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Resource Buying Games

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Abstract

In resource buying games a set of players jointly buys a subset of a finite resource set \(E\) (e.g., machines, edges, or nodes in a digraph). The cost of a resource \(e\) depends on the number (or load) of players using \(e\), and has to be paid completely by the players before it becomes available. Each player \(i\) needs at least one set of a predefined family \({\mathcal S}_i\subseteq 2^E\) to be available. Thus, resource buying games can be seen as a variant of congestion games in which the load-dependent costs of the resources can be shared arbitrarily among the players. A strategy of player \(i\) in resource buying games is a tuple consisting of one of \(i\)’s desired configurations \(S_i\in {\mathcal S}_i\) together with a payment vector \(p_i\in {\mathbb R}^E_+\) indicating how much \(i\) is willing to contribute towards the purchase of the chosen resources. In this paper, we study the existence and computational complexity of pure Nash equilibria (PNE, for short) of resource buying games. In contrast to classical congestion games for which equilibria are guaranteed to exist, the existence of equilibria in resource buying games strongly depends on the underlying structure of the families \({\mathcal S}_i\) and the behavior of the cost functions. We show that for marginally non-increasing cost functions, matroids are exactly the right structure to consider, and that resource buying games with marginally non-decreasing cost functions always admit a PNE.

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Notes

  1. Recall that \({\mathcal B}_i\subseteq 2^{E_i}\) is an anti-chain (w.r.t. \((2^{E_i}, \subseteq )\)) if \(B,B'\in {\mathcal B}_i,~ B\subseteq B'\) implies \(B=B'\).

References

  1. Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. J. ACM 55(6), 1–22 (2008)

    Article  MathSciNet  Google Scholar 

  2. Ackermann, H., Röglin, H., Vöcking, B.: Pure Nash equilibria in player-specific and weighted congestion games. Theor. Comput. Sci. 410(17), 1552–1563 (2009)

    Article  MATH  Google Scholar 

  3. Anshelevich, E., Caskurlu, B.: Exact and approximate equilibria for optimal group network formation. Theor. Comput. Sci. 412(39), 5298–5314 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Anshelevich, E., Caskurlu, B.: Price of stability in survivable network design. Theory Comput. Syst. 49(1), 98–138 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Anshelevich, E., Caskurlu, B., Hate, A.: Strategic multiway cut and multicut games. Theory Comput. Syst. 52(2), 200–220 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anshelevich, E., Dasgupta, A., Tardos, É., Wexler, T.: Near-optimal network design with selfish agents. Theory Comput. 4(1), 77–109 (2008)

    Article  MathSciNet  Google Scholar 

  7. Anshelevich, E., Karagiozova, A.: Terminal backup, 3d matching, and covering cubic graphs. SIAM J. Comput. 40(3), 678–708 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Caragiannis, I., Fanelli, A., Gravin, N., Skopalik. A.: Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure. In Boi Faltings, Kevin Leyton-Brown, and Panos Ipeirotis, editors, ACM Conference on Electronic Commerce, pp. 284–301, 2012

  9. Cardinal, J., Hoefer, M.: Non-cooperative facility location and covering games. Theor. Comput. Sci. 411, 1855–1876 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dunkel, J., Schulz, A.: On the complexity of pure-strategy Nash equilibria in congestion and local-effect games. Math. Oper. Res. 33(4), 851–868 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Epstein, A., Feldman, M., Mansour, Y.: Strong equilibrium in cost sharing connection games. Games Econ. Behav. 67(1), 51–68 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fabrikant, A., Papadimitriou, C., Talwar. K.: The complexity of pure Nash equilibria. In László Babai, editor, Proc. 36th Annual ACM Sympos. Theory Comput., pp. 604–612, (2004)

  13. Harks, T., Klimm, M.: On the existence of pure Nash equilibria in weighted congestion games. Math. Oper. Res. 37(3), 419–436 (2012)

    Article  MathSciNet  Google Scholar 

  14. Hoefer, M.: Non-cooperative tree creation. Algorithmica 53, 104–131 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hoefer, M.: Strategic cooperation in cost sharing games. Int. J. Game Theor. 42(1), 29–53 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hoefer, M., Skopalik. A.: On the complexity of Pareto-optimal Nash and strong equilibria. In S. Konogiannis, E. Koutsoupias, and P. Spirakis, editors, Proc. 3rd Internat. Sympos. Algorithmic Game Theory, volume 6386 of LNCS, pp. 312–322, (2010)

  17. Rosenthal, R.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theor. 2(1), 65–67 (1973)

    Article  MATH  Google Scholar 

  18. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, Germany (2003)

    Google Scholar 

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Correspondence to Britta Peis.

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Harks, T., Peis, B. Resource Buying Games. Algorithmica 70, 493–512 (2014). https://doi.org/10.1007/s00453-014-9876-6

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  • DOI: https://doi.org/10.1007/s00453-014-9876-6

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