IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/63538.html
   My bibliography  Save this paper

Invariance of the distribution of the maximum

Author

Listed:
  • Fosgerau, Mogens
  • Lindberg, Per Olov
  • Mattsson, Lars-Göran
  • Weibull, Jörgen

Abstract

Many models in economics involve probabilistic choices where each decision-maker selects the best alternative from a finite set. Viewing the value of each alternative as a random variable, the analyst is then interested in the choice probabilities, that is, the probability for an alternative to give the maximum value. Much analytical power can be gained, both for positive and normative analysis, if the maximum value is statistically independent of which alternative obtains the highest value. This note synthesizes and generalizes previous results on this invariance property. We provide characterizations of the property within a wide class of distributions that comprises the McFadden GEV class, show implications in several directions, and establish connections with copulas. We illustrate the usefulness of the invariance property by way of a few examples.

Suggested Citation

  • Fosgerau, Mogens & Lindberg, Per Olov & Mattsson, Lars-Göran & Weibull, Jörgen, 2015. "Invariance of the distribution of the maximum," MPRA Paper 63538, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:63538
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/63538/1/MPRA_paper_63538.pdf
    File Function: original version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Perloff, Jeffrey M & Salop, Steven, 1984. "Equilibrium with product differentiation," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt4cq0m6s3, Department of Agricultural & Resource Economics, UC Berkeley.
    2. P O Lindberg & E A Eriksson & L-G Mattsson, 1995. "Invariance of Achieved Utility in Random Utility Models," Environment and Planning A, , vol. 27(1), pages 121-142, January.
    3. Krishna, Vijay, 2009. "Auction Theory," Elsevier Monographs, Elsevier, edition 2, number 9780123745071.
    4. Resnick, Sidney I. & Roy, Rishin, 1990. "Leader and maximum independence for a class of discrete choice models," Economics Letters, Elsevier, vol. 33(3), pages 259-263, July.
    5. Michael Sattinger, 1984. "Value of an Additional Firm in Monopolistic Competition," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 51(2), pages 321-332.
    6. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 499-516, December.
    7. Mattsson, Lars-Göran & Weibull, Jörgen W. & Lindberg, Per Olov, 2014. "Extreme values, invariance and choice probabilities," Transportation Research Part B: Methodological, Elsevier, vol. 59(C), pages 81-95.
    8. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    9. de Palma, Andre & Kilani, Karim, 2007. "Invariance of conditional maximum utility," Journal of Economic Theory, Elsevier, vol. 132(1), pages 137-146, January.
    10. Train, Kenneth & Wilson, Wesley W., 2008. "Estimation on stated-preference experiments constructed from revealed-preference choices," Transportation Research Part B: Methodological, Elsevier, vol. 42(3), pages 191-203, March.
    11. Andriy Norets & Satoru Takahashi, 2013. "On the surjectivity of the mapping between utilities and choice probabilities," Quantitative Economics, Econometric Society, vol. 4(1), pages 149-155, March.
    12. Jeffrey M. Perloff & Steven C. Salop, 1985. "Equilibrium with Product Differentiation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 52(1), pages 107-120.
    13. Anas, Alex & Feng, Cheng Min, 1988. "Invariance of expected utilities in logit models," Economics Letters, Elsevier, vol. 27(1), pages 41-45.
    14. Baye, Michael R. & Hoppe, Heidrun C., 2003. "The strategic equivalence of rent-seeking, innovation, and patent-race games," Games and Economic Behavior, Elsevier, vol. 44(2), pages 217-226, August.
    Full references (including those not matched with items on IDEAS)

