The ultimatum game was first introduced to the literature by Güth, Schmittberger, and Schwarze [1982]. It is a one-shot two-stage sequential bargaining game. It is often used to illustrate the backward induction method of solving for a subgame perfect Nash equilibrium for monetary payoff maximizing players. However, the game involves salient fairness considerations and there are multiple reported results of equal-split or close to equal-split outcomes from experiments. Strategic considerations of the players include notions of fear, negative reciprocity, and other-regarding preferences.

In stage 1 of the ultimatum game the first player proposes a specific split of a fixed amount of money, say $10, to the second player. In stage 2 the second player can either accept the proposed split or reject it. If he/she accepts, the $10 is divided according to the first mover's proposal. If he/she rejects, both players get 0.

Interesting modifications: There exist various scenarios for the game; for example player 2 might also have an outside option or, instead of simply accepting/rejecting the offer, instead choose a number between 0 and 1 to scale the payoffs. Another variation of the ultimatum game is its repeated version with discounting.

The subgame perfect Nash equilibrium for agents with self-regarding preferences is for player 1 to propose keeping all the money for himself and by the tie-breaking rule for player 2 to accept because he/she will be indifferent between vetoing and accepting a proposal in which he/she receives a payoff of zero (or to pass the smallest possible positive amount of money, in this case $1 in the absence of the tie-breaking rule).

Dickinson [2002] in his classroom experiment reports that the players do not behave as predicted by the self-regarding preferences model. Instead, responders reject many positive offers and usually accept only close to equal-split proposals. The average offers to second movers in this classroom game vary from 27 to 37 percent of a pie. The results show that students? rejection frequency for the similar offer range goes up as the size of the pie increases.

Player 1 may propose a positive amount for player 2 because of:

• Altruistic other-regarding preferences
• Fear that player 2 might reject a "selfish" proposal

To test for quantitative effects of altruistic other-regarding preferences and fear of rejection of proposals one can use a dictator control treatment. For the description of Ultimatum Game Dictator Control go to the Dictator Game section.

One offer and a rejection or acceptance in the Ultimatum Game is similar to final-stage negotiations of various sorts:

• Firm - union negotiations: A firm offers the final contract and the union can either accept or reject it. Such an offer can be viewed as a percentage of the profit pie offered to the union. If the union rejects such situation may lead to a costly strike, which is represented by a zero payoff to each player.
• New automobile purchases: Informed buyers when making new automobile purchases often know or at least have a good estimate of the dealer?s profit on the vehicle. The last offer from either side may be understood as an offer of a percentage of the dealer?s profit pie.
• Bilateral trade negotiations: If the Bilateral trade negotiations break down, the gains from trade might get lost.
• Failure to pass legislation: Political coalition falls apart over the failure to agree on the distribution of economic rents.
• Negotiations for peace between two disputing countries: Such case can also be considered a pie-splitting game, take for example the dispute over the territory of Jerusalem between the Palestinian Liberation Organization and Israel. Rejection of the last offer can often lead to an escalation of violence or war.