We consider atomic congestion games on series-parallel networks, and study the structure of the sets of Nash equilibria and social local optima on a given network when the number of players varies. We establish that these sets are definable in Presburger arithmetic and that they admit semilinear representations whose all period vectors have a common direction. As an application, we prove that the prices of anarchy and stability converge to 1 as the number of players goes to infinity, and show how to exploit these semilinear representations to compute these ratios precisely for a given network and number of players.
@InProceedings{bertrand_et_al:LIPIcs.FSTTCS.2022.32, author = {Bertrand, Nathalie and Markey, Nicolas and Sadhukhan, Suman and Sankur, Ocan}, title = {{Semilinear Representations for Series-Parallel Atomic Congestion Games}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {32:1--32:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.32}, URN = {urn:nbn:de:0030-drops-174243}, doi = {10.4230/LIPIcs.FSTTCS.2022.32}, annote = {Keywords: congestion games, Nash equilibria, Presburger arithmetic, semilinear sets, price of anarchy} }
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