I think De Finetti’s justification is fine as far as it goes, but it doesn’t go quite as far as people think it does. Here’s a couple dialogues to illustrate my point.
Dialogue 1
A: I have secretly flipped a fair coin and looked at the result. What’s your probability that the coin came up heads?
B: I guess it’s 50%.
A: Great! Will you accept a bet against me that the coin came up heads, at 1:1 odds?
B: Hmm, no, that doesn’t seem fair because you already know the outcome of the coinflip and chose the bet accordingly.
A: So rational agents shouldn’t necessarily accept either side of a bet according to their stated beliefs?
B: I suppose so.
Dialogue 2
A: I believe the sky is green with probability 90% and also blue with probability 90%.
B: Great! I can Dutch book you now. Here’s a bet I want to make with you.
A: No, I don’t wanna accept that bet. The theory doesn’t force me to, as we learned in Dialogue 1.
The obvious steelman of dialogue participant A would keep the coin hidden but ready to inspect, so that A can offer bets having credible ignorance of the outcomes and B isn’t justified in updating on A offering the bet.
“One common objection to de Finetti’s theorem is that this betting game is rather contrived. For example, what if one refuses to bet? Does that end the argument? The answer is that the betting game is an abstract model for the decision-making situation in which every agent is unavoidably involved at every moment. Every action (including inaction) is a kind of bet, and every outcome can be seen as a payoff of the bet. Refusing to bet is like refusing to allow time to pass.”
I think a fair bet presupposes that both opponents will have access to the same amount of information, which is not the case in Dialogue 1. The bets in life are not always fair, but that has nothing to do with belief in probability axioms.
That Russell & Norvig quote doesn’t appear to be a very good response to the objection it’s addressing. De Finetti’s argument is supposed to be a pragmatic argument for probabilism. In response to someone asking “Why should my beliefs obey the probability calculus?”, de Finetti says “If you don’t, you’ll end up getting screwed (by being susceptible to dutch books).”
The response to de Finetti that Russell & Norvig are considering is “There are ways to get around susceptibility to dutch books other than accepting probabilism. For instance, I could formulate a policy of refusing to accept bets. Why is probabilism the right way to deal with susceptibility to dutch books?” Russell & Norvig are saying “Well, this is a thought experiment situation in which you are forced to bet.”
OK, but that completely ruins the pragmatic appeal of de Finetti’s theorem. I can feel the attraction of probabilism if it’s the only way I’m protected against being screwed in reality. But not if it’s the only way I’m protected against being screwed in an abstract model that doesn’t match reality. Why should I care about getting screwed in a thought experiment?
I think De Finetti’s justification is fine as far as it goes, but it doesn’t go quite as far as people think it does. Here’s a couple dialogues to illustrate my point.
Dialogue 1
A: I have secretly flipped a fair coin and looked at the result. What’s your probability that the coin came up heads?
B: I guess it’s 50%.
A: Great! Will you accept a bet against me that the coin came up heads, at 1:1 odds?
B: Hmm, no, that doesn’t seem fair because you already know the outcome of the coinflip and chose the bet accordingly.
A: So rational agents shouldn’t necessarily accept either side of a bet according to their stated beliefs?
B: I suppose so.
Dialogue 2
A: I believe the sky is green with probability 90% and also blue with probability 90%.
B: Great! I can Dutch book you now. Here’s a bet I want to make with you.
A: No, I don’t wanna accept that bet. The theory doesn’t force me to, as we learned in Dialogue 1.
In Dialogue 1 I adjust my probability estimate as the bet is offered, no?
That’s a reasonable thing to do, but can you obtain something like De Finetti’s justification of probability via Dutch books that way?
The obvious steelman of dialogue participant A would keep the coin hidden but ready to inspect, so that A can offer bets having credible ignorance of the outcomes and B isn’t justified in updating on A offering the bet.
Russel & Norvig:
“One common objection to de Finetti’s theorem is that this betting game is rather contrived. For example, what if one refuses to bet? Does that end the argument? The answer is that the betting game is an abstract model for the decision-making situation in which every agent is unavoidably involved at every moment. Every action (including inaction) is a kind of bet, and every outcome can be seen as a payoff of the bet. Refusing to bet is like refusing to allow time to pass.”
I think a fair bet presupposes that both opponents will have access to the same amount of information, which is not the case in Dialogue 1. The bets in life are not always fair, but that has nothing to do with belief in probability axioms.
That Russell & Norvig quote doesn’t appear to be a very good response to the objection it’s addressing. De Finetti’s argument is supposed to be a pragmatic argument for probabilism. In response to someone asking “Why should my beliefs obey the probability calculus?”, de Finetti says “If you don’t, you’ll end up getting screwed (by being susceptible to dutch books).”
The response to de Finetti that Russell & Norvig are considering is “There are ways to get around susceptibility to dutch books other than accepting probabilism. For instance, I could formulate a policy of refusing to accept bets. Why is probabilism the right way to deal with susceptibility to dutch books?” Russell & Norvig are saying “Well, this is a thought experiment situation in which you are forced to bet.”
OK, but that completely ruins the pragmatic appeal of de Finetti’s theorem. I can feel the attraction of probabilism if it’s the only way I’m protected against being screwed in reality. But not if it’s the only way I’m protected against being screwed in an abstract model that doesn’t match reality. Why should I care about getting screwed in a thought experiment?
That people often raise these objections is the reason why I prefer Savage’s theorem as a decision-making foundation for probability.