Experimental Design Problems and Nash Equilibrium Solutions

  • Egidio D’Amato
  • Elia Daniele
  • Lina MallozziEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 100)


In this paper we present a non-cooperative game theoretical model for the well-known problem of experimental design. Nash equilibrium solutions of a suitable game will be the optimal values of the design variables, given by the coordinates of points in a region in the spirit of the facility location model. Because of the dependency of the objective functions on the distance from the domain’s boundary, this problem has a strong analogy with the classical sphere packing problem. Theoretical and computational results are presented for this location problem by virtue of a genetic algorithm procedure for both two- and three-dimensional test cases.


Nash Equilibrium Location Problem Nash Equilibrium Problem Game Theoretical Model Admissible Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Başar, T., Olsder, G.J.: Dynamic noncooperative game theory. In: Classics in Applied Mathematics, vol. 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999). Reprint of the second (1995) editionGoogle Scholar
  2. 2.
    Benabbou, A., Borouchaki, H., Laug, P., Lu, J.: Sphere packing and applications to granular structure modeling. In: Garimella, R.V. (eds.) Proceedings of the 17th International Meshing Roundtable, 12–15 October. Springer, Berlin (2008)Google Scholar
  3. 3.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Springer, New York (1998)Google Scholar
  4. 4.
    D’Amato, E., Daniele, E., Mallozzi, L., Petrone G.: Equilibrium strategies via GA to Stackelberg games under multiple follower best reply. Int. J. Intell. Syst. 27, 74–85 (2012)CrossRefGoogle Scholar
  5. 5.
    D’Amato, E., Daniele, E., Mallozzi, L., Petrone, G., Tancredi, S.: A hierarchical multi-modal hybrid Stackelberg-nash GA for a leader with multiple followers game. In: Sorokin, A., Murphey, R., Thai, M.T., Pardalos, P.M. (eds.) Dynamics of Information Systems: Mathematical Foundations. Springer Proceedings in Mathematics & Statistics, vol. 20, pp. 267–280. Springer, New York (2012)Google Scholar
  6. 6.
    Dean, A., Voss, D.: Design and Analysis of Experiments. Springer Texts in Statistics. Springer, Dordrecht (1998)Google Scholar
  7. 7.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 81–197 (2002)CrossRefGoogle Scholar
  8. 8.
    Donev, A., Torquato, S., Stillinger, F.H., Connelly, R.: A linear programming algorithm to test for jamming in hard-sphere packings. J. Comput. Phys. 197(1), 139–166 (2004). doi:10.1016/ Scholar
  9. 9.
    Fudenberg, D., Tirole, J.: Game Theory. The MIT Press, Cambridge (1993)Google Scholar
  10. 10.
    Hales, T.C.: The sphere packing problem. J. Comput. Appl. Math. 42, 41–76 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mallozzi, L.: Noncooperative facility location games. Oper. Res. Lett. 35, 151–154 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mallozzi, L., D’Amato, E., Daniele E.: A game theoretical model for experiments design optimization. In T.M. Rassias, C.A. Floudas, S. Butenko (Eds.) Optimization in Science and Engineering, In Honor of the 60th Birthday of Panos M. Pardalos, Springer (2014)Google Scholar
  13. 13.
    Migdalas, A., Pardalos, P.M., Varbrand, P. (eds.) Multilevel Optimization: Algorithms and Applications. Kluwer Academic, Dordrecht (1997)Google Scholar
  14. 14.
    Nurmela, K.J.: Stochastic optimization methods in sphere packing and covering problems in discrete geometry and coding theory. Ph.D. thesis, Helsinki University of Technology, printed by Picaset Oy (1997)Google Scholar
  15. 15.
    Periaux, J., Chen, H.Q., Mantel, B., Sefrioui, M., Sui, H.T.: Combining game theory and genetic algorithms with application to DDM-nozzle optimization problems. Finite Elem. Anal. Des. 37, 417–429 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Sloane, N.J.A.: The Sphere Packing Problem. 1998 Shannon Lecture. AT&T Shannon Lab, Florham Park, NJ (1998)Google Scholar
  17. 17.
    Sorokin, A., Pardalos, P. (eds.): Dynamics of Information Systems: Algorithmics Approaches, Sorokin, A., Pardalos, P. (eds.): Springer Proceedings in Mathematics & Statistics, vol. 51 (2013)Google Scholar
  18. 18.
    Sutou, A., Dai, Y.: Global optimization approach to unequal sphere packing problems in 3D. J. Optim. Theory Appl. 114(3), 671–694 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Industrial Engineering - Aerospace SectionUniversity of Naples Federico IINaplesItaly
  2. 2.Department of Mathematics and ApplicationsUniversity of Naples Federico IINaplesItaly

Personalised recommendations