Assuming I'm a person and not an algorithm, which may not be what you meant, my X=Y=0 strategy is probably something like "Argue for half, settle for a third, don't settle for less than a third unless their argument is crazily persuasive."
If X and Y are both lots bigger than $1000, settle for anything.
If X dwarfs both Y and $1000, offer them all of it.
If Y dwarfs both X and $1000, refuse to settle for less than half.
You're definitely a person and not an algorithm, and you're expected to take the worth of a dollar into account.
If you prefer, you can also imagine the version where one player makes one proposal about how to cut the cake, and the other accepts it or rejects it on the spot.
So the size of the pot in the middle matters, presumably.
I notice that in the $1000 case, all your strategies are in terms of fractions of the pot. ("half of it," "all of it," etc.) Am I correct in assuming that as the pot gets life-changingly big, absolute dollar values start to matter more? For example, if there's a million dollars in the middle, and the other guy only offers you $50K of it, then it's still pretty expensive to say no, regardless of what X and Y are.
Well, in theory, but I've got a reasonably comfortable life at present, so my life doesn't need that much changing. It would be different if I was unemployed, my house was being foreclosed on, etc. Similarly, if the other guy was in the archtypical "kid has horrible disease, insurance has punted him, needs a million dollars" disasters, I'd be more willing to let him have it all. But if we were both people-like-me, I'd be willing to forgo $50K if the other guy was really being that much of a jerk.
(I suppose it would suck if there were hidden constraints that he had to offer only unfair deals...)
Well, tell you what. I happen to be the sole proprietor of a top-tier Jerk Scolding business. Give me the $50K, and I'll make that guy feel like even more of a jerk than if you had said no. $50K buys a lot of jerk scolding.
I am apparently an altruist who holds grudges. :)
It isn't quite that I want him to feel like a jerk - I want him to feel like being a jerk was not a winning strategy. (Maybe this ties into it seeming like an iterated puzzle? I'm not sure.)
Well, what if I frame it this way:
Your neighbor discovers oil on his property. If he builds a well in his back yard, he stands to make a million bucks. Due to an arcane law, you have veto power over the project. Your neighbor offers you $50K to not veto the project.
While I understand what you're trying to do with that reframing, my immediate response is "hell no, $50K isn't nearly enough for me to live with an Oil Well next door." :-)
If it was minimal impact (which is what I'm assuming that reframing is for, because like tirinian, I'm not sure about living next door to an oil well, not to mention my housemate already clearly doesn't want to live next door to an oil well), then I don't veto the project, and I'm squeamish about taking the 50K for it.
But if it is oil on my property, I damn well do not give my neighbor $950K in order for him to not veto it and he is a jerk for suggesting it. :)
And yet this is where game theory runs into the baked into human instinct of punishing members of the group who try to screw you over. I would bet until you get to people who basically need that $50K right now to avoid something horrible, most people will let their inner monkey out and prefer to punish the wrongdoer.
Game theory probably says you should take a buck, because that's a buck more than you had before. Game theory on the single game never makes any sense because humans always assume an iterated game where it's important to prevent people from screwing you again next time.
Yeah, pretty tough question.
If I'm proposing, my initial proposal is that we end up with the same amount of money, if possible. I'd accept somewhat less. I agree with the general observation that as the pot gets bigger, I accept a smaller fraction. Speaking roughly, this is because the anger I feel at getting an unfair deal will grow very slowly if at all in the size of the pot. For small pots, the anger wins and I refuse to accept a raw deal, for large pots, the utils of the money wins.
I would take $50K and let you have $950K in the ultimatum game. For $50, I'd say screw it to teach you the lesson.
I'm assuming this is a single-stage game, but the "person not an algorithm" caveat makes me think that my strategy would differ if it's a repeated game. If there is a good chance that I may have the opportunity to split another pool, I'd be willing to take a lower percentage for the chance of winning again. Doubly so if it's with the same player, but even if it's with a different player.
(Kinda like negotiating with publishers.)
So you'd be willing to teach the other player that they can screw you over? I would think in a repeated game you'd be willing to let a few rounds go by with no payout to prove a point.
It'd make sense (to me) for a small number of repeated games, but I agree it's not good for a large or infinite number of games, where 50% of X+Y seems optimal. As a person, it is unlikely I will play an infinite number of games.
My preference would be to practically guarantee a large payout each play instead of maximizing future payouts. I'm trying to maximize my own profit... I don't care how much the other person makes. Put another way, if I was playing fewer than 50 games, I'd rather make 49% every round than risk 0% any round, especially if the other player decides to adopt a tit-for-tat strategy.
(I'm also operating on the assumption that X+Y is less or equal to $1000, which isn't actually stated anywhere.)
X and Y can be any values. X could be zero and Y could be a million dollars.
I suppose they could be negative, but that's less interesting.
If Y was larger than $1000+X I'd probably only agree to an arrangement in which I would receive the full $1000+X.
It's easy to think of unstated factors that would affect that strategy, though. Do I get to play again if we arrive at a successful agreement? After an unsuccessful agreement? Does the other player get to choose who he/she plays with? That sort of thing, but that's well outside the scope of the problem you posed.
As I've been musing about this, I also note that my internal concept of fairness is entirely based in the framing of the numbers.
"I give each of you $1000" is fair.
"I push a magic button that multiplies each of your net worth by 101%" is fair.
"I give you $1000 and him $5000 because he has five times as much money already as you do" is not fair.
Yeah, when discussing "fairness" I find I really need to know who the other person is to figure out what is fair. As I have less and less information about the other person, fairness comes closer and closer to everything is divided equally, but I think that's a special case of being forced to assume the other person is exactly like me.
It's not really a game theory question at all. Game theory looks at this problem and says, any agreement is a Nash equilibrium, so talk or whatever for 59 seconds and then say "yes".
As a human, I will assume that the other person uses the same strategy I do, and within that constraint insist on the point of agreement that maximizes my money. In the absence of any information about the other person, like if it's Bill Gates or a Bangladeshi leper or a terminally ill Buddhist monk, I sit at the point of splitting the money as evenly as possible and I don't budge. I.e., as long as the absolute value of the difference between X and Y is <=$1000, name the split that results in us both winning the same amount, and if one of X or Y is more than $1000 bigger, then the person getting screwed gets all $1000.
Oh, I make sure to talk first. How I conduct myself during those 60 seconds depends entirely on a snap psych evaluation of the other player.
I concede that it's not really a game theory question. I just couldn't figure out what else to call it.
Although, it's certainly related to the sorts of things one talks about when learning game theory, if only to delineate what the field is and what it isn't.
And I was reminded o the answer I gave a game theory professor when he asked what the optimal strategy for playing Chicken was, and I said "point the car at the other guy, hit the accelerator, and then, when you know he's looking, throw the steering wheel out the window."