ultimatum game

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.

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.{{Cite journal |last1=Fudenberg |first1=Drew |last2=Tirole |first2=Jean |date=1991-04-01 |title=Perfect Bayesian equilibrium and sequential equilibrium |url=https://dx.doi.org/10.1016%2F0022-0531%2891%2990155-W |journal=Journal of Economic Theory |language=en |volume=53 |issue=2 |pages=236–260 |doi=10.1016/0022-0531(91)90155-W |issn=0022-0531}} 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.

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.

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 - $\backslash epsilon$ for some small number $\backslash epsilon$ and the receiver's best response would still be to accept. The weak equilibrium is an artifact of the strategy space being continuous.

The first experimental analysis of the ultimatum game was by Werner Güth, Rolf Schmittberger, and Bernd Schwarze:{{cite journal |last1=Güth |first1=Werner |last2=Schmittberger |first2=Rolf |last3=Schwarze |first3=Bernd |title=An experimental analysis of ultimatum bargaining |journal=Journal of Economic Behavior & Organization |date=1982 |volume=3 |issue=4 |pages=367–388 |doi=10.1016/0167-2681(82)90011-7|url=http://www.dklevine.com/archive/refs4291.pdf }} 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){{cite journal|last=Sanfey|first=Alan |author2=Rilling |author3=Aronson |author4=Nystrom |author5=Cohen|s2cid=7111382 |title=The Neural Basis of Economic Decision-Making in the Ultimatum Game|journal=Science|date=13 June 2003|volume=300|issue=5626|pages=1755–1758|jstor=3834595|doi=10.1126/science.1082976|pmid=12805551|bibcode=2003Sci...300.1755S }} people offer "fair" (i.e., 50:50) splits, and offers of less than 30% are often rejected.See {{cite book|author = Henrich, Joseph, Robert Boyd, Samuel Bowles, Colin Camerer, Ernst Fehr, and Herbert Gintis|year =2004| title = Foundations of Human Sociality: Economic Experiments and Ethnographic Evidence from Fifteen Small-Scale Societies|publisher = Oxford University Press}}{{cite journal| author = Oosterbeek, Hessel, Randolph Sloof, and Gijs van de Kuilen|year=2004| title = Cultural Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis|journal = Experimental Economics|volume = 7|pages = 171–188| doi = 10.1023/B:EXEC.0000026978.14316.74| issue = 2|s2cid=17659329}}

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.{{Cite journal | doi=10.1073/pnas.0710069105|pmid = 18316737| pmc=2268795| title=Heritability of cooperative behavior in the trust game| journal=Proceedings of the National Academy of Sciences| volume=105| issue=10| pages=3721–3726| year=2008| last1=Cesarini| first1=D.| last2=Dawes| first2=C. T.| last3=Fowler| first3=J. H.| last4=Johannesson| first4=M.| last5=Lichtenstein| first5=P.| last6=Wallace| first6=B.|bibcode = 2008PNAS..105.3721C|doi-access = free}} It has also been found that delaying the responder's decision leads to people accepting "unfair" offers more often.{{Cite journal|title = Emotions, rejections, and cooling off in the ultimatum game.|last1 = Bosman|first1 = Ronald|date = 2001|journal = Unpublished Manuscript, University of Amsterdam|last2 = Sonnemans|first2 = Joep|last3 = Zeelenberg|first3 = Marcel Zeelenberg}}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.{{Cite journal|title = Cooling Off in Negotiations: Does it Work?|journal = Journal of Institutional and Theoretical Economics|date = 2015|pages = 565–588|volume = 171|issue = 4|doi = 10.1628/093245615X14307212950056|first1 = Jörg|last1 = Oechssler|first2 = Andreas|last2 = Roider|first3 = Patrick W.|last3 = Schmitz|citeseerx = 10.1.1.333.5336|s2cid = 14646404}} Common chimpanzees behaved similarly to humans by proposing fair offers in one version of the ultimatum game involving direct interaction between the chimpanzees.{{cite journal|last=Proctor|first=Darby |author2=Williamson |author3=de Waal |author4=Brosnan|title=Chimpanzees play the ultimatum game|journal=PNAS|date=2013|volume=110|issue=6|pages=2070–2075|doi=10.1073/pnas.1220806110|pmid=23319633 |pmc=3568338 |doi-access=free }} 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.{{cite journal|last=Kaiser|first=Ingrid |author2=Jensen |author3=Call |author4=Tomasello|title=Theft in an ultimatum game: chimpanzees and bonobos are insensitive to unfairness|journal=Biology Letters|date=2012|volume=8|issue=6|pages=942–945|doi=10.1098/rsbl.2012.0519 |pmid=22896269 |pmc=3497113}}

