I'm not getting through here am I..... There are 3 things at play in this problem: maths/game theory, logic (in this case, the two are distinct) and psychology. The psychology side of it is not particularly interesting or difficult. Would I rather make more money, or screw over my opponent? Of course, any sane person would choose to make more money, rather than screw over his opponent. That's not even worthy of comment. The point here is that there is a mathematical, game-theoretically sound argument for $2 being the optimal choice. Logically, it's stupid - the aim of the game, as several people have noted, is not to screw over your opponent, and by taking the game-theoretical approach, we end up minimizing our payout. Simple logic trumps the maths. That, to me, is the interesting side.
In terms of the psychology, this, rather than the prisoners' dilemma, is more akin to all sorts of conflicts and negotiations, whereby we go in with mindsets of it being a zero-sum-game, and that it's more important (for reasons, perhaps, of personal pride, or political pressure) not to get the best deal for yourself, but to get a better deal than your opponents, or at least to feel like you've done them over. We can all see this, right? In terms of global politics, game theory suggests tactics very similar to north korea, and, currently, theresa mays' brexit negotiations stance, i.e., appear unpredictable, and make your opponents think you're ready to blow yourself up as well as them, if you don't get what you want. If you'll pardon my french, c'est évidemment bolloques. The psychology of the problem is only interesting insofar as to the parallels with real life.