    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. Fosgerau, Mogens & Lindberg, Per Olov & Mattsson, Lars-Göran & Weibull, Jörgen, 2018. "A note on the invariance of the distribution of the maximum," Journal of Mathematical Economics, Elsevier, vol. 74(C), pages 56-61.
    2. Mattsson, Lars-Göran & Weibull, Jörgen W. & Lindberg, Per Olov, 2014. "Extreme values, invariance and choice probabilities," Transportation Research Part B: Methodological, Elsevier, vol. 59(C), pages 81-95.
    3. Murata, Yasusada, 2007. "Taste heterogeneity and the scale of production: Fragmentation, unification, and segmentation," Journal of Urban Economics, Elsevier, vol. 62(1), pages 135-160, July.
    4. Ushchev, Philip & Zenou, Yves, 2018. "Price competition in product variety networks," Games and Economic Behavior, Elsevier, vol. 110(C), pages 226-247.
    5. Marco A. Haan & José L. Moraga‐González, 2011. "Advertising for Attention in a Consumer Search Model," Economic Journal, Royal Economic Society, vol. 121(552), pages 552-579, May.
    6. Berry, Steven & Levinsohn, James & Pakes, Ariel, 1995. "Automobile Prices in Market Equilibrium," Econometrica, Econometric Society, vol. 63(4), pages 841-890, July.
    7. Creane, Anthony & Manduchi, Agostino, 2022. "Informative advertising in monopolistically competitive markets," International Journal of Industrial Organization, Elsevier, vol. 83(C).
    8. Torshizi Mohammad & Fulton Murray E. & Gray Richard S., 2018. "Non-Linear Demand in a Linear Town," Journal of Agricultural & Food Industrial Organization, De Gruyter, vol. 16(2), pages 1-12, November.
    9. Clarissa Yeap, 2008. "Price markups in oligopoly models with differentiated products," Economics Bulletin, AccessEcon, vol. 12(15), pages 1-10.
    10. Verboven, F.L., 1994. "International price discrimination in the European car market : An econometric model of oligopoly behavior with product differentiation," Discussion Paper 1994-51, Tilburg University, Center for Economic Research.
    11. Bonatti, Alessandro, 2011. "Brand-specific tastes for quality," International Journal of Industrial Organization, Elsevier, vol. 29(5), pages 562-575, September.
    12. Gabaix, Xavier & Laibson, David & Li, Deyuan & Li, Hongyi & Resnick, Sidney & de Vries, Casper G., 2016. "The impact of competition on prices with numerous firms," Journal of Economic Theory, Elsevier, vol. 165(C), pages 1-24.
    13. repec:ebl:ecbull:v:12:y:2008:i:15:p:1-10 is not listed on IDEAS
    14. Simon P. Anderson & Andre de Palma, 1989. "The Logit as a Model of Product Differentiation: Further Results and Extensions," Discussion Papers 913, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    15. Raymond Deneckere & Michael Rothschild, 1986. "Monopolistic Competition and Preference Diversity," Discussion Papers 684, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    16. Osharin Alexander & Verbus Valery, 2015. "Heterogeneous consumers and market structure in a monopolistically competitive setting," EERC Working Paper Series 15/03e, EERC Research Network, Russia and CIS.
    17. Andrew Rhodes & Jidong Zhou, 2019. "Consumer Search and Retail Market Structure," Management Science, INFORMS, vol. 67(6), pages 2607-2623, June.
    18. Chi, Chang Koo & Murto, Pauli & Valimaki, Juuso, 2017. "All-Pay Auctions with Affiliated Values," MPRA Paper 80799, University Library of Munich, Germany.
    19. Yongmin Chen & Scott J Savage, 2011. "The Effects of Competition on the Price for Cable Modem Internet Access," The Review of Economics and Statistics, MIT Press, vol. 93(1), pages 201-217, February.
    20. Daniel McFadden, 2009. "The human side of mechanism design: a tribute to Leo Hurwicz and Jean-Jacque Laffont," Review of Economic Design, Springer;Society for Economic Design, vol. 13(1), pages 77-100, April.
    21. Au, Pak Hung & Kawai, Keiichi, 2020. "Competitive information disclosure by multiple senders," Games and Economic Behavior, Elsevier, vol. 119(C), pages 56-78.

    More about this item

    Keywords

    Choice; random utility; extreme value; leader-maximum; invariance; independence;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities
    • D01 - Microeconomics - - General - - - Microeconomic Behavior: Underlying Principles

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:pra:mprapa:63538. 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: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

    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.