Flaxman, Abraham, Gamarnik, David and Sorkin, Gregory B.
(2011)
*First-passage percolation on a ladder graph, and the path cost in a VCG auction.*
Random Structures and Algorithms, 38 (3).
pp. 350-364.
ISSN 1042-9832

## 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.

Item Type: | Article |
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Official URL: | http://eu.wiley.com/WileyCDA/WileyTitle/productCd-... |

Additional Information: | © 2010 Wiley Periodicals, Inc. |

Divisions: | Management |

Subjects: | H Social Sciences > HF Commerce Q Science > QA Mathematics |

Sets: | Research centres and groups > Management Science Group Departments > Management |

Date Deposited: | 13 Apr 2011 10:47 |

Last Modified: | 20 Jan 2020 04:20 |

URI: | http://eprints.lse.ac.uk/id/eprint/35055 |

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