Ultimatum game

Ultimatum game

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The ultimatum game is a game
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

 often played in economic experiments
Experimental economics
Experimental economics is the application of experimental methods to study economic questions. Data collected in experiments are used to estimate effect size, test the validity of economic theories, and illuminate market mechanisms. Economic experiments usually use cash to motivate subjects, in...

  in which two players interact to decide how to divide a sum of money that is given to them. The first player proposes how to divide the sum between the two players, and the second player can either accept or reject this proposal. If the second player rejects, neither player receives anything. If the second player accepts, the money is split according to the proposal. The game is played only once so that reciprocation is not an issue.

Equilibrium analysis

For illustration, we will suppose there is a smallest division of the good available (say 1 cent). Suppose that the total amount of money available is x.

The first player chooses some amount p in the interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 [0,x]. The second player chooses some function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 f: [0, x] → {"accept", "reject"} (i.e. the second chooses which divisions to accept and which to reject). We will represent the strategy profile as (p, f), where p is the proposal and f is the function. If f(p) = "accept" the first receives p and the second xp, otherwise both get zero. (p, f) is a Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

 of the ultimatum game if f(p) = "accept" and there is no y > p such that f(y) = "accept" (i.e. player 2 would reject all proposals in which player 1 receives more than p). The first player would not want to unilaterally increase his demand since the second will reject any higher demand. The second would not want to reject the demand, since he would then get nothing.

There is one other Nash equilibrium where p = x and f(y) = "reject" for all y>0 (i.e. the second rejects all demands that gives the first any amount at all). Here both players get nothing, but neither could get more by unilaterally changing his / her strategy.

However, only one of these Nash equilibria satisfies a more restrictive equilibrium concept
Solution concept
In game theory, a solution concept is a formal rule for predicting how the game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players, therefore predicting the result of the game...

, subgame perfection
Subgame perfect equilibrium
In game theory, a subgame perfect equilibrium is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game...

. Suppose that the first demands a large amount that gives the second some (small) amount of money. By rejecting the demand, the second is choosing nothing rather than something. So, it would be better for the second to choose to accept any demand that gives her any amount whatsoever. If the first knows this, he will give the second the smallest (non-zero) amount possible.

Experimental results

In many culture
Culture is a term that has many different inter-related meanings. For example, in 1952, Alfred Kroeber and Clyde Kluckhohn compiled a list of 164 definitions of "culture" in Culture: A Critical Review of Concepts and Definitions...

s, people offer "fair" (i.e., 50:50) splits, and offers of less than 20% are often rejected. One limited study on monozygotic and dizygotic twins claims that genetic variation can affect reactions to unfair offers, though the study failed to employ actual controls for environmental differences.


The highly mixed results (along with similar results in the Dictator game
Dictator game
The dictator game is a game in experimental economics, similar to the ultimatum game. Experimental results offer evidence against the rationally self-interested individual concept of economic behavior, though precisely what to conclude from the evidence is controversial.-Description:In the...

) have been taken to be both evidence for and against the so-called "Homo economicus
Homo economicus
Homo economicus, or Economic human, is the concept in some economic theories of humans as rational and narrowly self-interested actors who have the ability to make judgments toward their subjectively defined ends...

" assumptions of rational, utility-maximizing, individual decisions. Since an individual who rejects a positive offer is choosing to get nothing rather than something, that individual must not be acting solely to maximize his economic gain, unless one incorporates economic applications of social, psychological, and methodological factors (such as the observer effect). Several attempts to explain this behavior are available. Some authors suggest that individuals are maximizing their expected utility
Expected utility hypothesis
In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different utility...

, but money does not translate directly into expected utility. Perhaps individuals get some psychological benefit from engaging in punishment or receive some psychological harm from accepting a low offer. It could also be the case that the second player, by having the power to reject the offer, uses such power as leverage against the first player, thus motivating him to be fair.

The classical explanation of the ultimatum game as a well-formed experiment approximating general behaviour often leads to a conclusion that the rational behavior in assumption is accurate to a degree, but must encompass additional vectors of decision making. However, several competing models suggest ways to bring the cultural preferences of the players within the optimized utility function
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

 of the players in such a way as to preserve the utility maximizing agent as a feature of microeconomics. For example, researchers have found that Mongolia
Mongolia is a landlocked country in East and Central Asia. It is bordered by Russia to the north and China to the south, east and west. Although Mongolia does not share a border with Kazakhstan, its western-most point is only from Kazakhstan's eastern tip. Ulan Bator, the capital and largest...

n proposers tend to offer even splits despite knowing that very unequal splits are almost always accepted. Similar results from other small-scale societies players have led some researchers to conclude that "reputation
Reputation of a social entity is an opinion about that entity, typically a result of social evaluation on a set of criteria...

