The ultimatum game is a popular experimental economics game in which two players interact to decide how to divide a sum of money, first described by Nobel laureate John Harsanyi in 1961. The first player, the proposer, proposes a division of the sum with the second player, the responder. The responder can either accept the proposed division or reject it. If the responder accepts, the money is split according to the proposal; if the responder rejects, neither player receives anything. Both players know in advance the rules of the game.
The game is typically designed as a one-shot interaction to isolate immediate reactions to fairness, thereby minimizing the influence of potential future interactions. However, even within this one-shot context, participants' decision-making processes may implicitly involve considering the potential consequences of repeated interactions, due to the fact that humans have evolved within societies that interact repeatedly. This design is crucial for observing pure, unadulterated responses to the proposed division.
Equilibrium analysis For ease of exposition, the simple example illustrated above can be considered, where the proposer has two options: a fair split, or an unfair split. The argument given in this section can be extended to the more general case where the proposer can choose from many different splits.
A Nash equilibrium is a set of strategies (one for the proposer and one for the responder in this case), where no individual party can improve their reward by changing strategy. If the proposer always makes an unfair offer, the responder will do best by always accepting the offer, and the proposer will maximize their reward. Although it always benefits the responder to accept even unfair offers, the responder can adopt a strategy that rejects unfair splits often enough to induce the proposer to always make a fair offer. Any change in strategy by the proposer will lower their reward. Any change in strategy by the responder will result in the same reward or less. Thus, there are two sets of Nash equilibria for this game: The proposer always makes an unfair offer, and the responder always accepts an unfair offer. (The proposer never gives a fair offer so the responder can accept fair offers with any frequency without affecting the average reward.) The proposer always makes a fair offer. The responder rejects unfair offers often enough to make fair offers at least as profitable as unfair offers, and always accepts fair offers.
Finite horizon In a non-repeated or finite-horizon ultimatum game, the first Nash equilibria (unfair offer, always accept) are the only that satisfy a stricter condition called subgame perfection equilibrium (SPE). The game can be viewed as having two subgames that repeat themselves: the subgame where the proposer makes a fair offer, and the subgame where the proposer makes an unfair offer. An SPE occurs when there are Nash Equilibria in every subgame, that players have no incentive to deviate from. Using backward induction, we see that in the final stage, the responder will always accept any offer. Therefore, in previous stages, the proposer will always offer the minimum amount. Thus, the responder's threat to reject unfair offers in the second Nash equilibrium is not credible in a finite setting.
Infinite horizon However, in an infinite-horizon ultimatum game, the analysis changes significantly. Repeated interactions allow for strategies based on reputation and reciprocity. Discount factors become crucial, and the Folk Theorem suggests that many payoff distributions, including "fair" outcomes, can be supported as Nash equilibria, and potentially as subgame perfect equilibria. The one-shot deviation principle is used to verify SPE in these cases. Therefore, the conclusion that only the "unfair offer, always accept" equilibrium is SPE is specific to finite horizon games. Infinite horizon games can have many SPE.
Multi-valued or continuous strategies
The simplest version of the ultimatum game has two possible strategies for the proposer, Fair and Unfair. A more realistic version would allow for many possible offers. For example, the item being shared might be a dollar bill, worth 100 cents, in which case the proposer's strategy set would be all integers between 0 and 100, inclusive for their choice of offer, S. This would have two subgame perfect equilibria: (Proposer: S=0, Accepter: Accept), which is a weak equilibrium because the acceptor would be indifferent between their two possible strategies; and the strong (Proposer: S=1, Accepter: Accept if S>=1 and Reject if S=0).A strategy is an action plan, not an outcome or an action, so the Acceptor's strategy has to say what they would do in every possible circumstance, even though in equilibrium S=0 and S>1 do not happen.
The ultimatum game is also often modelled using a continuous strategy set. Suppose the proposer chooses a share S of a pie to offer the receiver, where S can be any real number between 0 and 1, inclusive. If the receiver accepts the offer, the proposer's payoff is (1-S) and the receiver's is S. If the receiver rejects the offer, both players get zero. The unique subgame perfect equilibrium is (S=0, Accept). It is weak because the receiver's payoff is 0 whether they accept or reject. No share with S > 0 is subgame perfect, because the proposer would deviate to S' = S - for some small number and the receiver's best response would still be to accept. The weak equilibrium is an artifact of the strategy space being continuous.
