1,634 research outputs found
Computing Equilibria in Anonymous Games
We present efficient approximation algorithms for finding Nash equilibria in
anonymous games, that is, games in which the players utilities, though
different, do not differentiate between other players. Our results pertain to
such games with many players but few strategies. We show that any such game has
an approximate pure Nash equilibrium, computable in polynomial time, with
approximation O(s^2 L), where s is the number of strategies and L is the
Lipschitz constant of the utilities. Finally, we show that there is a PTAS for
finding an epsilo
Sparse Covers for Sums of Indicators
For all , we show that the set of Poisson Binomial
distributions on variables admits a proper -cover in total
variation distance of size ,
which can also be computed in polynomial time. We discuss the implications of
our construction for approximation algorithms and the computation of
approximate Nash equilibria in anonymous games.Comment: PTRF, to appea
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
On the Complexity of Dynamic Mechanism Design
We introduce a dynamic mechanism design problem in which the designer wants
to offer for sale an item to an agent, and another item to the same agent at
some point in the future. The agent's joint distribution of valuations for the
two items is known, and the agent knows the valuation for the current item (but
not for the one in the future). The designer seeks to maximize expected
revenue, and the auction must be deterministic, truthful, and ex post
individually rational. The optimum mechanism involves a protocol whereby the
seller elicits the buyer's current valuation, and based on the bid makes two
take-it-or-leave-it offers, one for now and one for the future. We show that
finding the optimum deterministic mechanism in this situation - arguably the
simplest meaningful dynamic mechanism design problem imaginable - is NP-hard.
We also prove several positive results, among them a polynomial linear
programming-based algorithm for the optimum randomized auction (even for many
bidders and periods), and we show strong separations in revenue between
non-adaptive, adaptive, and randomized auctions, even when the valuations in
the two periods are uncorrelated. Finally, for the same problem in an
environment in which contracts cannot be enforced, and thus perfection of
equilibrium is necessary, we show that the optimum randomized mechanism
requires multiple rounds of cheap talk-like interactions
Cycles in adversarial regularized learning
Regularized learning is a fundamental technique in online optimization,
machine learning and many other fields of computer science. A natural question
that arises in these settings is how regularized learning algorithms behave
when faced against each other. We study a natural formulation of this problem
by coupling regularized learning dynamics in zero-sum games. We show that the
system's behavior is Poincar\'e recurrent, implying that almost every
trajectory revisits any (arbitrarily small) neighborhood of its starting point
infinitely often. This cycling behavior is robust to the agents' choice of
regularization mechanism (each agent could be using a different regularizer),
to positive-affine transformations of the agents' utilities, and it also
persists in the case of networked competition, i.e., for zero-sum polymatrix
games.Comment: 22 pages, 4 figure
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