Mislearning from censored data: The gambler's fallacy and other correlational mistakes in optimal‐stopping problems

Abstract

I study endogenous learning dynamics for people who misperceive intertemporal correlations in random sequences. Biased agents face an optimal‐stopping problem. They are uncertain about the underlying distribution and learn its parameters from predecessors. Agents stop when early draws are “good enough,” so predecessors' experiences contain negative streaks but not positive streaks. When agents wrongly expect systematic reversals (the “gambler's fallacy”), they understate the likelihood of consecutive below‐average draws, converge to overpessimistic beliefs about the distribution's mean, and stop too early. Agents uncertain about the distribution's variance overestimate it to an extent that depends on predecessors' stopping thresholds. I also analyze how other misperceptions of intertemporal correlation interact with endogenous data censoring.

Belief Convergence under Misspecified Learning: A Martingale Approach

“…• Optimal stopping under the gambler's fallacy: Similar reasoning yields the global stability result in He (2018), where m can again be seen to be increasing and admit a unique fixed point (see Online Appendix C.2.2).…”

“…Section 5.1 considers single-agent active learning in rich one-dimensional state spaces, as in many important applications in the literature. We show that the iterated elimination criterion in Theorem 2 is straightforward to verify in this setting and can be used to unify and generalize convergence results in applications such as monopoly pricing with a misspecified demand curve (Example 1), effort choice by an overconfident agent (Heidhues, Kőszegi, and Strack, 2018), and optimal stopping under the gambler's fallacy (He, 2018).…”

“…Several important earlier papers have examined the convergence of misspecified learning dynamics in a variety of single-agent (e.g., Nyarko, 1991;Schwartzstein, 2014;Fudenberg, Romanyuk, and Strack, 2017;Heidhues, Kőszegi, and Strack, 2018;He, 2018;Bushong and Gagnon-Bartsch, 2019;Cong, 2019;Heidhues, Kőszegi, and Strack, 2021) and social learning settings (e.g., Eyster and Rabin, 2010;Bohren, 2016;Gagnon-Bartsch, 2017;Bohren, Imas, and Rosenberg, 2019). The approaches in these papers are either tailored to particular environments and forms of misspecification or apply in more general settings but rely on specific parametric assumptions (e.g., Gaussian signals in Fudenberg, Romanyuk, and Strack, 2017;Heidhues, Kőszegi, and Strack, 2021).…”

“…In He (2018), each period consists of a two-stage decision problem. In the first stage, output x 1 follows N (m * 1 , σ 2 ).…”

“…Given current belief µ ∈ ∆(Ω), the agent chooses the threshold a ∈ R to maximize the expected value of u : R × (R ∪ {∅}) → R, where u(x 1 , x 2 ) denotes the utility when she draws (x 1 , x 2 ), and u(x 1 , ∅) denotes the utility when she only draws x 1 . Under the assumptions in He (2018) C.3 Preliminary results for Section 5.2…”