Algorithms for Competitive Division of Chores

Abstract

We study the problem of allocating divisible bads (chores) among multiple agents with additive utilities, when money transfers are not allowed. The competitive rule is known to be the best mechanism for goods under additive utilities and was recently extended to chores by Bogomolnaia et al. [BMSY17]. For both goods and chores, the rule produces Pareto optimal and envy-free allocations. In the case of goods, the outcome of the competitive rule can be easily computed. Competitive allocations solve the Eisenberg-Gale convex program; hence the outcome is unique and can be approximately found by standard gradient methods. An exact algorithm that runs in polynomial time in the number of agents and goods was given by Orlin [Orl10].In the case of chores, the competitive rule does not solve any convex optimization problem; instead, competitive allocations correspond to local minima, local maxima, and saddle points of the Nash Social Welfare on the Pareto frontier of the set of feasible utilities. The rule becomes multivalued and none of the standard methods can be applied to compute its outcome.In this paper, we show that all the outcomes of the competitive rule for chores can be computed in strongly polynomial time if either the number of agents or the number of chores is fixed. The approach is based on a combination of three ideas: all consumption graphs of Pareto optimal allocations can be listed in polynomial time; for a given consumption graph, a candidate for a competitive allocation can be constructed via explicit formula; and a given allocation can be checked for being competitive using a maximum flow computation as in Devanur et al. [DPSV02]. Our algorithm immediately gives an approximately-fair allocation of indivisible chores by the rounding technique of Barman and Krishnamurthy [BK19].

Polynomial Time Algorithms to Find an Approximate Competitive Equilibrium for Chores

“…Nevertheless, the CE with bads exhibits far less structure than the CE with goods as explained in the introduction. There are polynomial time enumerative algorithms known only when there are constant number of agents or chores [BS19,GM20]. Quite recently, [CGMM21] gave an LCP formulation for determining CEEI with mixed manna (goods and bads) when the utility functions are separable piecewise-linear and concave (SPLC) which includes linear.…”

“…Although goods and chores problems seem similar, results for chores are surprisingly contrasting: Even in the restricted case of linear disutilities, the set of CEEI can be non-convex and disconnected [BMSY17,BMSY19], and in the exchange model computing a CE is PPAD-hard [CGMM20]. No polynomial time algorithms are known to find CEEI with chores, except for when number of agents or number of chores is a constant [BS19,GM20]. 2 We note that the combinatorial approaches known for the goods case [DPSV08,Orl10,Vég12] seem to fail due to disconnectedness of the CEEI set (see Remark 1 for further explanation).…”

How to Fairly Allocate Easy and Difficult Chores

“…As mentioned earlier, we achieve our main result -an EF1+PO allocation of chores with bivalued utilities -using the framework of Fisher markets and competitive equilibria. Even when the items are divisible (i.e., if they can be portioned out between the agents), these remain well-defined and yield (exact) EF+PO [29]; for goods, they coincide with the maximum Nash welfare rule and can be computed in strongly polynomial time [13,27], while for chores, they have a more intricate structure [8] and their computation is an open question [10]. In case of indivisible items, Barman et al [6] use this framework to achieve EF1+PO in pseudo-polynomial time and Garg and Murhekar [18] improve the running time to polynomial when each agent has at most polynomially many utility levels across all bundles of goods.…”

Best of Both Worlds: Ex-Ante and Ex-Post Fairness in Resource Allocation

“…The following property is a relaxation of EF1 and enjoys strong algorithmic support in conjunction with PO, for goods (Section 5) as well as bads (Section 7) [9,16].…”

“…This is somewhat striking because it has been observed that many results for the case of goods do not easily carry over to the case of bads [14]. For example, while it is known that a competitive equilibrium with equal incomes (CEEI) fractional allocation exists for the case of bads, and such an allocation is envy-free and Pareto optimal, it is no longer obtained by maximizing or minimizing the product of (dis)utilities, and whether such an allocation can be computed in polynomial time is a major open question [16,29]. However, if such an allocation is given, Brânzei and Sandomirskiy [16] show that one can round it to obtain an integral Prop1+EF 1 1 +fPO allocation in strongly polynomial time.…”