# Acausal Trade and the Ultimatum Game

In the ultimatum game you have two participants. Let's call them Adam and Becky. You offer Adam 100 dollars and tell him he has to split it with Becky. Adam makes an offer to Becky, and Becky can either accept or reject the offer. If she rejects the offer they both get nothing, otherwise they split the cash in accordance with Adam's offer.

For example Adam might offer Becky $20. If Becky rejects that they both end up with 0. If Becky accepts it, Becky get's $20, and Adam gets $80.

What is the best strategy for both Becky and Adam?

Under [causal decision theory](https://en.wikipedia.org/wiki/Causal_decision_theory) the answer is simple. Whatever amount Adam offers Becky, Becky should accept, as the alternative is getting nothing. Therefore Adam should offer Becky a single cent, which Becky will accept, leaving Adam with $99.99, and Becky with 1 cent.

Surprise surprise, when you offer real humans this game they don't accept the single cent, and usually end up rejecting any offers below about 30%. In most cases Adam will offer Becky a 50/50 split, which she will accept.

Wikipedia has a [large section](https://en.wikipedia.org/wiki/Ultimatum_game#Explanations) dedicated to explaining why humans don't act like we would expect based on causal decision theory here. Perhaps humans are just irrational, or maybe they're being rational when you take into account other factors, like maintaining their image as a fair person.

For some reason nobody points out that what the humans are doing is absolutely 100% the correct thing to do! Our theoretical Homo Economicus who accepts 1 cent is clearly beaten by real life Becky who refuses to settle for less than 50/50, and so gets $50 every time! If your strategy loses, [that's a problem with your decision theory not the world](https://www.lesswrong.com/posts/6ddcsdA2c2XpNpE5x/newcomb-s-problem-and-regret-of-rationality)!

That's the main point of this post - I was shocked to see that nobody pointed that out in the article, so felt the need to shout it from the rooftops here.

The rest of this post will be dedicated to explaining how a rational agent might arrive at a 50/50 split.

Under causal decision theory, even though Becky should accept 1 cent when it's offered, if she can precommit to only accepting $99.99 up front, she should. For example, she could offer a friend $10,000 if she accepts any offer less than $99.99. Adam, knowing that Becky will be forced to reject any offer for under $99.99 will be forced to make such an offer, so that at least he gains 1 cent.

Of course Adam could counter by precommitting before Becky does to not make an offer under $99.99, so it becomes a race to see who can precommit first.

However the game is usually played in situations where being able to verbally unambiguously precommit in such a way as causal decision theory would accept the precommit isn't really possible. For example under causal decision theory, a precommittment to throw all your money away if you accepted 1 cent wouldn't actually work, since once you accepted 1 cent you would have no incentive to actually throw away your money. You need some kind of external entity to force you to carry out your precommitment.

Of course that's because causal decision theory is plain wrong. There's various better decision theories, for example [updateless decision theory](https://www.lesswrong.com/tag/updateless-decision-theory). One of the main outcomes of such decision theories is you should act in every situation as if you made all the best precommitments in all possible situations from the day you were born, and you should always carry out such precommitments.

So up front, right now what precommitments should you make for both Adam's situation and Becky's?

At first it might seem tempting to precommit to offering 1 cent as Adam, or rejecting any offer less than $99.99 as Becky. The problem with that is that your opponent is likely to have made pretty much the same precommitments as you, so you'll just come away with nothing.

If however you precommit to offering $50 as Adam, and rejecting any offer less than $50 as Becky, then if your opponent made the same precommitments you would make a decent offer, get it accepted, and thus maximize your payoff in both situations.

And there we have it - acuasal trade. Without Adam or Becky needing to communicate with each other, they were able to strike a fair deal. Both of them have agreed not to make outrageous precommitments, and in return for Adam making a fair offer now, Becky has precommitted to making a fair offer if she was in his place.

So I suppose the real question the wikipedia article should ask is, "how come humans aren't rational enough to always offer a 50/50 split?"