Isn’t it a threat to reject unfair offers in the Ultimatum Game?
TLDR: title.
In the Ultimatum Game, there are $10 to be divided between two players. First, Player 1 proposes a split, such as “I get $8 and you get $2” or “we both get $5″. Then, Player 2 either accepts this proposal or rejects it. If they reject it, both players get nothing.
Standard theory around here is that you should reject offers where you’d get less than $5. It’s a little counterintuitive at first, because if you’re offered $2, you might think “why not just take it? The alternative is to get nothing.” But the standard reason to reject it is that, if you make a consistent (or predictable-in-advance) policy of rejecting unfair offers, then people won’t make you unfair offers, because they know they’ll get nothing if they do.
This feels an awful lot like a threat to me, now that I think about it… and standard theory around here is also that nobody should make or give in to threats. The reason being: if you give in to threats, all you’re doing is showing people that threatening you is a good idea, and that might entice them into making threats when they wouldn’t otherwise have done that.
Think of a manipulative partner saying “if you don’t do these dishes right now, I’ll break up with you” — I think people say this when they know they’re talking to a spineless loser who’ll get up and go miserably do the dishes, and not when they know they’re talking to a well-spined person who will say “what? fuck you”. Because if they’re talking to someone who won’t give in, then they have to go ahead and break up with you over one load of dishes, and that’s not actually in their interests. So, goes standard theory, be the person who won’t be influenced by that nonsense, and the nonsense won’t come.
But wait a second. Suppose you’re playing the ultimatum game with someone, and you’re Player 1. Obviously, your own favorite outcome is to get all $10 for yourself, but that leaves the other person with $0 which makes them indifferent between accepting and rejecting. So maybe instead, you plan to offer them $1 while keeping $9 for yourself. That way, it’s in their own interests to accept the offer.
You’re not trying to hurt them; you’re trying to have more money.
But they look across the table at you, and say “if you offer me less than $5, I’ll make sure we both get nothing”.
Why is that not a threat? Why should you give in and change your plan away from what would otherwise be your default, in the absence of their self-harming commitment that they only made because they think it’ll work on you? Why should you do what they demand, when, as is the defining characteristic of a threat, it’s only good for them if you give in?
Not sure that is the standard theory: it was “around here” that I saw described the strategy of accepting low offers at reduced odds.
Essentially give them the choice of “100% chance of $5, or somewhat less than (5/6) chance of $6, or less than (5/7) chance of $7, etc”
That way the other party can’t expect to benefit by deviating from what you deem a fair division, but you’re also not destroying all the expected value available, in retaliation to an unfair offer.
Which will also more gracefully handle differences of opinion about the value being divided or the ‘fair’ midpoint, in cases where it’s less clear cut than just dividing $10 by two.
Well sure, but isn’t that still a threat?
If we use the word “threat” to encompass any possibility of declining the offer, then you reduce the game to having only two solutions: either total capitulation, or some kind of use of threats.
I think it’s more useful to draw more distinctions. Whereby one can negotiate (including the possibility of that negotiation resulting in no-deal) or otherwise have your actions be contingent on their offer, without calling it a “threat”.
Probabilistic acceptance at least lacks the element of spite that leads to senseless destruction of value.
I definitely agree that probabilistic rejection is a reasonable and useful strategy. I’m not saying it isn’t, I’m saying that the standard “don’t give in to threats” maxim seems to say people shouldn’t tolerate it. That puts two standard elements of our zeitgiest in conflict with each other, and I’m looking for a resolution.
If we say that rejection (even probabilistic rejection) in the Ultimatum Game is reasonable and other agents should negotiate around your tendency to do it, and we say nothing about why this doesn’t meet the definition of a threat when it prima facie seems to, then we’re saying agents should sometimes give in to threats, which we usually say they never should.
I think this gets at one of the central problems with the concept of game-theoretic threats. I don’t know if anyone ever figured out an especially solid formal solution — it sort of comes down to finding some reasonable conception of a privileged null (or default) action, as I understand it.
I think it comes down to notions of fairness. And it can’t be within game theory, it has to be outside of it, because of scaling issues.
Imagine Alice and Bob are dividing a bunch of apples. Alice says “let’s split them evenly”. Bob says “no, my utility scale assigns 2 utility to each apple while yours assigns only 1″ (because Bob scaled his utility function, keeping his behavior the same). Or Bob says “no, my utility function assigns 2 utility to each apple because I’m hungrier” and Alice says “no, it should be less because you’re heavier” and so on. That’s why the notion of fairness can’t be derived just from the numbers of the game, it has to pull in other considerations.
I tried to write more about this a couple years ago.
