Here’s a similar game with a potentially simpler setup:
I have a coin which is either fair or has heads on both sides. For some price P, you can ask me to flip the coin and tell you the outcome; we can do this as many times as you like. Then you guess which kind of coin I’m using, and if you guess right I’ll give you £1000.
Suppose you’re certain 1000:1 that the coin is fair. The only coin flip outcomes worth considering are runs of heads (obviously once you see a single outcome of tails, you’re done).
If you ask for k=5 or more coin flips, then you can only be wrong if the coin is fair but ended up doing HHH...HH anyway, which is long enough to convince you. This has probability less than 1/2^k (since Pr[fair coin] < 1). At that point, every additional coin flip is worth a ridiculously tiny amount you can solve for.
If you ask for fewer than 5 coin flips, then no sequence of coin flips you see will convince you that the coin isn’t fair with probability over 50%, and you’ll just end up betting on that no matter what you see. So these coin flips are worthless unless you will have the chance to buy more.
The “either 2⁄3 or 1/3” coin is actually worse in the scary-zeroes department. The nice feature of both the 2d6 and the double-heads coins is that they completely lack an outcome, so no matter what your prior is, if you are currently considering betting on these, there is a small chance you’ll be convinced not to.
On the other hand, if both coins can come up both heads and tails, then a single extra flip is worthless unless your prior odds are between 2:1 and 1:2 -- no matter what outcome you see, your belief will shift in one direction or the other by exactly one bit. It’s like a biased random walk on the number line, and you don’t know what the bias is. But no matter where you are, there’s always some number of coin flips that will be worth buying (all at once), although potentially the price you’d pay for them would be really tiny.
Here’s a similar game with a potentially simpler setup:
I have a coin which is either fair or has heads on both sides. For some price P, you can ask me to flip the coin and tell you the outcome; we can do this as many times as you like. Then you guess which kind of coin I’m using, and if you guess right I’ll give you £1000.
I think that’s a nice model for my model problem. Does it have the scary zeros in it, or is another coin flip always worth something?
I should probably also think about the ‘either 2⁄3 or 1⁄3’ coin.
I’ll go off and do so.
Suppose you’re certain 1000:1 that the coin is fair. The only coin flip outcomes worth considering are runs of heads (obviously once you see a single outcome of tails, you’re done).
If you ask for k=5 or more coin flips, then you can only be wrong if the coin is fair but ended up doing HHH...HH anyway, which is long enough to convince you. This has probability less than 1/2^k (since Pr[fair coin] < 1). At that point, every additional coin flip is worth a ridiculously tiny amount you can solve for.
If you ask for fewer than 5 coin flips, then no sequence of coin flips you see will convince you that the coin isn’t fair with probability over 50%, and you’ll just end up betting on that no matter what you see. So these coin flips are worthless unless you will have the chance to buy more.
The “either 2⁄3 or 1/3” coin is actually worse in the scary-zeroes department. The nice feature of both the 2d6 and the double-heads coins is that they completely lack an outcome, so no matter what your prior is, if you are currently considering betting on these, there is a small chance you’ll be convinced not to.
On the other hand, if both coins can come up both heads and tails, then a single extra flip is worthless unless your prior odds are between 2:1 and 1:2 -- no matter what outcome you see, your belief will shift in one direction or the other by exactly one bit. It’s like a biased random walk on the number line, and you don’t know what the bias is. But no matter where you are, there’s always some number of coin flips that will be worth buying (all at once), although potentially the price you’d pay for them would be really tiny.