when we assume the utility of an action can be modelled as a normal distribution, we are allowing for the possibility of negative and positive infinite utility. However, the expected value of the action is still finite (and equal to the mean of the distribution).
The first sentence here is not true. The formula below is the PDF of a normal distribution:
f(x)=1σ√2πexp(−12(x−μσ)2)
The limit of f(x) as x approaches either ∞ or −∞ is zero.
Moreover, if the first sentence I quoted from your comment were true, there would be no way for the second sentence to be true. This is the definition of expected utility:
∑outcomesU(outcome)P(outcome)
Where U(outcome) is the utility of an outcome and P(outcome) is its probability.
If you have an unbounded utility function, and you have any probability greater than zero (say, 10−101010) that the outcome of your action has infinitely positive utility, and a similarly nonzero probability (say, 10−10101010) that it has infinitely negative utility, then the formula for expected utility simplifies to
The limit of f(x) as x approaches either ∞ or −∞ is zero.
By “possibility of negative and positive infinite utility”, I meant there is a non-null probability of a negative or positive utility with arbitrarily large magnitude. I think infinite is often used as meaning arbitrarily large, but I see now that Michael was not using it that way. Sorry for my confusion, and thanks for clarifying!
Moreover, if the first sentence I quoted from your comment were true, there would be no way for the second sentence to be true.
I agree. In the 1st sentence, “infinite” was supposed to mean “arbitrarily large” (in which case the 2nd sentence would be true).
The first sentence here is not true. The formula below is the PDF of a normal distribution:
f(x)=1σ√2πexp(−12(x−μσ)2)
The limit of f(x) as x approaches either ∞ or −∞ is zero.
Moreover, if the first sentence I quoted from your comment were true, there would be no way for the second sentence to be true. This is the definition of expected utility:
∑outcomesU(outcome)P(outcome)
Where U(outcome) is the utility of an outcome and P(outcome) is its probability.
If you have an unbounded utility function, and you have any probability greater than zero (say, 10−101010) that the outcome of your action has infinitely positive utility, and a similarly nonzero probability (say, 10−10101010) that it has infinitely negative utility, then the formula for expected utility simplifies to
∞⋅10−101010−∞⋅10−10101010=∞−∞
which is undefined.
Hi Fermi,
By “possibility of negative and positive infinite utility”, I meant there is a non-null probability of a negative or positive utility with arbitrarily large magnitude. I think infinite is often used as meaning arbitrarily large, but I see now that Michael was not using it that way. Sorry for my confusion, and thanks for clarifying!
I agree. In the 1st sentence, “infinite” was supposed to mean “arbitrarily large” (in which case the 2nd sentence would be true).