Put simply, VOI is the difference between your expected value with and without the information.
So with Melatonin, let’s simplify to 2 possibilities:
A) Melatonin has no effect, costs $10 per year, for a value of −1
B) Saves you 15 minutes per day (+5 utilons), costs $10 per year (-1 utilon), for a net value of +4 utilons.
Now, let’s say you think that A and B are equally likely. Then the expected value of not taking Melatonin is 0, and the expected value of taking it is 0.5 −1 + 0.5 4 = 1.5. With only this information available, you will always take Melatonin, so your expected value is 1.5.
Then let’s say you are considering a definitive experiment (so you will know with p=1 whether A or B is true).
If A is true then you will not take Melatonin, so the value of that outcome is 0 utilons.
If B is true, then you will take Melatonin, for a value of 4 utilons.
And by conservation of expected evidence, it is equally likely that the experiment will decide for A or B.*
Then the expected value of your decision with perfect info is 0.5 0 + 0.5 4 = 2 > 1.5, so the VOI is 0.5 utilons..
*Equally likely only because of how I set up the problem. Conservation of expected evidence would also be satisfied if the experiment would probably favor one side weakly, but improbably favor the other side strongly.
VOI is higher when the experiment shifts your beliefs a lot, lower when the expected change in belief is small. For example, praying is sufficiently unlikely to work that it’s not worth my time to test it. There are other cases where my uncertainty is high, but I can’t think of sufficiently good cheap experiments.
VOI is higher when you would gain a lot if it told you to change your plans. For example, if you would have taken Adderall without an experiment, and Adderall is expensive, then finding out it doesn’t work saves you a lot of money. This is less true for melatonin.
UPDATE: Math corrected. thanks!
Put simply, VOI is the difference between your expected value with and without the information.
So with Melatonin, let’s simplify to 2 possibilities:
A) Melatonin has no effect, costs $10 per year, for a value of −1
B) Saves you 15 minutes per day (+5 utilons), costs $10 per year (-1 utilon), for a net value of +4 utilons.
Now, let’s say you think that A and B are equally likely. Then the expected value of not taking Melatonin is 0, and the expected value of taking it is 0.5 −1 + 0.5 4 = 1.5. With only this information available, you will always take Melatonin, so your expected value is 1.5.
Then let’s say you are considering a definitive experiment (so you will know with p=1 whether A or B is true).
If A is true then you will not take Melatonin, so the value of that outcome is 0 utilons.
If B is true, then you will take Melatonin, for a value of 4 utilons.
And by conservation of expected evidence, it is equally likely that the experiment will decide for A or B.*
Then the expected value of your decision with perfect info is 0.5 0 + 0.5 4 = 2 > 1.5, so the VOI is 0.5 utilons..
*Equally likely only because of how I set up the problem. Conservation of expected evidence would also be satisfied if the experiment would probably favor one side weakly, but improbably favor the other side strongly.
So what should you conclude from this?
VOI is higher when the experiment shifts your beliefs a lot, lower when the expected change in belief is small. For example, praying is sufficiently unlikely to work that it’s not worth my time to test it. There are other cases where my uncertainty is high, but I can’t think of sufficiently good cheap experiments.
VOI is higher when you would gain a lot if it told you to change your plans. For example, if you would have taken Adderall without an experiment, and Adderall is expensive, then finding out it doesn’t work saves you a lot of money. This is less true for melatonin.
Expected value of taking M without information is 0.5 −1 + 0.5 4 = 1.5, not 1. VoI in this case is 0.5 utilon.