56 research outputs found
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
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
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
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
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
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 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 appears to be even more difficult.Comment: 22 page
- …