The following is a probability paradox I've been thinking about. It involves Bayes' rule; if you're not familiar, a good starting example is a urn that has a 50% chance of containing one black ball and one white ball, and a 50% chance of having two black balls. If you reach in at random and pull out a black ball, it becomes more likely there were two black balls to begin with. Specifically, there is now a 1/3 chance of there being black and white balls in the urn, and a 2/3 chance of there being two black balls.
Alice and her husband Bob are kidnapped by an eccentric millionaire that performs probability experiments on people. Without them seeing the result, a coin is to be flipped.
- If heads, one of the two of them (at random) will be brought to an office with a big red button that, if pushed, will transfer \$5000 to their joint bank account. The other will be left in the holding cell.
- If tails, both them them will be brought to separate offices with big red buttons, each with which removes \$2000 from their joint bank account.
They can discuss strategy ahead of time, but after the experiment starts they will be kept separate from each other. What should they do?
Alice and Bob reason their only two strategies are to push the button or not. Not pushing the button is a net zero, and pushing the button yields an expected payoff of $1/2 \cdot (5000) + 1/2 \cdot(-4000) = 500$. Not being particularly risk averse, they decide to go ahead and press the button.
The experiment starts, and Alice is summoned to an office with a big red button. She is about to confidently press the button in accordance with their strategy, when she suddenly has second thoughts. The fact that she has been brought to an office and not left in the holding cell gives her new information. Using standard Bayes analysis, the probability the coin was heads is now $1/3$, and the probability the coin was tails is $2/3$. Thus, now the expected value of their strategy is $1/3 \cdot ( 5000) + 2/3 \cdot (-4000) = -1000$.
Suddenly, not pushing the button seems like a good strategy. What's going on? What should Alice do, which analysis was flawed, and what is wrong with the flawed analysis? If it matters to your answer, assume Bob thinks in a very similar way to Alice, and would likely make the same decision as she would.