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Oscar_Cunningham comments on Utility is unintuitive - Less Wrong Discussion

-5 Post author: jsteinhardt 09 December 2010 05:39AM

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Comment author: Oscar_Cunningham 10 December 2010 10:02:21AM *  0 points [-]

there is no reason as far as I can tell why u([E1 and E2]) has to have anything to do with u(E1)+u(E2), for instance. Among other things, u is only defined up to an additive constant and so not only is there no reason to be true, it will be completely false for almost all possible utility functions

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.

Multiplicative constants won't have an effect, but they don't change additivity, either.

Comment author: JGWeissman 10 December 2010 07:59:23PM 1 point [-]

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.

Then

EU(refuse bet) = 0
EU(take bet) = .5 * (-1) + .5 * (-2) = -1.5

So, using U, you would refuse the bet.

And

EV(refuse bet) = 1,000,000
EV(take bet) = .5 * (999,999) + .5*(999.998) = 999,998.5

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.

Comment author: nshepperd 10 December 2010 10:25:13AM *  0 points [-]

A sure loss compared to what? Not taking the bet has a utility too.

A bet that loses 1 util on heads and 2 on tails compared to the status quo would have utilities like:

  • u(heads) = -1
  • u(tails) = -2
  • u(status quo) = 0

Add a constant to all these and the status quo will still be higher.

Comment author: Oscar_Cunningham 10 December 2010 10:27:47AM 1 point [-]

Correct, my mistake.

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