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Encyclopedia > Ultimatum game

The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. The first player proposes how to divide a sum of money with the second party. If the second player rejects this division, neither gets anything. If the second accepts, the first gets their demand and the second gets the rest. Experimental Economics is the use of experimental methods to evaluate theoretical predictions of economic behaviour. ... Anonymity is the state of not being identifiable within a set, called the anonymity set. When referring to human beings, we say that a person is anonymous when the identity of that person is not known. ...

Contents


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 in the interval [0,x]. The second player chooses some function f: [0, x] -> {"accept", "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 x-p, otherwise both get zero. (p, f) is a Nash equilibrium of the Ultimatum game if f(p) = "accept" and there is no y > p such that f(y) = "accept". I.e. p is the largest amount the second will accept. The first player would not want to unilaterally increase their demand since the second will reject any higher demand. The second would not want to reject the demand, since he would then get nothing. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In general, a function is part of an answer to a question about why some object or process occurred in a system that evolved or was designed with some goal. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ...


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 their strategy.


However, only one of these Nash equilibria satisfies a more restrictive equilibrium concept subgame perfection. 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 him any amount whatsoever. If the first knows this, they will give the second the smallest (non-zero) amount possible[1]. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ...


Experimental results

In many cultures, people (except economics grad students) offer "fair" (50:50) splits, and offers of <20% are often rejected (Henrich et al. 2004; Oosterbeek et al. 2004). These results (along with similar results in the Dictator Game) are taken to be evidence against the homo economicus model of individual decisions. Since an individual who rejects a positive offer is choosing to get nothing rather than something, individuals must not be acting solely to maximize their economic gain. Several attempts to explain this behavior are available. Some authors suggest that individuals are maximizing their expected utility, but money does not translate directly into expected utility (Bolton 1991; Ochs and Roth 1989). Perhaps individuals get some psychological benefit from engaging in punishment or receive some psychological harm from accepting a low offer. The word culture comes from the Latin root colere (to inhabit, to cultivate, or to honor). ... Economics (deriving from the Greek words οίκω [okos], house, and νέμω [nemo], rules hence household management) is the social science that studies the allocation of scarce resources to satisfy unlimited wants. ... A graduate student (also, grad student or grad in American English, postgraduate student or postgrad in British English) is an individual who has completed a bachelors degree (B.A., B.S./B.Sc. ... The Dictator Game is a very simple game in experimental economics, similar to the Ultimatum Game. ... Homo economicus, or Economic man, is a term used for an approximation or model of homo sapiens that acts to obtain the highest possible well-being for himself given available information about opportunities and other constraints, both natural and institutional, on his ability to achieve his predetermined goals. ... The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. ...


Explanations

The classical explanation of the Ultimatum game as a well-formed experiment approximating general behaviour often leads to a conclusion that the homo economicus model of economic self-interest is incomplete. However, several competing models suggest ways to bring the cultural preferences of the players within the optimized utility function 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 Mongolian 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 lead some researchers to conclude that “reputation” is seen as more important than any economic reward.[2] Another way of integrating the conclusion with utility maximization is some form of Inequity aversion model (preference for fairness). Homo economicus, or Economic man, is a term used for an approximation or model of homo sapiens that acts to obtain the highest possible well-being for himself given available information about opportunities and other constraints, both natural and institutional, on his ability to achieve his predetermined goals. ... This article is about utility in economics and in game theory. ... Microeconomics is the study of the economic behaviour of individual consumers, firms, and industries and the distribution of production and income among them. ... Reputation is the general opinion of the public towards a person, a group of people, or an organization. ... Inequity aversion is the preference for fair rewards and fairplay in Anthropology (in the sub-disciplines sociology, economics, sociobiology, psychology, Evolutionary psychology, and primate behaviourology). ...


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) is less commonly offered now, in light of empirical evidence against it.[3] Many models of human behavior in the social sciences assume that humans can be reasonably approximated or described as rational entities, especially as conceived by rational choice theory. ...


It has been hypothesised (e.g. by James Surowiecki) 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 are the stakes the closer offers approach an even split, even in a 100 USD game played in Indonesia, where average 1995 per-capita income was 670 USD. Rejections are reportedly independent of the stakes as this level, with 30 USD offers being turned down in Indonesia, as in the United States, even though this equates to two week's wages in Indonesia[4]. James Surowiecki (born 1967) is a staff writer at The New Yorker magazine, where he writes a regular column on business. ... 1995 was a common year starting on Sunday of the Gregorian calendar. ...


