There's still more to say on this fascinating issue.
Try this variation on the Berger library example yesterday. There are 100 doors on the stage. I've placed $60,000 behind one of them; it's yours if you select it. You choose a door -- say, Door 37. The probability that the prize is behind that door is 1/100. The probability that it's behind one of the other 99 doors is 99/100. I, knowing where the prize lies, start opening doors to reveal goats and asking you to switch. After doing this for a while only Door 37 and Door 11 remain closed. Would you switch from your original choice to 11?
OK. Here's a variant to illustrate how crucial it is that Monty knows where the prize has been placed. Imagine a different but related game. Monty has a large black velvet sack in which there are three tiles. Each of the tiles has a number on it -- 1, 2, or 3. Monty puts the three tiles into the bag and shakes it. You can't see into the bag, nor can Monty. You are to choose a number -- 1, 2, or 3. Monty will have his assistant draw one tile from the bag and then another tile from the bag. If the number that you chose is the number of the last tile drawn from the bag, you win $60,000. Otherwise, you get nothing.
Suppose that you select tile 1. Monty shakes the bag containing the tiles. His assistant draws out tile 2. Tiles 1 and 3 remain in the bag.
Now suppose that at this point Monty turns to you and says, "You chose tile 1. That tile and tile 3 remain in the bag. Let me give you three options. (1) You stick with tile 1. (2) You switch to tile 3. (3) I give you $35,000 in cash." Assuming that you are risk-neutral, what should you do?
Now, the correct thing to do is to take the cash. The expected value of each of the remaining tiles is $30,000 (1.2 x $60,000, because each tile is equally likely to be the last drawn). Why is the likelihood of each tile's being a winner 1/2 here when it was 1/3 and 2/3 in the door example? Because Monty doesn't know which tile will be the last chosen or have any way of influencing which tile that will be.
I'll have only a couple more variants of this before I move on to other things.
TSU
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