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

The Edgeworth Conjecture with Small Coalitions and Approximate Equilibria in Large Economies

We revisit the connection between bargaining and equilibrium in exchange economies, and study its algorithmic implications. We consider bargaining outcomes to be allocations that cannot be blocked (i.e., profitably re-traded) by coalitions of small size and show that these allocations must be approximate Walrasian equilibria. Our results imply that deciding whether an allocation is approximately Walrasian can be done in polynomial time, even in economies for which finding an equilibrium is known to be computationally hard.Comment: 26 page

Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets

In 1979, Hylland and Zeckhauser \cite{hylland} gave a simple and general scheme for implementing a one-sided matching market using the power of a pricing mechanism. Their method has nice properties -- it is incentive compatible in the large and produces an allocation that is Pareto optimal -- and hence it provides an attractive, off-the-shelf method for running an application involving such a market. With matching markets becoming ever more prevalant and impactful, it is imperative to finally settle the computational complexity of this scheme. We present the following partial resolution: 1. A combinatorial, strongly polynomial time algorithm for the special case of $0/1$ utilities. 2. An example that has only irrational equilibria, hence proving that this problem is not in PPAD. Furthermore, its equilibria are disconnected, hence showing that the problem does not admit a convex programming formulation. 3. A proof of membership of the problem in the class FIXP. We leave open the (difficult) question of determining if the problem is FIXP-hard. Settling the status of the special case when utilities are in the set $\{0, {\frac 1 2}, 1 \}$ appears to be even more difficult.Comment: 22 page

The Edgeworth Conjecture with Small Coalitions and Approximate Equilibria in Large Economies

We revisit the connection between bargaining and equilibrium in exchange economies, and study its algorithmic implications. We consider bargaining outcomes to be allocations that cannot be blocked (i.e., profitably re-traded) by coalitions of small size and show that these allocations must be approximate Walrasian equilibria. Our results imply that deciding whether an allocation is approximately Walrasian can be done in polynomial time, even in economies for which finding an equilibrium is known to be computationally hard

Essays on the Computation of Economic Equilibria and Its Applications.

The computation of economic equilibria is a central problem in algorithmic game theory. In this dissertation, we investigate the existence of economic equilibria in several markets and games, the complexity of computing economic equilibria, and its application to rankings. It is well known that a competitive economy always has an equilibrium under mild conditions. In this dissertation, we study the complexity of computing competitive equilibria. We show that given a competitive economy that fully respects all the conditions of Arrow-Debreu's existence theorem, it is PPAD-hard to compute an approximate competitive equilibrium. Furthermore, it is still PPAD-Complete to compute an approximate equilibrium for economies with additively separable piecewise linear concave utility functions. Degeneracy is an important concept in game theory. We study the complexity of deciding degeneracy in games. We show that it is NP-Complete to decide whether a bimatrix game is degenerate. With the advent of the Internet, an agent can easily have access to multiple accounts. In this dissertation we study the path auction game, which is a model for QoS routing, supply chain management, and so on, with multiple edge ownership. We show that the condition of multiple edge ownership eliminates the possibility of reasonable solution concepts, such as a strategyproof or false-name-proof mechanism or Pareto efficient Nash equilibria. The stationary distribution (an equilibrium point) of a Markov chain is widely used for ranking purposes. One of the most important applications is PageRank, part of the ranking algorithm of Google. By making use of perturbation theories of Markov chains, we show the optimal manipulation strategies of a Web spammer against PageRank under a few natural constraints. Finally, we make a connection between the ranking vector of PageRank or the Invariant method and the equilibrium of a Cobb-Douglas market. Furthermore, we propose the CES ranking method based on the Constant Elasticity of Substitution (CES) utility functions.Ph.D.Computer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64821/1/duye_1.pd

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 Edgeworth Conjecture with Small Coalitions and Approximate Equilibria in Large Economies

We revisit the connection between bargaining and equilibrium in exchange economies, and study its algorithmic implications. We consider bargaining outcomes to be allocations that cannot be blocked (i.e., profitably re-traded) by coalitions of small size and show that these allocations must be approximate Walrasian equilibria. Our results imply that deciding whether an allocation is approximately Walrasian can be done in polynomial time, even in economies for which finding an equilibrium is known to be computationally hard

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

We introduce a new technique for proving membership of problems in FIXP - the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets. In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser