4,631 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 planar algebra of a semisimple and cosemisimple Hopf algebra
To a semisimple and cosemisimple Hopf algebra over an algebraically closed
field, we associate a planar algebra defined by generators and relations and
show that it is a connected, irreducible, spherical, non-degenerate planar
algebra with non-zero modulus and of depth two. This association is shown to
yield a bijection between (the isomorphism classes, on both sides, of) such
objects.Comment: 16 pages, 20 figures; content adde
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
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
Guionnet-Jones-Shlyakhtenko subfactors associated to finite-dimensional Kac algebras
We analyse the Guionnet-Jones-Shlyakhtenko construction for the planar
algebra associated to a finite-dimensional Kac algebra and identify the factors
that arise as finite interpolated free group factors.Comment: 18 pages, 21 figures, corrected typo
An Incentive Compatible, Efficient Market for Air Traffic Flow Management
We present a market-based approach to the Air Traffic Flow Management (ATFM)
problem. The goods in our market are delays and buyers are airline companies;
the latter pay money to the FAA to buy away the desired amount of delay on a
per flight basis. We give a notion of equilibrium for this market and an LP
whose solution gives an equilibrium allocation of flights to landing slots as
well as equilibrium prices for the landing slots. Via a reduction to matching,
we show that this equilibrium can be computed combinatorially in strongly
polynomial time. Moreover, there is a special set of equilibrium prices, which
can be computed easily, that is identical to the VCG solution, and therefore
the market is incentive compatible in dominant strategy.Comment: arXiv admin note: substantial text overlap with arXiv:1109.521
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