I think the disagreement boils down to what different people find would be a fair split of the money.

I don't think there is much disagreement that if you had a good reason to spite the other guy, spending some small amount of money to pay back would be warranted.  tomcollins admitted as much in his reply to the (admittedly extreme!) question of vetoing the guy who killed his family.

So the real question is: should you have any reason to spite the other guy?  Which comes down to: is he being an asshole by asking for 99.998% of the money, or is it fair for him to do so just because he "can"?

There is a mindset --let's call it carebear; just a label, no ridicule intended-- that expects humans to display some sensitivity towards each other's utilities.  We are social animals endowed with empathy.  Morals are almost innate, common sense rules of thumb that make life better for everyone.  Equality is seen as a sane default, even in non egalitarian cultures.  An inequality must be warranted: someone worked harder, was smarter, or got luckier.

For a carebear, the fair split in the Ultimatum Game is an equal one (althought others may be pragmatically accepted), and the Splitter is being a jerk for trying to abuse his position.  He's failing the basic rule of "do as you'd like to be done to you".  Being put in the position to accept $10 is doubly indignating.  Not only is he trying to get almost all of the money, but he's relying on you being nice about it (the utility of the $10 themselves barely registers, in this context).

There is a mindset --let's call it cutthroat-- by which, at least when money is involved, humans are expected to behave as selfish aggressive maximizers in a game with only the most basic ground rules: basically, respect for physical integrity and property (some would argue that real cutthroats won't respect anything that can't be defended, but again, it's just a label, take it as defined here).  For the cutthroat, morals are there mostly to avoid physical violence, and behaving morally consists of refraining from theft and aggression.  Behaviors that a carebear would consider "abusive" and "exploitive" are not immoral unless mediated by violence or threat thereof.

For a --um-- throatcutter, the Splitter is just being logical so it's wrong to spite him in first place.  The destructive spiteful reaction thus seems doubly irrational.

Of course I'm simplifying a bit.  Barring that, any big objections so far?

As another point of redundancy:

This guy advertises himself as "online backups for the truly paranoid".

http://www.tarsnap.com/

The guy is kind of a math and crypto whiz.  He used to be Security Officer of FreeBSD; don't know whether that's the case still.  He's cperciva at Hacker News.  His personal website:

http://www.daemonology.net

Given his pricing policy:


I bet he could be convinced to take bitcoins.

[Disclaimer: I don't know the man nor am associated with him in any way.  I have never used his service and I can't vouch for it first hand.  It just seems an obvious fit for this thread.]

I agree.  I brought up the link only for the man-bites-dog appeal.

Yes, it will certainly have some effect.  Now, if taxes didn't exist, some ability to fund research would be lost too.  I'm not convinced that the increased generosity would make up for that.

Just for your puzzlement/horror: 

http://news.bbc.co.uk/2/hi/europe/8321967.stm

I don't understand.  Guess something got lost in editing here.  Could you please complete or rephrase this?

I claim that those other two situations are actually different, and the fact that they reduce to the same mathematical model only attests to the insufficiency of the model for these applications.  The anchor you establish with the initial claims makes it a morally different situation.

You may go to further extremes: imagine that I sue you for bogus claims but you can't afford a proper defense.  I offer to settle for all your stuff minus one dollar.  You can refuse, but let's imagine that the legal system in your country is broken enough that in that case you lose *all* of your money to legal expenses.  If we stick to money, the Nash equilibrium is the same.  Is this the same situation, only camouflaged to trick you into feeling more ripped off?  Would you still give me all the money but one dollar?

The positions of the splitter and boolean are asymmetric.  That doesn't imply that their moral claims on the money need be different.  It's "free money" for both, zero versus zero claim.  Zero equals zero.

On what basis do you assume that you can derive moral claims from expected outcome in an idealized, partial, and demonstrably unrealistic model of the conflict?

You lament that our wetware is obsolete and that we don't do our best to conform to an abstract, reductionist, disembodied model of rationality.  Well, we are what we are!  And the very concepts of utility and value, that are at the center of mathematical models of rationality, ultimately derive from the obsolete craves and ambitions you despise.  

