Remind me which bookies count and which don’t, in the context of the proofs of properties?
If any computable bookie is allowed, a non-Bayesian is in trouble against a much larger bookie who can just (maybe through its own logical induction) discover who the bettor is and how to exploit them.
[EDIT: First version of this comment included “why do convergence bettors count if they don’t know the bettor will oscillate”, but then I realized the answer while Abram was composing his response, so I edited that part out. Editing it back in so that Abram’s reply has context.]
For me, the most general answer is the framework of logical induction, where the bookies are allowed so long as they have poly-time computable strategies. In this case, a bookie doesn’t have to be guaranteed to make money in order to count; rather, if it makes arbitrarily much money, then there’s a problem. So convergence traders are at risk of being stuck with a losing ticket, but, their existence forces convergence anyway.
If we don’t care about logical uncertainty, the right condition is instead that the bookie knows the agent’s beliefs, but doesn’t know what the outcome in the world will be, or what the agent’s future beliefs will be. In this case, it’s traditional to requite that bookies are guaranteed to make money.
(Puzzles of logical uncertainty can easily point out how this condition doesn’t really make sense, given EG that future events and beliefs might be computable from the past, which is why the condition doesn’t work if we care about logical uncertainty.)
In that case, I believe you’re right, we can’t use convergence traders as I described them.
Yet, it turns out we can prove convergence a different way.
To be honest, I haven’t tried to understand the details of those proofs yet, but you can read about it in the paper “It All Adds Up: Dynamic Coherence of Radical Probabilism” by Sandy Zabell.
Remind me which bookies count and which don’t, in the context of the proofs of properties?
If any computable bookie is allowed, a non-Bayesian is in trouble against a much larger bookie who can just (maybe through its own logical induction) discover who the bettor is and how to exploit them.
[EDIT: First version of this comment included “why do convergence bettors count if they don’t know the bettor will oscillate”, but then I realized the answer while Abram was composing his response, so I edited that part out. Editing it back in so that Abram’s reply has context.]
It’s a good question!
For me, the most general answer is the framework of logical induction, where the bookies are allowed so long as they have poly-time computable strategies. In this case, a bookie doesn’t have to be guaranteed to make money in order to count; rather, if it makes arbitrarily much money, then there’s a problem. So convergence traders are at risk of being stuck with a losing ticket, but, their existence forces convergence anyway.
If we don’t care about logical uncertainty, the right condition is instead that the bookie knows the agent’s beliefs, but doesn’t know what the outcome in the world will be, or what the agent’s future beliefs will be. In this case, it’s traditional to requite that bookies are guaranteed to make money.
(Puzzles of logical uncertainty can easily point out how this condition doesn’t really make sense, given EG that future events and beliefs might be computable from the past, which is why the condition doesn’t work if we care about logical uncertainty.)
In that case, I believe you’re right, we can’t use convergence traders as I described them.
Yet, it turns out we can prove convergence a different way.
To be honest, I haven’t tried to understand the details of those proofs yet, but you can read about it in the paper “It All Adds Up: Dynamic Coherence of Radical Probabilism” by Sandy Zabell.