If I have a bet that loses me 1 util on heads, and 2 utils on tails, I wont take it; it’s a sure loss. Add a million to those values and we get a bet that I would take. So clearly u changes on addition of constants.
If you were really adding a constant to the utility function, you would also add a million to the utility of not taking the bet.
Let U be your original utility function and V be U + 1,000,000, differring by an additive constant.
If you were really adding a constant to the utility function, you would also add a million to the utility of not taking the bet.
Let U be your original utility function and V be U + 1,000,000, differring by an additive constant.
Then
So, using U, you would refuse the bet.
And
So, using V, you would also refuse to take the bet.
As expected, adding a constant to the utility function did not change the decision.