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# Ultimatum game

Extensive form representation of a two proposal ultimatum game. Player 1 can offer a fair (F) or unfair (U) proposal; player 2 can accept (A) or reject (R).

The ultimatum game is a game often played in economic experiments 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 [0,x]. The second player chooses some function 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 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, 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 her any amount whatsoever. If the first knows this, he will give the second the smallest (non-zero) amount possible.[1]

## Experimental results

In industrialized[2] cultures,[which?] people offer "fair" (i.e., 50:50) splits, and offers of less than 20% are often rejected.[3] 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.[4]

## Explanations

The highly mixed results (along with similar results in the Dictator game) have been taken to be both evidence for and against the so-called "Homo economicus" 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, but money does not translate directly into expected utility.[5] 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 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 led some researchers to conclude that "reputation" is seen as more important than any economic reward.[6] Others have proposed the social status of the responder may be part of the payoff.[7] Another way of integrating the conclusion with utility maximization is some form of inequity aversion 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) is less commonly offered now, in light of empirical evidence against it.[8]

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.[citation needed] 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, 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, even though this equates to two week's wages in Indonesia.[9]

### Neurologic explanations

Generous offers in the ultimatum game (offers exceeding the minimum acceptable offer) are commonly made. Zak, Stanton & Ahmadi (2007)[10] showed that two factors can explain generous offers: empathy and perspective taking. They varied empathy by infusing participants with intranasal oxytocin 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.[11] 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, a region associated with visceral disgust. 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 meditators. 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.[12]

People whose serotonin levels have been artificially lowered will reject unfair offers more often than players with normal serotonin levels.[13]

This is true whether the players are on placebo or are infused with a hormone that makes them more generous in the ultimatum game.[14][15]

People who have ventromedial frontal cortex lesions were found to be more likely to reject unfair offers.[16] 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.[17]

### Evolutionary game theory

Other authors have used evolutionary game theory to explain behavior in the ultimatum game.[18] 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.

## Sociological applications

The ultimatum game is important from a sociological perspective, because it illustrates the human unwillingness to accept injustice. The tendency to refuse small offers may also be seen as relevant to the concept of honour.

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.

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 (Illinois)) 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.[19]

## History

The first ultimatum game was developed in 1982 as a stylized representation of negotiation, by Güth, Schmittberger, and Schwarze.[20] It has since become a popular economic experiment, and was said to be "quickly catching up with the Prisoner's Dilemma as a prime showpiece of apparently irrational behavior" in a paper by Martin Nowak, Karen M. Page, and Karl Sigmund.[21]

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

In the "ultimatum game with tipping", a tip is allowed from responder back to proposer, a feature of the trust game, and net splits tend to be more equitable.[23]

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.[24]

Robert Aumann's Blackmailer Paradox appears to be a repeated game in which the ultimatum game is played many times by the same players for high stakes.

The pirate game illustrates a variant with more than two participants with voting power, as illustrated in Ian Stewart's "A Puzzle for Pirates".[25]

## Notes

1. ^ 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; she 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.
2. ^ Sanfey, Alan; Rilling, Aronson, Nystrom, Cohen (13). "The Neural Basis of Economic Decision-Making in the Ultimatum Game". Science 300 (5626): 1755–1758. DOI:10.1126/science.1082976. JSTOR 3834595. PMID 12805551.
3. ^ See Joseph Henrich et al. (2004) and Oosterbeek et al. (2004).
4. ^ http://www.pnas.org/content/105/10/3721.full.pdf+html
5. ^ See Bolton (1991), and Ochs and Roth, A. E. (1989).
6. ^
7. ^ Social Role in the Ultimate Game
8. ^ 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.
9. ^ See "Do higher stakes lead to more equilibrium play?" (page 18) in 3. Bargaining experiments, Professor Armin Falk's summary at the Institute for the Study of Labor.
10. ^ Zak PJ, Stanton AA, Ahmadi S (2007), Oxytocin Increases Generosity in Humans. PloSONE 2(11):e1128. [1]
11. ^ Sanfey, et al. (2002)
12. ^ Kirk et al. (2011). "Interoception Drives Increased Rational Decision-Making in Meditators Playing the Ultimatum Game". Frontiers in Neuroscience 5:49. PMC 3082218. PMID 21559066. //www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3082218.
13. ^ Crockett, Molly J.; Luke Clark, Golnaz Tabibnia, Matthew D. Lieberman, Trevor W. Robbins (2008-06-05). "Serotonin Modulates Behavioral Reactions to Unfairness" (– Scholar search). Science 320 (5884): 1155577. DOI:10.1126/science.1155577. PMC 2504725. PMID 18535210. Retrieved 2008-06-22.
14. ^ Neural Substrates of Decision-Making in Economic Games Scientific Journals International [2]
15. ^ Oxytocin Increases Generosity in Humans PloSONE 2(11):e1128 [3]
16. ^ Koenigs, Michael; Daniel Tranel (January 2007). "Irrational Economic Decision-Making after Ventromedial Prefrontal Damage: Evidence from the Ultimatum Game". Journal of Neuroscience 27 (4): 951–956. DOI:10.1523/JNEUROSCI.4606-06.2007. PMC 2490711. PMID 17251437. //www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2490711.
17. ^ Moretti, Laura; Davide Dragone, Giuseppe di Pellegrino (2009). "Reward and Social Valuation Deficits following Ventromedial Prefrontal Damage". Journal of Cognitive Neuroscience 21 (1): 128–140. DOI:10.1162/jocn.2009.21011. PMID 18476758.
18. ^ See, for example, Gale et al. (1995), Güth and Yaari (1992), Huck and Oechssler (1999), Nowak & Sigmund (2000) and Skyrms (1996)
19. ^ 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.)
20. ^ Güth et al. (1982), page 367: the description of the game at Neuroeconomics cites this as the earliest example.
21. ^ Nowak, M. A.; Page, K. M.; Sigmund, K. (2000). "Fairness Versus Reason in the Ultimatum Game". Science 289 (5485): 1773–1775. DOI:10.1126/science.289.5485.1773. PMID 10976075.  edit
22. ^
23. ^ Ruffle (1998), p. 247.
24. ^ The reverse ultimatum game and the effect of deadlines is from Gneezy, Haruvy, & Roth, A. E. (2003).
25. ^ Stewart, Ian (May 1999). "A Puzzle for Pirates". Scientific American 05: 98–99. Retrieved 3/11/2011.