Yeah, I’m not totally sure but it does seem reasonably plausible to me that it requires caring about structural details that game theory tends to factor out. One might hope you could derive the relevant concept of “fairness” from the game theory, but… I’m not particularly sure you can.
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You’re massively and incorrectly overgeneralizing the standard theory. In ideal rational-actor perfect predictor scenarios, one should be known as the type of person who doesn’t take bad deals, and the only way to do that against perfect predictors is to BE the type of person who doesn’t take bad deals.
The two things which don’t apply in your post:
1) you have no reason to believe the other person knows you or that your intent has any impact on their offer. They’re not a perfect predictor, so turning down their $2 just costs you $2, without changing the probability that they offer more in other universes or future interactions.
2) It only applies to certain types of threat (where it’s not a direct causal clear action). Nobody’s advocating you withhold your wallet from an armed mugger.
I don’t understand either of your two corrections. For #1, is it really usually a requirement that the person making you the offer be a perfect predictor? What about a human-level predictor in an iterated game? Yudkowsky writes that on dath ilan, people use The Algorithm, which is to reject unfair offers with high enough probability to make it a bad gamble for the offer-maker. Dath ilan is populated by humans, not perfect predictors, so I think it’s fair to say standard theory says that this applies to humans.
Or, well, okay, maybe I should say “the zeitgeist of Lesswrong” instead of “standard theory”. In which case, my post is challenging our zeitgeist’s understanding of these two principles, not some theorem with much stricter requirements for who it applies to.
For #2, are you sure nobody’s advocating that? I think Planecrash has a plot element that demonstrates that people do advocate this (spoilers for Planecrash, obviously):
Keltham wants to destroy the entire universe because he’d rather do that than let Hell continue to exist. He realizes that the gods may bargain him down from that by offering to just fix Hell, but ~ignores this consideration because if he fixates on that and does anything differently in pursuit of such bargaining, then he’d be making a Threat instead of just maximizing his utility function, and the gods would predictably ignore him. Carissa is around too, but he rejects several possible ways she could influence his project on the grounds that, since she doesn’t prefer to destroy the universe and only wants Hell to be fixed, then if he lets her influence him in the wrong way, the gods might consider Keltham’s universe-destruction game to ultimately be a Threat made by Carissa, and predictably ignore it.
This seems very analogous to the mugging case. If it’s different, what’s the difference? Mind you that the person making the threat is a (very smart) human, not a perfect predictor.
Alternatively, what does the zeitgiest around here say about the human-governance principle “don’t negotiate with terrorists”? That one is unambiguously applied to humans. Do we generally disagree with it for that reason?
I’m just not convinced that my post frames either of these two things incorrectly. I think people around here do generally say that human beings should both reject unfair Ultimatum Game offers and not give in to threats.
50⁄50 is sort of an obvious Schelling point norm? If neither of you want to give into blackmail and have to agree to get any money at all, it seems like a relatively symmetric situation. But interesting question…
I think the point of the word ‘threat’ is that it is about a consequence which is illegitimate in some form, according to the wider culture. Like, if I say I’ll hit you if you don’t give me money, it is a threat, because I am not allowed to just randomly hit you. But if I say ‘If you break into my house, I’ll call the police’ that is not a threat, if that is normally what we do when a burglary happens.
So I would say that splitting the money evenly is in some way the fairest way to deal with this, and where we would want the culture to end up. So if I say that I’ll reject an unfavorable split, it is not a threat, but if I say I’ll reject anything under $9 it is a threat, because I am trying to get more than my fair share.
This also works nicely with ‘we shouldn’t give in to threats’. People shouldn’t give in to random threats of violence, but if the wider culture wants them to cooperate, giving in is prosocial. Not stealing because you’d get arrested is a good thing for society, in most cases.
Of course this does then require some sort of judgement on what the difference between a legitimate and illegitimate consequence is, and that is a really hard question. I don’t know how to answer it, but I’d gesture towards something like ‘those consequences that make society a better place to live in’.
Suppose that you made the Ultimatum of giving $1 out of $10 and your rival precommited to rejecting any unfair Ultimatums, while you know that the rival WILL uphold the precommitment. Then attempting not to give in would leave you worse off than actually giving in.
The main issue is that the rival’s attempt to give in would incentivise you to threaten the rival. On the other hand, if Player 1 was in such debt that only receiving at least $8 could save them, then I would expect Player 1 to be justified in setting the Ultimatum of $8, especially if we account for the standard theory being more complex than merely rejecting unfair offers on sight.
me and my friends’ll be picking up all the free $5 bills you and your enemies spite each other with.