6,680 research outputs found
Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games
We show that there is a polynomial-time approximation scheme for computing
Nash equilibria in anonymous games with any fixed number of strategies (a very
broad and important class of games), extending the two-strategy result of
Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a
probabilistic result of more general interest: The distribution of the sum of n
independent unit vectors with values ranging over {e1, e2, ...,ek}, where ei is
the unit vector along dimension i of the k-dimensional Euclidean space, can be
approximated by the distribution of the sum of another set of independent unit
vectors whose probabilities of obtaining each value are multiples of 1/z for
some integer z, and so that the variational distance of the two distributions
is at most eps, where eps is bounded by an inverse polynomial in z and a
function of k, but with no dependence on n. Our probabilistic result specifies
the construction of a surprisingly sparse eps-cover -- under the total
variation distance -- of the set of distributions of sums of independent unit
vectors, which is of interest on its own right.Comment: In the 49th Annual IEEE Symposium on Foundations of Computer Science,
FOCS 200
Optimum Statistical Estimation with Strategic Data Sources
We propose an optimum mechanism for providing monetary incentives to the data
sources of a statistical estimator such as linear regression, so that high
quality data is provided at low cost, in the sense that the sum of payments and
estimation error is minimized. The mechanism applies to a broad range of
estimators, including linear and polynomial regression, kernel regression, and,
under some additional assumptions, ridge regression. It also generalizes to
several objectives, including minimizing estimation error subject to budget
constraints. Besides our concrete results for regression problems, we
contribute a mechanism design framework through which to design and analyze
statistical estimators whose examples are supplied by workers with cost for
labeling said examples
Incremental Recompilation of Knowledge
Approximating a general formula from above and below by Horn formulas (its
Horn envelope and Horn core, respectively) was proposed by Selman and Kautz
(1991, 1996) as a form of ``knowledge compilation,'' supporting rapid
approximate reasoning; on the negative side, this scheme is static in that it
supports no updates, and has certain complexity drawbacks pointed out by
Kavvadias, Papadimitriou and Sideri (1993). On the other hand, the many
frameworks and schemes proposed in the literature for theory update and
revision are plagued by serious complexity-theoretic impediments, even in the
Horn case, as was pointed out by Eiter and Gottlob (1992), and is further
demonstrated in the present paper. More fundamentally, these schemes are not
inductive, in that they may lose in a single update any positive properties of
the represented sets of formulas (small size, Horn structure, etc.). In this
paper we propose a new scheme, incremental recompilation, which combines Horn
approximation and model-based updates; this scheme is inductive and very
efficient, free of the problems facing its constituents. A set of formulas is
represented by an upper and lower Horn approximation. To update, we replace the
upper Horn formula by the Horn envelope of its minimum-change update, and
similarly the lower one by the Horn core of its update; the key fact which
enables this scheme is that Horn envelopes and cores are easy to compute when
the underlying formula is the result of a minimum-change update of a Horn
formula by a clause. We conjecture that efficient algorithms are possible for
more complex updates.Comment: See http://www.jair.org/ for any accompanying file
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions
We show that the widely used homotopy method for solving fixpoint problems,
as well as the Harsanyi-Selten equilibrium selection process for games, are
PSPACE-complete to implement. Extending our result for the Harsanyi-Selten
process, we show that several other homotopy-based algorithms for finding
equilibria of games are also PSPACE-complete to implement. A further
application of our techniques yields the result that it is PSPACE-complete to
compute any of the equilibria that could be found via the classical
Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in
[Savani and von Stengel]. These results show that our techniques can be widely
applied and suggest that the PSPACE-completeness of implementing homotopy
methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc
First-order transition in small-world networks
The small-world transition is a first-order transition at zero density of
shortcuts, whereby the normalized shortest-path distance undergoes a
discontinuity in the thermodynamic limit. On finite systems the apparent
transition is shifted by . Equivalently a ``persistence
size'' can be defined in connection with finite-size
effects. Assuming , simple rescaling arguments imply that
. We confirm this result by extensive numerical simulation in one to
four dimensions, and argue that implies that this transition is
first-order.Comment: 4 pages, 3 figures, To appear in Europhysics Letter
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