Skip to main content
SpringerLink
Log in

Strategic Form Games on Digraphs

  • Chapter
  • First Online:
Book cover Pareto-Nash-Stackelberg Game and Control Theory

Part of the book series: Smart Innovation, Systems and Technologies ((SIST,volume 89))

Abstract

The chapter deals with strategic form games on digraphs, and examines maximin solution concepts based on different types of digraph substructures Ungureanu (Computer Science Journal of Moldova 6 3(18): 313–337, 1998, [1]), Ungureanu (ROMAI Journal 12(1): 133–161, 2016, [2]). Necessary and sufficient conditions for maximin solution existence in digraph matrix games with pure strategies are formulated and proved. Some particular games are considered. Algorithms for finding maximin substructures are suggested. Multi-player simultaneous games and dynamical/hierarchical games on digraphs are considered too.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ungureanu, V. 1998. Games on digraphs and constructing maximin structures. Computer Science Journal of Moldova 6 3(18): 313–337.

    Google Scholar 

  2. Ungureanu, V. 2016. Strategic games on digraphs. ROMAI Journal 12 (1): 133–161.

    MathSciNet  Google Scholar 

  3. Christofides, N. 1975. Graph Theory: An Algorithmic Approach, 415. London: Academic Press.

    MATH  Google Scholar 

  4. Papadimitriou, C., and K. Steiglitz. 1982. Combinatorial Optimization: Algorithms and Complexity, 510. Englewood Cliffs: Prentice-Hall Inc.

    MATH  Google Scholar 

  5. Berge, C. 1962. The Theory of Graphs (and its Applications). London: Methuen & Co.; New York: Wiley, 272pp.

    Google Scholar 

  6. Van den Nouweland, A., P. Borm, W. van Golstein Brouwers, R. Groot Bruinderink, and S. Tijs. 1996. A game theoretic approach to problems in telecommunications. Management Science 42 (2): 294–303.

    Article  MATH  Google Scholar 

  7. Altman, E., T. Boulogne, R. El-Azouzi, T. JimTnez, and L. Wynter. 2006. A survey on networking games in telecommunications. Computers & Operations Research 33 (2): 286–311.

    Article  MathSciNet  MATH  Google Scholar 

  8. Tardos, E. 2004. Network Games, Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing (STOC?04), 341–342. Chicago, Illinois, USA, June 13–15 2004.

    Google Scholar 

  9. Altman, E., and L. Wynter. 2004. Crossovers between transportation planning and telecommunications, networks and spatial economics. Editors 4 (1): 5–124.

    Google Scholar 

  10. Jackson, M.O., and Y. Zenou. 2015. Games on networks. In Handbook of Game Theory with Economic Applications, vol. 4, ed. H. Peyton Young, and Sh Zamir, 95–163. Amsterdam: North-Holland.

    Google Scholar 

  11. Roughgarden, T. 2002. The price of anarchy is independent of the network topology. Journal of Computer and System Sciences 67 (2): 341–364.

    Article  MathSciNet  MATH  Google Scholar 

  12. Suri, S., C. Toth, and Y. Zhou. 2005. Selfish load balancing and atomic congestion games, Proceedings of the 16th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), 188–195.

    Google Scholar 

  13. Christodoulou, G., and E. Koutsoupias. 2005. The price of anarchy of finite congestion games, Proceedings of the Thirty-Seven Annual ACM Symposium on Theory of Computing (STOC?05), 67–73. New York, USA.

    Google Scholar 

  14. Czumaj, A., and B. Vöcking. 2007. Tight bounds for worst-case equilibria. ACM Transactions on Algorithms 3 (1): 11. Article 4.

    Google Scholar 

  15. Vetta, A. 2002. Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions, Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’02), 416pp.

    Google Scholar 

  16. Akella, A., S. Seshan, R. Karp, S. Shenker, and C. Papadimitriou. 2002. Selfish behavior and stability of the Internet: A game-theoretic analysis of TCP, ACM SIGCOMM Computer Communication Review - Proceedings of the 2002 SIGCOMM Conference, 32(4): 117–130.

    Google Scholar 

  17. Dutta, D., A. Goel, and J. Heidemann. 2002. Oblivious AQM and nash equilibria, ACM SIGCOMM Computer Communication Review - Proceedings of the 2002 SIGCOMM conference, 32(3): 20.

