We give exponential lower bounds on the Price of Stability (PoS) of weighted congestion games with polynomial cost functions. In particular, for any positive integer $d$ we construct rather simple games with cost functions of degree at most $d$ which have a PoS of at least $\varOmega(\Phi_d)^{d+1}$, where $\Phi_d\sim d/\ln d$ is the unique positive root of the equation $x^{d+1}=(x+1)^d$. This almost closes the huge gap between $\varTheta(d)$ and $\Phi_d^{d+1}$. Our bound extends also to network congestion games. We further show that the PoS remains exponential even for singleton games. More generally, we provide a lower bound of $\varOmega((1+1/\alpha)^d/d)$ on the PoS of $\alpha$-approximate Nash equilibria for singleton games. All our lower bounds hold for mixed and correlated equilibria as well. On the positive side, we give a general upper bound on the PoS of $\alpha$-approximate Nash equilibria, which is sensitive to the range $W$ of the player weights and the approximation parameter $\alpha$. We do this by explicitly constructing a novel approximate potential function, based on Faulhaber's formula, that generalizes Rosenthal's potential in a continuous, analytic way. From the general theorem, we deduce two interesting corollaries. First, we derive the existence of an approximate pure Nash equilibrium with PoS at most $(d+3)/2$; the equilibrium's approximation parameter ranges from $\varTheta(1)$ to $d+1$ in a smooth way with respect to $W$. Second, we show that for unweighted congestion games, the PoS of $\alpha$-approximate Nash equilibria is at most $(d+1)/\alpha$.

  • 1.  M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., Dover, New York, 1964, http://people.math.sfu.ca/~cbm/aands/. Google Scholar

  • 2.  H. Ackermann H. Röglin and  B. Vöcking , On the impact of combinatorial structure on congestion games , J. ACM , 55 ( 2008 ), pp. 1 -- 22 , https://doi.org/10.1145/1455248.1455249. CrossrefISIGoogle Scholar

  • 3.  S. Aland D. Dumrauf M. Gairing B. Monien and  F. Schoppmann , Exact price of anarchy for polynomial congestion games , SIAM J. Comput. , 40 ( 2011 ), pp. 1211 -- 1233 , https://doi.org/10.1137/090748986. LinkISIGoogle Scholar

  • 4.  S. Albers , On the value of coordination in network design , SIAM J. Comput. , 38 ( 2009 ), pp. 2273 -- 2302 , https://doi.org/10.1137/070701376. LinkISIGoogle Scholar

  • 5.  E. Anshelevich A. Dasgupta J. Kleinberg É. Tardos T. Wexler and  T. Roughgarden , The price of stability for network design with fair cost allocation , SIAM J. Comput. , 38 ( 2008 ), pp. 1602 -- 1623 , https://doi.org/10.1137/070680096. LinkISIGoogle Scholar

  • 6.  B. Awerbuch Y. Azar and  A. Epstein , The price of routing unsplittable flow , SIAM J. Comput. , 42 ( 2013 ), pp. 160 -- 177 , https://doi.org/10.1137/070702370. LinkISIGoogle Scholar

  • 7.  J. Bernoulli, Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis, impensis Thurnisiorum, fratrum, Basile\ae, 1713, https://books.google.de/books?id=ebdcNfPz8l8C. Google Scholar

  • 8.  K. Bhawalkar M. Gairing and  T. Roughgarden , Weighted congestion games: The price of anarchy, universal worst-case examples, and tightness , ACM Trans. Econ. Comput. , 2 ( 2014 ), 14 , https://doi.org/10.1145/2629666. CrossrefGoogle Scholar

  • 9.  V. Bilò , A unifying tool for bounding the quality of non-cooperative solutions in weighted congestion games , Theory Comput. Syst. , 62 ( 2018 ), pp. 1288 -- 1317 , https://doi.org/10.1007/s00224-017-9826-1. CrossrefISIGoogle Scholar

  • 10.  V. Bilò I. Caragiannis A. Fanelli and  G. Monaco , Improved lower bounds on the price of stability of undirected network design games , Theory Comput. Syst. , 52 ( 2013 ), pp. 668 -- 686 , https://doi.org/10.1007/s00224-012-9411-6. CrossrefISIGoogle Scholar

  • 11.  V. Bilò and  C. Vinci , On the impact of singleton strategies in congestion games, in Proceedings of the 25th Annual European Symposium on Algorithms, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Wadern , Germany , 2017 , 17 , https://doi.org/ 10 .4230/lipics.esa.2017.17. Google Scholar

  • 12.  V. Bilò, M. Flammini, and L. Moscardelli, The price of stability for undirected broadcast network design with fair cost allocation is constant, Games Econom. Behavior, to appear, https://doi.org/10.1016/j.geb.2014.09.010. Google Scholar

  • 13.  I. Caragiannis M. Flammini C. Kaklamanis P. Kanellopoulos and  L. Moscardelli , Tight bounds for selfish and greedy load balancing , Algorithmica , 61 ( 2011 ), pp. 606 -- 637 , https://doi.org/10.1007/s00453-010-9427-8. CrossrefISIGoogle Scholar

