Ultimatum game
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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.
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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.
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 possibleTemplate:Ref.
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.
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.Template:Ref Another way of integrating the conclusion with utility maximization is some form of Inequity aversion model (preference for fairness).
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.Template:Ref
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 IndonesiaTemplate:Ref.
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.
Sociological applications
The split dollar game is important from a sociological perspective, because it illustrates the human willingness to accept injustice and social inequality.
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) 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.” Template:Ref
History
The first Ultimatum game was developed is 1982 as a stylized representation of negotation, by Güth, Werner, Schmittberger, and SchwarzeTemplate:Ref. 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"Template:Ref.
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 endowmentTemplate:Ref (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.Template:Ref
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.Template:Ref
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.
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
Template:Note 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 (http://www.ssc.wisc.edu/~andreoni/WorkingPapers/anblan.pdf) 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.
Template:Note See "Do higher stakes lead to more equilibrium play?" (page 18) in 3. Bargaining experiments (http://www.iza.org/teaching/falk_WS2003/falk_l3_bargaining.pdf), Professor Armin Falk's summary at the Instute for the Study of Labor (http://www.iza.org/index_html?mainframe=http%3A//www.iza.org/teaching/falk_WS2003).
Template:Note Mongolian/Kazakh study conclusion (http://www.psych.upenn.edu/~fjgil/Ultimatum.pdf) from University of Pennsylvania.
Template:Note A forthcoming paper “On the Behavior of Proposers in Ultimatum Games” Journal of economic behaviour and organization (http://www.elsevier.com/wps/find/journaldescription.cws_home/505559/description) has the thesis that learning will not cause NE-convergence: see the abstract (http://www.qmw.ac.uk/~ugte173/abs/abs.jebo2.html).
Template:Note See The Ultimatum game detailed description (http://www.fte.org/capitalism/activities/ultimatum/) as a class room plan from EconomicsTeaching.org. (This is a more thorough explanation of the practicalities of the game than is possible here.)
Template:Note 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 (http://neuroeconomics.typepad.com/neuroeconomics/2003/09/what_is_the_ult.html) cites this as the earliest example.)
Template:Note See the introduction of Fairness Versus Reason in the Ultimatum Game (http://www.ped.fas.harvard.edu/pdf_files_old/Science00.pdf) Martin A. Nowak, Karen M. Page, Karl Sigmund
Template:Note Ultimatum game with proposer competition (http://homepage.univie.ac.at/christoph.hauert/gamelab/ultiproposer.html) by the GameLab (http://homepage.univie.ac.at/christoph.hauert/gamelab/).
Template:Note 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.
Template:Note 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 (http://gsbwww.uchicago.edu/fac/uri.gneezy/vita/deadline.pdf) From Games and Economic Behavior 45.
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