30 research outputs found

    Equilibria, Fixed Points, and Complexity Classes

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    Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes

    Essays on the Computation of Economic Equilibria and Its Applications.

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    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

    The Complexity of Fairness through Equilibrium

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    Competitive equilibrium with equal incomes (CEEI) is a well known fair allocation mechanism; however, for indivisible resources a CEEI may not exist. It was shown in [Budish '11] that in the case of indivisible resources there is always an allocation, called A-CEEI, that is approximately fair, approximately truthful, and approximately efficient, for some favorable approximation parameters. This approximation is used in practice to assign students to classes. In this paper we show that finding the A-CEEI allocation guaranteed to exist by Budish's theorem is PPAD-complete. We further show that finding an approximate equilibrium with better approximation guarantees is even harder: NP-complete.Comment: Appeared in EC 201

    The Complexity of Non-Monotone Markets

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    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

    Proportional Dynamics in Exchange Economies

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    We study the Proportional Response dynamic in exchange economies, where each player starts with some amount of money and a good. Every day, the players bring one unit of their good and submit bids on goods they like, each good gets allocated in proportion to the bid amounts, and each seller collects the bids received. Then every player updates the bids proportionally to the contribution of each good in their utility. This dynamic models a process of learning how to bid and has been studied in a series of papers on Fisher and production markets, but not in exchange economies. Our main results are as follows: - For linear utilities, the dynamic converges to market equilibrium utilities and allocations, while the bids and prices may cycle. We give a combinatorial characterization of limit cycles for prices and bids. - We introduce a lazy version of the dynamic, where players may save money for later, and show this converges in everything: utilities, allocations, and prices. - For CES utilities in the substitute range [0,1)[0,1), the dynamic converges for all parameters. This answers an open question about exchange economies with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known. We also note that proportional response is a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.Comment: 25 pages, 6 figure

    Market Equilibrium with Transaction Costs

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    Identical products being sold at different prices in different locations is a common phenomenon. Price differences might occur due to various reasons such as shipping costs, trade restrictions and price discrimination. To model such scenarios, we supplement the classical Fisher model of a market by introducing {\em transaction costs}. For every buyer ii and every good jj, there is a transaction cost of \cij; if the price of good jj is pjp_j, then the cost to the buyer ii {\em per unit} of jj is p_j + \cij. This allows the same good to be sold at different (effective) prices to different buyers. We provide a combinatorial algorithm that computes ϵ\epsilon-approximate equilibrium prices and allocations in O(1ϵ(n+logm)mnlog(B/ϵ))O\left(\frac{1}{\epsilon}(n+\log{m})mn\log(B/\epsilon)\right) operations - where mm is the number goods, nn is the number of buyers and BB is the sum of the budgets of all the buyers

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

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    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
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