018;Bade 2019;Moulin 1980Moulin , 2017Savaglio and Vannucci 2019).…”

Submodular optimization views on the random assignment problem

“…We have presented a solution for the non-pricing allocation of divisible goods to agents with utility functions and submodular constraints and showed the adaptability of the probabilistic serial mechanism of Bogomolnaia and Moulin [5] to the nonpricing allocation of divisible goods to agents with utility functions. Our results reveal how the probabilistic serial mechanism of Bogomolnaia and Moulin [5] and its extensions [27,[18][19][20] are related to the optimal fair allocation algorithms of Megiddo [30], Fujishige [16], Gallo, Grigoriadis, and Tarjan [21], and Groenevelt [22] and to the greedy algorithm of Edmonds [11]. This also shows that it is natural to consider extensions of the original assignment problem to those with sets of available good vectors expressed by submodular functions as in [18][19][20].…”

“…We then show that the probabilistic serial mechanism of Bogomolnaia and Moulin [5] and its extensions [24,27,28,[18][19][20] can be interpreted as special cases of our scheme given here for appropriately chosen utility functions. This reveals the close relation between (i) the optimal fair allocation problem and its solution algorithms in [30,16,21,10,26] and (ii) the probabilistic serial mechanisms in [5,24,27,28,[18][19][20] and others, which seems to be worth further investigating. The results of the present paper furnish further insights into the behavior of the probabilistic serial mechanism and reveal the combinatorial structures that really validate it.…”

Extended Random Assignment Mechanisms on a Family of Good Sets

“…PS mechanism was extended by Fujishige et al by generalizing the constraint with the fixed quota on each good to a system of linear inequalities induced by a polytope called polymatroid [11,12]. The extension also included multi-unit demands by agents and a lottery on indivisible goods.…”