Evolutionary game theory

Other authors have used evolutionary game theory to explain behavior in the Ultimatum Game. (Gale, Binmore, and Samuelson 1995; Güth and Yarri 1992; Harms 1997; Huck and Oechssler 1999; Skyrms 1996; Zollman 2005). Simple evolutionary models, e.g. the replicator dynamics, cannot account for the evolution of fair proposals or for rejections. These authors have attempted to provide increasingly complex models to explain fair behavior. Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ... The replicator equation is a differential equation that defines the dynamics of evolutionary games. ...


Sociological applications

The split dollar game is important from a sociological perspective, because it illustrates the human willingness to accept injustice and social inequality. Social interactions of people and their consequences are the subject of sociology studies. ... Justice is a concept involving the fair and moral treatment of all persons, especially in law. ... Social inequality refers to disparities in the distribution of material wealth in a society. ...


The extent to which people are willing to tolerate different distributions of the reward from "cooperative" ventures results in inequality that is, measurably, exponential across the strata of management within large corporations. See also: Inequity Aversion within companies . Co-operation refers to the practice of people or greater entities working in common with commonly agreed-upon goals and possibly methods, instead of working separately in competition. ... Management (from Old French ménagement the art of conducting, directing, from Latin manu agere to lead by the hand) characterises the process of leading and directing all or part of an organization, often a business, through the deployment and manipulation of resources (human, financial, material, intellectual or intangible). ... Inequity aversion is the preference for fair rewards and fairplay in Anthropology (in the sub-disciplines sociology, economics, sociobiology, psychology, Evolutionary psychology, and primate behaviourology). ...


Some see the implications of the Ultimatum game as profoundly relevant to the relationship between society and the free market, with Prof. P.J. Hill, (Wheaton College) saying: A free market is an idealized market, where all economic decisions and actions by individuals regarding transfer of money, goods, and services are voluntary, and are therefore devoid of coercion and theft (some definitions of coercion are inclusive of theft). Colloquially and loosely, a free market economy is an economy... Wheaton College is the name of two colleges in the United States: Wheaton College, Illinois Wheaton College, Massachusetts External Links Wheaton College (Illinois) Wheaton College (Massachusetts) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

“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.” [5]

History

The first Ultimatum game was developed is 1982 as a stylized representation of negotation, by Güth, Werner, Schmittberger, and Schwarze[6]. It has since become the most popular of the standard Experiments in economics, and is said to be "catching up with the Prisoner's dilemma as a prime show-piece of apparently irrational behaviour"[7]. 1982 is a common year starting on Friday of the Gregorian calendar. ... From Latin ex- + -periri (akin to periculum attempt). ... Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? The prisoners dilemma is a type of non-zero-sum game (game in the sense of Game Theory). ...


Variants

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[8] (which would be the Nash Equilibrium assuming no collusion among proposers).


The “Ultimatum Game with tipping” – if a tip is allowed, from responder back to proposer the game includes a feature of the trust game, and splits tend to be (net) more equitable.[9] A tip (also known as a gratuity) is that amount of payment to certain service sector professionals which is in addition to the advertised bill or fee. ... Trust games in experimental economics extend the dictator game one step by having the reward that the dictator can (unilaterally) split between herself and a partner partially decided by an initial gift from that partner. ...


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.[10]


External links

  • Game-tree based analysis of the Ultimatum Game

References

Bolton, G.E. (1991) "A comparative Model of Bargaining: Theory and Evidence" American Economic Review 81:1096-1136


Gale, John, Kenneth G. Binmore, and Larry Samuelson (1995) “Learning to be Imperfect: The Ultimatum Game” Games and Economic Behavior 8: 56-90.


Güth, W. and M. Yaari (1992) “An Evolutionary Approach to Explain Reciprocal Behavior in a Simple Strategic Game” in (U. Witt ed) Explaining Process and Change – Aproaches to Evolutionary Economics Ann Arbor 23-34.


Henrich, Joseph, Robert Boyd, Samuel Bowles, Colin Camerer, Ernst Fehr, and Herbert Gintis (2004) Foundations of Human Sociality: Economic Experiments and Ethnographic Evidence from Fifteen Small-Scale Societies. Oxford University Press


Huck, Steffen and Jörg Oechssler (1999) “The Indirect Evolutionary Approach to Explaining Fair Allocations” Games and Economic Behavior 28: 13-24. 1999 is a common year starting on Friday of the Common Era, and was designated the International Year of Older Persons by the United Nations. ...


Ochs, J. and A.E. Roth (1989) "An Experimental Study of Sequential Bargaining" American Economic Review 79: 355-384.


