Welfare guarantees for proportional allocations

TL;DR

This paper improves the known bounds on the inefficiency of proportional allocation games, showing that equilibria can be at most 50% efficient, and extends these results to budget-constrained settings with a constant bound.

Contribution

It provides a tighter analysis of the price of anarchy for proportional allocation games, improving the lower bound from 26.8% to 50%, and extends the analysis to budget-constrained scenarios.

Findings

Improved the price of anarchy bound to 50%.

Simplified the analysis of equilibrium inefficiency.

Extended bounds to budget-constrained settings.

Abstract

According to the proportional allocation mechanism from the network optimization literature, users compete for a divisible resource -- such as bandwidth -- by submitting bids. The mechanism allocates to each user a fraction of the resource that is proportional to her bid and collects an amount equal to her bid as payment. Since users act as utility-maximizers, this naturally defines a proportional allocation game. Recently, Syrgkanis and Tardos (STOC 2013) quantified the inefficiency of equilibria in this game with respect to the social welfare and presented a lower bound of 26.8% on the price of anarchy over coarse-correlated and Bayes-Nash equilibria in the full and incomplete information settings, respectively. In this paper, we improve this bound to 50% over both equilibrium concepts. Our analysis is simpler and, furthermore, we argue that it cannot be improved by arguments that do not take the equilibrium structure into account. We also extend it to settings with budget constraints where we show the first constant bound (between 36% and 50%) on the price of anarchy of the corresponding game with respect to an effective welfare benchmark that takes budgets into account.