30 research outputs found
Equilibria, Fixed Points, and Complexity Classes
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.
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
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
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
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
Market Equilibrium with Transaction Costs
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 and every good , there is a
transaction cost of \cij; if the price of good is , then the cost to
the buyer {\em per unit} of 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 -approximate
equilibrium prices and allocations in
operations -
where is the number goods, is the number of buyers and is the sum
of the budgets of all the buyers
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