Optimal Sample Sizes and Statistical Decision Rules

Journal: Theoretical Economics (Forthcoming)

Abstract: Consider the retail chain CEO who is contemplating changing store-opening times from 9 am to 8 am. A manager, who wants this change implemented, is asked to run a pilot in a handful of stores and present the data on profits per store. The CEO will implement the change only if the profits increase in at least x% of stores in the data. In how many stores should the manager run the pilot to maximize the chance that the CEO approves the new store opening time? 

Such persuasion settings are ubiquitous in daily life where a decision maker (the CEO) is persuaded to take a particular action by a better-informed party (the manager), who has different incentives from the decision maker and controls the size of the data obtained by the decision maker. Yet another example of a persuasion setting is when a think tank decides how extensive surveys to administer to influence a politician’s decision-making.

This paper solves for data sizes that maximize the chance of persuasion for various decision threshold levels x. Returning to the CEO-manager example, if x=60, the manager will run the pilot in 5 stores. Instead, if x=55, the manager will run the pilot in 7 stores. In fact, as long as the CEO’s decision threshold x is above 50, the manager will find it optimal to run the pilot in an odd number of stores. Most importantly, for an outside observer, the CEO appears to follow the simple majority rule: implement the change if and only if profits increase in a simple majority of stores in the data. Intuitively, these results show that choice regularities in a large class of persuasion environments, indexed by decision threshold x, can be summarized using statistical decision rules à la Abraham Wald.