### First-Passage Percolation on a Width-2 Strip and the Path Cost in a VCG Auction [chapter]

Abraham Flaxman, David Gamarnik, Gregory B. Sorkin
2006 Lecture Notes in Computer Science
This paper studies the time constant for first-passage percolation and the Vickery-Clarke-Groves (VCG) payment for the shortest path on a width-2 strip with random edge costs. These statistics attempt to describe two seemingly unrelated phenomena, arising in physics and economics respectively: the first-passage percolation time predicts how long it takes for a fluid to spread through a random medium, while the VCG payment for the shortest path is the cost maximizing social welfare among selfish
more » ... agents. However, our analyses of the two are quite similar, and require solving (slightly different) recursive distributional equations. For first-passage percolation where the edge costs are independent Bernoulli random variables we find the time constant exactly. It turns out to be a rational function of the Bernoulli parameter; for example, for Be(1/2) the time constant is 9/28. For first-passage percolation where the edge costs are uniform random variables we present a reasonably efficient means for obtaining upper and lower bounds; our calculations show the time constant to be between 0.42 and 0.43. Using 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 distribution. For the VCG payment, we restrict our attention to the case where the edge costs are independent Bernoulli random variables. Here we find that the VCG payment for a strip of length n tends to n times a rational function of the Bernoulli parameter, which we determine explicitly. For Be(1/2), the VCG payment over n tends to 1829/1568 ≈ 1.166, which is about 3.63 times the length of the shortest path.