Oosterbeek, Hessel, Randolph Sloof, and Gijs van de Kuilen (2004) "Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis." Experimental Economics 7: 171-188.


Skyrms, Brian (1996) Evolution of the Social Contract. Cambridge: Cambridge Press.


Zollman, Kevin (2005) "Game Theoretic Explanations in Complex Environments" Manuscript


Notes

^  Technically, making a zero offer to the responder, and accepting this offer is also a Nash Equilibrium, as the responder's threat to reject the offer is no longer credible since they now gain nothing (materially) by refusing the zero amount offered. Normally, when a player is indifferent between various strategies the principle in Game Theory is that the strategy with an outcome which is Pareto optimally better for the other players is chosen (as a sort of tie-breaker to create a unique NE). However, it is generally assumed that this principle should not apply to an Ultimatum game player offered nothing; he is instead assumed to reject the offer although accepting it would be an equally subgame perfect NE. For instance, the University of Wisconsin summary: Testing Subgame Perfection Apart From Fairness in Ultimatum Games from 2002 admits the possibility that the proposer may offer nothing but qualifies the subgame perfect NE with the words (almost nothing) throughout the Introduction. In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ... Credibility is the believability of a statement, action, or source, and the ability of the observer to believe the above. ... Apathy is the lack of emotion, motivation, or enthusiasm. ... Game theory is a branch of applied mathematics that uses models to study interactions with formalised incentive structures (games). It has applications in a variety of fields, including economics, international relations, evolutionary biology, political science, and military strategy. ... Pareto efficiency, or Pareto optimality, is a central concept in economics with broad applications in game theory, engineering and the social sciences. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ... For the University of Wisconsin system, see University of Wisconsin System. ... 2002 is a common year starting on Tuesday of the Gregorian calendar. ...


^  See "Do higher stakes lead to more equilibrium play?" (page 18) in 3. Bargaining experiments, Professor Armin Falk's summary at the Instute for the Study of Labor.


^  Mongolian/Kazakh study conclusion from University of Pennsylvania. The University of Pennsylvania (commonly referred to as Penn or UPenn, although the former is the preferred and recognized nickname of the University) is a private university in Philadelphia, Pennsylvania and a member of the Ivy League. ...


^  A forthcoming paper “On the Behavior of Proposers in Ultimatum Games” Journal of economic behaviour and organization has the thesis that learning will not cause NE-convergence: see the abstract.


^  See The Ultimatum game detailed description as a class room plan from EconomicsTeaching.org. (This is a more thorough explanation of the practicalities of the game than is possible here.)


^  1982, Güth, Werner, Schmittberger, and Schwarze, “An Experimental Analysis of Ultimatum Bargaining,” Journal of Economic Behavior and Organization, 3:4 (December), page 367. (Description of the game at Neuroeconomics cites this as the earliest example.) 1982 is a common year starting on Friday of the Gregorian calendar. ...


^  See the introduction of Fairness Versus Reason in the Ultimatum Game Martin A. Nowak, Karen M. Page, Karl Sigmund


^  Ultimatum game with proposer competition by the GameLab.


^  Ruffle, Bradley J. (1998) “More is Better, but Fair is Fair: Tipping in Dictator and Ultimatum Games,” Games and Economic Behavior, 23:2 (May), page 247. 1998 is a common year starting on Thursday of the Gregorian calendar, and was designated the International Year of the Ocean. ...


^  The reverse ultimatum game and the effect of deadlines is from a 2003 study by Gneezy, Haruvy, & Roth, Bargaining under a deadline: evidence from the reverse ultimatum game From Games and Economic Behavior 45. 2003 is a common year starting on Wednesday of the Gregorian calendar. ...

Topics in game theory
Evolutionarily stable strategy - Mechanism design - No-win - Winner's curse - Zero-sum
Games: Prisoner's dilemma - Chicken - Stag hunt - Ultimatum game - Matching pennies ...
Related topics: Mathematics - Economics - Behavioral economics - Evolutionary biology - Evolutionary game theory - Population genetics - Behavioral ecology
[ edit ]

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The classical explanation of the Ultimatum game as a well-formed experiment approximating general behaviour often leads to a conclusion that the Homo economicus model of economic self-interest is incomplete.
The first Ultimatum game was developed in 1982 as a stylized representation of negotiation, by Güth, Werner, Schmittberger, and Schwarze.
^ The reverse ultimatum game and the effect of deadlines is from a 2003 study by Gneezy, Haruvy, and Roth, Bargaining under a deadline: evidence from the reverse ultimatum game From Games and Economic Behavior 45.
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