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.{{cite journal | last1 = Henrich | first1 = Joseph | last2 = Boyd | first2 = Robert | last3 = Bowles | first3 = Samuel | last4 = Camerer | first4 = Colin | last5 = Fehr | first5 = Ernst | last6 = Gintis | first6 = Herbert | last7 = McElreath | first7 = Richard | last8 = Alvard | first8 = Michael | last9 = Barr | first9 = Abigail | last10 = Ensminger | first10 = Jean | last11 = Henrich | first11 = Natalie Smith | last12 = Hill | first12 = Kim | last13 = Gil-White | first13 = Francisco | last14 = Gurven | first14 = Michael | last15 = Marlowe | first15 = Frank W. | last16 = Patton | first16 = John Q. | last17 = Tracer | first17 = David | title = "Economic man" in cross-cultural perspective: Behavioral experiments in 15 small-scale societies | journal = Behavioral and Brain Sciences | date = December 2005 | volume = 28 | issue = 6 | pages = 795–815 | issn = 0140-525X | eissn = 1469-1825 | doi = 10.1017/S0140525X05000142 | pmid = 16372952 | s2cid = 3194574 | url = https://authors.library.caltech.edu/2278/}} Proposers and responders from WEIRD (Western, educated, industrialized, rich, democratic) societies are most likely to settle on equal splits.{{cite journal | last1 = Henrich | first1 = Joseph | last2 = Heine | first2 = Steven J. | last3 = Norenzayan | first3 = Ara | title = The weirdest people in the world? | journal = Behavioral and Brain Sciences | date = June 2010 | volume = 33 | issue = 2–3 | pages = 61–83 | issn = 0140-525X | eissn = 1469-1825 | doi = 10.1017/S0140525X0999152X | pmid = 20550733 | s2cid = 219338876 | url = https://www2.psych.ubc.ca/~henrich/pdfs/WeirdPeople.pdf}}{{cite web |url=https://www.fresheconomicthinking.com/2010/10/weird-people-western-educated.html |title=WEIRD people: Western, Educated, Industrialised, Rich, Democratic... and unlike anyone else on the planet |last=Murray |first=Cameron |date=October 11, 2010 |access-date=July 1, 2022}}{{cite web |url=https://psmag.com/social-justice/joe-henrich-weird-ultimatum-game-shaking-up-psychology-economics-53135 |title=We Aren't The World |last=Watters |first=Ethan |date=June 14, 2017 |access-date=July 1, 2022}}

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,{{cite journal | last1 = Leliveld | first1 = Marijke C. | last2 = van Dijk | first2 = Eric | last3 = van Beest | first3 = Ilja | title = Initial Ownership in Bargaining: Introducing the Giving, Splitting, and Taking Ultimatum Bargaining Game | journal = Personality and Social Psychology Bulletin | date = September 2008 | volume = 34 | issue = 9 | pages = 1214–1225 | issn = 0146-1672 | eissn = 1552-7433 | doi = 10.1177/0146167208318600 | pmid = 18587058 | s2cid = 206443401 | url = }} or characterizing the game as a windfall game versus a routine transaction game.{{cite journal | last1 = Lightner | first1 = Aaron D. | last2 = Barclay | first2 = Pat | last3 = Hagen | first3 = Edward H. | title = Radical framing effects in the ultimatum game: the impact of explicit culturally transmitted frames on economic decision-making | journal = Royal Society Open Science | date = December 2017 | volume = 4 | issue = 12 | page = 170543 | eissn = 2054-5703 | doi = 10.1098/rsos.170543 | pmid = 29308218| pmc = 5749986 | bibcode = 2017RSOS....470543L | url = }}

