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Frugality in path auctions

Frugality in path auctions
Frugality in path auctions
We consider the problem of picking (buying) an inexpensive $s-t$ path in a graph where edges are owned by independent (selfish) agents, and the cost of an edge is known to its owner only. We study the problem of finding {\em frugal mechanisms} for this task, {\em i.e.} we investigate the payments the buyer must make in order to buy a path. First, we show that any mechanism with (weakly) dominant strategies (or, equivalently, any truthful mechanism) for the agents can force the buyer to make very large payments. Namely, for every such mechanism, the buyer can be forced to pay $c(P) + \frac{1}{2}k(c(Q)-c(P))$, where $c(P)$ is the cost of the shortest path, $c(Q)$ is the cost of the second-shortest path, and $k$ is the number of edges in $P$. This extends the previous work of Archer and Tardos~\cite{at}, who showed a similar lower bound for a subclass of truthful mechanisms called {\em min-function} mechanisms. Our lower bounds have no such limitations on the mechanism. Motivated by this lower bound, we study mechanisms for this problem providing Bayes--Nash equilibrium strategies for the agents. In this class, we identify the {\em optimal} mechanism with regard to total payment. We then demonstrate a separation in terms of average overpayments between the classical VCG mechanism and the optimal mechanism showing that under various natural distributions of edge costs, the optimal mechanism pays at most logarithmic factor more than the actual cost, whereas VCG pays $\sqrt{k}$ times the actual cost. On the other hand, we also show that the optimal mechanism does incur at least a constant factor overpayment in natural distributions of edge costs. Since our mechanism is optimal, this gives a lower bound on all mechanisms with Bayes--Nash equilibria.
701-709
Elkind, Edith
7a013473-5cd0-4e41-b907-66b30a04a400
Sahai, Amit
680d332c-3f45-4621-8548-e69ecca43212
Steiglitz, Ken
89c837f5-2b6e-402d-9a4a-afc56a7b8a0f
Munro, J. Ian
8b4f248f-2931-499e-bd37-ff1e2ba5fd90
Elkind, Edith
7a013473-5cd0-4e41-b907-66b30a04a400
Sahai, Amit
680d332c-3f45-4621-8548-e69ecca43212
Steiglitz, Ken
89c837f5-2b6e-402d-9a4a-afc56a7b8a0f
Munro, J. Ian
8b4f248f-2931-499e-bd37-ff1e2ba5fd90

Elkind, Edith, Sahai, Amit and Steiglitz, Ken (2004) Frugality in path auctions. Munro, J. Ian (ed.) Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, United States. 11 - 14 Jan 2004. pp. 701-709 .

Record type: Conference or Workshop Item (Paper)

Abstract

We consider the problem of picking (buying) an inexpensive $s-t$ path in a graph where edges are owned by independent (selfish) agents, and the cost of an edge is known to its owner only. We study the problem of finding {\em frugal mechanisms} for this task, {\em i.e.} we investigate the payments the buyer must make in order to buy a path. First, we show that any mechanism with (weakly) dominant strategies (or, equivalently, any truthful mechanism) for the agents can force the buyer to make very large payments. Namely, for every such mechanism, the buyer can be forced to pay $c(P) + \frac{1}{2}k(c(Q)-c(P))$, where $c(P)$ is the cost of the shortest path, $c(Q)$ is the cost of the second-shortest path, and $k$ is the number of edges in $P$. This extends the previous work of Archer and Tardos~\cite{at}, who showed a similar lower bound for a subclass of truthful mechanisms called {\em min-function} mechanisms. Our lower bounds have no such limitations on the mechanism. Motivated by this lower bound, we study mechanisms for this problem providing Bayes--Nash equilibrium strategies for the agents. In this class, we identify the {\em optimal} mechanism with regard to total payment. We then demonstrate a separation in terms of average overpayments between the classical VCG mechanism and the optimal mechanism showing that under various natural distributions of edge costs, the optimal mechanism pays at most logarithmic factor more than the actual cost, whereas VCG pays $\sqrt{k}$ times the actual cost. On the other hand, we also show that the optimal mechanism does incur at least a constant factor overpayment in natural distributions of edge costs. Since our mechanism is optimal, this gives a lower bound on all mechanisms with Bayes--Nash equilibria.

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Published date: 2004
Additional Information: Event Dates: January 11-14, 2004
Venue - Dates: Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, United States, 2004-01-11 - 2004-01-14
Organisations: Electronics & Computer Science

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Local EPrints ID: 263439
URI: http://eprints.soton.ac.uk/id/eprint/263439
PURE UUID: af593edc-9abf-4621-8578-38b118c84201

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Date deposited: 15 Feb 2007
Last modified: 25 Nov 2019 21:03

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