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Almost Everybody Disagrees Almost All the Time: The Genericity of Weakly Merging Nowhere


  • Ronald Miller

    (Columbia University - Department of Economics)

  • Chris Sanchirico

    (University of Pennsylvania Law School & Wharton School)


Suppose we randomly pull two agents from a population and ask them to observe an unfolding, infinite sequence of zeros and ones. If each agent starts with a prior belief about the true sequence and updates this belief on revelation of successive observations, what is the chance that the two agents will come to agree on the likelihood that the next draw is a one? In this paper we show that there is no chance. More formally, we show that under a very unrestrictive definition of what it means to draw priors "randomly," the probability that two priors have any chance of weakly merging is zero. Indeed, almost surely, the two measures will be singular--one prior will think certain a set of sequences that the other thinks impossible, and vice versa. Our result is meant as a critique of the "rational learning" literature, which seeks positive convergence results on infinite product spaces by augmenting the process of Bayesian updating with seeming regularity conditions, variously labeled "consistency" or "compatibility" assumptions. Our object is to investigate just how regular these assumption and results are when considered in the space of all possible prior distributions. Our results on the genericity of nowhere weak merging and singularity speak not just to the specific assumptions and results that appear in the literature, but to the "rational learning" approach generally. We call instead for a different approach to learning, one that recognizes the necessity of genuine, substantive restrictions on beliefs and proposes "extra rational" restrictions that are explicitly grounded in our best understanding of human behavior, ideally gleaned from experimental data.

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  • Ronald Miller & Chris Sanchirico, "undated". "Almost Everybody Disagrees Almost All the Time: The Genericity of Weakly Merging Nowhere," Scholarship at Penn Law upenn_wps-1001, University of Pennsylvania Law School.
  • Handle: RePEc:bep:upennl:upenn_wps-1001

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    References listed on IDEAS

    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702.
    2. Kalai, Ehud & Lehrer, Ehud, 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Econometric Society, vol. 61(5), pages 1019-1045, September.
    3. Miller, Ronald I. & Sanchirico, Chris William, 1999. "The Role of Absolute Continuity in "Merging of Opinions" and "Rational Learning"," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 170-190, October.
    4. Hurkens Sjaak, 1995. "Learning by Forgetful Players," Games and Economic Behavior, Elsevier, vol. 11(2), pages 304-329, November.
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    Cited by:

    1. Young, H. Peyton, 2002. "On the limits to rational learning," European Economic Review, Elsevier, vol. 46(4-5), pages 791-799, May.
    2. Dean Foster & H Peyton Young, 1999. "On the Impossibility of Predicting the Behavior of Rational Agents," Economics Working Paper Archive 423, The Johns Hopkins University,Department of Economics, revised Jun 2001.
    3. Foster, Dean P. & Young, H. Peyton, 2003. "Learning, hypothesis testing, and Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 45(1), pages 73-96, October.

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