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First‐passage percolation on a ladder graph, and the path cost in a VCG auction
Author(s) -
Flaxman Abraham,
Gamarnik David,
Sorkin Gregory B.
Publication year - 2011
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20328
Subject(s) - bernoulli's principle , bernoulli distribution , mathematics , random variable , random graph , shortest path problem , bounded function , constant (computer programming) , path (computing) , function (biology) , graph , mathematical optimization , combinatorics , computer science , mathematical analysis , physics , statistics , evolutionary biology , biology , thermodynamics , programming language
Abstract This paper studies the time constant for first‐passage percolation, and the Vickrey‐Clarke‐Groves (VCG) payment, for the shortest path on a ladder graph (a width‐2 strip) with random edge costs, treating both in a unified way based on recursive distributional equations. For first‐passage percolation where the edge costs are independent Bernoulli random variables we find the time constant exactly; it is a rational function of the Bernoulli parameter. For first‐passage percolation where the edge costs are uniform random variables we present a reasonably efficient means for obtaining arbitrarily close upper and lower bounds. Using properties of Harris chains we also show that the incremental cost to advance through the medium has a unique stationary distribution, and we compute stochastic lower and upper bounds. We rely on no special properties of the uniform distribution: the same methods could be applied to any well‐behaved, bounded cost distribution. For the VCG payment, with Bernoulli‐distributed costs the payment for an n ‐long ladder, divided by n , tends to an explicit rational function of the Bernoulli parameter. Again, our methods apply more generally. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 350‐364, 2011