00:01
Okay, this question says, given a lottery, we want to show a, that given the utilities of various outcomes of this lottery, we want to show that the lottery is strictly preferred to p, which is a certain outcome.
00:21
So that's just showing that the expected utility with a lottery, which is 0 .5 times u of b plus 0 .5 times u of m is equal to 0 .5 times 100 plus 0 .5, which is 55.
00:47
And that's strictly preferred to u of p, which is 50.
00:52
And now it says a linear transformation of the utility can serve as a utility function that preserves preferences.
01:07
So, f is defined in this case as a linear transformation.
01:17
And so, we can look at it as a plus a, which is to say a plus b times u of x should preserve the properties or preferences rather.
01:47
So let's look at what happens if we do that with a lottery.
01:50
We have 0 .5 times a plus b u of x or u of b, capital b, plus 0 .5 a plus b u of m, which is equal to a plus b times 0 .5 u of b plus 0 .5 u of m, and that's a plus b times 55.
02:34
And we want to know, is that strictly preferred to a plus b times u of p, which is a plus b times 50.
02:48
So yes, as long as a and b are positive, these preferences are preserved.
02:54
However, if we look at a nonlinear transformation, i .e.
02:58
G of x equals log of x, we're going to see that that's not the case...