## References

• Alvard, M. (2004). "The Ultimatum Game, Fairness, and Cooperation among Big Game Hunters". In Henrich, J., Boyd, R., Bowles, S., Gintis, H., Fehr, E., and Camerer, C.. Foundations of Human Sociality: Ethnography and Experiments in 15 small-scale societies. Oxford University Press. pp. 413–435.
•
• Bicchieri, Cristina and Jiji Zhang (2008). "An Embarrassment of Riches: Modeling Social Preferences in Ultimatum games", in U. Maki (ed) Handbook of the Philosophy of Economics, Elsevier
• Bolton, G.E. (1991). "A comparative Model of Bargaining: Theory and Evidence". American Economic Review 81: 1096–1136.
• Gale, J., Binmore, K.G., and Samuelson, L. (1995). "Learning to be Imperfect: The Ultimatum Game". Games and Economic Behavior 8: 56–90. DOI:10.1016/S0899-8256(05)80017-X.
• Gneezy, Haruvy, and Roth, A. E. (2003). "Bargaining under a deadline: evidence from the reverse ultimatum game" (PDF – Scholar search). Games and Economic Behavior 45 (2): 347. DOI:10.1016/S0899-8256(03)00151-9.
• Güth, W., Schmittberger, and Schwarze (1982). "An Experimental Analysis of Ultimatum Bargaining". Journal of Economic Behavior and Organization 3 (4): 367–388. DOI:10.1016/0167-2681(82)90011-7.
• Güth, W. and Yaari, M. (1992). "An Evolutionary Approach to Explain Reciprocal Behavior in a Simple Strategic Game". In U. Witt. Explaining Process and Change – Approaches to Evolutionary Economics. Ann Arbor. pp. 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, S. and Oechssler, J. (1999). "The Indirect Evolutionary Approach to Explaining Fair Allocations". Games and Economic Behavior 28: 13–24. DOI:10.1006/game.1998.0691.
• Ochs, J. and Roth, A. E. (1989). "An Experimental Study of Sequential Bargaining". American Economic Review 79: 355–384.
• Oosterbeek, Hessel, Randolph Sloof, and Gijs van de Kuilen (2004). "Cultural Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis". Experimental Economics 7 (2): 171–188. DOI:10.1023/B:EXEC.0000026978.14316.74.
• Ruffle, B.J. (1998). "More is Better, but Fair is Fair: Tipping in Dictator and Ultimatum Games". Games and Economic Behavior 23 (2): 247. DOI:10.1006/game.1997.0630. .
• Sanfey et al.; Rilling, JK; Aronson, JA; Nystrom, LE; Cohen, JD (2002). "The neural basis of economic decision-making in the ultimatum game". Science 300 (5626): 1755–1758. DOI:10.1126/science.1082976. PMID 12805551. .
• Skyrms, B. (1996). Evolution of the Social Contract. Cambridge University Press.
• Zak, P.J., Stanton, A.A., Ahmadi, S. (2007). Brosnan, Sarah. ed. "Oxytocin Increases Generosity in Humans" (PDF). Public Library of Science ONE 2 (11): e1128.. DOI:10.1371/journal.pone.0001128. PMC 2040517. PMID 17987115.
• Angela A. Stanton (2007). "Neural Substrates of Decision-Making in Economic Games" (PDF). Scientific Journals International 1 (1): 1–64..

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