The highly mixed results, along with similar results in the dictator game, have been taken as both evidence for and against the 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 their economic gain, unless one incorporates economic applications of social, psychological, and methodological factors (such as the observer effect).{{citation needed|date=January 2013}} Several attempts have been made to explain this behavior. Some suggest that individuals are maximizing their expected utility, but money does not translate directly into expected utility.{{cite journal|author = Bolton, G.E. |year =1991|title = A comparative Model of Bargaining: Theory and Evidence|journal = American Economic Review|volume = 81|pages = 1096–1136}}{{cite journal|author = Ochs, J. and Roth, A. E.|year =1989| title = An Experimental Study of Sequential Bargaining| journal = American Economic Review|volume = 79|pages = 355–384}} Perhaps individuals get some psychological benefit from engaging in punishment or receive some psychological harm from accepting a low offer.{{citation needed|date=January 2013}} 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 them to be fair.{{Cite journal|last1=Eriksson|first1=Kimmo|last2=Strimling|first2=Pontus|last3=Andersson|first3=Per A.|last4=Lindholm|first4=Torun|date=2017-03-01|title=Costly punishment in the ultimatum game evokes moral concern, in particular when framed as payoff reduction|url=http://www.sciencedirect.com/science/article/pii/S0022103116302098|journal=Journal of Experimental Social Psychology|language=en|volume=69|pages=59–64|doi=10.1016/j.jesp.2016.09.004|s2cid=151549755 |issn=0022-1031}}

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.{{Cite journal|last=Nowak|first=M. A.|date=2000-09-08|title=Fairness Versus Reason in the Ultimatum Game|url=https://www.science.org/doi/10.1126/science.289.5485.1773|journal=Science|volume=289|issue=5485|pages=1773–1775|doi=10.1126/science.289.5485.1773|pmid=10976075|bibcode=2000Sci...289.1773N|s2cid=6342076 }} Behavioral economic and psychological accounts suggest that second players who reject offers less than 50% of the amount at stake do so for one of two reasons. An altruistic punishment account suggests that rejections occur out of altruism: people reject unfair offers to teach the first player a lesson and thereby reduce the likelihood that the player will make an unfair offer in the future. Thus, rejections are made to benefit the second player in the future, or other people in the future. By contrast, a self-control account suggests that rejections constitute a failure to inhibit a desire to punish the first player for making an unfair offer. Morewedge, Krishnamurti, and Ariely (2014) found that intoxicated participants were more likely to reject unfair offers than sober participants.{{Cite journal|title = Focused on fairness: Alcohol intoxication increases the costly rejection of inequitable rewards|journal = Journal of Experimental Social Psychology|date = 2014-01-01|pages = 15–20|volume = 50|doi = 10.1016/j.jesp.2013.08.006|first1 = Carey K.|last1 = Morewedge|first2 = Tamar|last2 = Krishnamurti|first3 = Dan|last3 = Ariely}} As intoxication tends to exacerbate decision makers' prepotent response, this result provides support for the self-control account, rather than the altruistic punishment account. Other research from social cognitive neuroscience supports this finding.{{Cite journal|title = The Sunny Side of Fairness Preference for Fairness Activates Reward Circuitry (and Disregarding Unfairness Activates Self-Control Circuitry)|journal = Psychological Science|date = 2008-04-01|issn = 0956-7976|pmid = 18399886|pages = 339–347|volume = 19|issue = 4|doi = 10.1111/j.1467-9280.2008.02091.x|first1 = Golnaz|last1 = Tabibnia|first2 = Ajay B.|last2 = Satpute|first3 = Matthew D.|last3 = Lieberman|s2cid = 15454802}}