" is seen as more important than any economic reward. Others have proposed the social status of the responder may be part of the payoff. Another way of integrating the conclusion with utility maximization is some form of inequity aversion
Inequity aversion
Inequity aversion is the preference for fairness and resistance to incidental inequalities. The social sciences that study inequity aversion include sociology, economics, psychology, anthropology and ethology.-Human studies:...

 model (preference for fairness). Even in anonymous one-shot settings, the economic-theory suggested outcome of minimum money transfer and acceptance is rejected by over 80% of the players.

An explanation which was originally quite popular was the "learning" model, in which it was hypothesized that proposers’ offers would decay towards the sub game perfect NE (almost zero) as they mastered the strategy of the game. (This decay tends to be seen in other iterated games). However, this explanation (bounded rationality
Bounded rationality
Bounded rationality is the idea that in decision making, rationality of individuals is limited by the information they have, the cognitive limitations of their minds, and the finite amount of time they have to make a decision...

) is less commonly offered now, in light of empirical evidence against it.

It has been hypothesised (e.g. by James Surowiecki
James Surowiecki
James Michael Surowiecki is an American journalist. He is a staff writer at The New Yorker, where he writes a regular column on business and finance called "The Financial Page".-Background:...

) that very unequal allocations are rejected only because the absolute amount of the offer is low. The concept here is that if the amount to be split were ten million dollars a 90:10 split would probably be accepted rather than spurning a million dollar offer. Essentially, this explanation says that the absolute amount of the endowment is not significant enough to produce strategically optimal behaviour. However, many experiments have been performed where the amount offered was substantial: studies by Cameron and Hoffman et al. have found that the higher the stakes are the closer offers approach an even split, even in a 100 USD game played in Indonesia
Indonesia , officially the Republic of Indonesia , is a country in Southeast Asia and Oceania. Indonesia is an archipelago comprising approximately 13,000 islands. It has 33 provinces with over 238 million people, and is the world's fourth most populous country. Indonesia is a republic, with an...

, where average 1995 per-capita income was 670 USD. Rejections are reportedly independent of the stakes at this level, with 30 USD offers being turned down in Indonesia, as in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...

, even though this equates to two week's wages in Indonesia.

Neurologic explanations

Generous offers in the ultimatum game (offers exceeding the minimum acceptable offer) are commonly made. Zak, Stanton & Ahmadi (2007) showed that two factors can explain generous offers: empathy and perspective taking. They varied empathy by infusing participants with intranasal oxytocin
Oxytocin is a mammalian hormone that acts primarily as a neuromodulator in the brain.Oxytocin is best known for its roles in sexual reproduction, in particular during and after childbirth...

 or placebo (blinded). They affected perspective-taking by asking participants to make choices as both player 1 and player 2 in the ultimatum game, with later random assignment to one of these. Oxytocin increased generous offers by 80% relative to placebo. Oxytocin did not affect the minimum acceptance threshold or offers in the dictator game (meant to measure altruism). This indicates that emotions drive generosity.

Rejections in the ultimatum game have been shown to be caused by adverse physiologic reactions to stingy offers. In a brain imaging experiment by Sanfey et al., stingy offers (relative to fair and hyperfair offers) differentially activated several brain areas, especially the anterior insular cortex
Insular cortex
In each hemisphere of the mammalian brain the insular cortex is a portion of the cerebral cortex folded deep within the lateral sulcus between the temporal lobe and the frontal lobe. The cortical area overlying it towards the lateral surface of the brain is the operculum...

, a region associated with visceral disgust
Disgust is a type of aversion that involves withdrawing from a person or object with strong expressions of revulsion whether real or pretended. It is one of the basic emotions and is typically associated with things that are regarded as unclean, inedible, infectious, gory or otherwise offensive...

. If Player 1 in the ultimatum game anticipates this response to a stingy offer, they may be more generous.

An increase in rational decisions in the game has been found among experienced Buddhist
Buddhism is a religion and philosophy encompassing a variety of traditions, beliefs and practices, largely based on teachings attributed to Siddhartha Gautama, commonly known as the Buddha . The Buddha lived and taught in the northeastern Indian subcontinent some time between the 6th and 4th...

Meditation is any form of a family of practices in which practitioners train their minds or self-induce a mode of consciousness to realize some benefit....

. fMRI data show that meditators recruit the posterior insular cortex (associated with interoception) during unfair offers and show reduced activity in the anterior insular cortex compared to controls.

