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Computing Equilibria in Anonymous Games
[article]

2007
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arXiv
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pre-print

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 epsilon

arXiv:0710.5582v1
fatcat:hhsna67thnfddoaovmbjaki7oi
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On Oblivious PTAS's for Nash Equilibrium
[article]

2011
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arXiv
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pre-print

If a game has a Nash equilibrium with probability values that are either zero or Omega(1) then this equilibrium can be found exhaustively in polynomial time. Somewhat surprisingly, we show that there is a PTAS for the games whose equilibria are guaranteed to have small-O(1/n)-values, and therefore large-Omega(n)-supports. We also point out that there is a PTAS for games with sparse payoff matrices, which are known to be PPAD-complete to solve exactly. Both algorithms are of a special kind that

arXiv:1102.2280v1
fatcat:i4ev3zrfp5fy3hfepmsup5ehi4
## more »

... e call oblivious: The algorithm just samples a fixed distribution on pairs of mixed strategies, and the game is only used to determine whether the sampled strategies comprise an eps-Nash equilibrium; the answer is yes with inverse polynomial probability. These results bring about the question: Is there an oblivious PTAS for Nash equilibrium in general games? We answer this question in the negative; our lower bound comes close to the quasi-polynomial upper bound of [Lipton, Markakis, Mehta 2003]. Another recent PTAS for anonymous games is also oblivious in a weaker sense appropriate for this class of games (it samples from a fixed distribution on unordered collections of mixed strategies), but its runtime is exponential in 1/eps. We prove that any oblivious PTAS for anonymous games with two strategies and three player types must have 1/eps^c in the exponent of the running time for some c>1/3, rendering the algorithm in [Daskalakis 2008] essentially optimal within oblivious algorithms. In contrast, we devise a poly(n) (1/eps)^O(log^2(1/eps)) non-oblivious PTAS for anonymous games with 2 strategies and any bounded number of player types. Our algorithm is based on the construction of a sparse (and efficiently computable) eps-cover of the set of all possible sums of n independent indicators, under the total variation distance. The size of the cover is poly(n) (1/ eps^O(log^2 (1/eps)).##
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A Note on Approximate Nash Equilibria
[chapter]

2006
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Lecture Notes in Computer Science
*

In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03] , and no approximation better than 1 4 is possible by any algorithm that examines equilibria involving fewer than log n strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a 1 2 -approximate Nash equilibrium in any 2-player game. For the more

doi:10.1007/11944874_27
fatcat:slbpx6lum5f3xcwfghsfl4tmni
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... demanding notion of well-supported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0 − 1), and that an approximation of 5 6 is possible contingent upon a graph-theoretic conjecture. log n 2 ) by examining all supports of size log n 2 . It was pointed out in [Alt94] that, even for zero-sum games, no algorithm that examines supports smaller than about log n can achieve an approximation better than 1 4 . Can this gap between 1 4 and 3 4 be bridged by looking at small supports? And how can the barrier of 1 4 be broken in polynomial time?##
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A note on approximate Nash equilibria

2009
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Theoretical Computer Science
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For the more demanding notion of approximately well supported Nash equilibrium due to [

doi:10.1016/j.tcs.2008.12.031
fatcat:m4bm47yox5b4bab7apukx4nuna
*Constantinos*Daskalakis, Paul W. Goldberg, Christos H. ...*Papadimitriou*, The complexity of computing a Nash equilibrium, SIAM Journal on Computing (in press) Preliminary version appeared in STOC (2006)] no nontrivial bound is known; we show that the problem can ...##
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Approximate Nash equilibria in anonymous games

