1. Value of Information
Samu Mäntyniemi

2. Value of Information (VoI)
Maximum price that a decision maker would be willing to pay for the information prior to making the decision
If new information would re-rank the decision:
VoI > 0
If the rank does not change
VoI = 0

3. Example: harvesting a crop
Farmer’s problem
Profit of the harvest depends on the weather
If it rains, harvesting is inefficient
If it does not rain, harvesting slowly is better than having to hurry
Today is sunny for sure, but it may rain tomorrow
Harvest all today?
Or start today and continue tomorrow?
How much to pay for perfect weather forecast before decision?

4. Rain vs profit Farmer’s prior: P(Rain) = 0.22 Decision \ Rain tomorrow
No rain
Harvest all today
40000
Harvest today and tomorrow
22500
45000
Farmer’s prior:
P(Rain) = 0.22

5. Decision without new information
Harvest 1 day:
EU = 0.22 x x = 40000
Harvest 2 days:
EU = 0.22 x x = 40050
Decision: Harvest 2 days
Maximum Expected Utility (MEU) = 40050

6. Decisions with new information
Weather forecast: Rain
Weather forecast: No rain
Harvest 1 day
EU = 40000
Harvest 2 days
EU = 22500
Decision: Harvest 1 day
MEU | Rain = 40000
Harvest 1 day
EU = 40000
Harvest 2 days
EU = 45000
Decision: Harvest 2 days
MEU | No rain = 45000

7. Expected MEU EMEU = P(Rain) x MEU | Rain + P(No rain) x MEU | no rain
Before buying the weather forecast, farmer does not know what the forecast will say
Only thinks that P(Rain) = 0.22
Then the expected maximum utility after buying the forecast is
EMEU = P(Rain) x MEU | Rain + P(No rain) x MEU | no rain
EMEU = 0.22 x x 45000
EMEU = 43900

8. VoI VoI : how much the information increases expected profit
VoI = Expected profit with information – Expected profit without information
VoI = EMEU – MEU
VoI = – 4050
VoI = 3850
If the forecast is free: take it, expect to benefit 3850
If the forecast costs 3000: buy it, you expect to gain 850
If the forecast costs 3850: buy or not, no difference
If the forecast costs 4000: decide without the forecast