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.{{Cite journal|last=Gil-White|first=F.J.|date=2004|title=Ultimatum Game with an Ethnicity Manipulation|pages=260–304}}[http://www.psych.upenn.edu/~fjgil/Ultimatum.pdf Mongolian/Kazakh study conclusion] from University of Pennsylvania. Others have proposed the social status of the responder may be part of the payoff.{{Cite journal|last1=Harris|first1=Alison|last2=Young|first2=Aleena|last3=Hughson|first3=Livia|last4=Green|first4=Danielle|last5=Doan|first5=Stacey N.|last6=Hughson|first6=Eric|last7=Reed|first7=Catherine L.|date=2020-01-09|title=Perceived relative social status and cognitive load influence acceptance of unfair offers in the Ultimatum Game|journal=PLOS ONE|language=en|volume=15|issue=1|pages=e0227717|doi=10.1371/journal.pone.0227717|issn=1932-6203|pmc=6952087|pmid=31917806|bibcode=2020PLoSO..1527717H|doi-access=free}}[http://radoff.com/blog/2010/05/18/social-role-ultimatum-game/ Social Role in the Ultimate Game] 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.{{Cite journal|last1=Bellemare|first1=Charles|last2=Kröger|first2=Sabine|last3=Soest|first3=Arthur Van|date=2008|title=Measuring Inequity Aversion in a Heterogeneous Population Using Experimental Decisions and Subjective Probabilities|url=https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1468-0262.2008.00860.x|journal=Econometrica|language=en|volume=76|issue=4|pages=815–839|doi=10.1111/j.1468-0262.2008.00860.x|s2cid=55369709 |issn=1468-0262}}

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 Nash equilibrium (almost zero) as they mastered the strategy of the game; this decay tends to be seen in other iterated games.{{citation needed|date=January 2013}} However, this explanation (bounded rationality) is less commonly offered now, in light of subsequent empirical evidence.{{cite journal |last1=Brenner |first1=Thomas |last2=Vriend |first2=Nicholas J. |date=2006 |title=On the behavior of proposers in ultimatum games |journal=Journal of Economic Behavior & Organization |volume=61 |issue=4 |pages=617–631 |doi=10.1016/j.jebo.2004.07.014 |citeseerx=10.1.1.322.733 }}

It has been hypothesized (e.g. by James Surowiecki) that very unequal allocations are rejected only because the absolute amount of the offer is low.{{Cite book|last=Surowieki|first=James|title=The Wisdom of Crowds|publisher=Anchor|year=2005}} The concept here is that if the amount to be split were 10 million dollars, a 9:1 split would probably be accepted rather than rejecting a 1 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 higher stakes cause offers to approach closer to an even split, even in a US$100 game played in Indonesia, where average per-capita income is much lower than in the United States. Rejections are reportedly independent of the stakes at this level, with US$30 offers being turned down in Indonesia, as in the United States, even though this equates to two weeks' wages in Indonesia. However, 2011 research with stakes of up to 40 weeks' wages in India showed that "as stakes increase, rejection rates approach zero".{{Cite journal|title = Stakes Matter in Ultimatum Games|journal = American Economic Review|date = 2011|issn = 0002-8282|pages = 3427–3439|volume = 101|issue = 7|doi = 10.1257/aer.101.7.3427|first1 = Steffen|last1 = Andersen|first2 = Seda|last2 = Ertaç|first3 = Uri|last3 = Gneezy|first4 = Moshe|last4 = Hoffman|first5 = John A|last5 = List|citeseerx = 10.1.1.222.5059}} It is worth noting that the instructions offered to proposers in this study explicitly state, "if the responder's goal is to earn as much money as possible from the experiment, they should accept any offer that gives them positive earnings, no matter how low," thus framing the game in purely monetary terms.

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.{{cite journal| author = Zak, P.J., Stanton, A.A., Ahmadi, S. | url = http://www.neuroeconomicstudies.org/pdf/ZakGenerosity.pdf |year=2007| title= Oxytocin Increases Generosity in Humans| journal= PLOS ONE | volume =2| pages= e1128 | issue=11| pmid = 17987115| pmc = 2040517| doi = 10.1371/journal.pone.0001128| editor1-last = Brosnan| editor1-first = Sarah| bibcode = 2007PLoSO...2.1128Z | doi-access = free }}{{Cite journal|last1=Zak|first1=Paul J.|last2=Stanton|first2=Angela A.|last3=Ahmadi|first3=Sheila|date=2007-11-07|title=Oxytocin Increases Generosity in Humans|journal=PLOS ONE|language=en|volume=2|issue=11|pages=e1128|doi=10.1371/journal.pone.0001128|issn=1932-6203|pmc=2040517|pmid=17987115|bibcode=2007PLoSO...2.1128Z|doi-access=free}} 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.Sanfey, et al. (2002) 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.{{cite journal|author = Kirk|year=2011|title=Interoception Drives Increased Rational Decision-Making in Meditators Playing the Ultimatum Game|journal=Frontiers in Neuroscience|volume=5|pmc=3082218|pmid=21559066|doi=10.3389/fnins.2011.00049|pages=49|display-authors=etal|doi-access=free}}

