Ultimatum Game
What is Ultimatum Game?
The ultimatum game is an experiment where one player proposes how to split a sum and the other can accept it or reject it, leaving both with nothing.
One player, the proposer, is given a sum of money and offers some share of it to a second player, the responder. If the responder accepts, both keep the agreed amounts; if the responder rejects, both get zero and the game ends. Standard theory has a sharp prediction: a responder who cares only about money should accept any positive offer, so the proposer should offer the smallest possible unit and keep the rest. Real behavior looks nothing like that, since offers cluster near an even split and low offers are frequently rejected, meaning responders pay real money to punish a split they see as unfair. The pattern appears across many countries, though typical offers vary with the culture and setting of the experiment.
Ultimatum Game: a worked example
A proposer is handed $100 to split in $1 units. Pure self-interest predicts an offer of $1, because the responder's real choice is $1 or nothing and $1 is more. In practice an offer of $10 is rejected often, which costs the responder that $10 and costs the proposer the $90 he would have kept. Anticipating this, most proposers offer $40 to $50, keeping $50 to $60 instead of the $99 the theory recommends. Turning down $10 is irrational if money is the only thing being counted, and rational the moment fair treatment carries value of its own.
The mistake students make with ultimatum game
Students confuse the ultimatum game with the dictator game. In the dictator game the second player has no choice at all, so anything the first player gives away reflects generosity rather than fear of rejection, and dictator offers are positive but smaller. The second mistake is calling a rejection an error. Rejecting is consistent with maximizing utility once fairness enters the payoff, and it is money, not utility, that gets left on the table.
Ultimatum Game questions
What does game theory predict in the ultimatum game?
Game theory with purely money-motivated players predicts the proposer offers the smallest possible amount and the responder accepts it. Solving backwards, the responder compares a tiny positive sum with zero and should take the sum, so the proposer can safely keep almost everything. This is the subgame perfect equilibrium, and it is close to the opposite of what experiments find.
Why do people reject low offers in the ultimatum game?
People reject low offers because unfair treatment carries a cost they will pay money to avoid accepting. Rejection punishes the proposer at a price to yourself, which fits models where utility depends on relative payoffs and not only on your own cash. It is evidence that fairness and reciprocity belong inside the payoff function rather than outside the model.
What is the difference between the ultimatum game and the dictator game?
The ultimatum game gives the second player power to reject, while the dictator game gives them no say at all. That single difference separates generosity from strategic fear, since a proposer facing possible rejection has a self-interested reason to offer more. Comparing the two shows both motives are real, because dictator offers stay above zero while ultimatum offers run higher still.
Formula / Example
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