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Suppose there are two such groups. The First deals with the information by ignoring it (applying the indifference principle?).
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I would not call this the indifference principle. There are three Stages:
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[1] The Information. I see amount A.
[2] The Effect. ???
[3] The Decision. Switch or not. Call off bet or not.[/list]
The indifference principle assumes that there is no effect.
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Is that correct? This Indifference Principle is new to me. But from your description of the Bent Coin I don't see that it assumes there is no effect from Bending the Coin. I see it as saying that since as far as we know, the Bend can bias Heads just as well as Tails, for the purposes of making a Bet on the First Coin Flip we may as well figure the chances are still 50-50 even though we know they are probably not.
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What about the hypothetical group of Random Switchers who use their pocket calculator to generate an exponentially distributed random variable, and switch envelopes when A is less than the number they generated? They do better on average than the Always Switchers and the Never Switchers. This, by itself, should be convincing enough evidence that The Effect exists.
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I'm not arguing that there is no effect. Certainly this shows that there is. This raises an interesting question though. The Decision by Calculator improves the results of the Switchers. But if the $100 bettors use the same method to decide whether to continue their bets or call them off, does it improve their results? Will they also always get better results than by just ignoring the amount A, and always continuing the bet?
PairTheBoard