> The puzzle is to find the flaw in the very compelling line of reasoning above. This includes determining exactly why and under what conditions that step is not correct, in order to be sure not to make this mistake in a more complicated situation where the misstep may not be so obvious. In short, the problem is to solve the paradox. Thus, in particular, the puzzle is not solved by the very simple task of finding another way to calculate the probabilities that does not lead to a contradiction.
There are two flaws. One is the omission of counterfactual reasoning. This produces the logical contradiction 2=1 at step 7, but you get there a bit earlier because the steps before it are where you do case based reasoning. The scenario where this happens is when you are in imperfect information contexts. You can reason about subgames in perfect information games without the subgames influencing each other. You can't reason about them in imperfect information games without the subgames influencing each other.
The lessons is this - you have to consider counterfactuals. To give a concrete example of someone doing this, think Jeff Bezos when he talks about how he decided to found Amazon. He says he considered the situation in which he founded it and failed and the situation where he founded it and succeeded and the situation where he didn't found it. He didn't get his expectation conditioned on one state, but over an information set that contained many counterfactual outcomes.
The second flaw is much deeper and is what leads to the paradox. The failure is that the entire framing fails to recognize that EV is a function of policy. You can see this by setting the policy to always switch, as they do, and noting that the actual EV for switch is now undefined. Therefore in claiming knowledge of EV without having policy as a higher order function of EV you get a contradiction. 3/2A=0 and/or 3/2A=undefined.
The lesson in this is that policy function influences EV. This should hopefully be obvious to most people in more complicated situations. For example, slamming your head into your table has a lower EV than enjoying a drink of water. You have to account for this whenever your policy choice isn't defined.
There are two flaws. One is the omission of counterfactual reasoning. This produces the logical contradiction 2=1 at step 7, but you get there a bit earlier because the steps before it are where you do case based reasoning. The scenario where this happens is when you are in imperfect information contexts. You can reason about subgames in perfect information games without the subgames influencing each other. You can't reason about them in imperfect information games without the subgames influencing each other.
The lessons is this - you have to consider counterfactuals. To give a concrete example of someone doing this, think Jeff Bezos when he talks about how he decided to found Amazon. He says he considered the situation in which he founded it and failed and the situation where he founded it and succeeded and the situation where he didn't found it. He didn't get his expectation conditioned on one state, but over an information set that contained many counterfactual outcomes.
The second flaw is much deeper and is what leads to the paradox. The failure is that the entire framing fails to recognize that EV is a function of policy. You can see this by setting the policy to always switch, as they do, and noting that the actual EV for switch is now undefined. Therefore in claiming knowledge of EV without having policy as a higher order function of EV you get a contradiction. 3/2A=0 and/or 3/2A=undefined.
The lesson in this is that policy function influences EV. This should hopefully be obvious to most people in more complicated situations. For example, slamming your head into your table has a lower EV than enjoying a drink of water. You have to account for this whenever your policy choice isn't defined.