Liquid Price of Anarchy

TL;DR

This paper analyzes the efficiency of simple auction formats with budget constraints using the Liquid Price of Anarchy, establishing bounds on their worst-case equilibrium performance for various valuation classes.

Contribution

It introduces the Liquid Price of Anarchy as a new measure for budgeted auction settings and proves constant bounds for mixed Nash equilibria with additive bidders and divisible items.

Findings

LPoA is bounded by a constant for mixed Nash equilibria with additive bidders.

The results extend to simultaneous second price auctions and Bayesian equilibria.

Tight bounds are established for pure Nash equilibria with fractionally-subadditive valuations.

Abstract

Incorporating budget constraints into the analysis of auctions has become increasingly important, as they model practical settings more accurately. The social welfare function, which is the standard measure of efficiency in auctions, is inadequate for settings with budgets, since there may be a large disconnect between the value a bidder derives from obtaining an item and what can be liquidated from her. The Liquid Welfare objective function has been suggested as a natural alternative for settings with budgets. Simple auctions, like simultaneous item auctions, are evaluated by their performance at equilibrium using the Price of Anarchy (PoA) measure -- the ratio of the objective function value of the optimal outcome to the worst equilibrium. Accordingly, we evaluate the performance of simultaneous item auctions in budgeted settings by the Liquid Price of Anarchy (LPoA) measure -- the ratio of the optimal Liquid Welfare to the Liquid Welfare obtained in the worst equilibrium. Our main result is that the LPoA for mixed Nash equilibria is bounded by a constant when bidders are additive and items can be divided into sufficiently many discrete parts. Our proofs are robust, and can be extended to achieve similar bounds for simultaneous second price auctions as well as Bayesian Nash equilibria. For pure Nash equilibria, we establish tight bounds on the LPoA for the larger class of fractionally-subadditive valuations. To derive our results, we develop a new technique in which some bidders deviate (surprisingly) toward a non-optimal solution. In particular, this technique does not fit into the smoothness framework.