Value of Information

What is the "Value of Information"?

Imagine you are at a store, about to buy something. You know there is another store 10 km away that might sell the same item cheaper, or it might be more expensive. To find out, you would have to drive over there and burn some gas. So you weigh it up. Unless there is a good chance the second store is cheaper, or a chance it is a lot cheaper, the trip is not worth it.

Now picture this. A friend of yours happens to be standing in that second store right now. He calls you and says, "I will tell you what the item costs here, but you have to pay me a small fee." What is a fair fee to pay him? You would never pay him more than the most money that information could save you. That fair fee, the most you would sensibly hand over for the tip, is the value of the information.

In short, the value of information is a number, usually in money or in utility, that tells you how much it is worth paying for a piece of information that helps you make a better decision.

Here is a simple way to feel it. If a tip can save you at most $5, then $5 is the ceiling. Paying $2 for it is a good deal. Paying $6 for it is a bad deal. You are now paying more for the tip than the tip can ever give back. That is the whole idea in one sentence: never pay more for information than it can save you.

Expected Value of Perfect Information

The Expected Value of Perfect Information tells you, on average, the fair price to pay for perfect information about something uncertain. Perfect means it removes all the guesswork. If someone offered to wipe out the uncertainty for a fee, this number is the most that fee should be before it stops being worth it.

Let's go back to the shopping example. Say the item costs either $15 or $20 in Market 1, and either $15 or $20 in Market 2. You do not know which market has the cheaper price and which has the higher one. To keep it simple, assume there is a 50% chance of each price at each store.

Both markets are far away. So you start wondering: is it worth driving to both of them just to check the prices first? If you knew the price at both stores ahead of time, how much would you save on average? That average saving, the gain you get purely from knowing both prices in advance, is the Expected Value of Perfect Information.

How to calculate the Expected Value of Perfect Information

Let's work through the numbers for this little decision tree. Once you see the steps, the whole idea will click into place.

The expected cost in Market 1 = 0.5 * 15 + 0.5 * 20 = 17.5

The expected cost in Market 2 = 0.5 * 15 + 0.5 * 20 = 17.5

The lower of the two expected costs is the minimum of (17.5, 17.5) = 17.5.

So if you just pick a market at random, with no extra information, you will spend $17.50 on average for the item.

Let's call this the Expected Value Without Perfect Information, or EVwoPI for short. We call it that because, at this point, you still do not know which market charges what.

Now suppose you drive to both markets and check the prices yourself. Then you would have perfect information about both. But here is the puzzle: how do you measure the worth of that information before you actually have it? There is a simple way to think about it. Say your friend has already been to both markets and knows both prices. He wants to sell you that information. You think to yourself, "I do not know exactly what you found, but whatever it is, it has to be one of these four cases."

Case 1: Market 1 price is $15, and Market 2 price is $15.

Case 2: Market 1 price is $15, and Market 2 price is $20.

Case 3: Market 1 price is $20, and Market 2 price is $15.

Case 4: Market 1 price is $20, and Market 2 price is $20.

Each of these four cases is equally likely, so each has a 25% chance. Look at them closely. In three of them you can buy the item for $15, and in one of them you are stuck paying $20.

In Case 1, Case 2, or Case 3, you can pick the market that sells the item for $15. Only in Case 4 are you forced to pay $20, because both markets charge $20.

So there is a 75% chance you end up paying $15 and a 25% chance you pay $20. That means you can work out the average outcome with perfect information even before you ever hear the prices.

The Expected Value With Perfect Information = 0.75 * 15 + 0.25 * 20 = 16.25

So if you had that perfect information, on average you would spend $16.25 on the item.

Notice what just happened. Your friend has not told you a single price yet, but based on the four possible cases, you have already figured out the worth of the information he is holding back.

So how valuable is that information? Without it, you were going to spend $17.50 on average by picking a random market. With it, that average drops to $16.25.

How much did the perfect information save you on average?

The saving is $17.50 - $16.25 = $1.25.

That $1.25 is the Expected Value of Perfect Information for this decision. It is the most you should ever pay your friend for the prices. If he offers them to you for less than $1.25, take the deal, it is worth it. If he asks for more, walk away. Say he wants $3 for the tip. The tip can only save you $1.25 on average, so you would lose money paying $3 for it. Easy call.

EVPI in the Decision Tree software

The SpiceLogic Decision Tree Maker and Analyzer works this out for you automatically. You do not have to run any of the math by hand. Once you have built your decision tree, just open the "Options Analyzer" tab shown below. Whenever there is uncertainty in the tree, the Options Analyzer calculates the Expected Value of Perfect Information and displays it. As you can see, it reports $1.25, the exact same number we got by hand for this decision tree. That match is a good sanity check: the software is doing the same reasoning you just walked through, only faster.