Experimental results The first experimental analysis of the ultimatum game was by Werner Güth, Rolf Schmittberger, and Bernd Schwarze: Their experiments were widely imitated in a variety of settings. When carried out between members of a shared social group (e.g., a village, a tribe, a nation, humanity) people offer "fair" (i.e., 50:50) splits, and offers of less than 30% are often rejected.See
Despite the outcome predicted by Nash equilibrium, numerous experimental studies have shown that people behave quite differently.
Oosterbeek et al. (2004) reviewed 37 studies and found that the first player typically offers around 40% of the "pie," although the percentage tends to decrease with larger pie sizes and when the players lack experience. Similarly, Andersen et al. (2018) observed that the rejection rate of unfair offers declines as the size of the pie being divided increases. Cooper and Dutcher (2011) reviewed a wide range of studies and found that experienced players tend to accept higher offers and reject lower ones.
The effect of anonymity The absence of direct interaction among participants and the concealment of their identities serve to minimize the influence of social pressures. In real-world settings, however, individuals are frequently subject to both social and emotional pressures. Accordingly, it is essential to investigate the extent to which generosity and cooperation are affected by such contextual factors.
Bolton and Zwick (1995) conducted a controlled experiment using the ultimatum game, in which they systematically varied both the level of anonymity between players and the experimenter, as well as the players' capacity to impose punitive measures. Their findings indicated that increasing anonymity led to a rise in the proportion of outcomes consistent with the Nash equilibrium, from 30% to 46%. More strikingly, when the capacity to punish was eliminated, the rate of Nash-consistent outcomes increased from 30% to nearly 100%. The authors concluded that the ability to punish accounts for deviations from the Nash equilibrium to a greater extent than anonymity.
Similarly, Charness and Gneezy (2008) found that while the disclosure of recipients' surnames significantly increased generosity in the dictator game, it had no such effect in the ultimatum game. They inferred from this result that strategic motives tend to outweigh altruistic considerations in such interactions.
The effect of the participant origin Several studies have examined the impact of participants' cultural or national origin on their level of generosity in economic decision-making games. Oosterbeek et al. (2004) found that participants from more traditional societies tended to make lower offers in the ultimatum game. Chuah et al. (2007) conducted a cross-cultural ultimatum game experiment involving Malaysian and British participants. Their results revealed that Malaysian players were more generous toward compatriots than toward British players. In contrast, the offers made by British participants were not influenced by the recipient's nationality.
The effect of the participants gender Solnick (2001) investigated the role of gender in ultimatum game behavior. In the experiment, players were required to simultaneously decide on their offer (if they were the proposer) and the minimum amount they would accept (if they were the responder). The findings indicated that offers were influenced by the gender of the responder: when the responder was male and the proposer was female, the average offer tended to be more generous. Furthermore, both male and female responders set higher minimum acceptance thresholds when the proposer was female. García-Gallego et al. (2012) found that, although women generally exhibit greater risk aversion than men, they tended to make lower offers and were more likely to reject higher offers in the ultimatum game.
Fairness and genetic proximity One limited study of monozygotic and dizygotic twins claims that genetic variation can have an effect on reactions to unfair offers, though the study failed to employ actual controls for environmental differences. It has also been found that delaying the responder's decision leads to people accepting "unfair" offers more often.See Grimm, Veronika and F. Mengel (2011). "Let me sleep on it: Delay reduces rejection rates in Ultimatum Games'', Economics Letters, Elsevier, vol. 111(2), pages 113–115, May. Common chimpanzees behaved similarly to humans by proposing fair offers in one version of the ultimatum game involving direct interaction between the chimpanzees. However, another study also published in November 2012 showed that both kinds of chimpanzees (common chimpanzees and bonobos) did not reject unfair offers, using a mechanical apparatus.
Cross-cultural differences
Some studies have found significant differences between cultures in the offers most likely to be accepted and most likely to maximize the proposer's income. In one study of 15 small-scale societies, proposers in gift-giving cultures were more likely to make high offers and responders were more likely to reject high offers despite anonymity, while low offers were expected and accepted in other societies, which the authors suggested were related to the ways that giving and receiving were connected to social status in each group. Proposers and responders from WEIRD (Western, educated, industrialized, rich, democratic) societies are most likely to settle on equal splits.
Framing effects
Some studies have found significant effects of framing on game outcomes. Outcomes have been found to change based on characterizing the proposer's role as giving versus splitting versus taking, or characterizing the game as a windfall game versus a routine transaction game.
Explanations