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 her demand and the second gets the rest.

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 her 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 him any amount whatsoever. If the first knows this she will give the second the smallest possible.

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.

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 unjust distributions of the reward from "cooperative" ventures results in inequality that is, measurably, exponential across the strata of management within large corporations.

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