A Model of Non-Belief in the Law of Large Numbers Draft for distribution in February 2011 talks — Missing proofs

Psychological research suggests that people believe that even in very large samples proportions might depart significantly from the population mean. We model this “non-belief in the Law of Large Numbers” by assuming that a person believes that proportions in any given sample of binary signals might be determined by a rate different than the true rate. In inference, a non-believer attends too little to sample size, and remains uncertain even after observing an arbitrarily large sample. Non-belief is often a necessary enabler of other biases, such as the over-influence of vivid signals, that would otherwise be overwhelmed by the logic of the Law of Large Numbers. Because it prevents recognition that a large number of independent betterthan-fair bets is extremely unlikely to yield an aggregate loss, non-belief helps explain how loss aversion can induce reluctance to accept even negligible risk. We explore various assumptions about whether non-believers separately or jointly process signals that arrive in separate clumps. If a non-believer naively anticipates separating new signals but after the fact infers by pooling all signals, he can be caught in a “learning trap”— persistently paying for information that he thinks will let him reach a decision, but which in retrospect he finds inconclusive. In observational-learning settings, non-believers can end up in an information cascade where they ignore their private information, but (unlike believers) may also end up in an “informational eddy,” forever following their own signals. JEL Classification: B49, D03, D14, D83, G11 ∗We thank Herman Chernoff, John Phillips, and Daniel Read for helpful comments in the first decade of this paper’s production, and Don Moore and seminar participants at UC Berkeley, Cornell University, Koc University, and the LSE-WZB Conference on Behavioral IO for helpful comments during the second decade. We are grateful to Ahmed Jaber, Greg Muenzen, Desmond Ong, Nathaniel Schorr, Dennis Shiraev, Josh Tasoff, Mike Urbancic, and Xiaoyu Xu for research assistance. Raymond thanks the University of Michigan School of Information’s Socio-Technical Infrastructure for Electronic Transactions Multidisciplinary Doctoral Fellowship funded by NSF IGERT grant #0654014 for financial support. E-mail: [email protected], [email protected], [email protected].