There are four coins to share. The proposer makes one offer, take it or leave it: an even split of two each, or three for themselves and one for the responder.
If the responder accepts, that is the split. If they refuse, neither gets anything.
You are Proposer
Play down the tree. The responder decides at random here, like a coin toss.
Solve it from the end
Start at the last move and ask what the player there would do, then step back one move at a time. That is backward induction.
What it teaches
Proposing first in a one-shot split lets you keep nearly all of it, and the fact that real responders turn such offers down is the plainest evidence that payoffs are not money.
The usual mistakeReading responder rejections as irrationality, when what has actually been shown is that the payoff numbers in the model were the wrong numbers, because being treated unfairly carries a real cost.
The checkPut the responder's feelings about the split into the payoffs rather than outside them. The behaviour is then a best reply, and what failed was the measurement rather than the reasoning.
What people actually do
Dictators give away money when nothing forces them to
The theory says
With the recipient stripped of any move, keeping the whole sum strictly dominates every alternative, so the dictator keeps everything and anything given is unexplained by the game.
People actually
A majority of dictators give something, the modal positive gift is well short of half, and average giving is a fifth or so of the stake, substantially less than ultimatum proposers offer but clearly more than zero. Giving falls when the stakes are real rather than hypothetical, when anonymity from the experimenter is increased, and when the act is framed as taking rather than giving.
Why is still argued over.
People reject free money
The theory says
In a one-shot ultimatum game with money payoffs, the responder prefers any positive amount to nothing, so backward induction gives a unique subgame-perfect outcome in which the proposer offers the smallest positive amount and the responder accepts it.
People actually
Modal offers are at or near an even split, mean offers run between about a third and a half of the pie, and offers below a fifth are rejected most of the time, at a real cost to the rejecter. The pattern holds across stake sizes large enough to matter and across every industrialised subject pool tested.
Why is still argued over.
In the real world
Fiction: The Godfather Part II
My offer is this: nothing
The senator opens by demanding a large payment for the gaming licence and a share of the business. Michael answers with one take-it-or-leave-it counter: nothing, and the senator can pay the licence fee himself. There is no haggling after it, because the senator has no move that hurts Michael more than it hurts him.
A single offer, no counter-offer, and a refusal that leaves both parties with nothing, which is what allows the proposer to take almost the whole surplus; the scene is memorable because it shows exactly how little a responder with no next move can hold out for.
Real worldContested
A final dowry demand
In parts of South Asia dowry demands are made, and sometimes raised, close to the wedding, when the other family's only alternative is to call the match off and absorb the loss and the shame of having done so. The practice has been unlawful in India since 1961 and is still widely reported, and demands made at that point are hard to refuse precisely because the alternative on offer is so bad. Inference, not fact: raising the demand late is not a negotiating error but the point of the timing, since the proposer's leverage is created entirely by how much worse the responder's alternative becomes as the date approaches.
There is one offer, no counter-offer and no next round, and refusal destroys the gain for both sides rather than producing a better deal, which is what lets the demand be so lopsided; the refusals that do happen are the clearest sign that the payoff is not only money.
Next door on the wall
Trust gameThe second mover here can only destroy value, not create it. Not on the wall yet.
Dictator gameThe recipient has no move at all, so nothing given can be explained as buying off a rejection, which is exactly what makes it the control condition. Not on the wall yet.
Nash demand gameBoth sides commit at once, so neither can present the other with a take it or leave it offer. Not on the wall yet.
Pirate gameWhat a responder gets from refusing is set by the succession rather than by zero, so some can be bought with a single coin and others cannot be bought at all. Not on the wall yet.