Quick puzzle about utility functions under affine transformations
Here’s a puzzle based on something I used to be confused about:
It is known that utility functions are equivalent (i.e. produce the same preferences over actions) up to a positive affine transformation: u’(x) = au(x) + b where a is positive.
Suppose I have u(vanilla) = 3, u(chocolate) = 8. I prefer an action that yields a 50% chance of chocolate over an action that yields a 100% chance of vanilla, because 0.5(8) > 1.0(3).
Under the positive affine transformation a = 1, b = 4; we get that u’(vanilla) = 7 and u’(chocolate) = 12. Therefore I now prefer the action that yields a 100% chance of vanilla, because 1.0(7) > 0.5(12).
How to resolve the contradiction?