What's the point of sex, 99% of the time?  What's the point of art?  Of anything, really?  You can tell yourself any story about what you want in life, you can build an intricate symbolic structure representing a rationalization about why you even bother waking up in the morning.  No matter how high and imposing the towers, that castle will ultimately rest on your mostly obsolete emotions.

Rationality helped a fair bit bring us away from the caves, but what ultimately took us from there, and what made that an improvement, were the same ultimately pointless craves and ambitions that got us in there in the first place.

So I repeat: overriding our emotions is often a requisite of rationality.  But ignoring our emotions is not automatically rational.  There must be some worthy reward, current or expected, to warrant the sacrifice.

Even if it's just "practicing restraint for when I need it."

In your case (and sorry for gratuitous speculation, but even if I'm wrong this illustrates my reasoning), accepting the $0.01 allows you to conform to a respected model of rationality and thus tell yourself, and signal to others, that you are above most of the hairless monkeys.  Bet that feels good, doesn't it?  This could be traced back to various sorts of primitive psychosocial mechanisms (self-image, self-esteem, dominance, status, ...).

In my case, $0.01 or $10 are not nearly worth the outrage of being abused.  The delta between that suffering and the satisfaction of giving greedy pig Splitter the finger is worth more than that.  Other people pay comparable amounts to get a movie or book and experience a weak version of similar emotions by proxy.  Who are you to tell me that my action is irrational?

And immoral?  My choice passes most sniff tests: Golden Rule, check.  "Think global, act local", sure.  I wouldn't mind if everyone did as me.

[Emphasis mine]
Your terminology suggests you're seeing this as a game that you play to maximize your score.  The very title of the thread supports that framing.  If that was in the "social contract" of the interaction (as it is in a game of poker) then going cutthroat would be protected from social stigmas by the "magic circle" of the game (just like in poker you can deceive and bluff, in Diplomacy you can stab, or in boxing you can punch, with ideally no hard feelings).  I was intending this more as a metaphor for social interactions in the wild.

Both players have equal claim (or lack thereof) on the money.  There's no moral basis for anything but an even split, which is just a sane and socially ingrained default.  As he deviates from an even split, the splitter not only gets utility at the boolean's expense, but he causes more harm than value he gets (because of the added grief of abuse).  By increasing the risk of a veto, this disequilibrium further reduces the global expected value.

You may contest that there are "irrational" forces at play here, like the expectation of an equal return as the basis of what is "fair" in absence of other claims, and the drive to punish exploitive behavior.  The conviction with which you call these irrational and idiotic suggests that you hold a reductionistic belief that game theory is straightforwardly applicable to (even stylized) real life interactions, despite assumptions like unbounded "rationality" on all parts, perfect information, perfect selfishness of actors, constrained time domain, isolated strategy space.

Don't you acknowledge the possibility that these "irrational" assumptions and biases may be evolutionarily stable, advantageous features for individuals and societies?

You do acknowledge that if you were the splitter, you'd be more generous than you think you "should", in anticipation for "idiocy".  Of course, it's the prevalence of such actual "idiocy" that makes it a credible threat.  And that, in turn, gets "idiots" a better payoff in this game.  So isn't this "smart", in a way?

He gets to propose a split, and you get the final say.  That doesn't make the money his.  Until the game is concluded, the money remains the Faucet's property, and neither of you have any a priori claim on it.

That's a tautology.  Do you mean that only money needs be taken into account in order to calculate utility?

There is a strong hardwired social expectation for a fair (and I won't even use quotes here) split, and there is an emotional cost in overriding that.  You can decide it's worth it, but if you just ignore this cost, you are, by definition, not being rational.  And once you start factoring any non-monetary factors, the $0.01 vs $4,999.99 Nash equilibrium breaks down.

If you go to the wikipedia page for the Ultimatum Game you'll see that across cultures, almost everyone will reject any offer below some 20%-30%.  So, you see, the vast majority of humanity is very "weak" of character.

This and other experiments suggests that we are hardwired to tend towards collaboration and towards punishing defectors, because it has proven evolutionarily stable.

In a real world where almost nobody will take less than 20%, is asking for $4,999 rational?

In the extreme case you were presented before, after the Splitter has killed your family, would you still accept his $10 vs $4,990?  What do you mean he "deserves" revenge?  That's not any more Nash-rational.  What does the killing of your family have to do with the deal, at all?  Is there something wrong with your character?

