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# A convergence analysis of the price of anarchy in atomic congestion games

## Wu, Zijun; Möhring, Rolf H.; Ren, Chunying; Xu , Dachuan

We analyze the convergence of the price of anarchy (PoA) of Nash equilibria in atomic congestion games with growing total demand T. When the cost functions are polynomials of the same degree, we obtain explicit rates for a rapid convergence of the PoAs of pure and mixed Nash equilibria to 1 in terms of 1/T and dmax/T, where dmax is the maximum demand controlled by an individual. Similar convergence results carry over to the random inefficiency of the random flow induced by an arbitrary mixed Nash equilibrium. For arbitrary polynomial cost functions, we derive a related convergence rate for the PoA of pure Nash equilibria (if they exist) when the demands fulfill certain regularity conditions and dmax is bounded as T→∞. In this general case, also the PoA of mixed Nash equilibria converges to 1 as T→∞ when dmax is bounded. Our results constitute the first convergence analysis for the PoA in atomic congestion games and show that selfish behavior is well justified when the total demand is large.