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Academic Commons Search Resultsen-usThe Role of Absolute Continuity in "Merging of Opinions" and "Rational Learning"
http://academiccommons.columbia.edu/catalog/ac:100372
Miller, Ronald I.; Sanchirico, Chris Williamhttp://hdl.handle.net/10022/AC:P:15696Thu, 03 Mar 2011 00:00:00 +0000Two agents with different priors watch a sequence unfold over time, updating their priors about the future course of the sequence with each new observation. Blackwell and Dubins (1962) show that the agents’ opinions about the future will converge if their priors over the sequence space are absolutely continuous: i.e., if they agree on what events are possible. From this Kalai and Lehrer (1993) conclude that the players in a repeated game will eventually agree about the future course of play and thus that “rational learning leads to Nash equilibrium.” We provide an alternative proof on convergence that clarifies the role of absolute continuity and in doing so casts doubt on the relevance of the result. From the existence of continued disagreement we construct a sequence of mutually favorable, uncorrelated “bets.” By a law of large numbers, both agents are sure that they win these bets on average over the long run and this disagreement over what is possible violates absolute continuity.Economic theoryrm170, cs282EconomicsWorking papersAlmost Everybody Disagrees Almost All the Time: The Genericity of Weakly-Merging Nowhere
http://academiccommons.columbia.edu/catalog/ac:100430
Miller, Ronald I.; Sanchirico, Chris Williamhttp://hdl.handle.net/10022/AC:P:15713Thu, 03 Mar 2011 00:00:00 +0000Suppose 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 to occur 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.Economic theoryrm170, cs282EconomicsWorking papers