@tomcollins: It's not his money either.  The person who originally had it (let's call him the Faucet, for agility, and similarly call the other guy the Splitter and you the Boolean, since you only get to say yes or no) asked you to "split" it somehow.

Somehow you seem to imply that by putting the Splitter in the favorable side of a Nash Equilibrium, the Faucet has effectively given him the whole $5,000, so all that is left for you is to try and extort all you can from him, from a position of no legitimacy and hardly any force.  That is the game theoretical, abstractly "rational" framing, the one that ignores all human emotion and common sense conventions about fairness.  Very few people in the world would have seen it this way in pre-Nash times.

As we know, there are emotions and instincts at play.  Issues of self-image.  Empathy.  Perhaps genetically or culturally inherited notions and heuristics regarding property and fairness.  I can fight those notions, try to abstract away my own human nature, in pursuit of a rational ideal.  That, even to the extent we assume it's possible, doesn't come without a cost.  Those factors modify the real payoff matrix.  It's not purely about money.  Not even for those who act like it was (more on this later).

And what is my motivation to fight my own instincts?  The payoff I get for that effort is warping the game so I'm at the extreme sucker end.  Is that rational?

Repressing or overcoming emotion is often a requisite of rationality.  But that doesn't mean it is, in itself, automatically a rational thing to do.  For that, a benefit needs to be derived.

I argue that people who opt for --let's call it-- the Nash strategy do that self-restraint work in order to defend a self-image as "rational" beings, for a culturally established notion of rationality.  Nothing wrong with that, but it's a tradeoff, not inherently *the* rational thing to do.

Because it gets recursive and messy.  Once you convince yourself that going against primal emotions is the "good" thing to do, and if you manage to hide the costs under the carpet, then you won't feel as much of a sucker, so it will really "cost" less to you.  For example, by convincing yourself that the Splitter is not being a selfish greedy pig, you may actually dampen the primal outrage of being abused, and by doing that you may be justifying a greedy Splitter position.

This recursiveness means that beliefs on the game may be self-realising.

tl;dr:  Emotions alter not only what seems rational, but what is rational.  Beliefs about rationality, in turn, alter emotions, and thus, recursively, what is rational.  Any attempt at rationalizing the game transforms it, in some sort of Heisenberg uncertainty principle.  So a static game theoretical analysis that conflates money with utility doesn't settle the debate of what is rational.  

tl;dr's tl;dr: For all practical purposes, there is no rational answer.  What's right for you depends on your personality and cultural baggage.

I, who would reject the offer in the original formulation, would take these offers.

Perhaps.  Then you can call them envious and jealous, respectively.  It's less plausible, though, that the people who, like me, reject the original offer but accept these reformulations, are acting out of envy or jealousy, since if I was envious I would not accept either V1 or V2, and if I was jealous I wouldn't accept V2.

@deadlizard: the original "game" is simple in its terms, and for every one person answering, the reply is simple and even obvious.  But the psychological and social implications are varied and complex, as the variety of responses, and reactions to responses, attests.  The little variations on the game help analyze our motivations and explore that psychological space.

For example, my latest reformulations try to prove that what drives most of us who would reject the original deal is not envy of jealousy, on the basis that most of us would respond differently to the modified games.

I don't agree that rejecting the $10 deal can be characterized as an act of jealousy or envy.  Let me illustrate with a few examples less loaded than the grandma heritage one.

Variant 1:

Imagine that the mysterious stranger came with the following proposal instead:  "I have $5000 to give you two guys, and I will split it randomly.  You don't get to further negotiate your shares, only take it or leave it.  If both accept, you get your share.  If either rejects, neither gets money."  Seeing that your expected outcome is $2500 and you have nothing to lose, you both immediately accept.  You get to confirm that the randomization method (die rolls, roulette, computer PRN generator, whatever) is fair.

So, you both cross your fingers and hope for the best.  Unfortunately for you, the dice give $10 to you and $4990 to the other guy.  Uncomfortably concealing his mix of joy and worry, the other guy accepts his $4990.  What do you do?

