But this post isn't about people's feelings. It's about rational agents attempting to maximize wealth.
No, the entire game is about people's feelings, if you allow discussion before the offer and accept steps, and/or if it's humans making the decisions (where you cannot assume that self-identity and the habitual expectation of future impact don't dominate).
It's only about rational agents attempting to maximize wealth if you make it purely anonymous and hide the other player's actions (and ideally that the other player is even involved). Tell agent 1 they can pick an amount to keep, but not who their counterpart is (or even if it's human - just say it's an agent optimizing it's wealth). Tell player 2 they can accept the amount offered, or reject it, but don't add the psychology of what "might have been", by telling them that it has any effect on player 1 (or even that player 1 exists), or that the amount offered is variable based on some other agent.
But that's boring - perfectly rational agents maximizing their wealth and not considering any future interactions pretty much do what Nash says - offer the minimum, accept anything over 0. The game is interesting ONLY when those conditions don't hold.
When you allow discussion, all this goes out the window. It's just psychology - what do you think the other player will accept, rather than "punishing" you even at a cost to themselves. As you note, if precommittment is asymmetric (player 2 can use it, player 1 can't), that just reverses the places. If it's symmetrical, then it's back to pure psychology about where the line is set.
See also Altruistic Punishment (one reference https://www.nature.com/articles/415137a).
"The mathematical analysis is simple: Player 2 should always accept" - that is incorrect. As the game is defined, players are equal. Player Two wields the obvious veto power by not accepting a proposal he doesn't like. Player One has a no less effective veto power by not advancing a proposal he doesn't like in the first place. Players communicate about the proposals before the match, which effectively turns it into a infinitely repeated game.
Asymmetry only arises if there is no prior communication. Only in that case Player One has an advantage, even if we ignore any "feelings", play rationally, and not allow taking future rounds into consideration (i.e. only play once).
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The Ultimatum Game is a simple game in which two players attempts to split a $100 reward. They can communicate with each other for 10 minutes, after which:
At first glance, the mathematical analysis is simple: Player 2 should always accept (since anything is better than nothing), so Player 1 should offer a 99-to-1 split to maximize their winnings.
Much of the commentary around this game revolves around the fact that when you play this game with humans, Player 2's sense of "fairness" will cause them to "irrationally" reject sufficiently imbalanced splits.
But this post isn't about people's feelings. It's about rational agents attempting to maximize wealth. (I don't doubt that all these ideas have been discussed before, though in most LW posts I found with this game in it, the game itself is not discussed for more than a paragraph or two).
If you're Player 2 and you want to walk away with more than $1, what do you do?
It's pretty simple, actually - all you need to do is to immediately communicate to Player 1 that you've sworn an Unbreakable Vow that you will reject anything other than a 99-1 split in your favor. (Or, more practically, give Player 1 a cryptographic proof of a cryptographic contract that destroys $1000 if you accept anything other than 99-1.) And just like that, the tables are turned. Player 1 now gets to decide between walking away with $1 or walking away with nothing.
This style of play involves reducing your options and committing to throwing away money in a wide variety of scenarios. But against a Player 1 who's as naive as the original analysis's Player 2, it works. It's the madman theory of geopolitics - sometimes the best move is to declare yourself crazy.
This game corresponds fairly directly to the idea of economic surplus: in a positive-sum transaction, both sides want the transaction to go through, but there remains the lingering question of how to split the surplus.
I unfortunately read the news a lot, so I see a lot of big companies and governments getting into fights with this shape.
Let's imagine two types of mindsets: Compromiser and Hardliner. The Compromiser will accept the "I get $1, you get $99" deal, "unfair" though it is. The Hardliner will never accept or propose anything but "I get $99, you get $1". If two hardliners play, they both get $0; if two compromisers play, they each get $50.
You can now make the standard 2x2 box in your head, and notice that there are 2 equilibria - CH and HC. Compared to Prisoner's Dilemma, this game is really easy - 3 of the 4 boxes will end up maximizing total surplus, and the one that doesn't is not a stable equilibrium.
In terms of "moral takeaways", Prisoner's Dilemma has a vibe of, "if you have two people who can keep their promises, they'll do well for themselves in the world." This game's takeaway is a bit more complicated: "Take a hard line and stand up for yourself, otherwise the world will pass you by. But don't go too far beyond what's fair."
Adding iterations to this game is interesting:
From a total-surplus perspective, the relative "easiness" of this game is encouraging - it's a good thing that these dynamics, not those of prisoner's dilemma, are the ones that govern every supply chain, joint venture, and partnership agreement.
One last note: There's a fun way to combine this and prisoner's dilemma: namely, having there be 2 people who make an offer (but still 1 person who accepts/rejects).
The game theory now tells you that both people making the offer should offer a $1-$99 split, otherwise the other person will undercut them. If we imagine the offer-makers as companies and the offer-taker as the consumer, we've gone from total monopoly to total competition. In total competition, the companies capture almost none of the value they generate, and it almost all goes to the consumer - the miracle of capitalism.
In this scenario, the two companies should want to collude with each other or merge with each other. The Prisoner's Dilemma situation makes the former difficult, and antitrust law interferes with the latter.
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Here's a claim to close this piece: People overuse Prisoner's Dilemma as a mental model, when they should be using something more along these lines.
The almost-but-not-quite symmetry of it is a bit awkward - I wonder if the "extended haggling" version above basically resolves that issue. (The game theoretic prediction of the original case does come true sometimes - witness the $2/share Bear Stearns deal - and it's usually because there is no more time for negotiation.)