2015
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Journal of Economic Theory
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We study from an algorithmic viewpoint anonymous games [Mil96, Blo99, Blo05, Kal05] . In these games a large population of players shares the same strategy set and, while players may have different payoff functions, the payoff of each depends on her own choice of strategy and the number of the other players playing each strategy (not the identity of these players). We show that, the intractability results of [DGP09a] and [Das11] for general games notwithstanding, approximate mixed Nash

doi:10.1016/j.jet.2014.02.002
fatcat:vsecyrgf5vd3nhkcll4uhkcdwm
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... a in anonymous games can be computed in polynomial time, for any desired quality of the approximation, as long as the number of strategies is bounded by some constant. In addition, if the payoff functions have a Lipschitz continuity property, we show that an approximate pure Nash equilibrium exists, whose quality depends on the number of strategies and the Lipschitz constant of the payoff functions; this equilibrium can also be computed in polynomial time. Finally, if the game has two strategies, we establish that there always exists an approximate Nash equilibrium in which either only a small number of players randomize, or of those who do, they all randomize the same way. Our results make extensive use of certain novel Central Limit-type theorems for discrete approximations of the distributions of multinonial sums.##
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Continuous Local Search
[chapter]

2011
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Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
*

We introduce CLS, for continuous local search, a class of polynomial-time checkable total functions that lies at the intersection of PPAD and PLS, and captures a particularly benign kind of local optimization in which the domain is continuous, as opposed to combinatorial, and the functions involved are continuous. We show that this class contains several well known intriguing problems which were heretofore known to lie in the intersection of PLS and PPAD but were otherwise unclassifiable:

doi:10.1137/1.9781611973082.62
dblp:conf/soda/DaskalakisP11
fatcat:ffcwl76t5nbidjz5kbxpoheemu
## more »

... g fixpoints of contraction maps, the linear complementarity problem for P matrices, finding a stationary point of a low-degree polynomial objective, the simple stochastic games of Shapley and Condon, and finding a mixed Nash equilibrium in congestion, implicit congestion, and network coordination games. The last four problems belong to CCLS, for convex CLS, another subclass of PPAD ∩ PLS seeking the componentwise local minimum of a componentwise convex function. It is open whether any or all of these problems are complete for the corresponding classes.##
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Optimum Statistical Estimation with Strategic Data Sources
[article]

2015
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arXiv
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pre-print

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

arXiv:1408.2539v2
fatcat:ccs3b33sefgs5i2xibck3ttlku
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... ng 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.##
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A Note on Strictly Competitive Games
[chapter]

2009
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Lecture Notes in Computer Science
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Strictly competitive games are a class of 2-player games often quoted in the literature to be a proper generalization of zero-sum games. Other times it is claimed, e.g. by Aumann, that strictly competitive games are only payoff transformations of zero-sum games. But to the best of our knowledge there is no proof of such claim. We shed light to this point of confusion in the literature, showing that any strictly competitive game is indeed a payoff transformation of a zero sum-game; in fact, an

doi:10.1007/978-3-642-10841-9_44
fatcat:llj2texz6rhdtfb6o6o4hhh4mm
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... fine transformation. We offer two proofs of this fact, one combinatorial and one algebraic.##
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Sparse covers for sums of indicators

2014
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Probability theory and related fields
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For all n, ǫ > 0, we show that the set of Poisson Binomial distributions on n variables admits a proper ǫ-cover in total variation distance of size n 2 + n · (1/ǫ) O(log 2 (1/ǫ)) , 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. Covers such as the one provided by Theorem 1 are of interest in the design of algorithms, when one is searching a class of

doi:10.1007/s00440-014-0582-8
fatcat:m32wigpebnfcvd6fujfspu2apm
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... istributions C to identify an element of the class with some quantitative property, or in optimizing over a class with respect to some objective. If the metric used in the construction of the cover is relevant for the problem at hand, and the cover is discrete, relatively small and easy to construct, then one can provide a useful approximation to the sought distribution by searching the cover, instead of searching all of C. For example, it is shown in [DP07, DP09, DP13] that Theorem 1 implies efficient algorithms for computing approximate Nash equilibria in an important class of multiplayer games, called anonymous [Mil96, Blo99] . We proceed with a fairly detailed sketch of the proof of our main cover theorem, Theorem 1, stating two additional results, Theorems 2 and 3. The complete proofs of Theorems 1, 2 and 3 are deferred to Sections 3, 4 and 5 respectively. Section 1.4 discusses related work, while Section 2 provides formal definitions, as well as known approximations to the Poisson Binomial distribution by simpler distributions, which are used in the proof.##
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Sparse Covers for Sums of Indicators
[article]