Variant 2:

The mysterious stranger proposes: "I'm feeling generous and I'd like to give away $5,000.  But honestly, I like you [smiles and taps the other guy in the shoulder] better, so have $4,990$.  I hope you [looks at you as if counting the nanoseconds to take his eyes elsewhere] don't mind you get $10.  You don't get to negotiate, just say yes or no, now.  If either of you rejects the offer nobody gets a dollar."

What would you do?

Note that I'm not saying that the above reformulations are equivalent to the original one.  They're equivalent in the game theoretical sense, but very different psychologically.  Actually, I'm using tomcollings's technique here to argue the opposite point.  I would have a different response to these reformulations (and to the ones advanced by tomcollings) than I would have to the original one.  But I argue that's not because I'm inconsistent or hypocritical, but because there are legitimate reasons why the differences in framing matter.

Formally, this assumes that first you try to maximize your own payoff, and that being equal, you try to maximize global utility.

It can be argued that the "unknown person offering" the money has signaled a lower preference for keeping it than the "greedy unknown person" taking it.

This is a minor point, though, and somewhat besides the question.  Just got curious about whether the folks that would take one cent would go all the way to accepting nothing at all.

What if you consider that the utility measure already accounts for all these psychological factors?  So the numbers in the matrix don't represent raw dollars,  but an accurate estimation¹ of how happy would it make you, in relative terms, to pick that option, all things considered.  Then rationality is accurate enough a name for optimization of that, isn't it?

That is formally equivalent to calling the utility money and assuming (again, for the sake of mathematical simplification) that that is all you care about.

1: Don't ask me how would you go about attributing a scalar value to the hairy mess of actual human preferences and biases, though.  Game theory doesn't even try to do this.

You can play all sort of mind games to make her believe that you won't take an "unfair" offer, but the formulation states that she'll offer you $10 nevertheless.  Your option (or not) to communicate before she makes her call only matters as far as that would affect your decision.

I mean, would you accept more or less money if you had a chance to state your position first?  If you swore you wouldn't take anything but half and she offered $10, would you be any more or less likely to accept it, in relation to the case where you didn't get a chance to talk before her offer?

If not, then all this talk about mind games is irrelevant to the question, isn't it?

If you can make such a binding decision you are effectively reversing the roles.  You are now the one calling the quantity and the other guy is the one that only gets to accept or veto.  From your perspective, you are not playing, and therefore not winning, the original game.

In the problem as formulated, you cannot make such a binding decision.

Now, imagine you could make a non-binding declaration of how much you will accept, and that the other guy offers you $10 nevertheless.  Does that change your choice?

Very precisely.  You think that shouldn't matter?

That is also why db's framing ("they give you 5000$ and I get to say how much you pay me to keep the rest") makes the problem subtly different, even though the underlying reward matrix remains identical in monetary terms.

I have insisted you don't necessarily treat this as a game theoretical abstract problem where utility is perfectly encoded in the monetary payoff and your job is just to maximize that.  That is, not unless that's honestly how you would frame it in real life.

The purpose of the thought experiment is exactly to explore how real human emotions and perceptions, and how real social dynamics, transform a problem which otherwise (that is, in an abstract game theoretical formulation) would have a clear unambiguous solution, invariant from all the different framings that maintain the same monetary reward structure.  That is, to explore the difference between rationality as formally defined in game theory and what makes sense to you in real life.

Whether you choose to call the latter "rational" or not is a matter of definitional nuance and not the main point.

I thought of a poll, but I'm more interested in the explanations than the breakdown.

This is besides the question, but in game theoretical terms, an iterated version of the game would promote more equality.  If we're going to play repeatedly, I (proposer) know that you have a rational reason to veto an "unfair" proposal --that is, so I'll learn to be fairer next time.  So I'll probably offer a more equal split to begin with.

In a one-shot game, there is never any purely financial incentive to reject whatever offer.  As I said, not even a 5000$/0$ one.

@vladimir: It's a one-off.

@vuce: So you would accept that the other guy keep $5000 and you $0?  Technically (and again, assuming we identify utility with monetary payoff, as you seem to have done), that is a Nash equilibrium too: rejecting the deal would hurt the other guy while netting you nothing.

@db: That would be the exact same deal, with a different framing.  My guess is that if you did a parallel study, dividing the subjects in two halves and asking either question, you'd get significantly different results.  Ironically, I think your framing favors the first guy.  Guess why?