The Combinatorial World (of Auctions) According to GARP

TL;DR

This paper introduces a combinatorial algorithm for finding optimal virtual valuation functions in auctions that satisfy rationality constraints, enabling more flexible and efficient mechanism design with quantifiable welfare guarantees.

Contribution

It presents a novel combinatorial approach to compute virtual valuation functions with optimal rationality guarantees, including methods for approximate rationality and relaxed constraints in auctions.

Findings

Efficient algorithm for optimal virtual valuation functions.

Existence of minimally rational virtual valuation functions.

Framework for relaxed rationality constraints in auction design.

Abstract

Revealed preference techniques are used to test whether a data set is compatible with rational behaviour. They are also incorporated as constraints in mechanism design to encourage truthful behaviour in applications such as combinatorial auctions. In the auction setting, we present an efficient combinatorial algorithm to find a virtual valuation function with the optimal (additive) rationality guarantee. Moreover, we show that there exists such a valuation function that both is individually rational and is minimum (that is, it is component-wise dominated by any other individually rational, virtual valuation function that approximately fits the data). Similarly, given upper bound constraints on the valuation function, we show how to fit the maximum virtual valuation function with the optimal additive rationality guarantee. In practice, revealed preference bidding constraints are very demanding. We explain how approximate rationality can be used to create relaxed revealed preference constraints in an auction. We then show how combinatorial methods can be used to implement these relaxed constraints. Worst/best-case welfare guarantees that result from the use of such mechanisms can be quantified via the minimum/maximum virtual valuation function.