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 with additive utilities and was recently extended to chores by (2017). 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. 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 (2002). Our algorithm immediately gives an approximately-fair allocation of indivisible chores by the rounding technique of Barman and Krishnamurthy (2018).
The polynomial time algorithms in these works are designed for economies that satisfy implicit or explicit convexity assumptions. For example, in the case of Fisher markets, the competitive equilibrium solves the Eisenberg-Gale convex program @cite_67 for a large class of utilities, maximizing the Nash product (i.e., the geometric mean of the utilities weighted by the budgets of the agents). Moreover, the equilibrium is unique, robust @cite_10 (i.e., small errors in the observation of the market parameters do not change the competitive allocation by much), and admits polynomial time approximation algorithms based on gradient descent methods as well as exact algorithms (see, e.g., Chapters 5 and 6 in @cite_12 ). In contrast, in the case of chores, there is a multiplicity of equilibria and no robustness guarantee: the set of equilibria admits no continuous selections @cite_39 .
@cite_67 @cite_10 @cite_12 @cite_39
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We study Fisher markets that admit equilibria wherein each good is integrally assigned to some agent. While strong existence and computational guarantees are known for equilibria of Fisher markets with additive valuations, such equilibria, in general, assign goods fractionally to agents. Hence, Fisher markets are not directly applicable in the context of indivisible goods. In this work we show that one can always bypass this hurdle and, up to a bounded change in agents' budgets, obtain markets that admit an integral equilibrium. We refer to such markets as pure markets and show that, for any given Fisher market (with additive valuations), one can efficiently compute a "near-by," pure market with an accompanying integral equilibrium. Our work on pure markets leads to novel algorithmic results for fair division of indivisible goods. Prior work in discrete fair division has shown that, under additive valuations, there always exist allocations that simultaneously achieve the seemingly incompatible properties of fairness and efficiency; here fairness refers to envy-freeness up to one good (EF1) and efficiency corresponds to Pareto efficiency. However, polynomial-time algorithms are not known for finding such allocations. Considering relaxations of proportionality and EF1, respectively, as our notions of fairness, we show that fair and Pareto efficient allocations can be computed in strongly polynomial time. We provide the first polynomial time exact algorithm for computing an Arrow-Debreu market equilibrium for the case of linear utilities. Our algorithm is based on solving a convex program using the ellipsoid algorithm and simultaneous diophantine approximation. As a side result, we prove that the set of assignments at equilibrium is convex and the equilibrium prices themselves are log-convex. Our convex program is explicit and intuitive, which allows maximizing a concave function over the set of equilibria. On the practical side, Ye developed an interior point algorithm [Lecture Notes in Comput. Sci. 3521, Springer, New York, 2005, pp. 3-5] to find an equilibrium based on our convex program. We also derive separate combinatorial characterizations of equilibrium for Arrow-Debreu and Fisher cases. Our convex program can be extended for many nonlinear utilities and production models. Our paper also makes a powerful theorem (Theorem 6.4.1 in [M. Grotschel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 1993]) even more powerful (in Theorems 12 and 13) in the area of geometric algorithms and combinatorial optimization. The main idea in this generalization is to allow ellipsoids to contain not the whole convex region but a part of it. This theorem is of independent interest. We give the rst strongly polynomial time algorithm for computing an equilibrium for the linear utilities case of Fisher’s market model. We consider a problem with a set B of buyers and a set G of divisible goods. Each buyer i starts with an initial integral allocation ei of money. The integral utility for buyer i of good j is Uij. We rst develop a weakly polynomial time algorithm that runs in O(n 4 logUmax +n 3 emax) time, where n =jBj +jGj. We further modify the algorithm so that it runs in O(n 4 logn) time. These algorithms improve upon the previous best running time of O(n 8 logUmax +n 7 logemax), due to [5] We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ--approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5 2 + Oγ for arbitrarily small γ > 0; for latency functions with maximum degree d≥ 2, it is d2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1 γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ-games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications.