IDEAS home Printed from https://ideas.repec.org/a/kap/theord/v90y2021i3d10.1007_s11238-021-09800-9.html
   My bibliography  Save this article

The evolutionary stability of optimism, pessimism, and complete ignorance

Author

Listed:
  • Burkhard C. Schipper

    (University of California)

Abstract

We seek an evolutionary explanation for why in some situations humans maintain either optimistic or pessimistic attitudes toward uncertainty and are ignorant to relevant aspects of their environment. Players in strategic games face Knightian uncertainty about opponents’ actions and maximize individually their Choquet expected utility with respect to neo-additive capacities (Chateauneuf et al. 2007) allowing for both an optimistic or pessimistic attitude toward uncertainty as well as ignorance to strategic dependencies. An optimist (resp. pessimist) overweighs good (resp. bad) outcomes. A complete ignorant never reacts to opponents’ changes of actions. We focus on sub- and supermodular aggregative games and provide monotone comparative statics w.r.t. optimism/pessimism. With qualifications, we show that in finite populations, optimistic (resp. pessimistic) complete ignorance is evolutionary stable and yields a strategic advantage in submodular (resp. supermodular) games with aggregate externalities. Moreover, this evolutionary stable preference leads to Walrasian behavior in these classes of games.

Suggested Citation

  • Burkhard C. Schipper, 2021. "The evolutionary stability of optimism, pessimism, and complete ignorance," Theory and Decision, Springer, vol. 90(3), pages 417-454, May.
    • Handle: RePEc:kap:theord:v:90:y:2021:i:3:d:10.1007_s11238-021-09800-9
    DOI: 10.1007/s11238-021-09800-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11238-021-09800-9
    File Function: Abstract
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11238-021-09800-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Wakker, Peter P, 2001. "Testing and Characterizing Properties of Nonadditive Measures through Violations of the Sure-Thing Principle," Econometrica, Econometric Society, vol. 69(4), pages 1039-1059, July.
    2. Bade, Sophie, 2011. "Ambiguous act equilibria," Games and Economic Behavior, Elsevier, vol. 71(2), pages 246-260, March.
    3. William H. Sandholm, 2001. "Preference Evolution, Two-Speed Dynamics, and Rapid Social Change," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 4(3), pages 637-679, July.
    4. William Novshek, 1985. "On the Existence of Cournot Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 52(1), pages 85-98.
    5. Jürgen Eichberger & David Kelsey & Burkhard C. Schipper, 2009. "Ambiguity and social interaction," Oxford Economic Papers, Oxford University Press, vol. 61(2), pages 355-379, April.
    6. Steffen Huck & Georg Kirchsteiger & Jörg Oechssler, 2005. "Learning to like what you have - explaining the endowment effect," Economic Journal, Royal Economic Society, vol. 115(505), pages 689-702, July.
    7. Schipper, Burkhard C., 2009. "Imitators and optimizers in Cournot oligopoly," Journal of Economic Dynamics and Control, Elsevier, vol. 33(12), pages 1981-1990, December.
    8. Jürgen Eichberger & David Kelsey & Burkhard Schipper, 2008. "Granny Versus Game Theorist: Ambiguity in Experimental Games," Theory and Decision, Springer, vol. 64(2), pages 333-362, March.
    9. Dubey, Pradeep & Haimanko, Ori & Zapechelnyuk, Andriy, 2006. "Strategic complements and substitutes, and potential games," Games and Economic Behavior, Elsevier, vol. 54(1), pages 77-94, January.
    10. Heifetz, Aviad & Shannon, Chris & Spiegel, Yossi, 2007. "What to maximize if you must," Journal of Economic Theory, Elsevier, vol. 133(1), pages 31-57, March.
    11. Guth, Werner, 1995. "An Evolutionary Approach to Explaining Cooperative Behavior by Reciprocal Incentives," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(4), pages 323-344.
    12. Ted To, 1999. "Risk and evolution," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 13(2), pages 329-343.
    13. Ingela Alger & Jörgen W. Weibull, 2019. "Evolutionary Models of Preference Formation," Annual Review of Economics, Annual Reviews, vol. 11(1), pages 329-354, August.
    14. Ok, Efe A. & Vega-Redondo, Fernando, 2001. "On the Evolution of Individualistic Preferences: An Incomplete Information Scenario," Journal of Economic Theory, Elsevier, vol. 97(2), pages 231-254, April.
    15. Dow James & Werlang Sergio Ribeiro Da Costa, 1994. "Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction," Journal of Economic Theory, Elsevier, vol. 64(2), pages 305-324, December.
    16. Eichberger, Jurgen & Kelsey, David, 2002. "Strategic Complements, Substitutes, and Ambiguity: The Implications for Public Goods," Journal of Economic Theory, Elsevier, vol. 106(2), pages 436-466, October.
    17. Cooper, Arnold C. & Woo, Carolyn Y. & Dunkelberg, William C., 1988. "Entrepreneurs' perceived chances for success," Journal of Business Venturing, Elsevier, vol. 3(2), pages 97-108.
    18. Dekel, Eddie & Scotchmer, Suzanne, 1999. "On the Evolution of Attitudes towards Risk in Winner-Take-All Games," Journal of Economic Theory, Elsevier, vol. 87(1), pages 125-143, July.
    19. Acemoglu, Daron & Jensen, Martin Kaae, 2013. "Aggregate comparative statics," Games and Economic Behavior, Elsevier, vol. 81(C), pages 27-49.
    20. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-180, January.
    21. Fernando Vega-Redondo, 1997. "The Evolution of Walrasian Behavior," Econometrica, Econometric Society, vol. 65(2), pages 375-384, March.
    22. Alex Possajennikov, 2004. "Two-Speed Evolution of Strategies and Preferences In Symmetric Games," Theory and Decision, Springer, vol. 57(3), pages 227-263, November.
    23. Gilboa, Itzhak, 1987. "Expected utility with purely subjective non-additive probabilities," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 65-88, February.
    24. Camerer, Colin & Weber, Martin, 1992. "Recent Developments in Modeling Preferences: Uncertainty and Ambiguity," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 325-370, October.
    25. Hans-Werner Sinn, 2002. "Weber's Law and the Biological Evolution of Risk Preferences: The Selective Dominance of the Logarithmic Utility Function," CESifo Working Paper Series 770, CESifo.
    26. Robson, Arthur J., 1996. "The Evolution of Attitudes to Risk: Lottery Tickets and Relative Wealth," Games and Economic Behavior, Elsevier, vol. 14(2), pages 190-207, June.
    27. Hehenkamp, B. & Leininger, W. & Possajennikov, A., 2004. "Evolutionary equilibrium in Tullock contests: spite and overdissipation," European Journal of Political Economy, Elsevier, vol. 20(4), pages 1045-1057, November.
    28. Samuelson, Larry, 2001. "Introduction to the Evolution of Preferences," Journal of Economic Theory, Elsevier, vol. 97(2), pages 225-230, April.
    29. Ebbe Groes & Hans Jørgen Jacobsen & Birgitte Sloth & Torben Tranaes, 1998. "Nash Equilibrium with Lower Probabilities," Theory and Decision, Springer, vol. 44(1), pages 37-66, January.
    30. Thomas Jungbauer & Klaus Ritzberger, 2011. "Strategic games beyond expected utility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(2), pages 377-398, October.
    31. Jürgen Eichberger & Ani Guerdjikova, 2018. "Do markets prove pessimists right ?," Post-Print hal-01869548, HAL.
    32. Carlos Alós-Ferrer & Ana Ania, 2005. "The evolutionary stability of perfectly competitive behavior," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(3), pages 497-516, October.
    33. Amir, R., 1996. "Cournot oligopoly and theory of supermodular games," LIDAM Reprints CORE 1228, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    34. Amir, Rabah, 1996. "Cournot Oligopoly and the Theory of Supermodular Games," Games and Economic Behavior, Elsevier, vol. 15(2), pages 132-148, August.
    35. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    36. Skaperdas, Stergios, 1991. "Conflict and Attitudes toward Risk," American Economic Review, American Economic Association, vol. 81(2), pages 116-120, May.
    37. repec:spr:thdchp:978-3-540-68437-4_12 is not listed on IDEAS
    38. Chateauneuf, Alain & Eichberger, Jurgen & Grant, Simon, 2007. "Choice under uncertainty with the best and worst in mind: Neo-additive capacities," Journal of Economic Theory, Elsevier, vol. 137(1), pages 538-567, November.
    39. Ely, Jeffrey C. & Yilankaya, Okan, 2001. "Nash Equilibrium and the Evolution of Preferences," Journal of Economic Theory, Elsevier, vol. 97(2), pages 255-272, April.
    40. Federico Echenique, 2003. "The equilibrium set of two-player games with complementarities is a sublattice," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 22(4), pages 903-905, November.
    41. Jürgen Eichberger & David Kelsey & Burkhard C. Schipper, 2009. "Ambiguity and social interaction," Oxford Economic Papers, Oxford University Press, vol. 61(2), pages 355-379, April.
    42. Jürgen Eichberger & Ani Guerdjikova, 2018. "Do Markets Prove Pessimists Right?," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 59(4), pages 2259-2295, November.
    43. Kin Chung Lo, 1999. "Nash equilibrium without mutual knowledge of rationality," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 14(3), pages 621-633.
    44. Matthew J. Ryan, 2002. "What do uncertainty-averse decision-makers believe?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 20(1), pages 47-65.
    45. Burkhard C. Schipper, 2004. "Submodularity and the evolution of Walrasian behavior," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(4), pages 471-477, August.
    46. Schipper, Burkhard C., 2009. "Imitators and optimizers in Cournot oligopoly," Journal of Economic Dynamics and Control, Elsevier, vol. 33(12), pages 1981-1990, December.
    47. Xavier Vives, 2001. "Oligopoly Pricing: Old Ideas and New Tools," MIT Press Books, The MIT Press, edition 1, volume 1, number 026272040x, April.
    48. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2012. "Unbeatable imitation," Games and Economic Behavior, Elsevier, vol. 76(1), pages 88-96.
    49. Eddie Dekel & Jeffrey C. Ely & Okan Yilankaya, 2007. "Evolution of Preferences -super-1," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 74(3), pages 685-704.
    50. Hans-Werner Sinn, 2003. "Weber's Law and the Biological Evolution of Risk Preferences: The Selective Dominance of the Logarithmic Utility Function, 2002 Geneva Risk Lecture," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 28(2), pages 87-100, December.
    51. Alex Possajennikov, 2003. "Evolutionary foundations of aggregate-taking behavior," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(4), pages 921-928, June.
    52. Lo, Kin Chung, 1996. "Equilibrium in Beliefs under Uncertainty," Journal of Economic Theory, Elsevier, vol. 71(2), pages 443-484, November.
    53. Kockesen, Levent & Ok, Efe A. & Sethi, Rajiv, 2000. "The Strategic Advantage of Negatively Interdependent Preferences," Journal of Economic Theory, Elsevier, vol. 92(2), pages 274-299, June.
    54. Schaffer, Mark E., 1989. "Are profit-maximisers the best survivors? : A Darwinian model of economic natural selection," Journal of Economic Behavior & Organization, Elsevier, vol. 12(1), pages 29-45, August.
    55. Pierpaolo Battigalli & Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci, 2015. "Self-Confirming Equilibrium and Model Uncertainty," American Economic Review, American Economic Association, vol. 105(2), pages 646-677, February.
    56. Zhou Lin, 1994. "The Set of Nash Equilibria of a Supermodular Game Is a Complete Lattice," Games and Economic Behavior, Elsevier, vol. 7(2), pages 295-300, September.
    57. Jürgen Eichberger & David Kelsey, 2014. "Optimism And Pessimism In Games," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 55, pages 483-505, May.
    58. P. Battigalli & S. Cerreia‐Vioglio & F. Maccheroni & M. Marinacci, 2016. "A Note on Comparative Ambiguity Aversion and Justifiability," Econometrica, Econometric Society, vol. 84, pages 1903-1916, September.
    59. Karni, Edi & Schmeidler, David, 1986. "Self-preservation as a foundation of rational behavior under risk," Journal of Economic Behavior & Organization, Elsevier, vol. 7(1), pages 71-81, March.
    60. Daughety,Andrew F. (ed.), 2005. "Cournot Oligopoly," Cambridge Books, Cambridge University Press, number 9780521022842.
    61. Kockesen, Levent & Ok, Efe A. & Sethi, Rajiv, 2000. "Evolution of Interdependent Preferences in Aggregative Games," Games and Economic Behavior, Elsevier, vol. 31(2), pages 303-310, May.
    62. Marinacci, Massimo, 2000. "Ambiguous Games," Games and Economic Behavior, Elsevier, vol. 31(2), pages 191-219, May.
    63. Bulow, Jeremy I & Geanakoplos, John D & Klemperer, Paul D, 1985. "Multimarket Oligopoly: Strategic Substitutes and Complements," Journal of Political Economy, University of Chicago Press, vol. 93(3), pages 488-511, June.
    64. Warneryd, Karl, 2002. "Rent, risk, and replication: Preference adaptation in winner-take-all markets," Games and Economic Behavior, Elsevier, vol. 41(2), pages 344-364, November.
    65. R Cornes & R Hartley, 2005. "The Geometry of Aggregative Games," Economics Discussion Paper Series 0514, Economics, The University of Manchester.
    66. Kukushkin, Nikolai S., 1994. "A fixed-point theorem for decreasing mappings," Economics Letters, Elsevier, vol. 46(1), pages 23-26, September.
    67. Robson, Arthur J., 1996. "A Biological Basis for Expected and Non-expected Utility," Journal of Economic Theory, Elsevier, vol. 68(2), pages 397-424, February.
    68. Daniel Ellsberg, 1961. "Risk, Ambiguity, and the Savage Axioms," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 75(4), pages 643-669.
    69. Sarin, Rakesh K & Wakker, Peter, 1992. "A Simple Axiomatization of Nonadditive Expected Utility," Econometrica, Econometric Society, vol. 60(6), pages 1255-1272, November.
    70. Rubin, Paul H & Paul, Chris W, II, 1979. "An Evolutionary Model of Taste for Risk," Economic Inquiry, Western Economic Association International, vol. 17(4), pages 585-596, October.
    71. Corchon, Luis C., 1994. "Comparative statics for aggregative games the strong concavity case," Mathematical Social Sciences, Elsevier, vol. 28(3), pages 151-165, December.
    72. Walker, James M. & Gardner, Roy & Ostrom, Elinor, 1990. "Rent dissipation in a limited-access common-pool resource: Experimental evidence," Journal of Environmental Economics and Management, Elsevier, vol. 19(3), pages 203-211, November.
    73. Jean-Yves Jaffray & Fabrice Philippe, 1997. "On the Existence of Subjective Upper and Lower Probabilities," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 165-185, February.
    74. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-1277, November.
    75. Eichberger, Jurgen & Kelsey, David, 2000. "Non-Additive Beliefs and Strategic Equilibria," Games and Economic Behavior, Elsevier, vol. 30(2), pages 183-215, February.
    76. Aviad Heifetz & Chris Shannon & Yossi Spiegel, 2007. "The Dynamic Evolution of Preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 32(2), pages 251-286, August.
    77. John B. Van Huyck & Raymond C. Battalio & Richard O. Beil, 1991. "Strategic Uncertainty, Equilibrium Selection, and Coordination Failure in Average Opinion Games," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 106(3), pages 885-910.
    78. Epstein, Larry G., 1997. "Preference, Rationalizability and Equilibrium," Journal of Economic Theory, Elsevier, vol. 73(1), pages 1-29, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Martin Jensen, 2010. "Aggregative games and best-reply potentials," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 43(1), pages 45-66, April.
    2. Péter Bayer & Ani Guerdjikova, 2020. "Optimism leads to optimality: Ambiguity in network formation," Working Papers hal-03005107, HAL.
    3. Guarino, Pierfrancesco & Ziegler, Gabriel, 2022. "Optimism and pessimism in strategic interactions under ignorance," Games and Economic Behavior, Elsevier, vol. 136(C), pages 559-585.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dominiak, Adam & Eichberger, Jürgen, 2021. "Games in context: Equilibrium under ambiguity for belief functions," Games and Economic Behavior, Elsevier, vol. 128(C), pages 125-159.
    2. Jürgen Eichberger & David Kelsey, 2011. "Are the treasures of game theory ambiguous?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(2), pages 313-339, October.
    3. David Kelsey & Tigran Melkonyan, 2018. "Contests with ambiguity," Oxford Economic Papers, Oxford University Press, vol. 70(4), pages 1148-1169.
    4. Jürgen Eichberger & David Kelsey & Burkhard Schipper, 2008. "Granny Versus Game Theorist: Ambiguity in Experimental Games," Theory and Decision, Springer, vol. 64(2), pages 333-362, March.
    5. Jürgen Eichberger & David Kelsey & Burkhard C. Schipper, 2009. "Ambiguity and social interaction," Oxford Economic Papers, Oxford University Press, vol. 61(2), pages 355-379, April.
    6. Schipper, Burkhard C., 2009. "Imitators and optimizers in Cournot oligopoly," Journal of Economic Dynamics and Control, Elsevier, vol. 33(12), pages 1981-1990, December.
    7. Dominiak, Adam & Lee, Min Suk, 2017. "Coherent Dempster–Shafer equilibrium and ambiguous signals," Journal of Mathematical Economics, Elsevier, vol. 68(C), pages 42-54.
    8. Jürgen Eichberger & David Kelsey & Burkhard C. Schipper, 2009. "Ambiguity and social interaction," Oxford Economic Papers, Oxford University Press, vol. 61(2), pages 355-379, April.
    9. Acemoglu, Daron & Jensen, Martin Kaae, 2013. "Aggregate comparative statics," Games and Economic Behavior, Elsevier, vol. 81(C), pages 27-49.
    10. repec:awi:wpaper:0448 is not listed on IDEAS
    11. Jürgen Eichberger & David Kelsey, 2014. "Optimism And Pessimism In Games," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 55(2), pages 483-505, May.
    12. Calford, Evan M., 2020. "Uncertainty aversion in game theory: Experimental evidence," Journal of Economic Behavior & Organization, Elsevier, vol. 176(C), pages 720-734.
    13. Norman, Thomas W.L., 2012. "Equilibrium selection and the dynamic evolution of preferences," Games and Economic Behavior, Elsevier, vol. 74(1), pages 311-320.
    14. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2012. "Unbeatable imitation," Games and Economic Behavior, Elsevier, vol. 76(1), pages 88-96.
    15. David Kelsey & Sara Roux, 2015. "An experimental study on the effect of ambiguity in a coordination game," Theory and Decision, Springer, vol. 79(4), pages 667-688, December.
    16. Lo, Kin Chung, 2009. "Correlated Nash equilibrium," Journal of Economic Theory, Elsevier, vol. 144(2), pages 722-743, March.
    17. Ronald Stauber, 2019. "A strategic product for belief functions," ANU Working Papers in Economics and Econometrics 2019-668, Australian National University, College of Business and Economics, School of Economics.
    18. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2012. "Unbeatable imitation," Games and Economic Behavior, Elsevier, vol. 76(1), pages 88-96.
    19. Frank Riedel & Linda Sass, 2014. "Ellsberg games," Theory and Decision, Springer, vol. 76(4), pages 469-509, April.
    20. Yang, Jian, 2018. "Game-theoretic modeling of players’ ambiguities on external factors," Journal of Mathematical Economics, Elsevier, vol. 75(C), pages 31-56.
    21. Aviad Heifetz & Chris Shannon & Yossi Spiegel, 2007. "The Dynamic Evolution of Preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 32(2), pages 251-286, August.

    More about this item

    Keywords

    Ambiguity; Knightian uncertainty; Choquet expected utility; Neo-additive capacity; Hurwicz criterion; Minimax; Supermodularity; Aggregative games; Monotone comparative statics; Playing the field; Evolution of preferences;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D01 - Microeconomics - - General - - - Microeconomic Behavior: Underlying Principles
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:kap:theord:v:90:y:2021:i:3:d:10.1007_s11238-021-09800-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.