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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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$d_{max}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>max<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is bounded as <jats:inline-formula><jats:alternatives><jats:tex-math>$$T\\rightarrow \\infty .$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>T<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>\u221e<\/mml:mi>\n                    <mml:mo>.<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> In this general case, also the PoA of mixed Nash equilibria converges to 1 as <jats:inline-formula><jats:alternatives><jats:tex-math>$$T\\rightarrow \\infty $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>T<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>\u221e<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> when <jats:inline-formula><jats:alternatives><jats:tex-math>$$d_{max}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>max<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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.<\/jats:p>","DOI":"10.1007\/s10107-022-01853-0","type":"journal-article","created":{"date-parts":[[2022,6,28]],"date-time":"2022-06-28T13:03:47Z","timestamp":1656421427000},"page":"937-993","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A convergence analysis of the price of anarchy in atomic congestion games"],"prefix":"10.1007","volume":"199","author":[{"given":"Zijun","family":"Wu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3766-8728","authenticated-orcid":false,"given":"Rolf H.","family":"M\u00f6hring","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunying","family":"Ren","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dachuan","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,6,28]]},"reference":[{"key":"1853_CR1","volume-title":"Studies in the economics of transportation","author":"M Beckmann","year":"1956","unstructured":"Beckmann, M., McGuire, C., Winsten, C.: Studies in the economics of transportation. 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