118 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
Ascending-Price Algorithms for Unknown Markets
We design a simple ascending-price algorithm to compute a
-approximate equilibrium in Arrow-Debreu exchange markets with
weak gross substitute (WGS) property, which runs in time polynomial in market
parameters and . 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
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
Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives
A well-studied nonlinear extension of the minimum-cost flow problem is to
minimize the objective over feasible flows ,
where on every arc of the network, is a convex function. We give
a strongly polynomial algorithm for the case when all 's are convex
quadratic functions, settling an open problem raised e.g. by Hochbaum [1994].
We also give strongly polynomial algorithms for computing market equilibria in
Fisher markets with linear utilities and with spending constraint utilities,
that can be formulated in this framework (see Shmyrev [2009], Devanur et al.
[2011]). For the latter class this resolves an open question raised by Vazirani
[2010]. The running time is for quadratic costs,
for Fisher's markets with linear utilities and
for spending constraint utilities.
All these algorithms are presented in a common framework that addresses the
general problem setting. Whereas it is impossible to give a strongly polynomial
algorithm for the general problem even in an approximate sense (see Hochbaum
[1994]), we show that assuming the existence of certain black-box oracles, one
can give an algorithm using a strongly polynomial number of arithmetic
operations and oracle calls only. The particular algorithms can be derived by
implementing these oracles in the respective settings
Online Bipartite Matching and Adwords
The simplicity of the recently obtained proof of the optimal algorithm for
online bipartite matching (OBM), called RANKING \cite{KVV}, naturally raises
the possibility of extending this algorithm to the adwords problem, or its
special case called SMALL, in which bids are small compared to budgets; the
latter has been of considerable practical significance in ad auctions
\cite{MSVV}.
The attractive feature of our approach, in contrast to \cite{MSVV}, was that
it would yield a {\em budget-oblivious algorithm}, i.e., during its execution,
the algorithm would not need to know what fraction of budget each bidder has
spent -- only whether there is budget left-over. This would immediately render
the algorithm useful for autobidding platforms.
Our attempt met several hurdles, which are described in detail in the paper.
In particular, a substantial probabilistic development led us to obtain an
optimal, online, budget-oblivious algorithm for SINGLE-VALUED, which is
intermediate between OMB and the adwords problem; this algorithm is a natural
generalization of RANKING. For SMALL, we managed to overcome all but one
hurdle, namely failure of a property, called the no-surpassing property.
Interestingly enough, this property plays a minor role in the proofs of RANKING
as well as our algorithm for SINGLE-VALUED.Comment: 27 page
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