How do you design a Dutch book to take advantage of someone whose estimations sum to less than one, instead of more?
You buy the wagers from them instead of sell to them.
Likewise, what does it look like to have negative b?
So if you have −4(h) you will sell me a unit wager on h for negative $4 (in other words you will pay me to take the bet). Perhaps I need to rephrase the Will-to-wager assumption differently to make this possibility more explicit? (Edit: I have done so)
The Will-to-wager assumption feels too strong for me. I would like, for instance, to be able to say “I will wager up to $0.30 on H, or up to $0.60 on ~H. Likewise, I will sell you a wager on H for $0.70 or more, and on ~H for $0.40 or more.”
Of course, that’s effectively setting up my own Dutch book, but it feels very natural to me to associate uncertainty in an outcome with “gaps” that I don’t want to commit to either hypothesis without more data. Then again, I’m a fan of DST, and that’s sort of the point.
I would say that the reason for your intuition that uncertainty ⇒ gaps (which seems separate from risk-aversion-induced gaps) is that the person on the other end of the bet may have information you don’t, and so them offering to bet you counts as Bayesian evidence that the side they’re betting on is correct.
However, e.g., a simple computer program can commit to not knowing anything about the world, and solve this problem.
The Will-to-wager assumption feels too strong for me. I would like, for instance, to be able to say “I will wager up to $0.30 on H, or up to $0.60 on ~H. Likewise, I will sell you a wager on H for $0.70 or more, and on ~H for $0.40 or more.”
Well, this is sound betting strategy. As I say, you shouldn’t take bets with 0 expected return unless you just enjoy gambling; it’s a waste of your time. The question we need to answer is whether or not this principle can be given a more abstract or idealized interpretation that says something important about why Bayesianism is rational- the argument certainly doesn’t prove that non-Bayesians are going to get bilked all the time.
I think this misses the point, somewhat. There are important norms on rational action that don’t apply only in the abstract case of the perfect bayesian reasoner. For example, some kinds of nonprobabilistic “bid/ask” betting strategies can be Dutch-booked and some can’t. So even if we don’t have point-valued will-to-wager values, there are still sensible and not sensible ways to decide what bets to take.
What the Dutch book theorem gives you are restrictions on the kinds of will-to-wager numbers you can exhibit and still avoid sure loss. It’s a big leap to claim that these numbers perfectly reflect what your degrees of belief ought to be.
But that’s not really what’s at issue. The point I was making is that even among imperfect reasoners, there are better and worse ways to reason. We’ve sorted out the perfect case now. It’s been done to death. Let’s look at what kind of imperfect reasoning is best.
What the Dutch book theorem gives you are restrictions on the kinds of will-to-wager numbers you can exhibit and still avoid sure loss. It’s a big leap to claim that these numbers perfectly reflect what your degrees of belief ought to be.
Yes. This was the subject of half the post.
But that’s not really what’s at issue. The point I was making is that even among imperfect reasoners, there are better and worse ways to reason. We’ve sorted out the perfect case now. It’s been done to death. Let’s look at what kind of imperfect reasoning is best.
It actually is what was at issue in this year old post and ensuing discussion. There is no consensus justification for Bayesian epistemology. If you would rather talk about imperfect reasoning strategies than the philosophical foundations of ideal reasoning than you should go ahead and write a post about it. It isn’t all that relevant as a reply to my comment.
I’d always thought “What bets do I take” was the justification for Bayesian epistemology. Every policy decision (every decision of any kind) is a statement of the form “I’m prepared to accept these costs to receive these outcomes given these events”, this is a bet. If Bayesian epistemology lets you win bets then that’s all the justification it could ever need.
The above discussion about “what bets do I take?” is about literal, monetary wager-making. The sense in which any decision can be described in a way that is equivalent to such a wager is precisely the question being discussed here.
You buy the wagers from them instead of sell to them.
So if you have −4(h) you will sell me a unit wager on h for negative $4 (in other words you will pay me to take the bet). Perhaps I need to rephrase the Will-to-wager assumption differently to make this possibility more explicit? (Edit: I have done so)
Thanks for the revision.
The Will-to-wager assumption feels too strong for me. I would like, for instance, to be able to say “I will wager up to $0.30 on H, or up to $0.60 on ~H. Likewise, I will sell you a wager on H for $0.70 or more, and on ~H for $0.40 or more.”
Of course, that’s effectively setting up my own Dutch book, but it feels very natural to me to associate uncertainty in an outcome with “gaps” that I don’t want to commit to either hypothesis without more data. Then again, I’m a fan of DST, and that’s sort of the point.
I would say that the reason for your intuition that uncertainty ⇒ gaps (which seems separate from risk-aversion-induced gaps) is that the person on the other end of the bet may have information you don’t, and so them offering to bet you counts as Bayesian evidence that the side they’re betting on is correct.
However, e.g., a simple computer program can commit to not knowing anything about the world, and solve this problem.
Well, this is sound betting strategy. As I say, you shouldn’t take bets with 0 expected return unless you just enjoy gambling; it’s a waste of your time. The question we need to answer is whether or not this principle can be given a more abstract or idealized interpretation that says something important about why Bayesianism is rational- the argument certainly doesn’t prove that non-Bayesians are going to get bilked all the time.
I think this misses the point, somewhat. There are important norms on rational action that don’t apply only in the abstract case of the perfect bayesian reasoner. For example, some kinds of nonprobabilistic “bid/ask” betting strategies can be Dutch-booked and some can’t. So even if we don’t have point-valued will-to-wager values, there are still sensible and not sensible ways to decide what bets to take.
The question that needs answering isn’t “What bets do I take?” but “What is the justification for Bayesian epistemology?”.
What the Dutch book theorem gives you are restrictions on the kinds of will-to-wager numbers you can exhibit and still avoid sure loss. It’s a big leap to claim that these numbers perfectly reflect what your degrees of belief ought to be.
But that’s not really what’s at issue. The point I was making is that even among imperfect reasoners, there are better and worse ways to reason. We’ve sorted out the perfect case now. It’s been done to death. Let’s look at what kind of imperfect reasoning is best.
Yes. This was the subject of half the post.
It actually is what was at issue in this year old post and ensuing discussion. There is no consensus justification for Bayesian epistemology. If you would rather talk about imperfect reasoning strategies than the philosophical foundations of ideal reasoning than you should go ahead and write a post about it. It isn’t all that relevant as a reply to my comment.
I’d always thought “What bets do I take” was the justification for Bayesian epistemology. Every policy decision (every decision of any kind) is a statement of the form “I’m prepared to accept these costs to receive these outcomes given these events”, this is a bet. If Bayesian epistemology lets you win bets then that’s all the justification it could ever need.
The above discussion about “what bets do I take?” is about literal, monetary wager-making. The sense in which any decision can be described in a way that is equivalent to such a wager is precisely the question being discussed here.