The following dilemma may be posed to you at the beginning of Baldur's Gate II:

Suppose that an evil wizard has captured you and someone else and locked you in separate rooms, each with a button. You can either press the button within the next hour or not. If you press the button and the other person doesn't, they will live and you will die. If they press the button and you don't, you will live and they will die. If you both press the button, or if neither of you presses it, you will both die. You do not have a chance to discuss your choice with each other beforehand. What do you do?

In Baldur's Gate II, the other person was your sibling, but it's slightly easier to analyse if you assume that they are a random stranger and you are indifferent to whether they live or die. Then you might think that you should not press the button, because if you do, then you will die no matter what the other person does. But if they think that way as well, you will both refrain from pressing your buttons and you will both be killed. On the other hand, if you both decide to be heroic and lay down your life for the other one, you will still both be killed.

We can look at this using the concept of super-rational thinking described in Metamagical Themas. Super-rational thinking means that you reason:

"There is a right thing to do here (as in maximising the expected value of the result, not as in ethical). I don't know what it is yet, but because the other person and I are both reasoning beings, and the situation is symmetrical (nothing changes if you swap our roles), I know that we'll come to the same conclusion about what it is."

That doesn't seem to work here, because if you both do the same thing you get the worst possible result. That's because we've ignored the possibility that rather than choosing to press the button or not, you might use a randomising device and elect to press it with probability p. Then, given that you both use this strategy, your chance of survival is p(1-p), which is maximised when p=0.5.

Therefore if you are both focused on your own survival, you would do best to flip a coin. This is also the best strategy if you each believe the other person's survival to be infinitely more important than your own. It even works if you both think something like "I value her life more than mine, but I am indifferent between a 90% chance of survival for her and a 100% chance of survival for me", so long as it's the same something and you both know it's the same something. (If the other person is your enemy, you want her to die and you know she feels the same way about you, you can still apply super-rational thinking but the results will be different.)

Unfortunately, this strategy breaks down if you value her life more than she values yours or vice versa, because then the situation is no longer symmetrical. You might also wonder whether she knows about super-rational thinking - it doesn't work unless you both do it.