140 research outputs found
Resource Competition on Integral Polymatroids
We study competitive resource allocation problems in which players distribute
their demands integrally on a set of resources subject to player-specific
submodular capacity constraints. Each player has to pay for each unit of demand
a cost that is a nondecreasing and convex function of the total allocation of
that resource. This general model of resource allocation generalizes both
singleton congestion games with integer-splittable demands and matroid
congestion games with player-specific costs. As our main result, we show that
in such general resource allocation problems a pure Nash equilibrium is
guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure
Nash equilibrium.Comment: 17 page
Matroids are Immune to Braess Paradox
The famous Braess paradox describes the following phenomenon: It might happen
that the improvement of resources, like building a new street within a
congested network, may in fact lead to larger costs for the players in an
equilibrium. In this paper we consider general nonatomic congestion games and
give a characterization of the maximal combinatorial property of strategy
spaces for which Braess paradox does not occur. In a nutshell, bases of
matroids are exactly this maximal structure. We prove our characterization by
two novel sensitivity results for convex separable optimization problems over
polymatroid base polyhedra which may be of independent interest.Comment: 21 page
Quantized VCG Mechanisms for Polymatroid Environments
Many network resource allocation problems can be viewed as allocating a
divisible resource, where the allocations are constrained to lie in a
polymatroid. We consider market-based mechanisms for such problems. Though the
Vickrey-Clarke-Groves (VCG) mechanism can provide the efficient allocation with
strong incentive properties (namely dominant strategy incentive compatibility),
its well-known high communication requirements can prevent it from being used.
There have been a number of approaches for reducing the communication costs of
VCG by weakening its incentive properties. Here, instead we take a different
approach of reducing communication costs via quantization while maintaining
VCG's dominant strategy incentive properties. The cost for this approach is a
loss in efficiency which we characterize. We first consider quantizing the
resource allocations so that agents need only submit a finite number of bids
instead of full utility function. We subsequently consider quantizing the
agent's bids
Entropic Inequalities and Marginal Problems
A marginal problem asks whether a given family of marginal distributions for
some set of random variables arises from some joint distribution of these
variables. Here we point out that the existence of such a joint distribution
imposes non-trivial conditions already on the level of Shannon entropies of the
given marginals. These entropic inequalities are necessary (but not sufficient)
criteria for the existence of a joint distribution. For every marginal problem,
a list of such Shannon-type entropic inequalities can be calculated by
Fourier-Motzkin elimination, and we offer a software interface to a
Fourier-Motzkin solver for doing so. For the case that the hypergraph of given
marginals is a cycle graph, we provide a complete analytic solution to the
problem of classifying all relevant entropic inequalities, and use this result
to bound the decay of correlations in stochastic processes. Furthermore, we
show that Shannon-type inequalities for differential entropies are not relevant
for continuous-variable marginal problems; non-Shannon-type inequalities are,
both in the discrete and in the continuous case. In contrast to other
approaches, our general framework easily adapts to situations where one has
additional (conditional) independence requirements on the joint distribution,
as in the case of graphical models. We end with a list of open problems.
A complementary article discusses applications to quantum nonlocality and
contextuality.Comment: 26 pages, 3 figure
Designing Coalition-Proof Reverse Auctions over Continuous Goods
This paper investigates reverse auctions that involve continuous values of
different types of goods, general nonconvex constraints, and second stage
costs. We seek to design the payment rules and conditions under which
coalitions of participants cannot influence the auction outcome in order to
obtain higher collective utility. Under the incentive-compatible
Vickrey-Clarke-Groves mechanism, we show that coalition-proof outcomes are
achieved if the submitted bids are convex and the constraint sets are of a
polymatroid-type. These conditions, however, do not capture the complexity of
the general class of reverse auctions under consideration. By relaxing the
property of incentive-compatibility, we investigate further payment rules that
are coalition-proof without any extra conditions on the submitted bids and the
constraint sets. Since calculating the payments directly for these mechanisms
is computationally difficult for auctions involving many participants, we
present two computationally efficient methods. Our results are verified with
several case studies based on electricity market data
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