First of all, Alice is correct in that the expected value of the strategy (from her point of view) is now -1000. It would have been best if she had not been summoned to an office, in which case the couple would have won 5000. Since she has been summoned to an office, that possibility is ruled out. Let's look at the various possibilities. I will shorten the cases to A, B, and AB, where A means Alice was picked alone, B means Bob was picked alone, and AB means both were picked. I will use the term equity to mean the expected value of a case multiplied by the probability of that case.
A summary of the original possibilities:
A : +5000 (25% chance) (+1250 equity)
B : +5000 (25% chance) (+1250 equity)
AB: -4000 (50% chance) (-2000 equity)
Total equity: +500 (as stated in "The Plan")
A summary of the remaining possibilities once Alice is selected:
A : +5000 (33% chance) (+1667 equity)
AB: -4000 (67% chance) (-2667 equity)
Total equity: -1000 (as stated in "The Paradox")
This all makes sense so far. There was a 75% chance of Alice being selected, with expected value -1000, and a 25% chance of Alice not being selected, with expected value +5000.
-1000 * 0.75 + 5000 * 0.25 = 500 equity, which is the original equity.
So far so good. But Alice should still press the button. Why? Because her equity from pressing the button is positive:
A: +5000 (33% chance) (+1667 equity)
AB: -2000 (67% chance) (-1333 equity)
Total equity: +333
Note that in the AB case, her pressing the button only loses 2000 dollars and not 4000. Bob pressing his button and losing the other 2000 dollars is "a done deal" which results in -1333 equity, but that doesn't affect whether Alice should press her button or not.
In other words, Bob is going to press the button and lose 1333 equity. If Alice presses the button, she will regain 333 equity to make the final result -1000. If she does not press the button, the result will remain -1333 which is worse.
Edit: A commenter said, why doesn't Alice just not press the button and end up with 0 instead of -1000?
Response: Because it's always correct to press the button. Suppose Bob decided to "wussy out" and not press the button. That means that instead of Bob losing 1333 equity, he will have lost 0 equity. Alice should still press the button and gain 333 equity. Her decision is independent of Bob's decision.
Bob deciding not to press his button would be a bad one. It looks like a good decision (saving 1333 equity) because we are only looking at the cases where Alice was selected. But he would be losing 5000 dollars in the case where Alice was not selected.
#Another way to think about it#
Here's another way to think about it that might clear up the "paradox" for some people. Suppose Alice and Bob were strangers instead of a couple. And the rules were that if you press the button in heads case, you alone get 5000 dollars. And in the tails case, anyone who presses the button loses 2000 dollars from their own bank account. Now it is clear that you actually want to be picked. Because if you are, your expected return from pressing the button is (1/3) * 5000 - (2/3) * 2000 = +333 dollars. It doesn't matter that the other person might lose 2000 dollars. Their best strategy overall is also to press the button. It just happens that if you get picked, you know they will be losing 2000 dollars by pressing the button.
