Advertisement

Stackelberg Equilibrium Sets in Polymatrix Mixed-Strategy Generalized Stackelberg Games

  • Valeriu UngureanuEmail author
Chapter
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 89)

Abstract

By modifying the principle of simultaneity in strategy/simultaneous games to a hierarchical/sequential principle according to which players select their strategies in a known order, we obtain other class of games called Generalised Stackelberg Games or simply Stackelberg Games. Such games are called sequential games to especially highlight a sequential process of decision making. At every stage of Stackelberg games one player selects his strategy. To ensure choosing of his optimal strategy he solves an optimization problem. For such games a Stackelberg equilibrium concept is considered as a solution concept (Von Stackelberg in Marktform und Gleichgewicht (Market Structure and Equilibrium). Springer, Vienna, XIV+134, 1934, [1], Chenet al., in IEEE Transactions on Automatic Control AC-17, 791–798, 1972, [2], Simaan and Cruz in Journal of Optimization Theory and Applications 11, 613–626, 1973, [3], Simaan and Cruz in Journal of Optimization Theory and Applications 11, 533–555, 1973, [4], Leitmann in Journal of Optimization Theory and Applications 26, 637–648, 1978, [5], Blaquière in Une géneralisation du concept d’optimalité et des certains notions geometriques qui’s rattachent, No. 1–2, Bruxelles, 49–61, 1976, [6], Başar and Olsder in Dynamic noncooperative game theory. SIAM, Philadelphia, [7], Ungureanu in ROMAI Journal 4(1), 225–242, 2008, [8], Ungureanu in Mathematical Modelling, Optimization and Information Technologies, International Conference Proceedings, ATIC, 181–189, Evrica, Chişinău, [9], Peters in Game theory: A multi-leveled approach, p. XVII+494, Springer, Berlin, 2015, [10], Ungureanu and Lozan in Mathematical Modelling, Optimization and Information Technologies, International Conference Proceedings, ATIC, 370–382, Evrica, Chişinău, 2016, [11], Korzhyk et al. in Journal of Artificial Intelligence Research, 41, 297–327, 2011, [12]). The set of all Stackelberg equilibria is described/investigated as a set of optimal solutions of an optimization problem. The last problem is obtained by considering results of solving a sequence of optimization problems that reduce all at once the graph of best response mapping of the last player to the Stackelberg equilibrium set. Namely the problem of Stackelberg equilibrium set computing in bimatrix and polymatrix finite mixed-strategy games is considered in this chapter. A method for Stackelberg equilibrium set computing is exposed.

