IDEAS home Printed from https://ideas.repec.org/p/hal/pseptp/hal-00463265.html
   My bibliography  Save this paper

Axiomatization of an exponential similarity function

Author

Listed:
  • Antoine Billot

    (LEM - Laboratoire d'Économie Moderne - UP2 - Université Panthéon-Assas, PJSE - Paris-Jourdan Sciences Economiques - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Itzhak Gilboa

    (TAU - Tel Aviv University, GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

  • David Schmeidler

Abstract

An individual is asked to assess a real-valued variable y based on certain characteristics x = (x1,..., xm), and on a database consisting of n observations of (x1,..., xm, y). A possible approach to combine past observations of x and y with the current values of x to generate an assessment of y is similarity-weighted averaging. It suggests that the predicted value of y, yn+1s, be the weighted average of all previously observed values yi, where the weight of yi is the similarity between the vector xn+11,..., xn+1m, associated with yn+1, and the previously observed vector, xi1,..., xim. This paper axiomatizes, in terms of the prediction yn+1, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared.

Suggested Citation

  • Antoine Billot & Itzhak Gilboa & David Schmeidler, 2008. "Axiomatization of an exponential similarity function," PSE-Ecole d'économie de Paris (Postprint) hal-00463265, HAL.
  • Handle: RePEc:hal:pseptp:hal-00463265
    DOI: 10.1016/j.mathsocsci.2007.08.002
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Itzhak Gilboa & David Schmeidler, 2003. "Inductive Inference: An Axiomatic Approach," Econometrica, Econometric Society, vol. 71(1), pages 1-26, January.
    2. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2012. "Probabilities as Similarity-Weighted Frequencies," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 7, pages 169-184, World Scientific Publishing Co. Pte. Ltd..
    3. Itzhak Gilboa & Offer Lieberman & David Schmeidler, 2012. "Empirical Similarity," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 9, pages 211-243, World Scientific Publishing Co. Pte. Ltd..
    4. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2003. "Probabilities: Frequencies Viewed in Perspective," Levine's Bibliography 666156000000000295, UCLA Department of Economics.
    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. Gilboa, Itzhak & Lieberman, Offer & Schmeidler, David, 2011. "A similarity-based approach to prediction," Journal of Econometrics, Elsevier, vol. 162(1), pages 124-131, May.
    2. Todd Guilfoos & Andreas Duus Pape, 2020. "Estimating Case-Based Learning," Games, MDPI, vol. 11(3), pages 1-25, September.
    3. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2012. "Probabilities as Similarity-Weighted Frequencies," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 7, pages 169-184, World Scientific Publishing Co. Pte. Ltd..
    4. Golosnoy, Vasyl & Hamid, Alain & Okhrin, Yarema, 2014. "The empirical similarity approach for volatility prediction," Journal of Banking & Finance, Elsevier, vol. 40(C), pages 321-329.
    5. Itzhak Gilboa & Offer Lieberman & David Schmeidler, 2012. "Empirical Similarity," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 9, pages 211-243, World Scientific Publishing Co. Pte. Ltd..
    6. Todd Guilfoos & Andreas Pape, 2016. "Predicting human cooperation in the Prisoner’s Dilemma using case-based decision theory," Theory and Decision, Springer, vol. 80(1), pages 1-32, January.
    7. Gayer, Gabrielle, 2010. "Perception of probabilities in situations of risk: A case based approach," Games and Economic Behavior, Elsevier, vol. 68(1), pages 130-143, January.
    8. Huirong Zhang & Zhenyu Zhang & Lixin Zhou & Shuangsheng Wu, 2021. "Case-Based Reasoning for Hidden Property Analysis of Judgment Debtors," Mathematics, MDPI, vol. 9(13), pages 1-17, July.
    9. Robert F. Bordley, 2011. "Using Bayes' Rule to Update an Event's Probabilities Based on the Outcomes of Partially Similar Events," Decision Analysis, INFORMS, vol. 8(2), pages 117-127, June.
    10. Wolfgang Ossadnik & Dirk Wilmsmann & Benedikt Niemann, 2013. "Experimental evidence on case-based decision theory," Theory and Decision, Springer, vol. 75(2), pages 211-232, August.
    11. Pape, Andreas & Kurtz, Kenneth, 2013. "Evaluating Case-based Decision Theory: Predicting Empirical Patterns of Human Classification Learning (Extensions)," MPRA Paper 45206, University Library of Munich, Germany.
    12. Guerdjikova, Ani, 2008. "Case-based learning with different similarity functions," Games and Economic Behavior, Elsevier, vol. 63(1), pages 107-132, May.
    13. Teitelbaum, Joshua C., 2013. "Asymmetric empirical similarity," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 346-351.
    14. Bleile, Jörg, 2016. "Cautious Belief Formation," Center for Mathematical Economics Working Papers 507, Center for Mathematical Economics, Bielefeld University.
    15. Pape, Andreas Duus & Kurtz, Kenneth J., 2013. "Evaluating case-based decision theory: Predicting empirical patterns of human classification learning," Games and Economic Behavior, Elsevier, vol. 82(C), pages 52-65.
    16. Guerdjikova, Ani, 2004. "Preference for diversification with similarity considerations," Papers 04-48, Sonderforschungsbreich 504.
    17. Minjie Huang & Shunan Zhao & Andreas Pape, 2023. "Estimating Case‐based Individual and Social Learning in Corporate Tax Avoidance," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 85(2), pages 403-434, April.

