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fool comments on The Ellsberg paradox and money pumps - Less Wrong
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Comments (72)
I'm not sure what you mean. If it's because the situation was too symetrical, I think I adressed that.
For example, you could add or remove a couple of red balls. I still choose red over green, and green-or-blue over red-or-blue. I think the fact that it still can't lead to being dutch booked is going to be a surprise to many LW readers.
I agree, I don't think there is any way to dutch-book someone for being wrong but consistent with the laws of probability (that is, still assigning 1/3 probabilities to r,g,b even when that's wrong). They simply lose money on average. But this is an extra fact, unrelated to the triviality that is not being able to dutch-book someone based on an arbitrary choice between two equivalent options. Once they start paying for equivalent options, then they get money-pumped.
Okay. Suppose there is an urn with 31 red balls, and 60 balls that are either green or blue. I choose to bet on red over green, and green-or-blue over red-or-blue. These are no longer equivalent options, and this is definitely not consistent with the laws of probability. Agreed?
(My prior probability interval is P(red) = 31/91 exactly, P(green) = (1/2 +- 1/6)(60/91), P(blue) = (1/2 -+ 1/6)(60/91).)
It sounds like you expected (and continue to expect!) to be able to money-pump me.
I'm confused what your notation means. Let's drop the asymmetry for now and just focus on the fact that you appear to be violating the laws of probability. Does your (1/2 +- 1/6) notation mean that if I would give you a dollar if you drew a green ball, you would be willing to pay 1/3 of a dollar for that bet (bet 1)? Ditto for red (bet 2)? But then if you paid me a dollar if the ball came up (green-or-red), you would be willing to accept 1/2 of a dollar for that bet (bet 3)?
In that case, the dutch book consists of bets like (bet 1) + (bet 2) + (bet 3): you pay me 1/3, you pay me 1/3, I pay you 1/2 (so you paid me 1/6th of a dollar total). Then if the ball's green I pay you a dollar, if it's red I pay you a dollar, and if it's (green-or-red) you pay me a dollar.
If the bet pays $273 if I drew a red ball, I'd buy or sell that bet for $93. For green, I'd buy that bet for $60 and sell it for $120. For red-or-green, I would buy that for $153 and sell it for $213. Same for blue and red-or-blue. For green-or-blue, I'd buy or sell that for $180.
(Appendix A has an exact specification, and you may wish to (re-)read the boot dialogue.)
[ADDED: sorry, I missed "let's drop the asymmetry" .. then, if the bet pays $9 on red, buy or sell for $3; green, buy $2 sell $4; red-or-green, buy $5 sell $7; blue, red-or-blue same, green-or-blue, buy or sell $6. Assuming risk neutrality for $, etc etc no purchase necessary must be over 18 void in Quebec.]
Ah, I see. But now you'll get type 2 dutch booked - you'll pass up on certain money if someone offers you a winning bet that requires you to buy.
I guess you mean: you offer me a bet on green for $2.50 and a bet on blue for $2.50, and I'd refuse either. But I'd take both, which would be a bet on green-or-blue for $5. So no, no dutch book here either.
Or do you have something else in mind?
I mean that I could offer you $9 on green for 2.50, $9 on blue for 2.50, and $9 on red for 3.01, and you wouldn't take any of those bets, despite, in total, having a certainty of making 99 cents. This "type 2" dutch book argument (not really a dutch book, but it's showing a similar thing for the same reasons) is based on the principle that if you're passing up free money, you're doing something wrong :P
I wouldn't take any of them individually, but I would take green and blue together. Why would you take the red bet in this case?
I intentionally designed the bets so that your agent would take none of them individually, but that together they would be free money. If it has a correct belief, naturally a bet you won't take might look a little odd. But to an agent that honestly thinks P(green | buying) = 2/9, the green and blue bets will look just as odd.
And yes, your agent would take a bet about (green or blue). That is beside the point, since I merely first offered a bet about green, and then a bet about blue.
I wouldn't take any of them individually (except red), but I'd take all of them together. Why is that not allowed?
I would not make this prediction. I would think anyone who would understand that claim without having to look up "dutch book" should find that obvious.