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Intrinsic Robustness of the Price of Anarchy (2009)
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| Venue: | STOC'09 |
| Citations: | 101 - 12 self |
BibTeX
@MISC{Roughgarden09intrinsicrobustness,
author = {Tim Roughgarden},
title = {Intrinsic Robustness of the Price of Anarchy},
year = {2009}
}
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Abstract
The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a “canonical sufficient condition ” for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or “price of total anarchy”). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games — most notably, congestion games with cost functions restricted to an arbitrary fixed set — that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first
Keyphrases
smoothness argument nash equilibrium intrinsic robustness congestion game pure nash equilibrium bicriteria bound average objective function value objective function value optimal outcome polynomial-length best-response sequence non-polynomial cost function quantitative degradation inefficiency bound natural experimentation strategy arbitrary fixed total anarchy bayesian-nash equilibrium first tight bound simultaneous regret-minimization regret-minimizing player sum objective mixed nash optimal worst-case upper bound mild additional assumption worst-case measure canonical sufficient condition automatic implication selfish behavior fundamental connection upper bound cost function
