One of you proposes how to split a sum. The other accepts, and it stands, or refuses, and you both walk away with nothing. No second round, no negotiation. Something beats nothing, so a rational responder accepts anything at all and a proposer who knows that offers the smallest amount there is. Almost nobody does either.
Güth, Schmittberger and Schwarze ran this in 1982 and it has been run thousands of times since, across cultures, with stakes up to several months' wages. Offers cluster near half. Offers below about a fifth are refused perhaps half the time, by people who are, at that moment, paying real money to refuse them.
Refusing is not a mistake in any ordinary sense. The people who do it are not confused about the arithmetic, they do not change their minds when it is explained, and they do it more when the offer is deliberate than when it was randomly generated.
The population below has no feelings about any of this. Each agent carries two numbers: what it offers, and the least it will take. Everyone plays both roles, so the two numbers evolve against each other.
Most of this site is about populations that reach somewhere the reasoning did not predict. This page is the one where they land exactly where it said, and people do not.
A responder who turns down a fifth of the money is choosing nothing over something. If you take rational to mean maximising what you walk out with, that is irrational, and the standard reading is that the person is enforcing a fairness norm at their own expense.
There is a competing reading worth taking seriously. What is being maximised may not be the money. Refusing an insulting offer is a purchase: it buys the refusal, and possibly a reputation. Binmore has argued for years that people import expectations from a lifetime of repeated dealings into a game the experimenter insists is one-shot, and that this is not confusion so much as a sensible default.
With nobody knowing anybody, the population converges on the game-theoretic answer: minimal offers, minimal thresholds. That is a real finding, because it means the experimental result cannot be explained by evolution alone. Something else is doing the work.
Nobody here has any preference for fairness. What they have is an interest in being known as hard to push around.
Add reputation and fair offers appear. Nowak, Page and Sigmund (2000) showed this: if a proposer sometimes knows what a particular responder will accept, then holding out becomes worth something, and offers rise to meet it. Fairness comes out as a by-product of information, not as a value anyone holds.
Push reputation to the maximum and the population does not become fairer. Thresholds climb to nearly the whole sum and proposers hand it over, because they know exactly what will be refused. The mechanism that produced an even split at a quarter information produces near-total extraction at full information.
That is worth noticing before drawing anything comforting from the middle of the range. Information created the even split, and information can just as easily walk past it.
Reputation here is a single number the proposer either sees or does not. Real reputation is partial, transferable, and fakeable, and people invest in managing it. There is no communication, no promises, no anger, and no sense in which anyone is insulted. The model can show that information is sufficient to produce fair offers. It cannot show that information is what actually produces them in people.