  • 14.  I. Caragiannis, A. Fanelli, N. Gravin, and A. Skopalik, Efficient computation of approximate pure Nash equilibria in congestion games, in Proceedings of the 52nd IEEE Annual Symposium on Foundations of Computer Science, Palm Springs, CA, 2011, https://doi.org/10.1109/focs.2011.50. Google Scholar

  • 15.  I. Caragiannis A. Fanelli N. Gravin and  A. Skopalik , Approximate pure Nash equilibria in weighted congestion games: Existence, efficient computation, and structure , ACM Trans. Econ. Comput. , 3 ( 2015 ), 2 , https://doi.org/10.1145/2614687. CrossrefGoogle Scholar

  • 16.  H.-L. Chen and  T. Roughgarden , Network design with weighted players , Theory Comput. Syst. , 45 ( 2008 ), 302 , https://doi.org/10.1007/s00224-008-9128-8. CrossrefISIGoogle Scholar

  • 17.  G. Christodoulou and  M. Gairing , Price of stability in polynomial congestion games , ACM Trans. Econ. Comput. , 4 ( 2016 ), 10 , https://doi.org/10.1145/2841229. CrossrefGoogle Scholar

  • 18.  G. Christodoulou and  E. Koutsoupias , The price of anarchy of finite congestion games, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC '05), ACM , New York , 2005 , pp. 67 -- 73 , https://doi.org/10.1145/1060590.1060600. Google Scholar

  • 19.  G. Christodoulou and  E. Koutsoupias , On the price of anarchy and stability of correlated equilibria of linear congestion games, in Algorithms---ESA 2005, Springer , Berlin , 2005 , pp. 59 -- 70 . Google Scholar

  • 20.  G. Christodoulou E. Koutsoupias and  P. G. Spirakis , On the performance of approximate equilibria in congestion games , Algorithmica , 61 ( 2011 ), pp. 116 -- 140 , https://doi.org/10.1007/s00453-010-9449-2. CrossrefISIGoogle Scholar

  • 21.  G. Christodoulou M. Gairing Y. Giannakopoulos and  P. G. Spirakis , The price of stability of weighted congestion games, in 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, Prague , Czech Republic , 2018 , 150 , https://doi.org/ 10 .4230/LIPIcs.ICALP.2018.150. Google Scholar

  • 22.  J. H. Conway and R. K. Guy, The Book of Numbers, Springer, New York, 1996, https://doi.org/10.1007/978-1-4612-4072-3. Google Scholar

  • 23.  J. Dunkel and  A. S. Schulz , On the complexity of pure-strategy Nash equilibria in congestion and local-effect games , Math. Oper. Res. , 33 ( 2008 ), pp. 851 -- 868 . CrossrefISIGoogle Scholar

  • 24.  A. Fabrikant C. Papadimitriou and  K. Talwar , The complexity of pure Nash equilibria, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, STOC '04, ACM , New York , 2004 , pp. 604 -- 612 , https://doi.org/10.1145/1007352.1007445. Google Scholar

  • 25.  J. Faulhaber, Academia Algebr\ae: Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden, Johann Ulrich Schönigs, Augspurg, 1631, https://books.google.de/books?id=0pw_AAAAcAAJ. Google Scholar

  • 26.  M. Feldotto M. Gairing G. Kotsialou and  A. Skopalik , Computing approximate pure Nash equilibria in Shapley value weighted congestion games, in Web and Internet Economics 2017, Springer , Cham , 2017 , pp. 191 -- 204 , https://doi.org/10.1007/978-3-319-71924-5_14. Google Scholar

  • 27.  A. Fiat H. Kaplan M. Levy S. Olonetsky and  R. Shabo , On the price of stability for designing undirected networks with fair cost allocations, in the 33rd International Colloquium on Automata, Languages and Programming (ICALP), M. Bugliesi, B. Preneel, V. Sassone, and I. Wegener, eds., Springer , Berlin , 2006 , pp. 608 -- 618 , https://doi.org/10.1007/11786986_53. Google Scholar

  • 28.  D. Fotakis S. Kontogiannis and  P. Spirakis , Selfish unsplittable flows , Theoret. Comput. Sci. , 348 ( 2005 ), pp. 226 -- 239 . CrossrefISIGoogle Scholar

  • 29.  D. Fotakis S. Kontogiannis E. Koutsoupias M. Mavronicolas and  P. Spirakis , The structure and complexity of Nash equilibria for a selfish routing game , Theoret. Comput. Sci. , 410 ( 2009 ), pp. 3305 -- 3326 , https://doi.org/10.1016/j.tcs.2008.01.004. CrossrefISIGoogle Scholar

  • 30.  M. Goemans V. Mirrokni and  A. Vetta , Sink equilibria and convergence, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), Pittsburgh , PA , 2005 , pp. 142 -- 151 , https://doi.org/10.1109/SFCS.2005.68. Google Scholar

  • 31.  R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, Reading, MA, 1989. Google Scholar

