# First-passage percolation on a ladder graph, and the path cost in a VCG auction

## 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.