I'm taking a Probability and Epistemology class with Kenny Easwaran and Jim Van Cleve this semester. We've been looking at various representation theorems, and examining the axioms. One seemingly plausible axiom is from (Joyce's presentation of) Savage's theory, called "Independence". Joyce (from Foundations of Causal Decision Theory, pp. 85-85) tells us that this axiom tells us that "a rational agent's preference between [actions] A and A* should not depend on what happens in circumstances where the two yield identical outcomes." Joyce gives brief intuitive motivation for this axiom: "Given that A and A* produce equally good results when E is false it is plausible to think that any reason for preferring one to the other would have to be based on an assessment of their relative merits when E is true."

The idea is that any rational agent's preferences will satisfy the axioms. So, a counterexample to an axiom would be an agent who is rational, but whose preferences do not satisfy the axiom. As a possible counterexample to Independence, Kenny presented the Allais Paradox. It goes like this:
Imagine you have a (well-shuffled) deck of 100 cards, numbered 1-100. In the first scenario, you choose between A1 and B1:

A1:
If card number 1 is drawn, you win $0 (1% chance)
If any of cards 2-10 are drawn, you win $5 million (9% chance)
If any of cards 11-100 are drawn, you win $0 (90% chance)

B1:
If card 1 is drawn, you win $1 million
If any of cards 2-10 are drawn, you win $1 million
If any of cards 11-100 are drawn, you win $0

Next, you are asked to choose between A2 and B2:

A2:
If 1 is drawn, you win $0
If 2-10 are drawn, you win $5 million
If 11-100 are drawn, you win $1 million

B2:
If 1 is drawn, you win $1 million
If 2-10 are drawn, you win $1 million
If 11-100 are drawn, you win $1 million

Apparently, most people will choose A1 in the first game, and B2 in the second. This isn't supposed to seem irrational. But, since each of the pairs A1 & B1, and A2 & B2, are identical with respect to what happens when cards 11-100 are drawn, all that should matter, according to Independence, when it comes time to choose is what happens when 1-10 are drawn. But, in the first game, people want to pick the first option, while in the second game, people want to pick the second option, even though, once cards 11-100 are taken out of consideration, the pairs become identical. Satisfying Independence would require choosing either both A's or both B's. So, if Independence is a good axiom, rationality should require the same. But, it doesn't seem to. So, we have a counterexample to Independence.

I'm not sure what to think about this. There are reasons to think that the choice of first A1, and then B2, may be irrational, or at least motivated by non-rational motives. One of these may be risk-aversion. In the second game, choosing B2 guarantees you $1 million, while choosing A2 gives you a (small) chance of walking away with nothing. In the first game, you have a good chance of walking away with nothing with either option. So, if card 1 gets drawn and you chose A1, you may not feel so bad. But, if you could have had a guaranteed million dollars in the second game, and end up with nothing, you may feel worse. Also, I bet that the results would differ if we were dealing with smaller amounts of money, say $0, $1, and $5, or even $0, $100, $500. Is this irrational? I don't know, but it may cast a little doubt on the Paradox's status as a true counterexample.