Approximate well-supported Nash equilibria in symmetric bimatrix games

The $\varepsilon$-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than $\varepsilon$ to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant $\varepsilon$ currently known for which there is a polynomial-time algorithm that computes an $\varepsilon$-well-supported Nash equilibrium in bimatrix games is slightly below $2/3$. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a $(1/2+\delta)$-well-supported Nash equilibrium, for an arbitrarily small positive constant $\delta$

A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium

We present a direct reduction from k-player games to 2-player games that preserves approximate Nash equilibrium. Previously, the computational equivalence of computing approximate Nash equilibrium in k-player and 2-player games was established via an indirect reduction. This included a sequence of works defining the complexity class PPAD, identifying complete problems for this class, showing that computing approximate Nash equilibrium for k-player games is in PPAD, and reducing a PPAD-complete problem to computing approximate Nash equilibrium for 2-player games. Our direct reduction makes no use of the concept of PPAD, thus eliminating some of the difficulties involved in following the known indirect reduction.Comment: 21 page

ICGA Journal 24(2): special issue on Ken Thompson

This special issue of the ICGA_J focuses on Ken Thompson on the occasion of his retirement from Bell Labs in December 2000. The contents touch on Unix and cover ken's contribution to computer games, computer chess and 'ICCA', the International Computer Chess Association

The Asymptotic Behavior of Diameters in the Average

AbstractIn 1975 R. Ahlswede and G. Katona posed the following average distance problem (Discrete Math.17 (1977), 10): For every cardinality a ∈ {1, ..., 2n} determine subsets A of {0, 1}n with # A = a, which have minimal average inner Hamming distance. Recently I. Althöfer and T. Sillke (J. Combin. Theory Ser. B56 (1992), 296-301) gave an exact solution of this problem for the central value a = 2n − 1. Here we present nearly optimal solutions for a = 2λn with 0 < λ < 1: Asymptotically it is not possible to do better than choosing An = {(x1, ..., xn)|∑nt = 1xt = ⌊αn⌋}, where λ = −αlog α − (1 − α) log(1 − α). Next we investigate the following more general problem, which occurs, for instance, in the construction of good write-efficient-memories (WEMs). Given any finite set M with an arbitrary cost function d: M × M → R, the corresponding sum type cost function dn: Mn × Mn → R is defined by dn((x1, ..., xn, (y1, ..., yn) = ∑nt = 1d(xt, yt). The task is to find sets An, of a given cardinality, which minimize the average inner cost (1/(#An)2)∑a∈An∑a′∈Andn(a, a′). We prove that asymptotically optimal sets can be constructed by using "mixed typical sequences" with at most two different local configurations. As a non-trivial example we look at the Hamming distance for M = {1, ..., m} with m ≥ 3

Graph spanners in the streaming model: an experimental study

This article reports the results of an extensive experimental analysis of efficient algorithms for computing graph spanners in the data streaming model, where an (alpha, beta)-spanner of a graph G is a subgraph S subset of G such that for each pair of vertices the distance in S is at most a times the distance in G plus beta. To the best of our knowledge, this is the first computational study of graph spanner algorithms in a streaming setting. We compare experimentally the randomized algorithms proposed by Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and by Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716-727, 9-13 July 2007) for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007), Engineering and Applications Track, Eilat, Israel, 8-10 October 2007. LNCS, vol. 4698, pp. 605-617, 2007), designed for building small stretch spanners. All the algorithms we implemented work in a data streaming model where the input graph is given as a stream of edges in arbitrary order, and all of them need a single pass over the data. Differently from the algorithm in Ausiello et al., the algorithms in Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716-727, 9-13 July 2007) need to know in advance the number of vertices in the graph. The results of our experimental investigation on several input families confirm that all these algorithms are very efficient in practice, finding spanners with stretch and size much smaller than the theoretical bounds and comparable to those obtainable by off-line algorithms. Moreover, our experimental findings confirm that small values of the stretch factor are the case of interest in practice, and that the algorithm by Ausiello et al. tends to produce spanners of better quality than the algorithms by Baswana and Elkin, while still using a comparable amount of time and space resources