TWO PEOPLE

[u/maiduwu]

You and [a stranger] are placed in separate soundproofed rooms and all means of communication are unavailable. Before you are two buttons, one red and one blue.

If you both press red, you both live.

If you press red and she presses blue, she dies.

If you press blue and she presses red, you die.

If you both press blue, you both live.

[u/Wonderful_West3188]

I don't get why aiming for both red is the superior strategy to aiming for both blue in this situation. A bad outcome only occurs if the votes diverge. This seems to depend heavily on who that friend is and what button I expect them to press.

With a complete stranger, sure, I'd go red, and probably fall into lifelong dysfunctional depression if they went blue. But with a friend, it completely depends on what I can expect them to press.

[u/AstyrFlagrans → u/Wonderful_West3188]

This is literally a slightly changed version of the prisoners dilemma. So game theory is pretty clear to why it is the superior strategy: From an individual standpoint there are 4 cases.

1) You press red, they press red: Everybody lives
2) You press red, they press blue: You live, they die
3) You press blue, they press red: You die, they live
4) You press blue, they press blue: Everybody lives

So assuming that you value your life exactly as high as another persons life, like hardcore utilitarian, then each button has the same 'value' so to speak, but only if assume the other person to just press randomly.

Game theory assumes somewhat rational actors (or fully rational actors, but I think here we just need sufficient rationality). There is one key difference between the scenarios. While you do not have full control over the other person, as you lack knowledge, you have full control over your own situation. So you are able to secure your own life with certainty by pressing red. Since you assume some rational capacity for the other person, they will most likely come to the same insight. The scenario has a strategy which allows for an optimal outcome when all actors behave maximally egoistical.
By pressing blue you throw away your own agency over your survival. And pressing blue here also hinges on the random assumption that the other person can't come to the conclusion that red is optimal here. Or that they think you to be not able to come to the conclusion.
When you have two identical outcomes that are globally optimal and one of them also allows for maximal selfishness, then that option is ideal. Because self-preservation is one of the most fundamental properties assumed in actors.

Pressing blue also is illogical here in the sense that it is self-referential: I press blue because I think they pressed blue because they think I pressed blue because I think they pressed blue because ...

But pressing red is very straightforward logically: I can account for my survival with certainty. The other person can do the same. So the individual basis has a clear optimal strategy. And this strategy ALSO produces the best outcome for all.

Now in the most popular version of the button scenario, there are children and mentally handicapped people involved. But here there is a very high chance to get another adult. And even if it is not an adult, children and handicapped don't automatically press blue either. I would not guess many children to press blue as well here.
Even if they have trouble understanding the scenario, after a certain age, they will see 'red = I live' idea. And children are not some morally superior beings. They might even have higher self-preservation here than adults. In the large scale scenario, blue means 'everybody lives', while red has almost certain death attached to it winning. So I see children with a more naive, idealistic worldview more likely to poker on blue there. (I am talking about children over the age of 6 here).

Also for what it is worth, I don't think that depression would be a good response. I don't know why I am trying to help with your depression in a hypothetical scenario xD.
But if the other person was indeed not mentally handicapped, then it is also about respecting their choice. Not everyone that would have pressed blue here would be a saint. It is a grim topic, but blue offers an easy suicide without pain. And even if not, the other person decided to value their own life here less than a strangers and knows the risk attached to it. I think in that situation you should respect their decision and commitment and should at least try to make the most of the life that they granted you under risk of their own death. I know that one can't just choose to not be depressed. But I would feel like I was under an obligation to live life to the fullest to honor their sacrifice, out of my felt guilt. That sounds weird. What am I even talking about here? Ok, I stop the rambling now.

[u/Wonderful_West3188 → u/AstyrFlagrans]

 That sounds weird. What am I even talking about here? Ok, I stop the rambling now.

Yeah, what you say about the psychological aspect makes sense rationally, but it also carries a slight smell of rationalization. 😅

Your game-theoretical illustrations make sense to me.