References

  1. 1.
    Von Stackelberg, H. 1934. Marktform und Gleichgewicht (Market Structure and Equilibrium). Vienna: Springer, XIV+134 pp. (in German).Google Scholar
  2. 2.
    Chen, C.I., and J.B. Cruz, Jr. 1972. Stackelberg solution for two-person games with biased information patterns. IEEE Transactions on Automatic Control AC-17: 791–798.Google Scholar
  3. 3.
    Simaan, M., and J.B. Cruz Jr. 1973. Additional aspects of the Stackelberg strategy in nonzero sum games. Journal of Optimization Theory and Applications 11: 613–626.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Simaan, M., and J.B. Cruz Jr. 1973. On the Stackelberg strategy in nonzero sum games. Journal of Optimization Theory and Applications 11: 533–555.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Leitmann, G. 1978. On generalized Stackelberg strategies. Journal of Optimization Theory and Applications 26: 637–648.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blaquière, A. 1976. Une géneralisation du concept d’optimalité et des certains notions geometriques qui’s rattachent, Institute des Hautes Etudes de Belgique, Cahiers du centre d’études de recherche operationelle, Vol. 18, No. 1–2, Bruxelles, 49–61.Google Scholar
  7. 7.
    Başar, T., and G.J. Olsder. 1999. Dynamic noncooperative game theory. Society for Industrial and Applied Mathematics, vol. 536. Philadelphia: SIAM.Google Scholar
  8. 8.
    Ungureanu, V. 2008. Solution principles for simultaneous and sequential games mixture. ROMAI Journal 4 (1): 225–242.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ungureanu, V. 2008. Solution principles for generalized Stackelberg games. In Mathematical Modelling, Optimization and Information Technologies, International Conference Proceedings, ATIC, March 19–21. Chişinău. Evrica, 181–189.Google Scholar
  10. 10.
    Peters, H. 2015. Game theory: A multi-leveled approach, 2nd edn. Berlin: Springer, XVII+494 pp.Google Scholar
  11. 11.
    Ungureanu, V., and V. Lozan, 2016. Stackelberg equilibrium sets in bimatrix mixed-strategy games. In Mathematical Modelling, Optimization and Information Technologies, International Conference Proceedings, ATIC, March 22–25, 2016, Chişinău: Evrica, 370–382 (in Romanian).Google Scholar
  12. 12.
    Korzhyk, D., Z. Yin, C. Kiekintveld, V. Conitzer, and M. Tambe. 2011. Stackelberg versus Nash in security games: An extended investigation of interchangeability, equivalence, and uniqueness. Journal of Artificial Intelligence Research 41: 297–327.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ungureanu, V. 2006. Nash equilibrium set computing in finite extended games. Computer Science Journal of Moldova, 14, No. 3 (42): 345–365.Google Scholar
  14. 14.
    Ungureanu, V. 2013. Linear discrete-time Pareto-Nash-Stackelberg control problem and principles for its solving. Computer Science Journal of Moldova 21, No. 1 (61): 65–85.Google Scholar
  15. 15.
    Ungureanu, V., and V. Lozan, 2013. Stackelberg equilibria set in multi-matrix mixed strategy games. In Proceedings IIS: International Conference on Intelligent Information Systems, Chişinău: Institute of Mathematics and Computer Science, August 20–23, 2013, 114–117.Google Scholar
  16. 16.
    Arkhangel’skii, A.V., and V.V. Fedorchuk. 1990. The basic concepts and constructions of general topology. In General topology I: Basic concepts and constructions, vol. 17, ed. A.V. Arkhangel’skii, and L.S. Pontrjagin, 1–90. Berlin: Dimension Theory, Encyclopaedia of the Mathematical Sciences Springer.CrossRefGoogle Scholar
  17. 17.
    Rockafellar, R.T., and R.J.-B. Wets, 2009. Variational analysis, 3rd edn. Berlin: Springer, XII+726 pp.Google Scholar
  18. 18.
    Kolmogorov, A.N., and S.V. Fomin. 1957. Elements of the theory of functions and functional analysis: Metric and normed spaces, vol. 1, 141. Rochester, New York: Graylock Press.Google Scholar
  19. 19.
    Kolmogorov, A.N., and S.V. Fomin. 1961. Elements of the theory of functions and functional analysis: Measure, the lebesgue integral, hilbert space, vol. 2, 139. Rochester, New York: Graylock Press.Google Scholar
  20. 20.
    Kolmogorov , A.N., and S.V. Fomin, 1989. Elements of the Theory of Functions and Functional Analysis (Elementy teorii funczii i funczionalnogo analiza) Moskva: Nauka, 6th edn. 632 pp. (in Russian).Google Scholar
  21. 21.
    Kantorovich, L.V., and G.P. Akilov. 1977. Functional analysis, 742. Moskva: Nauka.Google Scholar
  22. 22.
    Dantzig, G.B., and M.N. Thapa. 2003. Linear programming 2: Theory and extensions, 475. New York: Springer.zbMATHGoogle Scholar
  23. 23.
    Rockafellar, T. 1970. Convex analysis, 468. Princeton: Princeton University Press.CrossRefzbMATHGoogle Scholar
  24. 24.
    Curtain, R.F., and A.J. Pritchard, 1977. Functional analysis in modern applied mathematics. London: Academic Press Inc., IX+339 pp.Google Scholar
  25. 25.
    Collatz, L. 1966. Functional analysis and numerical mathematics. New York: Academic Press, X+473 pp.Google Scholar
  26. 26.
    Göpfert, A., Tammer, C., Riahi, H., and C. Zălinescu, 2003. Variational methods in partially ordered spaces. New York: Springer, XIV+350 pp.Google Scholar
  27. 27.
    Berge, C. 1963. Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. Edinburgh and London: Oliver and Boyd Ltd, XIII+270 pp.Google Scholar
  28. 28.
