13 research outputs found
Market Equilibrium in Exchange Economies with Some Families of Concave Utility Functions
We present explicit convex programs which characterize the equilibrium for certain additively separable utility functions and CES functions. These include some CES utility functions that do not satisfy weak gross substitutability.Exchange economy, computation of equilibria, convex feasibility problem
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
Combinatorial Algorithms for General Linear Arrow-Debreu Markets
We present a combinatorial algorithm for determining the market clearing prices of a general linear Arrow-Debreu market, where every agent can own multiple goods. The existing combinatorial algorithms for linear Arrow-Debreu markets consider the case where each agent can own all of one good only. We present an O~((n+m)^7 log^3(UW)) algorithm where n, m, U and W refer to the number of agents, the number of goods, the maximal integral utility and the maximum quantity of any good in the market respectively. The algorithm refines the iterative algorithm of Duan, Garg and Mehlhorn using several new ideas. We also identify the hard instances for existing combinatorial algorithms for linear Arrow-Debreu markets. In particular we find instances where the ratio of the maximum to the minimum equilibrium price of a good is U^{Omega(n)} and the number of iterations required by the existing iterative combinatorial algorithms of Duan, and Mehlhorn and Duan, Garg, and Mehlhorn are high. Our instances also separate the two algorithms
Proportional Dynamics in Exchange Economies
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 , 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
Asynchronous Proportional Response Dynamics in Markets with Adversarial Scheduling
We study Proportional Response Dynamics (PRD) in linear Fisher markets where
participants act asynchronously. We model this scenario as a sequential process
in which in every step, an adversary selects a subset of the players that will
update their bids, subject to liveness constraints. We show that if every
bidder individually uses the PRD update rule whenever they are included in the
group of bidders selected by the adversary, then (in the generic case) the
entire dynamic converges to a competitive equilibrium of the market. Our proof
technique uncovers further properties of linear Fisher markets, such as the
uniqueness of the equilibrium for generic parameters and the convergence of
associated best-response dynamics and no-swap regret dynamics under certain
conditions
A strongly polynomial algorithm for linear exchange markets
We present a strongly polynomial algorithm for computing an equilibrium in Arrow-Debreu exchange markets with linear utilities. Our algorithm is based on a variant of the weakly-polynomial Duan–Mehlhorn (DM) algorithm. We use the DM algorithm as a subroutine to identify revealed edges, i.e. pairs of agents and goods that must correspond to best bang-per-buck transactions in every equilibrium solution. Every time a new revealed edge is found, we use another subroutine that decides if there is an optimal solution using the current set of revealed edges, or if none exists, finds the solution that approximately minimizes the violation of the demand and supply constraints. This task can be reduced to solving a linear program (LP). Even though we are unable to solve this LP in strongly polynomial time, we show that it can be approximated by a simpler LP with two variables per inequality that is solvable in strongly polynomial time
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