People whose serotonin
Serotonin or 5-hydroxytryptamine is a monoamine neurotransmitter. Biochemically derived from tryptophan, serotonin is primarily found in the gastrointestinal tract, platelets, and in the central nervous system of animals including humans...

 levels have been artificially lowered will reject unfair offers more often than players with normal serotonin levels.

This is true whether the players are on placebo or are infused with a hormone that makes them more generous in the ultimatum game.

People who have ventromedial frontal cortex lesions were found to be more likely to reject unfair offers. This was suggested to be due to the abstractness and delay of the reward, rather than an increased emotional response to the unfairness of the offer.

Evolutionary game theory

Other authors have used evolutionary game theory
Evolutionary game theory
Evolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...

 to explain behavior in the ultimatum game. Simple evolutionary models, e.g. the replicator dynamics
Replicator equation
In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness landscape...

, cannot account for the evolution of fair proposals or for rejections. These authors have attempted to provide increasingly complex models to explain fair behavior.

Sociological applications

The ultimatum game is important from a sociological
Sociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...

 perspective, because it illustrates the human unwillingness to accept injustice
Justice is a concept of moral rightness based on ethics, rationality, law, natural law, religion, or equity, along with the punishment of the breach of said ethics; justice is the act of being just and/or fair.-Concept of justice:...

. The tendency to refuse small offers may also be seen as relevant to the concept of honour
Honour or honor is an abstract concept entailing a perceived quality of worthiness and respectability that affects both the social standing and the self-evaluation of an individual or corporate body such as a family, school, regiment or nation...


The extent to which people are willing to tolerate different distributions of the reward from "cooperative
Cooperation or co-operation is the process of working or acting together. In its simplest form it involves things working in harmony, side by side, while in its more complicated forms, it can involve something as complex as the inner workings of a human being or even the social patterns of a...

" ventures results in inequality that is, measurably, exponential across the strata of management
Management in all business and organizational activities is the act of getting people together to accomplish desired goals and objectives using available resources efficiently and effectively...

 within large corporations. See also: Inequity aversion within companies.

Some see the implications of the ultimatum game as profoundly relevant to the relationship between society and the free market
Free market
A free market is a competitive market where prices are determined by supply and demand. However, the term is also commonly used for markets in which economic intervention and regulation by the state is limited to tax collection, and enforcement of private ownership and contracts...

, with Prof. P.J. Hill, (Wheaton College (Illinois)
Wheaton College (Illinois)
Wheaton College is a private, evangelical Protestant liberal arts college in Wheaton, Illinois, a suburb west of Chicago in the United States...

) saying:
“I see the [ultimatum] game as simply providing counter evidence to the general presumption that participation in a market economy (capitalism) makes a person more selfish.”


The first ultimatum game was developed in 1982 as a stylized representation of negotiation, by Güth, Schmittberger, and Schwarze. It has since become the most popular of the standard Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

s in economics, and is said to be "catching up with the Prisoner's dilemma
Prisoner's dilemma
The prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...

 as a prime show-piece of apparently irrational behaviour


In the “Competitive ultimatum game” there are many proposers and the responder can accept at most one of their offers: With more than three (naïve) proposers the responder is usually offered almost the entire endowment (which would be the Nash Equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

 assuming no collusion among proposers).

The “Ultimatum game with tipping” – if a tip
A tip is an extra payment made to certain service sector workers in addition to the advertised price of the transaction. Such payments and their size are a matter of social custom. Tipping varies among cultures and by service industry...

 is allowed, from responder back to proposer the game includes a feature of the trust game, and splits tend to be (net) more equitable.

The “Reverse ultimatum game” gives more power to the responder by giving the proposer the right to offer as many divisions of the endowment as they like. Now the game only ends when the responder accepts an offer or abandons the game, and therefore the proposer tends to receive slightly less than half of the initial endowment.

For a complete review of the ultimatum game in experiments, see "Evolving Economics: Synthesis" by Angela A. Stanton.

Robert Aumann
Robert Aumann
Robert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...

's Blackmailer Paradox appears to be a repeated game
Repeated game
In game theory, a repeated game is an extensive form game which consists in some number of repetitions of some base game . The stage game is usually one of the well-studied 2-person games...

 in which the ultimatum game is played many times by the same players for high stakes.

The Pirate game
Pirate game
The pirate game is a simple mathematical game. It illustrates how, if assumptions conforming to a homo economicus model of human behaviour hold, outcomes may be surprising. It is a multi-player version of the ultimatum game.-The game:...

illustrates a variant with more than two participants with voting power, as illustrated in Stewart's "A Puzzle for Pirates".

External links