First-Passage Percolation on a Width-2 Strip and the Path Cost in a VCG Auction
We study both 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 of maximizing social welfare among selfish agents. However, our analyses of the two are quite similar, and require solving (slightly different) recursive distributional equations. Using Harris chains, we can characterize distributions, not just expectations.
KeywordsMarkov Chain Short Path Edge Weight Edge Cost Unique Stationary Distribution
Unable to display preview. Download preview PDF.
- [AS04]Aldous, D.J., Steele, J.M.: The objective method: probabilistic combinatorial optimization and local weak convergence. In: Probability on discrete structures, Encyclopaedia Math. Sci., vol. 110, pp. 1–72. Springer, Berlin (2004)Google Scholar
- [AT02]Archer, A., Tardos, É.: Frugal path mechanisms. In: Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms (San Francisco, California), January 06-08, pp. 991–999 (2002)Google Scholar
- [CR04]Czumaj, A., Ronen, A.: On the expected payment of mechanisms for task allocation (extended abstract). In: Proceedings of the fifth ACM Conference on Electronic Commerce, pp. 252–253 (2004)Google Scholar
- [Dur96]Durrett, R.: Probability: theory and examples, 2nd edn. Duxbury Press, Belmont (1996)Google Scholar
- [Elk05]Elkind, E.: True costs of cheap labor are hard to measure: edge deletion and VCG payments in graphs. In: Proceedings of the sixth ACM Conference on Electronic Commerce, pp. 108–117 (2005)Google Scholar
- [ESS04]Elkind, E., Sahai, A., Steiglitz, K.: Frugality in path auctions. In: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 694–702 (2004)Google Scholar
- [GNSar]Gamarnik, D., Nowicki, T., Swirscsz, G.: Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method, Random Structures and Algorithms (to appear)Google Scholar
- [HW65]Hammersley, J.M., Welsh, J.D.: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. Bernoulli-Bayes-Laplace Anniversary, pp. 61–110. Springer, Heidelberg (1965)Google Scholar
- [KN05]Karger, D., Nikolova, E.: Brief announcement: on the expected overpayment of VCG mechanisms in large networks. In: PODC 2005: Proceedings of the 24th annual ACM symposium on Principles of distributed computing, pp. 126–126. ACM Press, New York (2005)Google Scholar
- [NR99]Nisan, N., Ronen, A.: Algorithmic mechanism design (extended abstract). In: Proceedings of the thirty-first annual ACM symposium on Theory of computing (Atlanta, Georgia, United States), pp. 129–140 (1999)Google Scholar