Liquid Price of Anarchy
Yossi Azar, Michal Feldman, Nick Gravin, Alan Roytman

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…
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