Committee, Expert Advice, and the Weighted Majority Algorithm: An Application to the Pricing Decision of a Monopolist

Abstract

We consider a repeated pricing decision problem of a monopolist (the decision-maker) who does not know the demand function of some new product, and hence the profit function. To decide, she is helped by a committee of N experts. Each expert has an estimation of the unknown demand function and use it to advise the decision-maker on how she should modify the current price. Decisions are taken with a weighted majority rule, where the weight of each expert, which may be interpreted as her decision power, evolves as a function of its accuracy. When a perfect exists, i.e., who always gives the correct advice, we show that she ends up with all the decision power in the long-run so that the decision-maker finds the optimal price. When such a perfect does not exist, the decision-maker is actually unable to consistently select an expert over time so that the sequences of prices and weights describe a limit cycle. Interestingly enough, if the decision-maker takes a large sample of the stationary behavior of prices, the empirical mean turns out to be arbitrarily close to the optimal price, independently of the "quality" of the experts, as long as there experts are "diverse" enough. This result gives thus support to the thesis developed in the books of Surowiecki (The wisdom of the crowd. Why the many are smarter than the few and how collective wisdom shapes business, economies, societies and nations, 2005) and Page (The difference: How the power of diversity creates better groups, firms, schools, and societies, 2007).