    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. Itzhak Gilboa & Offer Lieberman & David Schmeidler, 2012. "On the Definition of Objective Probabilities by Empirical Similarity," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 11, pages 259-280, World Scientific Publishing Co. Pte. Ltd..
    2. Itzhak Gilboa, 2010. "Questions in Decision Theory," Annual Review of Economics, Annual Reviews, vol. 2(1), pages 1-19, September.
    3. Gilboa, Itzhak & Schmeidler, David, 2010. "Simplicity and likelihood: An axiomatic approach," Journal of Economic Theory, Elsevier, vol. 145(5), pages 1757-1775, September.
    4. Gilboa, Itzhak & Lieberman, Offer & Schmeidler, David, 2011. "A similarity-based approach to prediction," Journal of Econometrics, Elsevier, vol. 162(1), pages 124-131, May.
    5. Gayer, Gabrielle, 2010. "Perception of probabilities in situations of risk: A case based approach," Games and Economic Behavior, Elsevier, vol. 68(1), pages 130-143, January.
    6. Bleile, Jörg, 2016. "Cautious Belief Formation," Center for Mathematical Economics Working Papers 507, Center for Mathematical Economics, Bielefeld University.
    7. Rossella Argenziano & Itzhak Gilboa, 2012. "History as a coordination device," Theory and Decision, Springer, vol. 73(4), pages 501-512, October.
    8. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2012. "Probabilities as Similarity-Weighted Frequencies," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 7, pages 169-184, World Scientific Publishing Co. Pte. Ltd..
    9. Itzhak Gilboa & Andrew Postlewaite & David Schmeidler, 2007. "Probabilities in Economic Modeling," Levine's Bibliography 843644000000000357, UCLA Department of Economics.
    10. Eichberger, Jurgen & Guerdjikova, Ani, 2007. "Multiple Priors as Similarity Weighted Frequencies," Working Papers 07-03, Cornell University, Center for Analytic Economics.
    11. Bleile, Jörg, 2016. "Limited Attention in Case-Based Belief Formation," Center for Mathematical Economics Working Papers 518, Center for Mathematical Economics, Bielefeld University.
    12. Mohlin, Erik, 2014. "Optimal categorization," Journal of Economic Theory, Elsevier, vol. 152(C), pages 356-381.
    13. repec:awi:wpaper:0470 is not listed on IDEAS
    14. Chiang, Mi-Hsiu & Chiu, Hsin-Yu & Kuo, Wei-Yu, 2021. "Predictive ability of similarity-based futures trading strategies," Pacific-Basin Finance Journal, Elsevier, vol. 68(C).
    15. Eichberger, Jürgen & Guerdjikova, Ani, 2013. "Ambiguity, data and preferences for information – A case-based approach," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1433-1462.
    16. Eichberger, Jürgen & Guerdjikova, Ani, 2010. "Case-based belief formation under ambiguity," Mathematical Social Sciences, Elsevier, vol. 60(3), pages 161-177, November.
    17. Teitelbaum, Joshua C., 2013. "Asymmetric empirical similarity," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 346-351.
    18. Eichberger, Jürgen & Guerdjikova, Ani, 2008. "Multiple Priors as Similarity Weighted Frequencies," Papers 08-07, Sonderforschungsbreich 504.
    19. Charness, Gary & Levin, Dan & Schmeidler, David, 2019. "An experimental study of estimation and bidding in common-value auctions with public information," Journal of Economic Theory, Elsevier, vol. 179(C), pages 73-98.
    20. Itzhak Gilboa & Andrew Postlewaite & Larry Samuelson & David Schmeidler, 2011. "Economic Models as Analogies," PIER Working Paper Archive 12-001, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    21. Stefania D'Amico, 2004. "Density Estimation and Combination under Model Ambiguity," Computing in Economics and Finance 2004 273, Society for Computational Economics.

    More about this item

    Keywords

    Similarity function; Axiom; Exponential decay;
    All these keywords.

    JEL classification:

    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs
    • D8 - Microeconomics - - Information, Knowledge, and Uncertainty

    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:hal:pseptp:hal-00463265. 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: Caroline Bauer (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.