Random Game Theory Question

[algorithmancy]

Here's the setup:

You and another contestant are given a prize pool of $1000, and the two of you have one minute to come to an agreement as to how to divide the money between yourselves. If you come to an agreement within the time limit, you each get your agreed-upon share, plus, you get an extra prize of $X and the other contestant gets an

Comments

[firstfrost]

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.

[algorithmancy]

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.

[firstfrost]

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...)

[algorithmancy]

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.

[rifmeister]

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.

[bakedweasels]

This.

[philiptan]

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.)

[arcanology]

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.

[philiptan]

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.)

[algorithmancy]

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.

[firstfrost]

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.

[mjperson]

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.

[kirisutogomen]

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.

[algorithmancy]

I concede that it's not really a game theory question. I just couldn't figure out what else to call it.

[kirisutogomen]

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."