Computational Mechanism Design Chapter 3

Nisan observes that other combinations such as xor of

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Unformatted text preview: ting the value for all possible bundles Par99 . In comparison, an OR bidding language S1 ; p1  or S2 ; p2 , which states that the agent wants S1 or S2 or both, has a linear-space representation of this valuation function. Nisan observes that other combinations, such as XOR-of-OR languages and OR-ofXOR languages, allow compact representations of certain preference structures and make tradeo s across expressiveness and compactness. Nisan proposes an OR* bidding language, which is expressive enough to be able to represent arbitrary preferences over discrete items, and as compact a representation as both OR-of-XOR and XOR-of-OR representations. However, Nisan provides an example with no polynomial-size representation even with the OR* language. The expressiveness of a bidding language, or the compactness of representations that it permits, becomes even more important when one considers the agent's underlying valuation problem. Suppose that an agent must solve an NP-hard constrained optimization problem P to compute its value for a set of items, with objective function g and constraints C . In the 74 XOR representation the agent must solve this problem P once for every possible input S G , i.e. requiring an exponential number of solutions to an NP-hard problem. Now consider an alternative bidding language, that allows the agent to send the speci cation of its optimization problem, i.e. P = g; C  directly to the auctioneer. Strategy-proofness is not a ected assuming the agent can trust the mechanism to interpret this bidding language faithfully, but the agent saves a lot of value computation. In general, we might consider a language in which the agent can send a bidding program" to the auctioneer, that the auctioneer will then execute as necessary to compute an agent's value for di erent subsets of items Nis00 . This is really just the extreme limit of the revelation principle: rather than requiring an agent to solve its local problem and compute its value for all possible outcomes, simply allow the agent to send the local problem speci cation directly to the auctioneer. From the perspective of the bidding agent this approach simpli es its valuation problem whenever the speci cation of its local problem is simpler than actually computing its value for all possible outcomes. A bidding program allows an agent to feed that speci cation directly to the auctioneer. From the perspective of the auctioneer, this is an even more centralized solution than providing a complete valuation function, and has worse-still privacy implications. The bidding program approach shifts the valuation computational burden from agents to the auctioneer. Notice for example that if the bidding program provides only black box" functionality, e.g. b : 2G ! R, the mechanism must compute bS  for all S G unless other consistency rules such as free disposal apply to an agents' values to compute the e cient solution. However, if the bidding program, or language, provides a richer functionality| for example allowing e cient pruning the value bS 0  on all bundles S 0 S is less than bS "; or computing approximate...
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This document was uploaded on 03/19/2014 for the course COMP CS286r at Harvard.

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