Abstract
Congestion games provide a model of human’s behavior of choosing an optimal strategy while avoiding congestion. In the past decade, matroid congestion games have been actively studied and their good properties have been revealed. In most of the previous work, the cost functions are assumed to be univariate or bivariate. In this paper, we discuss generalizations of matroid congestion games in which the cost functions are n-variate, where n is the number of players. First, we prove the existence of pure Nash equilibria in matroid congestion games with monotone cost functions, which extends that for weighted matroid congestion games by Ackermann, Röglin, and Vöcking (2009). Second, we prove the existence of pure Nash equilibria in matroid resource buying games with submodular cost functions, which extends that for matroid resource buying games with marginally nonincreasing cost functions by Harks and Peis (2014). Finally, motivated from polymatroid congestion games with \(\mathrm {M}^\natural \)-convex cost functions, we conduct sensitivity analysis for separable \(\mathrm {M}^\natural \)-convex optimization, which extends that for separable convex optimization over base polyhedra by Harks, Klimm, and Peis (2018).
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References
Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. J. ACM 55(6), 25:1–25:22 (2008). https://doi.org/10.1145/1455248.1455249
Ackermann, H., Röglin, H., Vöcking, B.: Pure Nash equilibria in player-specific and weighted congestion games. Theor. Comput. Sci. 410(17), 1552–1563 (2009). https://doi.org/10.1016/j.tcs.2008.12.035
Bhaskar, U., Fleischer, L., Hoy, D., Huang, C.-C.: On the uniqueness of equilibrium in atomic splittable routing games. Math. Oper. Res. 40(3), 634–654 (2015). https://doi.org/10.1287/moor.2014.0688
Cominetti, R., Correa, J.R., Moses, N.E.S.: The impact of oligopolistic competition in networks. Oper. Res. 57(6), 1421–1437 (2009). https://doi.org/10.1287/opre.1080.0653
Fujishige, S.: Submodular Functions and Optimization. Annals of Discrete Mathematics, vol. 58, 2nd edn. Elsevier, Amsterdam (2005)
Fujishige, S., Goemans, M.X., Harks, T., Peis, B., Zenklusen, R.: Congestion games viewed from M-convexity. Oper. Res. Lett. 43(3), 329–333 (2015). https://doi.org/10.1016/j.orl.2015.04.002
Fujishige, S., Goemans, M.X., Harks, T., Peis, B., Zenklusen, R.: Matroids are immune to Braess’ paradox. Math. Oper. Res. 42(3), 745–761 (2017). https://doi.org/10.1287/moor.2016.0825
Harks, T., Klimm, M., Peis, B.: Sensitivity analysis for convex separable optimization over integeral polymatroids. SIAM J. Optim. 28, 2222–2245 (2018). https://doi.org/10.1137/16M1107450
Harks, T., Peis, B.: Resource buying games. Algorithmica 70(3), 493–512 (2014). https://doi.org/10.1007/s00453-014-9876-6
Harks, T., Peis, B.: Resource buying games. In: Schulz, A.S., Skutella, M., Stiller, S., Wagner, D. (eds.) Gems of Combinatorial Optimization and Graph Algorithms, pp. 103–111. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24971-1_10
Harks, T., Timmermans, V.: Uniqueness of equilibria in atomic splittable polymatroid congestion games. J. Comb. Optim. 36(3), 812–830 (2018). https://doi.org/10.1007/s10878-017-0166-5
Huang, C.-C.: Collusion in atomic splittable routing games. Theory Comput. Syst. 52(4), 763–801 (2013). https://doi.org/10.1007/s00224-012-9421-4
Moriguchi, S., Shioura, A., Tsuchimura, N.: \(\rm M\)-convex function minimization by continuous relaxation approach: proximity theorem and algorithm. SIAM J. Optim. 21(3), 633–668 (2011). https://doi.org/10.1137/080736156
Murota, K.: Convexity and Steinitz’s exchange property. Adv. Math. 125, 272–331 (1996). https://doi.org/10.1006/aima.1996.0084
Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)
Murota, K.: Discrete convex analysis: a tool for economics and game theory. J. Mech. Inst. Des. 1, 151–273 (2016). https://doi.org/10.22574/jmid.2016.12.005
Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999). https://doi.org/10.1287/moor.24.1.95
Murota, K., Tamura, A.: Proximity theorems of discrete convex functions. Math. Program. 99(3), 539–562 (2004). https://doi.org/10.1007/s10107-003-0466-7
Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973). https://doi.org/10.1007/BF01737559
Schrijver, A.: Combinatorial Optimization - Polyhedra and Eciency. Springer, Heidelberg (2003)
Tran-Thanh, L., Polukarov, M., Chapman, A., Rogers, A., Jennings, N.R.: On the existence of pure strategy Nash equilibria in integer–splittable weighted congestion games. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 236–253. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24829-0_22
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This work is partially supported by JSPS KAKENHI Grant Numbers JP16K16012, JP26280001, JP26280004, Japan.
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Takazawa, K. (2019). Generalizations of Weighted Matroid Congestion Games: Pure Nash Equilibrium, Sensitivity Analysis, and Discrete Convex Function. In: Gopal, T., Watada, J. (eds) Theory and Applications of Models of Computation. TAMC 2019. Lecture Notes in Computer Science(), vol 11436. Springer, Cham. https://doi.org/10.1007/978-3-030-14812-6_37
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