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Affiliation and Dependence in Economic Models

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  • Luciano I. de Castro

Abstract

Affiliation has been a prominent assumption in the study of economic models with statistical dependence. Despite its large number of applications, especially in auction theory, affiliation has limitations that are important to be aware of. This paper shows that affiliation is a restrictive condition and the intuition usually given for its adoption may be misleading. Also, other usual justifications for affiliation are not compelling. Moreover, some implications of affiliation do not generalize to other definitions of positive dependence. These results show the need to consider alternatives to affiliation. The results of this paper suggest new directions for the study of dependence in economics. The main result classifies economic models of information and proves the existence of a minimally informative random variable that makes types conditionally independent. If this variable is known, then all results that are valid under independence are also valid for these models with statistically dependent types. Complementing this result, we describe a method to study general forms of dependence using grid distributions, which are distributions whose densities are constant in squares. This method allows a comprehensive investigation on the revenue ranking of auctions under general dependence.

Suggested Citation

  • Luciano I. de Castro, 2009. "Affiliation and Dependence in Economic Models," Discussion Papers 1479, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1479
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    File URL: http://www.kellogg.northwestern.edu/research/math/papers/1479.pdf
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    References listed on IDEAS

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    1. Bergin, James, 2001. "Common Knowledge with Monotone Statistics," Econometrica, Econometric Society, vol. 69(5), pages 1315-1332, September.
    2. Hubbard, Timothy P. & Li, Tong & Paarsch, Harry J., 2012. "Semiparametric estimation in models of first-price, sealed-bid auctions with affiliation," Journal of Econometrics, Elsevier, vol. 168(1), pages 4-16.
    3. Van Zandt, Timothy & Vives, Xavier, 2007. "Monotone equilibria in Bayesian games of strategic complementarities," Journal of Economic Theory, Elsevier, vol. 134(1), pages 339-360, May.
    4. Paul Klemperer, 2004. "Auctions: Theory and Practice," Online economics textbooks, SUNY-Oswego, Department of Economics, number auction1.
    5. Olivier Gossner & Ehud Kalai & Robert Weber, 2009. "Information Independence and Common Knowledge," Econometrica, Econometric Society, vol. 77(4), pages 1317-1328, July.
    6. Monteiro, Paulo Klinger & Moreira, Humberto, 2006. "First-price auctions without affiliation," Economics Letters, Elsevier, vol. 91(1), pages 1-7, April.
    7. Castro, Luciano I. De & Paarsch, Harry J., 2009. "Testing Affiliation in Private-values Models of First-price Auctions Using Grid Distributions," CEI Working Paper Series 2009-11, Center for Economic Institutions, Institute of Economic Research, Hitotsubashi University.
    8. John McMillan, 1994. "Selling Spectrum Rights," Journal of Economic Perspectives, American Economic Association, vol. 8(3), pages 145-162, Summer.
    9. Castro, Luciano I. de, 2007. "Affiliation, equilibrium existence and the revenue ranking of auctions," UC3M Working papers. Economics we074622, Universidad Carlos III de Madrid. Departamento de Economía.
    10. Athey, Susan, 2001. "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information," Econometrica, Econometric Society, vol. 69(4), pages 861-889, July.
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    Cited by:

    1. Oriol Carbonell-Nicolau & Richard McLean, 2014. "On the existence of Nash equilibrium in Bayesian games," Departmental Working Papers 201402, Rutgers University, Department of Economics.

    More about this item

    Keywords

    affiliation; positive dependence; statistical dependence of types; conditional independence; de Finetti’s theorem; minimally informative random variable; auctions; pure strategy equilibrium; revenue ranking;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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