2014
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arXiv
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pre-print

For all n, ϵ >0, we show that the set of Poisson Binomial distributions on n variables admits a proper ϵ-cover in total variation distance of size n^2+n · (1/ϵ)^O(^2 (1/ϵ)), 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.

arXiv:1306.1265v3
fatcat:4wihd7dl7bfijpgjjgzhdjn73q
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On a Network Generalization of the Minmax Theorem
[chapter]

2009
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Lecture Notes in Computer Science
*

We consider graphical games in which the edges are zero-sum games between the endpoints/players; the payoff of a player is the sum of the payoffs from each incident edge. Such games are arguably very broad and useful models of networked economic interactions. We give a simple reduction of such games to two-person zero-sum games; as a corollary, a mixed Nash equilibrium can be computed efficiently by solving a linear program and rounding off the results. Our results render polynomially efficient, and simplify considerably, the approach in [3] .

doi:10.1007/978-3-642-02930-1_35
fatcat:zlm63emfr5as5mm7pbudzps5ve
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Common solar wind drivers behind magnetic storm-magnetospheric substorm dependency
[article]

2018
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arXiv
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pre-print

The dynamical relationship between magnetic storms and magnetospheric substorms presents one of the most controversial problems of contemporary geospace research. Here, we tackle this issue by applying a causal inference approach to two corresponding indices in conjunction with several relevant solar wind variables. We demonstrate that the vertical component of the interplanetary magnetic field is the strongest and common driver of both, storms and substorms, and explains their the previously

arXiv:1802.02477v1
fatcat:i2xow6qwljgtnoxgyfepsl2nbi
## more »

... ported association. These results hold during both solar maximum and minimum phases and suggest that, at least based on the analyzed indices, there is no statistical evidence for a direct or indirect dependency between substorms and storms. A physical mechanism by which substorms drive storms or vice versa is, therefore, unlikely.##
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Euthanasia and suicide in antiquity: viewpoint of the dramatists and philosophers

2007
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Journal of the Royal Society of Medicine
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Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons
[article]

2018
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arXiv
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pre-print

We analyze linear independence of rank one tensors produced by tensor powers of randomly perturbed vectors. This enables efficient decomposition of sums of high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider range of perturbation models, including discrete ones. We give an application to recovering assemblies of neurons. Assemblies are large sets of neurons representing specific memories or concepts. The size of the intersection of two assemblies has been shown in

arXiv:1810.11896v1
fatcat:m6kpnq25tzgjtinm3cykwgzv2i
## more »

... ments to represent the extent to which these memories co-occur or these concepts are related; the phenomenon is called association of assemblies. This suggests that an animal's memory is a complex web of associations, and poses the problem of recovering this representation from cognitive data. Motivated by this problem, we study the following more general question: Can we reconstruct the Venn diagram of a family of sets, given the sizes of their ℓ-wise intersections? We show that as long as the family of sets is randomly perturbed, it is enough for the number of measurements to be polynomially larger than the number of nonempty regions of the Venn diagram to fully reconstruct the diagram.##
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Computing Equilibria in Anonymous Games

2007
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Foundations of Computer Science (FOCS), IEEE Annual Symposium on
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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 λ), where s is the number of strategies and λ is the Lipschitz constant of the utilities. Finally,

doi:10.1109/focs.2007.4389482
fatcat:jzcpmfin5bdwpfva7o5sal3u2a
## more »

... e show that there is a PTAS for finding an -approximate Nash equilibrium when the number of strategies is two.
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