Skip to main content
SpringerLink
Log in

Multi-player flow games

  • Published:
Autonomous Agents and Multi-Agent Systems Aims and scope Submit manuscript

Abstract

In the traditional maximum-flow problem, the goal is to transfer maximum flow in a network by directing, in each vertex in the network, incoming flow to outgoing edges. The problem corresponds to settings in which a central authority has control on all vertices of the network. Today’s computing environment, however, involves systems with no central authority. In particular, in many applications of flow networks, the vertices correspond to decision-points controlled by different and selfish entities. For example, in communication networks, routers may belong to different companies, with different destination objectives. This suggests that the maximum-flow problem should be revisited, and examined from a game-theoretic perspective. We introduce and study multi-player flow games (MFGs, for short). Essentially, the vertices of an MFG are partitioned among the players, and a player that owns a vertex directs the flow that reaches it. Each player has a different target vertex, and the objective of each player is to maximize the flow that reaches her target vertex. We study the stability of MFGs and show that, unfortunately, an MFG need not have a Nash Equilibrium. Moreover, the price of anarchy and even the price of stability of MFGs are unbounded. That is, the reduction in the flow due to selfish behavior is unbounded. We study the problem of deciding whether a given MFG has a Nash Equilibrium and show that it is \(\Sigma _2^P\)-complete, as well as the problem of finding optimal strategies for the players (that is, best-response moves), which we show to be NP-complete. We continue with some good news and consider a variant of MFGs in which flow may be swallowed. For example, when routers in a communication network may drop messages. We show that, surprisingly, while this model seems to incentivize selfish behavior, a Nash Equilibrium that achieves the maximum flow always exists, and can be found in polynomial time. Finally, we consider MFGs in which the strategies of the players may use non-integral flows, which we show to be stronger.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. Not to confuse with games in which players cooperate in order to construct a sub-graph that maximizes the flow in the traditional setting, which are also termed flow games (c.f., [16]).

  2. Throughout this paper, we consider pure strategies. Unlike mixed strategies, pure strategies are not random nor drawn from a distribution.

  3. Different aspects of networks have already been extensively studied from the perspectives of algorithmic game theory. This includes, for example, network formation games [4] or incentive issues in interdomain routing and the BGP protocol [11]. We are the first, however, to consider the maximum-flow problem from this perspective.

References

  1. Agarwal, S., Kodialam, M. S., & Lakshman, T. V. (2013). Traffic engineering in software defined networks. In Proceedings of 32nd IEEE international conference on computer communications, pp. 2211–2219.

  2. Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: Theory, algorithms, and applications. Englewood Cliffs: Prentice Hall.

    MATH  Google Scholar 

  3. Almagor, S., Avni, G., & Kupferman, O. (2015). Automatic generation of quality specifications. In Proceedings of 26th international conference on concurrency theory, vol. 42, pp. 325–339.

  4. Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T., & Roughgarden, T. (2008). The price of stability for network design with fair cost allocation. SIAM Journal on Computing, 38(4), 1602–1623.

    Article  MathSciNet  Google Scholar 

  5. Armbruster, A., Gosnell, M., McMillin, B., & Crow, M. L. (2005). Power transmission control using distributed max-flow. In Computer software and applications conference, 2005. COMPSAC 2005. 29th annual international, vol. 1, pp. 256–263. IEEE.

  6. Cheung, T. Y. (1983). Graph traversal techniques and the maximum flow problem in distributed computation. IEEE Transactions on Software Engineering, 4, 504–512.

    Article  Google Scholar 

  7. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms. Cambridge: MIT Press.

    MATH  Google Scholar 

  8. Dinic, E. A. (1970). Algorithm for solution of a problem of maximum flow in a network with power estimation. Soviet Mathematics Doklady, 11(5), 1277–1280. (English translation by RF. Rinehart).

    Google Scholar 

  9. Edmonds, J., & Karp, R. M. (1972). Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM, 19(2), 248–264.

    Article  Google Scholar 

  10. Even, S., Itai, A., & Shamir, A. (1976). On the complexity of timetable and multicommodity flow problems. SIAM Journal on Computing, 5(4), 691–703.

    Article  MathSciNet  Google Scholar 

  11. Feigenbaum, J., Papadimitriou, C. H., Sami, R., & Shenker, S. (2005). A BGP-based mechanism for lowest-cost routing. Distributed Computing, 18(1), 61–72.

    Article  Google Scholar 

  12. Ford, L. R., & Fulkerson, D. R. (1956). Maximal flow through a network. Canadian Journal of Mathematics, 8(3), 399–404.

    Article  MathSciNet  Google Scholar 

  13. Ford, L. R., & Fulkerson, D. R. (1962). Flows in networks. Princeton: Princeton University Press.

    MATH  Google Scholar 

  14. Goldberg, A. V., Tardos, É., & Tarjan, R. E. (1989). Network flow algorithms. Technical report, DTIC Document.

  15. Goldberg, A. V., & Tarjan, R. E. (1988). A new approach to the maximum-flow problem. Journal of the ACM, 35(4), 921–940.

    Article  MathSciNet  Google Scholar 

  16. Kalai, E., & Zemel, E. (1982). Totally balanced games and games of flow. Mathematics of Operations Research, 7(3), 476–478.

    Article  MathSciNet  Google Scholar 

  17. Koutsoupias, E., & Papadimitriou, C. (2009). Worst-case equilibria. Computer Science Review, 3(2), 65–69.

    Article  Google Scholar 

  18. Kupferman, O., Vardi, G., & Vardi, M. Y. (2017). Flow games. In Proceedings of 37th conference on foundations of software technology and theoretical computer science, vol. 93 of Leibniz international proceedings in informatics (LIPIcs), pp. 38:38–38:16.

  19. Lichtenstein, O., & Pnueli, A. (1985). Checking that finite state concurrent programs satisfy their linear specification. In Proceedings of 12th ACM symposium on principles of programming languages, pp. 97–107.

  20. Nash, J. F. (1950). Equilibrium points in n-person games. In Proceedings of the National Academy of Sciences of the USA.

  21. Nisan, N., Roughgarden, T., Tardos, E., & Vazirani, V. V. (2007). Algorithmic game theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  22. Papadimitriou, C. H. (2001). Algorithms, games, and the internet. In Proceedings of 33rd ACM symposium on theory of computing, pp. 749–753.

    Google Scholar 

  23. Pnueli, A., & Rosner, R. (1989). On the synthesis of a reactive module. In Proceedings of 16th ACM symposium on principles of programming languages, pp. 179–190.

  24. Tardos, É. (1985). A strongly polynomial minimum cost circulation algorithm. Combinatorica, 5(3), 247–255.

    Article  MathSciNet  Google Scholar 

  25. Vissicchio, S., Vanbever, L., & Bonaventure, O. (2014). Opportunities and research challenges of hybrid software defined networks. Computer Communication Review, 44(2), 70–75.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Université Libre de Bruxelles, Brussels, Belgium

    Shibashis Guha

  2. School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel

    Orna Kupferman & Gal Vardi

Authors
  1. Shibashis Guha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Orna Kupferman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gal Vardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gal Vardi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version has appeared in the 17th International Conference on Autonomous Agents and Multi-Agent Systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guha, S., Kupferman, O. & Vardi, G. Multi-player flow games. Auton Agent Multi-Agent Syst 33, 798–820 (2019). https://doi.org/10.1007/s10458-019-09420-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10458-019-09420-2

Keywords

Search

Navigation