Comment author:Wei_Dai
24 October 2009 11:29:16AM
3 points
[-]

Your example shows that we can't assign utilities to events within a single world, like acquiring game systems and games, and then add them up into a utility for that world, but it's not a counterexample to Independence, because of this part:

A and B are what happens in one possible world, and C and D are what happens in another.

Independence is necessary to assign utilities to possible world histories and aggregate those utilities linearly into expected utility. Consider the apples/oranges example again. There,

A = I get an apple in the world where coin is heads

B = I get an orange in the world where coin is heads

C = I get an apple in the world where coin is tails

D = I get an orange in the world where coin is tails

Then, according to Independence, my preferences must be either

A&C > B&C and A&D > B&D, or

A&C < B&C and A&D < B&D

If case 1, I should pick the transparent box with the apple, and if case 2, I should pick the transparent box with the orange.

(I just realized that technically, my example is wrong, because in case 1, it's possible that A&D > A&C and B&D > B&C. Then, I should most prefer an opaque box that contains an apple if the coin is heads and an orange if the coin is tails, since that gives me outcome A&D, and least prefer an opaque box that contains the opposite (gives me B&C). So unless I introduce other assumptions, I can only derive that I shouldn't simultaneously prefer both kinds of opaque boxes to transparent boxes.)

## Comments (153)

BestYour example shows that we can't assign utilities to events within a single world, like acquiring game systems and games, and then add them up into a utility for that world, but it's not a counterexample to Independence, because of this part:

Independence is necessary to assign utilities to possible

world historiesand aggregate those utilities linearly into expected utility. Consider the apples/oranges example again. There,Then, according to Independence, my preferences must be either

If case 1, I should pick the transparent box with the apple, and if case 2, I should pick the transparent box with the orange.

(I just realized that technically, my example is wrong, because in case 1, it's possible that A&D > A&C and B&D > B&C. Then, I should most prefer an opaque box that contains an apple if the coin is heads and an orange if the coin is tails, since that gives me outcome A&D, and least prefer an opaque box that contains the opposite (gives me B&C). So unless I introduce other assumptions, I can only derive that I shouldn't simultaneously prefer both kinds of opaque boxes to transparent boxes.)