    Google Scholar 

  18. Fabrikant, A., A. Luthra, E. Maneva, C.H. Papadimitriou and S. Shenker. 2003. On a network creation game, Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing (PODC ’03), 347–351.

    Google Scholar 

  19. Anshelevich, E., A. Dasgupta, E. Tardos, and T. Wexler. 2008. Near-optimal network design with selfish agents. Theory of Computing 4: 77–109.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kodialam, M., and T.V. Lakshman. 2003. Detecting network intrusions via sampling: A game theoretic approach. IEEE INFOCOM 1–10.

    Google Scholar 

  21. Han, Z., D. Niyato, W. Saad, T. Başar, and A. Hjørungnes. 2012. Game Theory in Wireless and Communication Networks: Theory, Models, and Applications. Cambridge: Cambridge University Press, XVIII+535pp.

    Google Scholar 

  22. Zhang, Y., and M. Guizani (eds.). 2011. Game Theory for Wireless Communications and Networking. Boca Raton: CRC Press, XIV+571pp.

    Google Scholar 

  23. Kim, S. 2014. Game Theory Applications in Network Design. Hershey: IGI Global, XXII+500pp.

    Google Scholar 

  24. Mazalov, V. 2014. Mathematical Game Theory and its Applications. Tokyo: Wiley, XIV+414pp.

    Google Scholar 

  25. Antoniou, J., and A. Pitsillides. 2013. Game Theory in Communication Networks: Cooperative Resolution of Interactive Networking Scenarios. Boca Raton: CRC Press, X+137pp.

    Google Scholar 

  26. Aubin, J.-P. 2005. Dynamical connectionist network and cooperative games. In Dynamic Games: Theory and Applications, ed. A. Haurie, and G. Zaccour, 1–36. US: Springer.

    Google Scholar 

  27. El Azouzi, R., E. Altman, and O. Pourtallier. 2005. Braess paradox and properties of wardrop equilibrium in some multiservice networks. In Dynamic Games: Theory and Applications, ed. A. Haurie, and G. Zaccour, 57–77. US: Springer.

    Chapter  Google Scholar 

  28. Pavel, L. 2012. Game Theory for Control of Optical Networks. New York: Birkhäuser, XIII+261pp.

    Google Scholar 

  29. Gurvitch, V., A. Karzanov, and L. Khatchiyan. 1988. Cyclic games: Finding of minimax mean cycles in digraphs. Journal of Computational Mathematics and Mathematical Physics 9 (28): 1407–1417. (in Russian).

    MathSciNet  Google Scholar 

  30. Von Stackelberg, H. 1934. Marktform und Gleichgewicht (Market Structure and Equilibrium). Vienna: Springer, (in German), XIV+134pp.

    Google Scholar 

  31. Wolfram, S. 2002. A New Kind of Science, 1197. Champaign, IL: Wolfram Media Inc.

    MATH  Google Scholar 

  32. Wolfram, S. 2016. An Elementary Introduction to the Wolfram Language. Champaign, IL: Wolfram Media, Inc., XV+324pp.

    Google Scholar 

  33. Boliac, R., and D. Lozovanu. 1996. Finding of minimax paths tree in weighted digraph. Buletinul Academiei de Ştiinţe a Republicii Moldova 3 (66): 74–82. (in Russian).

    MATH  Google Scholar 

  34. Biggs, N.L., E.K. Lloyd, and R.J. Wilson. 1976. Graph Theory, 1736–1936, 255. Oxford: Clarendon Press.

    MATH  Google Scholar 

  35. Schrijver, A. 2005. On the history of combinatorial optimization (Till 1960). Handbooks in Operations Research and Management Science 12: 1–68.

    Article  MATH  Google Scholar 

  36. Gutin, G., and P.P. Abraham (eds.). 2004. The Traveling Salesman Problem and its Variation, vol. 2, New-York: : Kluwer Academic Publishers, 831pp.

    Google Scholar 

  37. Lawler, E.L., J.K. Lenstra, A.H.G. Rinooy Kan, and D.B. Shmoys (eds.). 1985. The Traveling Salesman Problem. Chichester, UK: Wiley.

