446 research outputs found
Discrepancy convergence for the drunkard's walk on the sphere
We analyze the drunkard's walk on the unit sphere with step size theta and
show that the walk converges in order constant/sin^2(theta) steps in the
discrepancy metric. This is an application of techniques we develop for
bounding the discrepancy of random walks on Gelfand pairs generated by
bi-invariant measures. In such cases, Fourier analysis on the acting group
admits tractable computations involving spherical functions. We advocate the
use of discrepancy as a metric on probabilities for state spaces with isometric
group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at
http://www.math.hmc.edu/~su/papers.htm
A simplicial algorithm approach to Nash equilibria in concave games
In this paper we demonstrate a new method for computing approximate Nash equilibria in n-person games. Strategy spaces are assumed to be represented by simplices, while payoff functions are assumed to be concave. Our procedure relies on a simplicial algorithm that traces paths through the set of strategy profiles using a new variant of Sperner's Lemma for labelled triangulations of simplotopes, which we prove in this paper. Our algorithm uses a labelling derived from the satisficing function of Geanakoplos (2003) and can be used to compute approximate Nash equilibria for payoff functions that are not necessarily linear. Finally, in bimatrix games, we can compare our simplicial algorithm to the combinatorial algorithm proposed by Lemke and Howson (1964).simplicial algorithm, Nash equilibria, strategy labelling
The Shapley Value of Phylogenetic Trees
Every weighted tree corresponds naturally to a cooperative game that we call
a "tree game"; it assigns to each subset of leaves the sum of the weights of
the minimal subtree spanned by those leaves. In the context of phylogenetic
trees, the leaves are species and this assignment captures the diversity
present in the coalition of species considered. We consider the Shapley value
of tree games and suggest a biological interpretation. We determine the linear
transformation M that shows the dependence of the Shapley value on the edge
weights of the tree, and we also compute a null space basis of M. Both depend
on the "split counts" of the tree. Finally, we characterize the Shapley value
on tree games by four axioms, a counterpart to Shapley's original theorem on
the larger class of cooperative games.Comment: References added, and a section (calculating the Shapley value of a
tree game from its subtrees) was removed for length reasons (request of
referee) and may appear in another paper. 16 pages; related work at
http://www.math.hmc.edu/~su/papers.html. Journal of Mathematical Biology, to
appear. The original article is available at http://www.springerlink.co
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