A Bridge from Monty Hall to the (Anti-)Hot Hand: Restricted Choice, Selection Bias, and Empirical Practice

Abstract

We show how classic conditional probability puzzles, such as the Monty Hall problem, are intimately related to a novel instance of selection bias found in Miller and Sanjurjo (2015). We explain the connection by way of the principle of restricted choice, an inferential rule from the game of contract bridge, which makes clear how information is revealed from the action of an optimizing opponent/partner. We propose that the principle is most naturally formalized with the odds form representation of Bayes rule, and show how it can make puzzles that are typically seen as paradoxical, intuitive. We illustrate how knowledge of such puzzles, and the principle of restricted choice, may help researchers avoid particular critical errors when designing experiments and analyzing data. Further, we provide examples in research domains that include the hot hand and Psi (ESP). Finally, we use restricted choice to illustrate the conceptual, and quantitative, relationship between a version of the well-known Berkson's Paradox and the aforementioned selection bias.