6 research outputs found
Computing Equilibrium in Matching Markets
Market equilibria of matching markets offer an intuitive and fair solution
for matching problems without money with agents who have preferences over the
items. Such a matching market can be viewed as a variation of Fisher market,
albeit with rather peculiar preferences of agents. These preferences can be
described by piece-wise linear concave (PLC) functions, which however, are not
separable (due to each agent only asking for one item), are not monotone, and
do not satisfy the gross substitute property-- increase in price of an item can
result in increased demand for the item. Devanur and Kannan in FOCS 08 showed
that market clearing prices can be found in polynomial time in markets with
fixed number of items and general PLC preferences. They also consider Fischer
markets with fixed number of agents (instead of fixed number of items), and
give a polynomial time algorithm for this case if preferences are separable
functions of the items, in addition to being PLC functions.
Our main result is a polynomial time algorithm for finding market clearing
prices in matching markets with fixed number of different agent preferences,
despite that the utility corresponding to matching markets is not separable. We
also give a simpler algorithm for the case of matching markets with fixed
number of different items
Approximating equilibrium under constrained piecewise linear concave utilities with applications to matching markets
We study the equilibrium computation problem in the Fisher market model with constrained piecewise linear concave (PLC) utilities. This general class captures many well-studied special cases, including markets with PLC utilities, markets with satiation, and matching markets. For the special case of PLC utilities, although the problem is PPAD-hard, Devanur and Kannan (FOCS 2008) gave a polynomial-time algorithm when the number of items is constant. Our main result is a fixed parameter approximation scheme for computing an approximate equilibrium, where the parameters are the number of agents and the approximation accuracy. This provides an answer to an open question by Devanur and Kannan for PLC utilities, and gives a simpler and faster algorithm for matching markets as the one by Alaei, Jalaly and Tardos (EC 2017). The main technical idea is to work with the stronger concept of thrifty equilibria, and approximating the input utility functions by ‘robust’ utilities that have favorable marginal properties. With some restrictions, the results also extend to the Arrow–Debreu exchange market model
Constant inapproximability for Fisher markets
We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD = P. In fact, we prove that computing any approximation better than 1/11 is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility function
Algorithms for Competitive Division of Chores
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 Bogomolnaia et al (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 Devanur et al (2002).
Our algorithm immediately gives an approximately-fair allocation of
indivisible chores by the rounding technique of Barman and Krishnamurthy
(2018).Comment: 38 pages, 4 figure