    MATH  Google Scholar 

  38. Reinelt, G. 1994. The Traveling Salesman: Computational Solutions for TSP Applications, 230. Berlin: Springer.

    Google Scholar 

  39. Fleishner, H. 2000. Traversing graphs: The Eulerian and Hamiltonian theme. In Arc Routing: Theory, Solutions, and Applications, ed. M. Dror, 19–87. The Netherlands: Kluwer Academic Publishers.

    Chapter  Google Scholar 

  40. Golden, B., S. Raghavan, and E. Wasil (eds.). 2008. The Vehicle Routing Problem: Latest Advances and New Challenges. New York: Springer.

    MATH  Google Scholar 

  41. Ball, M.O., T.L. Magnanti, C.L. Monma, and G.L. Nemhauser (eds.). 1995. Network Routing, 779. Elsevier: Amsterdam.

    Google Scholar 

  42. Garey, M.R., and D.S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP Completeness, 351. San Francisco: W.H. Freeman.

    MATH  Google Scholar 

  43. Sipser, M. 2006. Introduction to the Theory of Computation, 2nd ed, Boston, Massachusetts: Thomson Course Technology, XIX+431pp.

    Google Scholar 

  44. Nielsen, M.A., and I.L. Chuang. 2010. Quantum Computation and Quantum Information, 10th Anniversary ed. Cambridge, UK: Cambridge University Press, XXXII+676pp.

    Google Scholar 

  45. Applegate, D.L., R.E. Bixby, V. Chvtal, J. William, and W.J. Cook. 2006. The Traveling Salesman Problem: A Computational Study, 606. Princeton: Princeton University Press.

    Google Scholar 

  46. Hitchcock, F.L. 1941. The distribution of product from several sources to numerous localities. Journal of Mathematical Physics 20 (2): 217–224.

    MathSciNet  MATH  Google Scholar 

  47. Díaz-Parra, O., J.A. Ruiz-Vanoye, B.B. Loranca, A. Fuentes-Penna, and R.A. and Barrera-Cámara. 2014. A survey of transportation problems. Journal of Applied Mathematics 2014: 17. Article ID 848129.

    Google Scholar 

  48. Hoffman, K.L., and M. Padberg. 1991. LP-based combinatorial problem solving. Annals of Operations Research 4: 145–194.

    Article  MathSciNet  MATH  Google Scholar 

  49. Padberg, M., and G. Rinaldi. 1991. A branch-and-cut algorithm for the resolution of large-scale traveling salesman problem. SIAM Review 33: 60–100.

    Article  MathSciNet  MATH  Google Scholar 

  50. Villani, C. 2003. Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, 382. Providence: American Mathematical Society.

    Google Scholar 

  51. Villani, C. 2008. Optimal Transport, Old and New, 1000. Berlin: Springer.

    Google Scholar 

  52. Ungureanu, V. 2006. Traveling salesman problem with transportation. Computer Science Journal of Moldova 14 2(41): 202–206.

    Google Scholar 

  53. Caric, T., and H. Gold (eds.). 2008. Vehicle Routing Problem, 152. InTech: Croatia.

    Google Scholar 

  54. Toth, P., and D. Vigo. 2002. The Vehicle Routing Problem, Society for Industrial and Applied Mathematics, 386pp.

    Google Scholar 

  55. Labadie, N., and C. Prodhon. 2014. A survey on multi-criteria analysis in logistics: Focus on vehicle routing problems. In Applications of Multi-Criteria and Game Theory Approaches: Manufacturing and Logistics, ed. L. Benyoucef, J.-C. Hennet, and M.K. Tiwari, 3–29. New Jersey: Springer.

    Chapter  Google Scholar 

  56. Bellman, R. 1957. Dynamic Programming, 365. New Jersey: Princeton University Press.

    MATH  Google Scholar 

  57. Golshtein, E., and D. Yudin. 1966. New Directions in Linear Programming. Moscow: Sovetskoe Radio, 527pp. (in Russian).

    Google Scholar 

  58. Ungureanu, V. 1997. Minimizing a concave quadratic function over a hypercube. Buletinul Academiei de Ştiinţe a Republicii Moldova: Mathematics Series 2 (24): 69–76. (in Romanian).

