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

On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.Comment: An extended abstract will appear in SODA 201

A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market

We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities.Comment: Preliminary version in ICALP 201

Non-Separable, Quasiconcave Utilities are Easy -- in a Perfect Price Discrimination Market Model

Recent results, establishing evidence of intractability for such restrictive utility functions as additively separable, piecewise-linear and concave, under both Fisher and Arrow-Debreu market models, have prompted the question of whether we have failed to capture some essential elements of real markets, which seem to do a good job of finding prices that maintain parity between supply and demand. The main point of this paper is to show that even non-separable, quasiconcave utility functions can be handled efficiently in a suitably chosen, though natural, realistic and useful, market model; our model allows for perfect price discrimination. Our model supports unique equilibrium prices and, for the restriction to concave utilities, satisfies both welfare theorems

Ascending-Price Algorithms for Unknown Markets

We design a simple ascending-price algorithm to compute a $(1+\varepsilon)$-approximate equilibrium in Arrow-Debreu exchange markets with weak gross substitute (WGS) property, which runs in time polynomial in market parameters and $\log 1/\varepsilon$. This is the first polynomial-time algorithm for most of the known tractable classes of Arrow-Debreu markets, which is easy to implement and avoids heavy machinery such as the ellipsoid method. In addition, our algorithm can be applied in unknown market setting without exact knowledge about the number of agents, their individual utilities and endowments. Instead, our algorithm only relies on queries to a global demand oracle by posting prices and receiving aggregate demand for goods as feedback. When demands are real-valued functions of prices, the oracles can only return values of bounded precision based on real utility functions. Due to this more realistic assumption, precision and representation of prices and demands become a major technical challenge, and we develop new tools and insights that may be of independent interest. Furthermore, our approach also gives the first polynomial-time algorithm to compute an exact equilibrium for markets with spending constraint utilities, a piecewise linear concave generalization of linear utilities. This resolves an open problem posed by Duan and Mehlhorn (2015).Comment: 33 page

The Complexity of Non-Monotone Markets

We introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory. We then prove that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. Building on this result, we settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utility functions, by proving that it is PPAD-complete when the Constant Elasticity of Substitution parameter \rho is any constant less than -1