Now I have another question: how does logical induction arbitrage against contradiction? The bet on a pays $1 if a is proved. The bet on ~a pays $1 if not-a is proved. But the bet on ~a isn’t “settled” when a is proved—why can’t the market just go on believing its .7? (Likely this is related to my confusion with the paper).
Again, my view may have drifted a bit from the LI paper, but the way I think about this is that the market maker looks at the minimum amount of money a trader has “in any world” (in the sense described in my other comment). This excludes worlds which the deductive process has ruled out, so for example if A∨B has been proved, all worlds will have either A or B. So if you had a bet which would pay $10 on A, and a bet which would pay $2 on B, you’re treated as if you have $2 to spend. It’s like a bookie allowing a gambler to make a bet without putting down the money because the bookie knows the gambler is “good for it” (the gambler will definitely be able to pay later, based on the bets the gambler already has, combined with the logical information we now know).
Of course, because logical bets don’t necessarily ever pay out, the market maker realistically shouldn’t expect that traders are necessarily “good for it”. But doing so allows traders to arbitrage logically contradictory beliefs, so, it’s nice for our purposes. (You could say this is a difference between an ideal prediction market and a mere betting market; a prediction market should allow arbitrage of inconsistency in this way.)
This excludes worlds which the deductive process has ruled out, so for example if A∨B has been proved, all worlds will have either A or B. So if you had a bet which would pay $10 on A, and a bet which would pay $2 on B, you’re treated as if you have $2 to spend.
I agree you can arbitrage inconsistencies this way, but it seems very questionable. For one, it means the market maker needs to interpret the output of the deductive process semantically. And it makes him go bankrupt if that logic is inconsistent. And there could be a case where a proposition is undecidable, and a meta-proposition about it is undecidable, and a meta-meta-propopsition about it is undecidable, all the way up, and then something bad happens, though I’m not sure what concretely.
Again, my view may have drifted a bit from the LI paper, but the way I think about this is that the market maker looks at the minimum amount of money a trader has “in any world” (in the sense described in my other comment). This excludes worlds which the deductive process has ruled out, so for example if A∨B has been proved, all worlds will have either A or B. So if you had a bet which would pay $10 on A, and a bet which would pay $2 on B, you’re treated as if you have $2 to spend. It’s like a bookie allowing a gambler to make a bet without putting down the money because the bookie knows the gambler is “good for it” (the gambler will definitely be able to pay later, based on the bets the gambler already has, combined with the logical information we now know).
Of course, because logical bets don’t necessarily ever pay out, the market maker realistically shouldn’t expect that traders are necessarily “good for it”. But doing so allows traders to arbitrage logically contradictory beliefs, so, it’s nice for our purposes. (You could say this is a difference between an ideal prediction market and a mere betting market; a prediction market should allow arbitrage of inconsistency in this way.)
I agree you can arbitrage inconsistencies this way, but it seems very questionable. For one, it means the market maker needs to interpret the output of the deductive process semantically. And it makes him go bankrupt if that logic is inconsistent. And there could be a case where a proposition is undecidable, and a meta-proposition about it is undecidable, and a meta-meta-propopsition about it is undecidable, all the way up, and then something bad happens, though I’m not sure what concretely.