Caring About Probabilities of Probabilities
Here are some notes on expected utility theory that I wrote up after a stimulating lunch. Suppose Jack might earn $100, $200, $300, or $400, with equal probabilities, an expected wealth of $250. His information partition is (100,200,300,400)– he can’t rule out anything.
Jack is very risk averse, so his utilities from known wealths are
U(100) = 0 - e
U(200) = 100 - e
U(300) = 120 - e
U(400) = 128 - e,
where e is the number of hours he works painting a house.
Tom offers to tell Jack whether his wealth will either low or high, but not the precise level, if in exchange Jack will spend 2 hours painting Tom’s house. Thus, Jack can get the information partition (100,200), (300,400). Should he accept?
Originally, his expected utility is
.25(0) + .25(100) + .25(120) + .25 (128) = 81
If he pays the 2 hours, his expected utility is
-2 + .5[.5U(100)+ .5 U(200) ]+ .5[.5U(300)+ .5 U(400) ]
which equals
-2 + .25U(100) + .25 U(200) .25U(300) + .25 U(400)
which equals
-2 + .25(0) + .25(100) + .25(120) + .25 (128) = -2 + 81 = 79.
Thus, accepting the offer to reduce his uncertainty simply reduces his expected utility.
This is the feature of Expected Utility Theory that the Ellsberg Paradox contradicts:
that people should not care about their knowledge of the future, only about the true
probabilities of each future event.
Resolving subjective uncertainty in beliefs is different from reducing risk.
Suppose Jack could pay 2 hours to change the probability distribution to
(.5 probability of $200, .5 probability of $300)
His expected utility would be
-2 + .5 (100) + .5 (120) = -2 + 110 = 108
But that is not just a change in his beliefs (though they would change too). It is a
change in the probabilities of various amounts of cash he might receive. He has ruled
out getting a mere $100.
Another way to look at all this is that using Expected Utility Theory, Jack cares only about end states, not about his beliefs along the way. His utility from getting $400 on Friday is entirely pleasure that he gets on Friday when he receives the cash (or, better, when he spends the cash). He get no pleasure on Tuesday, Wednesday, or Thursday from anticipating his happiness on Friday.
We could model anxiety earlier in the week about the uncertainty, but it would take a different model. We could, in the example earlier, say that today, before Jack knows his future wealth, he gets immediate utility of -20*(4-N) if his information partition has N information sets. Then, it would make sense for Jack to pay 20 hours of labor to increase N from 1 to 2, because it would cost 2 in utility and gain 20.
More generally, we could model the happiness along the way of looking forward to Friday, but that needs a more complicated model. This would not necessarily lead to Jack being willing to pay to resolve uncertainty. For example, having low wealth on Friday might be even worse in anticipation than in result (empirically, getting crippled is like that). In addition to the Friday utilities above, Jack might have Tuesday utility of
V(100) = -400
V(200) = 100
V(300) = 120
V(400) = 126
from perfect information on Tuesday, and
V(100,200) = -300
V(300,400) = 122
from moderately fine information, and
V(100,200,300,400) = 0
from perfectly coarse information.
The expected utility gain as a result of getting better information would then be
0 + .5(-300) + .5 (122) = -150+ 61 = -91.
He’d rather delay learning about the possibility of getting just $100.