# fool comments on A summary of Savage's foundations for probability and utility. - Less Wrong

33 22 May 2011 07:56PM

You are viewing a comment permalink. View the original post to see all comments and the full post content.

Sort By: Best

You are viewing a single comment's thread.

Comment author: 29 June 2011 11:16:58PM 0 points [-]

Here is a small counterexample to P2. States = { Red, Green, Blue }. Outcomes = { Win, Lose }. Since there are only two outcomes, we can write actions as the subset of states that Win. My preferences are: {} < { Green } = { Blue } < { Red } < { Red,Green } = { Red,Blue } < { Green,Blue } < { Red,Green,Blue }

Here is a situation where this may apply: There is an urn with 300 balls. 100 of them are red. The rest are either green or blue. You draw a ball from this urn.

So Red represents definite probability 1/3, while Green and Blue are unknowns. Depending on context, it sure looks like these are the right preferences to have. This is called the Ellsberg paradox.

Even if you insist this is somehow wrong, it is not going to be Dutch booked. Even if we extend the state space to include arbitrarily many fair coins (as P6 may require), and even if we extend the result space to allow for multiple draws or other payouts, we can define various consistent objective functions (that are not expected utility) which show this behaviour.