Skip to main content
SpringerLink
Log in

Variational Analysis in Psychological Modeling

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper develops some mathematical models arising in psychology and some other areas of behavioral sciences that are formalized via general preferences with variable ordering structures. Our considerations are based on the recent variational rationality approach, which unifies numerous theories in different branches of behavioral sciences using, in particular, worthwhile change and stay dynamics and variational traps. In the mathematical framework of this approach, we derive a new variational principle, which can be viewed as an extension of the Ekeland variational principle to the case of set-valued mappings on quasimetric spaces with cone-valued ordering variable structures. Such a general setting is proved to be appropriate for broad applications to the functioning of goal systems in psychology, which are developed in the paper. In this way, we give a certain answer to the following striking question: in the world, where all things change (preferences, motivations, resistances, etc.), where goal systems drive a lot of entwined course pursuits between means and ends, what can stay fixed for a while? The obtained mathematical results and new insights open the door to developing powerful models of adaptive behavior, which strongly depart from pure static general equilibrium models of the Walrasian type, which are typical in economics.

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

Similar content being viewed by others

Notes

  1. We particularly refer the reader to the book [8] and the more recent paper [14] with the vast bibliographies therein for applications of modern techniques of variational analysis and set-valued optimization to models of welfare economics, which are typical in microeconomics modeling.

References

  1. Walras, L.: Eléments d’Économie Politique Pure. Corbaz, Lausanne (1874)

    Google Scholar 

  2. Bridges, W.: Managing Transitions: Making the Most of Change, 3rd edn. Nicholas Brealey, Boston (2009)

    Google Scholar 

  3. Bao, T.Q., Mordukhovich, B.S., Soubeyran, A.: Variational principles in behavioral sciences. http://arxiv.org/abs/1311.6017 (2013)

  4. Soubeyran, A.: Variational Rationality, A Theory of Individual Stability and Change, Worthwhile and Ambidextry Behaviors. GREQAM, Aix-Marseille University, Provence (2009)

    Google Scholar 

  5. Soubeyran, A.: Variational Rationality and the Unsatisfied Man: Routines and the Course Pursuit Between Aspirations, Capabilities and Beliefs. GREQAM, Aix-Marseille University, Provence (2010)

    Google Scholar 

  6. Simon, H.: A behavioral model of rational choice. Q. J. Econom. 69, 99–188 (1955)

    Article  Google Scholar 

  7. Rockafellar, R.T., Wets, J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  8. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Applications. Springer, Berlin (2006)

    Google Scholar 

  9. Attouch, H., Soubeyran, A.: Local search proximal algorithms as decision dynamics with costs to move. Set Valued Var. Anal. 19, 157–177 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cruz Neto, J.X., Oliveira, P.R., Soares Jr, P.A., Soubeyran, A.: Learning how to play nash, potential games and alternating minimization method for structured nonconvex problems on riemannian manifolds. J. Convex Anal. 20, 395–438 (2013)

    MATH  MathSciNet  Google Scholar 

  11. Flores-Bazan, F., Luc, D.T., Soubeyran, A.: Maximal elements under reference-dependent preferences with applications to behavioral traps and games. J. Optim. Theory Appl. 155, 883–901 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Luc, D.T., Soubeyran, A.: Variable preference relations: existence of maximal elements. J. Math. Econ. 49, 251–262 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Moreno, F.G., Oliveira, P.R., Soubeyran, A.: A proximal algorithm with quasidistance. application to habit’s formation. Optimization 61, 1383–1403 (2011)

    Article  MathSciNet  Google Scholar 

  14. Bao, T.Q., Mordukhovich, B.S.: Set-valued optimization in welfare economics. Adv. Math. Econ. 13, 113–153 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ekeland, I.: Sur les problémes variationnels. C. R. Acad. Sci. Paris 275, 1057–1059 (1972)

    MATH  MathSciNet  Google Scholar 

  16. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1, 432–467 (1979)

    Article  MathSciNet  Google Scholar 

  18. Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern. 36, 531–562 (2007)

    MATH  MathSciNet  Google Scholar 

  19. Bao, T.Q., Mordukhovich, B.S.: Relative pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Progr. 122, 301–347 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Yu, P.L.: Habitual domains. Oper. Res. 39, 869–876 (1991)

    Article  Google Scholar 

  21. Yu, P.L., Larbani, M.: Two-person second-order games, part 1: formulation and transition anatomy. J. Optim. Theory Appl. 141, 619–639 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Larbani, M., Yu, P.L.: Two-person second-order games, part ii: restructuring operations to reach a win-win profile. J. Optim. Theory Appl. 141, 641–659 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yu, P.L., Chen, Y.C., Ehrgott, M., Salvatore Greco, J.R.F.: Dynamic MCDM, habitual domains and competence set analysis for effective decision making in changeable spaces. Trends in Multiple Criteria Decision Analysis (Chap. 1). Springer, Berlin (2010)

    Google Scholar 

  24. Larbani, M., Yu, P.L.: \(n\)-person second-order games: a paradigm shift in game theory. J. Optim. Theory Appl. 149, 447–473 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Larbani, M., Yu, P.L.: Decision making and optimization in changeable spaces, a new paradigm. J. Optim. Theory Appl. 155, 727–761 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Bento, G., Soubeyran, A.: Generalized inexact proximal algorithms: habit/routine formation with resistance to change, following worthwhile changes. J. Optim. Theory Appl., second revision (2014)

  27. Kremer, M.: The o-ring theory of economic development. Q. J. Econ. 108, 551–575 (1993)

    Article  Google Scholar 

  28. Alber, S., Heward, W.: Twenty-five behavior traps guaranteed to extend your students’ academic and social skills. Interv. Sch. Clin. 31, 285–289 (1996)

    Article  Google Scholar 

  29. Baer, D., Wolf, M.: The entry into natural communities of reinforcement. In: Ulrich, R., Stachnick, T., Mabry, J. (eds.): Control of Human Behavior, pp. 319–324. Glenview, IL, Scott Foresman (1970)

  30. Plous, C.: The Psychology of Judgment and Decision Making. Mcgraw-Hill Book Company, New York (1993)

    Google Scholar 

  31. Stephen, F.: The Power of Reinforcement. SUNY, New York (2004)

    Google Scholar 

  32. Baumeister, R.: Ego-depletion and self-control failure: an energy model of the self’s executive function. Self Identity 1, 129–136 (2002)

    Article  Google Scholar 

  33. Hammond, J.S., Keeney, R.L., Raiffa, H.: The hidden traps in decision making. Harv. Bus. Rev. 76, 47 (1998)

  34. Levinthal, D., March, J.: The myopia of learning. Strateg. Manag. J. 14, 95–112 (1993)

    Article  Google Scholar 

  35. March, J.G.: Exploration and exploitation in organizational learning. Organ. Sci. 2, 71–87 (1991)

    Article  Google Scholar 

  36. Appadurai, A.: The capacity to aspire: culture and the terms of recognition. In: Rao, V., Walton, M. (eds.) Culture and Public Action, pp. 59–84. The World Bank, Washington, D.C. (2004)

    Google Scholar 

  37. Heifetz, A., Minelli, E.: Aspiration traps. Discussion Paper 0610, Universitá degli Studi di Brescia (2006)

  38. Ray, D.: Aspirations, poverty and economic change. In: Banerjee, A., Benabou, R., Mookherjee, D. (eds.) What We Have Learnt about Poverty. Oxford University Press, Oxford (2006)

    Google Scholar 

  39. Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)

    MATH  Google Scholar 

  40. Gutiérrez, C., Jiménez, B., Novo, V.: A set-valued ekeland’s variational principle in vector optimization. SIAM J. Control Optim. 47, 883–903 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Khanh, P.Q., Quy, D.N.: On generalized ekeland’s variational principle and equivalent formulations for set-valued mappings. J. Glob. Optim. 49, 381–396 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  42. Liu, C.G., Ng, K.F.: Ekeland’s variational principle for set-valued functions. SIAM J. Optim. 21, 41–56 (2011)

    Article  MathSciNet  Google Scholar 

  43. Bao, T.Q., Mordukhovich, B.S.: Necessary nondomination conditions for set and vector optimization with variable structures. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0332-6

    Google Scholar 

  44. Chen, G.Y., Yang, X.Q.: Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112, 97–110 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  45. Eichfelder, G.: Optimal elements in vector optimization with a variable ordering structure. J. Optim. Theory Appl. 151, 217–240 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  46. Eichfelder, G., Ha, T.X.D.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62, 597–627 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  47. Bello Cruz, J., Bouza Allende, G.: A steepest descent-like method for variable order vector optimization problems. J. Optim. Theory Appl., to appear (2014)

  48. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin Heidelberg (1989)

    Book  Google Scholar 

  49. Howard, J.A., Sheth, J.N.: Theory of Buyer Behavior (Marketing). Wiley, New York (1969)

    Google Scholar 

  50. Brisoux, J.: Le Phénomène des Ensembles éVoqués: Une Étude Empirique des Dimensions Contenu et Taille. Université Laval, Thèse de doctorat (1980)

    Google Scholar 

  51. Brisoux, J.E., Laroche, M.: Evoked set formation and composition: an empirical investigation under a routinized response behavior situation. In: Monroe, K.B. (ed.) Advances in Consumer Research, pp. 357–361. Association for Consumer Research, Ann Arbor (1981)

    Google Scholar 

  52. Bhatia, S.: Associations and the accumulation of preference. Psychol. Rev., to appear (2014)

  53. Bhatia, S., Golman, R.: Attention and reference dependence. Department of Social & Decision Sciences, Carnegie Mellon University, Pittsburgh (2012)

    Google Scholar 

  54. Barg, J.: The four horsemen of automaticity: awareness, intention, efficiency, and control in social cognition. Handb. Soc. Cogn. 1, 1–40 (1994)

    MathSciNet  Google Scholar 

  55. Tversky, A.: Intransitivity of preferences. Psychol. Rev. 76, 31–48 (1969)

    Article  Google Scholar 

  56. Sen, A.: Commodities and Capabilities. Oxford University Press, Oxford (1985)

    Google Scholar 

Download references

Acknowledgments

The authors are gratefully indebted to the anonymous referee for his/her helpful remarks, which allowed us to improve the original presentation. Research of the second author was partially supported by the USA National Science Foundation under Grant DMS-1007132.

Author information

Authors and Affiliations

  1. Department of Mathematics & Computer Science, Northern Michigan University, Marquette, MI, 49855, USA

    T. Q. Bao

  2. Department of Mathematics, Wayne State University, Detroit, MI, 49855, USA

    B. S. Mordukhovich

  3. King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    B. S. Mordukhovich

  4. CNRS & EHESS, Aix-Marseille School of Economics, Aix-Marseille University, 13002 , Marseille, France

    A. Soubeyran

Authors
  1. T. Q. Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. S. Mordukhovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Soubeyran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Q. Bao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bao, T.Q., Mordukhovich, B.S. & Soubeyran, A. Variational Analysis in Psychological Modeling. J Optim Theory Appl 164, 290–315 (2015). https://doi.org/10.1007/s10957-014-0569-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0569-8

Keywords

Search

Navigation