The Ultimatum Game
The setup is simple: a host introduces you to a stranger. Both of you get shown a ten-dollar bill. The ten-dollar bill, the host explains, gets divided between you and the stranger however you, player one, chooses. (It could be a 50/50 split; it could be a 90/10.) There's a catch, though: the stranger, player two, can reject your proposal, leaving both you and him empty handed.
In a world where rational self-interest–the prevailing model of microeconomic decisionmaking–reduces to sheer profitmaking, there wouldn't be a reason to offer him a more equitable split than $9.99/$0.01. Player two seeks to maximise profit, so he shouldn't turn you down–deciding to "reject" would cost him a penny. And you, player one, should take advantage of that insight, offering only as much as makes his rejection unprofitable.
But who wouldn't reject in player two's shoes? And who wouldn't account for that in player one's? A penny seems little price to pay for the satisfaction of depriving an asshole of $9.99, right? And so player two assigns some sort of value to rejection–in this case, at least more than a penny. And player one does too: by foreseeing that threat of rejection, he's complicit in ascribing it some worth.
The Utility of Revenge
So what is that rejection worth? On what offers would you exercise it? Does it matter if the sum to be divided is $100? A million? Is it even remotely possible that you'd reject one percent of a million dollars ($10,000) to deny someone else $990,000? What if you already had billions?
The questions that I'm asking, basically, are these:
(Assuming that the "revenge" we're taking about is the sense of righteousness you get by penalising unfairness in ultimatum games,)
a) Is it price inelastic to income? (Would the billionaire still grumblingly accept a 99/1 payout of $10,000?) If not, how elastic is it? Is it a luxury good (100% price elastic to income)? Surely there's no one so pinched for a penny that denying 999 of them to an asinine co-player wouldn't still be worth it? Maybe there is, but I doubt it: at some point, "revenge" has to have a price floor–a price at which its exchange for such a sum isn't worth it to anyone. (A price where devaluing it any further would be an affront to human dignity.)
b) What's its demand elasticity to price? Imagine 1000 pairings of player ones and player twos. Further, imagine b) all the parties were in the same economic straits, and c) all the pairings played with $10. If we had 10 player-ones choosing a split of $9.99/$0.01, and 10 choosing $9.90/$0.10, would we expect more of the former rejected than the latter? Or put differently, would we expect an increase in rejections as the cost dropped? Intuition tells us "yes." Purchasing revenge for a penny will always be more popular than buying it for a dime.
But is that true for all values x < y? Say that we compare splits of $5.01/$4.99 and $6.00/$4.00. It wouldn't surprise me one bit if rejection were more common at $4.99. Why? Player one seems to be taunting player two–equality wouldn't cost him but one cent beyond his original proposal. But he's signalled it's worth it: he's ascribed an independent value to inequality that no one can refute. That signal, then, causes player two to reevaluate his own deck. Whereas before, the $4.99 loss might have outweighed revenge, now that same revenge seems sweeter: it's reinforced by the equality it ensures. Basically, as equality becomes more salient, so does the cost of acceptance. And in fact, acceptance may have grown costlier as a result. Yes, we've still got a demand elastic to price. But at somewhere around the $4.90 mark, its elasticity seems to have reversed! Revenge becomes more in demand at each higher price level. (At least, until hitting an even $5.00–at which point I'd imagine it'd plummet.)
Just a bit of reverie on a Sunday afternoon, I guess.
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