    MathSciNet  Google Scholar 

  59. Yanovskaya, E.B. 1968. Equilibrium points in polymatrix games. Lithuanian Mathematical Collection (Litovskii Matematicheskii Sbornik) 8 (2): 381–384. (in Russian).

    MATH  Google Scholar 

  60. Howson Jr., J.T. 1972. Equilibria of polymatrix games. Management Science 18: 312–318.

    Article  MathSciNet  MATH  Google Scholar 

  61. Eaves, B.C. 1973. Polymatrix games with joint constraints. SIAM Journal of Applied Mathematics 24: 418–423.

    Article  MathSciNet  MATH  Google Scholar 

  62. Von Neumann, J. 1928. Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100: 295–320. (in German).

    Article  Google Scholar 

  63. Kuhn, H.W. 1950. Extensive games, Proceedings of the National Academy of Sciences U.S.A., Vol. 36, 570–576.

    Google Scholar 

  64. Kuhn, H.W. 1953. Extensive games and the problem of information. Contributions to the Theory of Games, Vol. II, vol. 28, 217–243., Annals of Mathematics Study Princeton: Princeton University Press.

    Google Scholar 

  65. Kuhn, H.W. 2003. Lectures on the Theory of Games, vol. 37, 118. Annals of Mathematics Study Princeton: Princeton University Press.

    Book  Google Scholar 

  66. Alós-Ferrer, C., and K. Ritzberger. 2016. The Theory of Extensive Form Games. Berlin: Springer, XVI+239pp.

    Google Scholar 

  67. Nisan, N., T. Roughgarden, E. Tardos, and V.V. Vazirani (eds.). 2007. Algorithmic Game Theory. Cambridge, UK: Cambridge University Press, 775pp.

    Google Scholar 

  68. Shoham, Y., and K. Leyton-Brown. 2009. Multi-Agent Systems: Algorithmic, Game-Theoretic, and Logical Foundations, 532. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  69. Easley, D., and D. Kleinberg. 2010. Networks, Crowds, and Markets: Reasoning about a Highly Connected World, 833. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  70. Menache, I., and A. Ozdaglar. 2011. Network Games: Theory, Models, and Dynamics. Synthesis Lectures on Communication Networks California: Morgan & Claypool Publishers, XIV+143pp.

    Google Scholar 

  71. Lozovanu, D. 1993. A strongly polynomial time algorithm for finding minimax paths in network and solving cyclic games. Cybernetics and System Analysis 5: 145–151. (in Russian).

    MATH  Google Scholar 

  72. Mertens, J.F. 1987. Repeated Games, Proceedings of the International Congress of Mathematicians, Berkeley, Providence: American Mathematical Society, 1528–1577.

    Google Scholar 

  73. Aumann, R.J., M.B. Maschler, and R.E. Stearns. 1995. Repeated Games with Incomplete Information, 360. Cambridge, Massachusetts: MIT Press.

    Google Scholar 

  74. Fudenberg, D., and J. Tirole. 1991. Game Theory, 579. Cambridge: MIT Press.

    MATH  Google Scholar 

  75. Osborne, M.J., and A. Rubinstein. 1994. A Course in Game Theory, 373. Cambridge, Massachusetts: The MIT Press.

    MATH  Google Scholar 

  76. Mailath, G.J., and L. Samuelson. 2006. Repeated Games and Reputations: Long-Run Relationships. New York: Oxford University Press, XVIII+645pp.

    Google Scholar 

  77. Sorin, S. 2002. A First Course on Zero-Sum Repeated Games. Berlin: Springer, XV+204pp.

    Google Scholar 

  78. Osborne, M.J. 2009. An Introduction to Game Theory, International ed, 685. Oxford: Oxford University Press.

    Google Scholar 

  79. Brams, S.J. 1994. Theory of Moves. Cambridge: Cambridge University Press, XII+248pp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Moldova State University, Chișinău, Moldova

    Valeriu Ungureanu

Authors
  1. Valeriu Ungureanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valeriu Ungureanu .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ungureanu, V. (2018). Strategic Form Games on Digraphs. In: Pareto-Nash-Stackelberg Game and Control Theory. Smart Innovation, Systems and Technologies, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-75151-1_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-75151-1_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-75150-4

  • Online ISBN: 978-3-319-75151-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation