Tom, if CA’s allotment of points is generous enough that the limit makes little difference then it’s no longer true that “you never learn anything new about how many points he has remaining” because he’ll still stop if he runs out.
If he knows that he’s addressing Eliezer and that Eliezer will lower his probability estimate when CA stops, then indeed he’ll carry on until reaching the limit (if he can), but in that case what happens is that as he approaches the limit without having made any really strong arguments Eliezer will reason “if the diamond really were in box B then he’d probably be doing better than this” and lower his probability.
Suppose you meet CA, and he says “I think you should think the diamond is in box B, and here’s why”, and at that instant he’s struck by lightning and dies. Ignoring for the sake of argument any belief you might have that liars are more likely to be smitten by the gods, it seems to me that your estimate of the probability that the diamond is in box B should be almost exactly 1⁄2. (Very slightly higher, perhaps, because you’ve ruled out the case where there’s no evidence for that at all and CA is at least minimally honest.)
Therefore, your suggestion that you lower your probability estimate as soon as you know CA is going to argue his case must be wrong.
What actually happens is: after he’s presented evidence A1, A2, …, Ak, you know not only that A1, …, Ak are true but also that those are the bits of evidence CA chose to present. And you have some idea of what he’d choose to present if the actually available evidence were of any given strength. If A1, …, Ak are exactly as good as you’d expect given CA’s prowess and perfectly balanced evidence for the diamond’s location, then your probability estimate should remain at 1⁄2. If they’re better, it should go up; if they’re worse, it should go down.
Note that if you expect a profusion of evidence on each side regardless, k will have to be quite large before good evidence A1 … Ak increases your estimate much. If that’s the case, and if the evidence really does strongly favour box B, then a really clever CA will try to find a way to aggregate the evidence rather than presenting it piecemeal; so in such situations the presentation of piecemeal evidence is itself evidence against CA’s claim.
Tom, if CA’s allotment of points is generous enough that the limit makes little difference then it’s no longer true that “you never learn anything new about how many points he has remaining” because he’ll still stop if he runs out.
If he knows that he’s addressing Eliezer and that Eliezer will lower his probability estimate when CA stops, then indeed he’ll carry on until reaching the limit (if he can), but in that case what happens is that as he approaches the limit without having made any really strong arguments Eliezer will reason “if the diamond really were in box B then he’d probably be doing better than this” and lower his probability.
Suppose you meet CA, and he says “I think you should think the diamond is in box B, and here’s why”, and at that instant he’s struck by lightning and dies. Ignoring for the sake of argument any belief you might have that liars are more likely to be smitten by the gods, it seems to me that your estimate of the probability that the diamond is in box B should be almost exactly 1⁄2. (Very slightly higher, perhaps, because you’ve ruled out the case where there’s no evidence for that at all and CA is at least minimally honest.)
Therefore, your suggestion that you lower your probability estimate as soon as you know CA is going to argue his case must be wrong.
What actually happens is: after he’s presented evidence A1, A2, …, Ak, you know not only that A1, …, Ak are true but also that those are the bits of evidence CA chose to present. And you have some idea of what he’d choose to present if the actually available evidence were of any given strength. If A1, …, Ak are exactly as good as you’d expect given CA’s prowess and perfectly balanced evidence for the diamond’s location, then your probability estimate should remain at 1⁄2. If they’re better, it should go up; if they’re worse, it should go down.
Note that if you expect a profusion of evidence on each side regardless, k will have to be quite large before good evidence A1 … Ak increases your estimate much. If that’s the case, and if the evidence really does strongly favour box B, then a really clever CA will try to find a way to aggregate the evidence rather than presenting it piecemeal; so in such situations the presentation of piecemeal evidence is itself evidence against CA’s claim.