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Axiomatization of an Exponential Similarity Function

Author

Listed:
  • Antoine Billot

    (Universite de Paris II)

  • Itzhak Gilboa

    (School of Economics, Tel Aviv Univ.)

  • David Schmeidler

    (School of Mathematical Sciences, Tel Aviv Univ.)

Abstract

An agent is asked to assess a real-valued variable y based on certain characteristics x=(x^{1},...,x^{m}), and on a database consisting of n observations of (x^{1},...,x^{m},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, y_{n+1}^{s}, be the weighted average of all previously observed values y_{i}, where the weight of y_{i} is the similarity between the vector x_{n+1}^{1},...,x_{n+1}^{m}, associated with y_{n+1}, and the previously observed vector, x_{i}^{1},...,x_{i}^{m}. This paper axiomatizes, in terms of the prediction y_{n+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, 2004. "Axiomatization of an Exponential Similarity Function," Cowles Foundation Discussion Papers 1485, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1485
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d14/d1485.pdf
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    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, 2005. "Probabilities as Similarity-Weighted Frequencies," Econometrica, Econometric Society, vol. 73(4), pages 1125-1136, July.
    3. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2003. "Probabilities: Frequencies Viewed in Perspective," Levine's Bibliography 666156000000000295, UCLA Department of Economics.
    4. Itzhak Gilboa & Offer Lieberman & David Schmeidler, 2006. "Empirical Similarity," The Review of Economics and Statistics, MIT Press, vol. 88(3), pages 433-444, August.
    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.
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    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, 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.
    3. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2005. "Probabilities as Similarity-Weighted Frequencies," Econometrica, Econometric Society, vol. 73(4), pages 1125-1136, July.
    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, 2006. "Empirical Similarity," The Review of Economics and Statistics, MIT Press, vol. 88(3), pages 433-444, August.
    6. 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.
    7. 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.
    8. 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.
    9. Guerdjikova, Ani, 2008. "Case-based learning with different similarity functions," Games and Economic Behavior, Elsevier, vol. 63(1), pages 107-132, May.
    10. Teitelbaum, Joshua C., 2013. "Asymmetric empirical similarity," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 346-351.
    11. 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.
    12. Guerdjikova, Ani, 2004. "Preference for diversification with similarity considerations," Papers 04-48, Sonderforschungsbreich 504.

    More about this item

    Keywords

    Similarity; exponential;

    JEL classification:

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

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