People whose serotonin levels have been artificially lowered will reject unfair offers more often than players with normal serotonin levels.{{Cite journal

| doi = 10.1126/science.1155577

| pages = 1155577

| last = Crockett

| first = Molly J.

|author2= Luke Clark |author3=Golnaz Tabibnia | author4-link = Matthew Lieberman

|author4=Matthew D. Lieberman |author5=Trevor W. Robbins

| title = Serotonin Modulates Behavioral Reactions to Unfairness

| journal = Science

| date = 2008-06-05

| volume = 320

| pmid = 18535210

| issue = 5884

| pmc = 2504725

| bibcode = 2008Sci...320.1739C

}}

People who have ventromedial frontal cortex lesions were found to be more likely to reject unfair offers.{{cite journal|last=Koenigs|first=Michael|author2=Daniel Tranel |title=Irrational Economic Decision-Making after Ventromedial Prefrontal Damage: Evidence from the Ultimatum Game|journal=Journal of Neuroscience|date=January 2007|volume=27|issue=4|pages=951–956|doi=10.1523/JNEUROSCI.4606-06.2007|pmid=17251437|pmc=2490711}} 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.{{cite journal|last=Moretti|first=Laura|author2=Davide Dragone |author3=Giuseppe di Pellegrino |title=Reward and Social Valuation Deficits following Ventromedial Prefrontal Damage|journal=Journal of Cognitive Neuroscience|year=2009|volume=21|issue=1|pages=128–140|doi=10.1162/jocn.2009.21011|pmid=18476758|s2cid=42852077|url=https://zenodo.org/record/1068237}}

Other authors have used evolutionary game theory to explain behavior in the ultimatum game.{{cite journal|author = Gale, J., Binmore, K.G., and Samuelson, L.|year = 1995| title = Learning to be Imperfect: The Ultimatum Game|journal = Games and Economic Behavior|volume = 8|pages = 56–90|doi = 10.1016/S0899-8256(05)80017-X}}{{cite book|author1=Güth, W. |author2=Yaari, M. |year = 1992|chapter = An Evolutionary Approach to Explain Reciprocal Behavior in a Simple Strategic Game| editor = U. Witt| title = Explaining Process and Change – Approaches to Evolutionary Economics|publisher = Ann Arbor|pages = 23–34}}{{cite journal|author1=Huck, S. |author2=Oechssler, J. |s2cid=18398039 |year =1999| title = The Indirect Evolutionary Approach to Explaining Fair Allocations| journal = Games and Economic Behavior|volume = 28|pages = 13–24|doi = 10.1006/game.1998.0691}}{{cite book|author = Skyrms, B.|year = 1996|title = Evolution of the Social Contract|publisher =Cambridge University Press}} Simple evolutionary models, e.g. the replicator dynamics, cannot account for the evolution of fair proposals or for rejections.{{cite journal |last1=Forber |first1=Patrick |last2=Smead |first2=Rory |title=The evolution of fairness through spite |journal=Proceedings of the Royal Society B: Biological Sciences |date=7 April 2014 |volume=281 |issue=1780 |pages=20132439 |doi=10.1098/rspb.2013.2439 |pmid=24523265 |pmc=4027385 |s2cid=11410542 }} These authors have attempted to provide increasingly complex models to explain fair behavior.

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.

An early description of the ultimatum game is by Nobel laureate John Harsanyi in 1961, who footnotes Thomas Schelling's 1960 book, The Strategy of Conflict on its solution by dominance methods. Harsanyi says,{{cite journal |last1=Harsanyi |first1=John C. |title=On the Rationality Postulates underlying the Theory of Cooperative Games |journal=The Journal of Conflict Resolution |date=1961 |volume=5 |issue=2 |pages=367–196 |doi=10.1177/002200276100500205 |s2cid=220642229 |url=https://journals-sagepub-com.proxyiub.uits.iu.edu/doi/pdf/10.1177/002200276100500205}}

: "An important application of this principle is to ultimatum games, i.e., to bargaining games where one of the players can firmly commit himself in advance under a heavy penalty that he will insist under all conditions upon a certain specified demand (which is called his ultimatum).... Consequently, it will be rational for the first player to commit himself to his maximum demand, i.e., to the most extreme admissible demand he can make."

Josh Clark attributes modern interest in the game to Ariel Rubinstein,{{cite web |last1=Rubinstein |first1=Ariel |title=What's the ultimatum game? |date=1961 |website=HowStuffWorks |url= https://money.howstuffworks.com/ultimatum-game.htm}} but the best-known article is the 1982 experimental analysis of Güth, Schmittberger, and Schwarze.{{cite journal| year =1982|author = Güth, W., Schmittberger, and Schwarze|title = An Experimental Analysis of Ultimatum Bargaining|journal = Journal of Economic Behavior and Organization|volume= 3|issue =4|pages = 367–388| doi =10.1016/0167-2681(82)90011-7|url = http://www.dklevine.com/archive/refs4291.pdf}}, page 367: the description of the game at [http://neuroeconomics.typepad.com/neuroeconomics/2003/09/what_is_the_ult.html Neuroeconomics] cites this as the earliest example. Results from testing the ultimatum game challenged the traditional economic principle that consumers are rational and utility-maximising.{{Cite journal|last1=van Damme|first1=Eric|last2=Binmore|first2=Kenneth G.|last3=Roth|first3=Alvin E.|last4=Samuelson|first4=Larry|last5=Winter|first5=Eyal|last6=Bolton|first6=Gary E.|last7=Ockenfels|first7=Axel|last8=Dufwenberg|first8=Martin|last9=Kirchsteiger|first9=Georg|last10=Gneezy|first10=Uri|last11=Kocher|first11=Martin G.|date=2014-12-01|title=How Werner Güth's ultimatum game shaped our understanding of social behavior|url=http://www.sciencedirect.com/science/article/pii/S0167268114002741|journal=Journal of Economic Behavior & Organization|language=en|volume=108|pages=292–293|doi=10.1016/j.jebo.2014.10.014|issn=0167-2681}} This started a variety of research into the psychology of humans.{{Cite journal|last1=van Damme|first1=Eric|last2=Binmore|first2=Kenneth G.|last3=Roth|first3=Alvin E.|last4=Samuelson|first4=Larry|last5=Winter|first5=Eyal|last6=Bolton|first6=Gary E.|last7=Ockenfels|first7=Axel|last8=Dufwenberg|first8=Martin|last9=Kirchsteiger|first9=Georg|last10=Gneezy|first10=Uri|last11=Kocher|first11=Martin G.|date=2014-12-01|title=How Werner Güth's ultimatum game shaped our understanding of social behavior|url=http://www.sciencedirect.com/science/article/pii/S0167268114002741|journal=Journal of Economic Behavior & Organization|language=en|volume=108|pages=310–313|doi=10.1016/j.jebo.2014.10.014|issn=0167-2681}} Since the ultimatum game's development, it has 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.{{Cite journal | last1 = Nowak | first1 = M. A. | last2 = Page | first2 = K. M. | last3 = Sigmund | first3 = K. | title = Fairness Versus Reason in the Ultimatum Game | journal = Science | volume = 289 | issue = 5485 | pages = 1773–1775 | year = 2000 | pmid = 10976075 | doi = 10.1126/science.289.5485.1773| url = http://pure.iiasa.ac.at/6189/1/IR-00-057.pdf | bibcode = 2000Sci...289.1773N | s2cid = 6342076 }}

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[http://homepage.univie.ac.at/christoph.hauert/gamelab/ultiproposer.html Ultimatum game with proposer competition] by the [http://homepage.univie.ac.at/christoph.hauert/gamelab/ GameLab]. (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.{{cite journal|author = Ruffle, B.J.|s2cid = 9091550|year = 1998|title = More is Better, but Fair is Fair: Tipping in Dictator and Ultimatum Games|journal = Games and Economic Behavior|volume = 23|issue =2 |pages = 247–265|doi = 10.1006/game.1997.0630}}, p. 247.

