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mrow84 | 3 years ago

Inserting A=60 into the original line of reasoning (just to show that it doesn't immediately resolve the paradox):

1. Denote by A=60 the amount in the player's selected envelope.

2. The probability that A=60 is the smaller amount is 1/2, and that it is the larger amount is also 1/2.

3. The other envelope may contain either 2A=120 or A/2=30.

4. If A=60 is the smaller amount, then the other envelope contains 2A=120.

5. If A=60 is the larger amount, then the other envelope contains A/2=30.

6. Thus the other envelope contains 2A=120 with probability 1/2 and A/2=30 with probability 1/2.

7. So the expected value of the money in the other envelope is: 1/2(120) + 1/2(30) = 75

8. This is greater than A=60 so, on average, the person reasons that they stand to gain by swapping.

9. ...

edit: If anything, to me this obscures things further, because it makes it non-obvious that we are actually talking about ratios (i.e. 2A, 1/2A) rather than absolute numbers, and so the expectation calculated in step 7 is inappropriate - a geometric mean recovers the correct answer.

discuss

order

JoshCole|3 years ago

A can't collapse to sixty. It could still be 30, 60 or 120. The expected value calculation includes the value of A as defined in two different worlds: the world in which A/2 (30) and the world in which 2A (120).

But these two worlds never exist together. They are different realities. We can't determine which of them we are in without more information.

Trying to do so is an error, because we are in multiple subgames simultaneously. In imperfect information games you are already in more than one world. You are in the reality A and counterfactual 2A and you are in the reality 2A and counterfactual A. You can't tell which. You're in both.

In perfect information, you're not in both, so you can do this case based reasoning and not run into as much trouble. But when you're already in both worlds? Well doing the case based reasoning implies that the counterfactual world = the case based world. Its overwriting that part of the equations. Which means you just accidentally declared that either A/2=A or that 2A=A. That 2=1. You do this because you're still in the counterfactual reality, but you're pretending you are not.