    Aubin, J.-P., and H. Frankowska. 1990. Set-valued analysis, systems and control series, vol. 2, 480. Boston: Birkhäuser.Google Scholar
  29. 29.
    Burachik, R.S., and A.N. Iusem. 2008. Set-valued Mappings and Enlargements of Monotone Operators, 305. New York: Springer Science.Google Scholar
  30. 30.
    Zelinskii, J.B. 1993. Multi-valued mappings in analysis. Kiev: Naukova Dumka, 1993, 362 pp. (in Russian).Google Scholar
  31. 31.
    Chen, G., Huang, X., and Yang, X. Vector optimization: set-valued and variational analysis. Berlin: Springer, X+308 pp.Google Scholar
  32. 32.
    Bourbaki, N. 1995. General topology. Chapters 1–4, Elements of Mathematics, vol. 18, New York: Springer, VII+437 pp.Google Scholar
  33. 33.
    Bourbaki, N. 1998. General topology, Chapters 5–10, Elements of Mathematics. New York: Springer, IV+363 pp.Google Scholar
  34. 34.
    Arzelà, C. 1895. Sulle funzioni di linee. Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat., 5(5): 55–74 (in Italian).Google Scholar
  35. 35.
    Arzelà, C. 1882–1883. Un’osservazione intorno alle serie di funzioni. Rend. Dell’ Accad. R. Delle Sci. Dell’Istituto di Bologna, 142–159 (in Italian).Google Scholar
  36. 36.
    Ascoli, G. 1882–1883. Le curve limiti di una variet data di curve. Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat., 18(3): 521–586 (in Italian).Google Scholar
  37. 37.
    Dunford, N., and J.T. Schwartz, 1958. Linear operators, vol. 1: General Theory, New York: Wiley Interscience Publishers, XIV+858 pp.Google Scholar
  38. 38.
    Kelley, J.L. 1991. General topology. New York: Springer-Verlag, XIV+298 pp.Google Scholar
  39. 39.
    Arkhangel’skii, A.V. 1995. Spaces of mappings and rings of continuous functions. In General topology III: Paracompactness, Function Spaces, Descriptive Theory, ed. A.V. Arkhangel’skii. Encyclopaedia of the Mathematical Sciences, vol. 51, 1–70. Berlin: Springer.Google Scholar
  40. 40.
    Nash, J. 1951. Noncooperative games. Annals of Mathematics 54 (2): 280–295.CrossRefGoogle Scholar
  41. 41.
    Kakutani, S. 1941. A generalization of Brouwer’s fixed point theorem. Duke Mathematical Journal 8: 457–459.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Brouwer, L.E.J. 1911. Über Abbildungen von Mannigfaltigkeiten. Mathematische Annalen 71: 97–115. (in German).MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Berinde, V. 2007. Iterative approximation of fixed points, 2nd edn. Berlin: Springer, XV+322 pp.Google Scholar
  44. 44.
    Agarwal, R.P., M. Meehan, and D. O’Regan, 2001. Fixed point theory and applications. Cambridge: Cambridge University Press, X+170 pp.Google Scholar
  45. 45.
    Górniewicz, L. 2006. Topological fixed point theory of multivalued mappings. Dordrecht: Springer, XIII+538 pp.Google Scholar
  46. 46.
    Carl, S., and S. Heikkilä, 2011. Fixed point theory in ordered sets and applications: from differential and integral equations to game theory. New York: Springer Science + Business Media, XIV+477 pp.Google Scholar
  47. 47.
    Border, K.C. 1985. Fixed point theorems with applications to economics and game theory. Cambridge: Cambridge University Press, VIII+129 pp.Google Scholar
  48. 48.
    Brown, R.F., M. Furi, L. Górniewicz, and E. Jing, 2005. Handbook of topological fixed point theory, Dordrecht, Netherlands: Springer, IX+971 pp.Google Scholar
  49. 49.
    Granas, A., and J. Dugundji, 2003. Fixed point theory. New York: Springer, XVI+690 pp.Google Scholar
  50. 50.
    Conitzer, V., and T. Sandholm, 2006. Computing the optimal strategy to commit to, Proceedings of the 7th ACM Conference on Electronic Commerce, Ann Arbor, MI, USA, June 11–15, 2006, 82–90.Google Scholar
  51. 51.
    Letchford, J., V. Conitzer, and K. Munagala. 2009. Learning and approximating the optimal strategy to commit to, algorithmic game theory. Lecture Notes in Computer Science 5814: 250–262.CrossRefzbMATHGoogle Scholar
  52. 52.
    Marhfour, A. 2000. Mixed solutions for weak Stackelberg problems: existence and stability results. Journal of Optimization Theory and Applications 105 (2): 417–440.MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Ungureanu, V. 2001. Mathematical programming. Chişinău: USM, 348 pp. (in Romanian).Google Scholar
  54. 54.
    Dantzig, G.B., and M.N. Thapa. 1997. Linear programming 1: Introduction, 474. New York: Springer.zbMATHGoogle Scholar
  55. 55.
    Demyanov, V., and V. Malozemov, 1972. Introduction in minimax. Moscow: Nauka, 368 pp. (in Russian).Google Scholar
  56. 56.
    Murty, K.G. 2014. Computational and algorithmic linear algebra and n-dimenshional geometry, 480. Singapore: World Scientific Publishing Company.Google Scholar
  57. 57.
    Trefethen, L.N., and D. Bau, III, 1997. Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics, XII+361 pp.Google Scholar
  58. 58.
    Shoham, Y., and K. Leyton-Brown. 2009. Multi-agent systems: Algorithmic, game-theoretic, and logical foundations, 532. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  59. 59.
    Jukna, S. 2011. Extremal combinatorics: With applications in computer science, 2nd edn. Berlin: Springer, XXIII+411 pp.Google Scholar
  60. 60.
    Karmarkar, N. 1984. New polynomial-time algorithm for linear programming. Combinatorica 4 (4): 373–395.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceMoldova State UniversityChișinăuMoldova

Personalised recommendations