Information is valuable because it allows us to make better decisions.

Suppose I am considering two careers, and I’m earning to give, so I’m trying to earn as much as possible through my career to give to charity (in expectation, ignoring risk aversion). If I select job A, there’s a 50% chance that I’ll earn $2 million over the course of my career, and a 50% chance I’ll earn $2.2 million. If I select job B, there’s a 50% chance I’ll earn $3 million, and 50% chance I’ll earn $0.4 million. Now the expected value of job A is 0.5 x $2 million + 0.5 x $2.2 million = $2.1 million, and the expected value of job B is 0.5 x $3 million + 0.5 x $0.4 million = $1.7 million, so job A has a higher expected value, and if I have no further information I should select job A.

The value of information is the amount I would be willing to pay for information. Suppose someone offers to tell me for certain what I’ll earn in job B. There’s a 50% chance that they will tell me I’ll earn $3 million, which will give me expected earnings $0.9 million higher than my current option. But there’s a 50% chance my earnings will be low, and I won’t change my decision, so won’t gain any value from the information. So the information is worth 0.5 x $0.9 million + 0.5 x $0 = $0.45 million. This is how much I should be willing to pay to get the information. As the example illustrates, information can be very valuable.

However, it is not always so: suppose someone offers to tell me for certain what I’ll earn in job A. But regardless of what I earn in job A ($2 million or $2.2 million), I’ll earn more than the expected value of job B ($1.7 million), so this information is not valuable to me. This highlights that information is only valuable when it might allow the agent to improve his or her to decision.

The larger the difference in value between the options, the higher is the value of information: to take an extreme example, if I would earn $10 million in job A, and nothing in job B, I’d be willing to pay $5 million for the information about which was best. Since some interventions seem much better than most others, it is plausible that information on intervention effectiveness is generally very valuable.

## Further reading

Wikipedia. 2016. Value of information.