  • 32.  C. Hansknecht, M. Klimm, and A. Skopalik, Approximate pure Nash equilibria in weighted congestion games, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014), Leibniz International Proceedings in Informatics (LIPIcs) 28, K. Jansen, J. D. P. Rolim, N. R. Devanur, and C. Moore, eds., Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Wadern, Germany, 2014, pp. 242--257, https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.242. Google Scholar

  • 33.  T. Harks and  M. Klimm , On the existence of pure Nash equilibria in weighted congestion games , Math. Oper. Res. , 37 ( 2012 ), pp. 419 -- 436 , https://doi.org/10.1287/moor.1120.0543. CrossrefISIGoogle Scholar

  • 34.  T. Harks M. Klimm and  R. H. Möhring , Characterizing the existence of potential functions in weighted congestion games , Theory Comput. Syst. , 49 ( 2011 ), pp. 46 -- 70 , https://doi.org/10.1007/s00224-011-9315-x. CrossrefISIGoogle Scholar

  • 35.  T. Harks M. Klimm and  R. H. Möhring , Strong equilibria in games with the lexicographical improvement property , Internat. J. Game Theory , 42 ( 2012 ), pp. 461 -- 482 , https://doi.org/10.1007/s00182-012-0322-1. CrossrefISIGoogle Scholar

  • 36.  C. G. . Jacobi, De usu legitimo formulae summatoriae Maclaurinianae , J. Reine Angew. Math. , 12 ( 1834 ), pp. 263 -- 272 , http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00214008X. Google Scholar

  • 37.  D. E. Knuth Faulhaber and sums of powers , Math. Comp. , 61 ( 1993 ), pp. 277 -- 277 , https://doi.org/10.1090/s0025-5718-1993-1197512-7. CrossrefISIGoogle Scholar

  • 38.  E. Koutsoupias and  C. Papadimitriou , Worst-case equilibria, in Proceedings of the 16th Annual ACM Symposium on Theoretical Aspects of Computer Science (STACS '99), ACM , New York , 1999 , pp. 404 -- 413 . Google Scholar

  • 39.  E. Lee and  K. Ligett , Improved bounds on the price of stability in network cost sharing games, in Proceedings of the 14th ACM Conference on Electronic Commerce (EC '13), ACM , New York , 2013 , pp. 607 -- 620 , https://doi.org/10.1145/2492002.2482562. Google Scholar

  • 40.  D. H. Lehmer , On the maxima and minima of Bernoulli polynomials , Amer. Math. Monthly , 47 ( 1940 ), pp. 533 -- 538 , https://doi.org/10.2307/2303833. CrossrefGoogle Scholar

  • 41.  L. Libman and  A. Orda , Atomic resource sharing in noncooperative networks , Telecommunication Syst. , 17 ( 2001 ), pp. 385 -- 409 , https://doi.org/10.1023/A:1016770831869. CrossrefISIGoogle Scholar

  • 42.  D. S. Mitrinović, Analytic Inequalities, Springer, Berlin, Heidelberg, 1970, https://doi.org/10.1007/978-3-642-99970-3. Google Scholar

  • 43.  D. Monderer and  L. S. Shapley , Potential games , Games Econom. Behavior , 14 ( 1996 ), pp. 124 -- 143 . CrossrefISIGoogle Scholar

  • 44.  N. Nisan, T. Roughgarden, É. Tardos, and V. Vazirani, eds., Algorithmic Game Theory, Cambridge University Press, Cambridge, UK, 2007. Google Scholar

  • 45.  P. N. Panagopoulou and  P. G. Spirakis , Algorithms for pure Nash equilibria in weighted congestion games , J. Experimental Algorithmics , 11 ( 2007 ), 2 .7, https://doi.org/10.1145/1187436.1216584. CrossrefGoogle Scholar

  • 46.  R. W. Rosenthal , A class of games possessing pure-strategy Nash equilibria , Internat. J. Game Theory , 2 ( 1973 ), pp. 65 -- 67 . CrossrefGoogle Scholar

  • 47.  R. W. Rosenthal , The network equilibrium problem in integers , Networks , 3 ( 1973 ), pp. 53 -- 59 . CrossrefGoogle Scholar

  • 48.  T. Roughgarden , Intrinsic robustness of the price of anarchy , J. ACM , 62 ( 2015 ), 32 , https://doi.org/10.1145/2806883. CrossrefISIGoogle Scholar

  • 49.  T. Roughgarden and  É. Tardos , How bad is selfish routing? , J. ACM , 49 ( 2002 ), pp. 236 -- 259 , https://doi.org/10.1145/506147.506153. CrossrefISIGoogle Scholar

  • 50.  T. Roughgarden and  É. Tardos , Bounding the inefficiency of equilibria in nonatomic congestion games , Games Econom. Behavior , 47 ( 2004 ), pp. 389 -- 403 , https://doi.org/10.1016/j.geb.2003.06.004. CrossrefISIGoogle Scholar

  • 51.  A. S. Schulz and  N. S. Moses , On the performance of user equilibria in traffic networks, in Proceedings of the 14th Annual ACM--SIAM Symposium on Discrete Algorithms (Baltimore, MD), ACM, New York, SIAM , Philadelphia , 2003 , pp. 86 -- 87 . Google Scholar