Liquid Price of Anarchy

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.

The Efficiency of Resource Allocation Mechanisms for Budget-Constrained Users

“…Finally, we believe that the liquid welfare is an appropriate e ciency benchmark for auctions with budget-constrained players. e recent paper by Azar et al [2017] studies the LPoA of simultaneous rstprice auctions; obtaining similar results for other auction formats (e.g., see the recent survey of Roughgarden et al [2017]) is certainly important. Needless to say, we do not expect that the liquid welfare is unique as a measure of e ciency in se ings with budgets.…”

“…As an e ciency benchmark, liquid welfare has been studied recently in di erent contexts such as in the design of truthful mechanisms (see [Dobzinski and Paes Leme, 2014;Xiao, 2015, 2017]) and in the analysis of combinatorial Walrasian equilibria with budgets [Dughmi et al, 2016]. In the context of the price of anarchy, it was considered recently in simultaneous rst price auctions by Azar et al [2017] and in position auctions by Voudouris [2019]. Caragiannis and Voudouris [2016] were the rst to prove that the liquid price of anarchy of the Kelly mechanism is constant.…”

The Efficiency of Resource Allocation Mechanisms for Budget-Constrained Users

“…Informally, the liquid welfare is the total value of the players for the resource fraction they receive, with the value of each player capped by her budget. Following the recent paper [1], we use the term liquid price of anarchy (and abbreviate it as LPoA) to refer to the price of anarchy with respect to the liquid welfare, i.e., the ratio between the optimal liquid welfare of a game induced by a resource allocation mechanism and the worst liquid welfare over all equilibria of the game.…”

A note on the efficiency of position mechanisms with budget constraints

“…VCG. Like in the case of GSP, consider the bid vector b = (1 + ǫ, 1) which induces the assignment (1,2). The payments of the players are p 1 (b, VCG) = 1 − 1 λ and p 2 (b, VCG) = 0, yielding utilities of u 1 (b, VCG) = λ−1+ 1 λ and u 2 (b, VCG) = 1 λ .…”

“…For pure equilibria in particular, by exploiting the structure of worst-case equilibria, Caragiannis and Voudouris [5] were able to characterize the liquid price of anarchy of all mechanisms for the allocation of a single divisible resource, leading to tight bounds. Finally, Azar et al [2] refined the definition of the liquid welfare for randomized allocations and proved constant liquid price of anarchy bounds over general equilibrium concepts for simultaneous first price auctions.…”