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.The reverse ultimatum game and the effect of deadlines is from {{cite journal|year=2003 |author=Uri Gneezy, Ernan Haruvy, and Roth, A. E. |url=http://gsbwww.uchicago.edu/fac/uri.gneezy/vita/deadline.pdf |title=Bargaining under a deadline: evidence from the reverse ultimatum game |journal=Games and Economic Behavior |volume=45 |pages=347–368 |doi=10.1016/S0899-8256(03)00151-9 |issue=2 |url-status=dead |archive-url=https://web.archive.org/web/20040731061444/http://gsbwww.uchicago.edu/fac/uri.gneezy/vita/deadline.pdf |archive-date=July 31, 2004 }}

Incomplete information ultimatum games: Some authors have studied variants of the ultimatum game in which either the proposer or the responder has private information about the size of the pie to be divided.{{Cite journal|title = Experimental results on ultimatum games with incomplete information|journal = International Journal of Game Theory|date = 1993|issn = 0020-7276|pages = 171–198|volume = 22|issue = 2|doi = 10.1007/BF01243649|first1 = Michael|last1 = Mitzkewitz|first2 = Rosemarie|last2 = Nagel|author2-link=Rosemarie Nagel|s2cid = 120136865}}{{Cite journal|title = Contracting under Incomplete Information and Social Preferences: An Experimental Study|url = http://restud.oxfordjournals.org/content/80/4/1516|journal = The Review of Economic Studies|date = 2013|issn = 0034-6527|pages = 1516–1544|volume = 80|issue = 4|doi = 10.1093/restud/rdt010|first1 = Eva I.|last1 = Hoppe|first2 = Patrick W.|last2 = Schmitz}} These experiments connect the ultimatum game to principal-agent problems studied in contract theory.

The pirate game illustrates a variant with more than two participants with voting power, as illustrated in Ian Stewart's "A Puzzle for Pirates".{{cite journal | last = Stewart | first = Ian | title = A Puzzle for Pirates | journal = Scientific American | date = May 1999 | volume = 280 | issue = 5 | pages = 98–99 | url = http://euclid.trentu.ca/math/bz/pirates_gold.pdf | bibcode = 1999SciAm.280e..98S | doi = 10.1038/scientificamerican0599-98 | url-status = dead | archive-url = https://web.archive.org/web/20110927094648/http://euclid.trentu.ca/math/bz/pirates_gold.pdf | archive-date = 2011-09-27 }}

{{Div col|colwidth=20em}}

• Dictator game
• Fair division experiments
• Gift-exchange game
• Impunity game
• Neuroeconomics
• Public goods game
• Prisoner's dilemma
• Social preferences

{{Div col end}}

{{Reflist}}

• {{cite journal | url=http://mpra.ub.uni-muenchen.de/2369/ | title=Evolving Economics: Synthesis | author=Stanton, Angela | year=2006}}
• {{cite book |first=Ken |last=Binmore |author-link=Ken Binmore |chapter=The Ultimatum Game |pages=103–117 |title=Does Game Theory Work? |publisher=MIT Press |location=Cambridge |year=2007 |isbn=978-0-262-02607-9 }}
• {{cite book|author = Alvard, M.|year = 2004|chapter = The Ultimatum Game, Fairness, and Cooperation among Big Game Hunters|editor =Henrich, J. |editor2=Boyd, R. |editor3=Bowles, S. |editor4=Gintis, H. |editor5=Fehr, E. |editor6=Camerer, C.| title = Foundations of Human Sociality: Ethnography and Experiments in 15 small-scale societies|pages = 413–435|publisher = Oxford University Press|chapter-url = http://anthropology.tamu.edu/faculty/alvard/downloads/ultimatum.pdf}}
• {{cite SSRN |author=Bearden, J. Neil|year = 2001|title=Ultimatum Bargaining Experiments: The State of the Art |ssrn=626183}}
• 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

• [https://www.youtube.com/watch?v=xpkxLKV_3d0 Video lecture on the ultimatum game]

{{Game theory}}

{{Authority control}}

Category:Non-cooperative games

Category:Moral psychology

Category:Social science experiments

Category:Ultimata