A proposer splits 100 pounds any way she likes. The responder either accepts the split, and both are paid, or rejects it, and both get nothing. There is one offer and no negotiation. What does game theory predict, and what do real people do?
Short answer: Theory says offer the smallest positive amount and the responder accepts; real responders often reject offers below about a fifth of the pot, so real proposers offer 40 to 50 percent.
Step 1: Solve from the last move
The responder moves last, so start there. Faced with any positive offer x, accepting pays x and rejecting pays 0. If the responder cares only about money, she accepts every positive offer.
The proposer knows this. She therefore offers the smallest positive amount, say 1 pound, and keeps 99. That pair of strategies, offer the minimum and accept anything positive, is the subgame-perfect equilibrium: every decision is optimal at the point it is taken.
Step 2: Why the equilibrium is fragile
Subgame perfection rules out empty threats. A responder who announces she will reject anything below 40 is making a threat that costs her money to carry out, so in the purely monetary game it is not credible. That is the formal reason the theory lands on the minimum offer.
But the threat is credible if the responder's payoff includes something other than money. If rejecting an insulting offer feels worth more than one pound, then rejecting is optimal for her, and the equilibrium changes.
Step 3: What happens in the lab
Responders reject. Offers of about 20 percent of the pot or less are turned down a large share of the time across many countries and stake sizes. The responder is paying a little money to punish a split she sees as unfair.
Proposers anticipate this, which is why typical offers land between 40 and 50 percent. The equilibrium is not wrong; it is the equilibrium of a game with only monetary payoffs. The game people actually play has fairness inside the payoff function, and modelling that correctly is the point.
Step 4: The answer to give
Give both halves. First the clean backward-induction result, then the observation that rejections are real and what they imply about the payoff model. In a trading context, the same logic explains why a counterparty sometimes walks away from a trade that would have made them money: the price told them something about you.
Why interviewers ask it
Giving only the theory answer reads as untrained in how people behave; giving only the behavioural answer reads as untrained in game theory. Interviewers want both, in that order.
Knowing the proof is half of it. Work Ultimatum Game interactively, then take it under a clock with the rest of the Outcry simulators.