by Dr. Sam L. Savage
A classic example of the value of information involves the decision of whether or not to purchase a $25 umbrella in the face of a known 10% chance of rain. If it does rain and you do not buy the umbrella, you will do $100 damage to your suit. This is displayed in the decision tree below, in which, in keeping with tradition, a square represents a decision and a circle represents an uncertainty.
If you purchase the umbrella, you incur a $25 cost regardless of weather (upper branch). If you take your chances, there is a 10% chance of a $100 penalty, for an average of -$10. The decision tree software is no fool and goes for an average of losing $10 over a sure loss of $25 (green path).
The tree above was created with the XLTree Excel add-in (a beta version of which will be available with documentation to all participants in my upcoming webinar before we post it on our Tools page). This software was developed for my textbook, Decision Making with Insight, which was published in 2003. I donated XLTree to ProbabilityManagement.org, and updated it to use the Excel ribbon interface as shown below.
But back to the value of information. You have already decided not to buy the umbrella when a fairy walks down a moon beam and says, “I can tell you whether or not it will rain tomorrow.” So you say, “Cool, give me the scoop.” And the fairy says, “We haven’t worked for free since the 20th century, that’ll be two bucks.” Is it worth it? Here’s how to think about the value of information.
Without the information, you need to DECIDE what to do and then FIND OUT whether or not it will rain.
With the information, you will FIND OUT whether or not it will rain and then DECIDE what to do.
This is called flipping the tree, which you can do with the software. And when you flip the tree above you get the figure below.
The fairy will tell you that it will rain with 10% probability and that it won’t rain with 90% probability because those are the true probabilities and fairies don’t lie. Now, 10% of the time you will buy the umbrella for $25 and save your suit, while 90% of the time you will do nothing (see the green lines representing your decision under each case). On average you are out $2.50. So, what’s the information worth? You went from an average of average cost of $10 down to $2.50, an information value of $7.50. The Fairy’s offer of two bucks is a steal.
The case above describes what is known as the value of perfect information. Had you received the tip from a gnome (who are known to lie occasionally), we would have applied the value of imperfect information and it would have involved a more complicated tree.