Think Bayes

Chapter 6. Decision Analysis

The Price is Right problem

On November 1, 2007, contestants named Letia and Nathaniel appeared on The Price is Right, an American game show. They competed in a game called The Showcase, where the objective is to guess the price of a showcase of prizes. The contestant who comes closest to the actual price of the showcase, without going over, wins the prizes.

Nathaniel went first. His showcase included a dishwasher, a wine cabinet, a laptop computer, and a car. He bid $26,000.

Letia’s showcase included a pinball machine, a video arcade game, a pool table, and a cruise of the Bahamas. She bid $21,500.

The actual price of Nathaniel’s showcase was $25,347. His bid was too high, so he lost.

The actual price of Letia’s showcase was $21,578. She was only off by $78, so she won her showcase and, because her bid was off by less than $250, she also won Nathaniel’s showcase.

For a Bayesian thinker, this scenario suggests several questions:

1. Before seeing the prizes, what prior beliefs should the contestant have about the price of the showcase?

2. After seeing the prizes, how should the contestant update those beliefs?

3. Based on the posterior distribution, what should the contestant bid?

The third question demonstrates a common use of Bayesian analysis: decision analysis. Given a posterior distribution, we can choose the bid that maximizes the contestant’s expected return.

This problem is inspired by an example in Cameron Davidson-Pilon’s book, Bayesian Methods for Hackers. The ...