4. EXTENSIVE-FORM GAMES

The Ultimatum Game

The proposer makes a single offer to a responder about how to split some amount of money (e.g., $10 in $1 units). The responder must either i) accept the proposed split or ii) reject the proposal, which results in zero earnings for both parties.

Güth, W., Schmittberger, and Schwarze (1982). "An Experimental Analysis of Ultimatum Bargaining". Journal of Economic Behavior and Organization: 367–388.

Ultimatum game: Standard Prediction

Proposer

0 10

x

Responder

10-x, x 0, 0

Accept Reject

Goal: solve for proposer’s equilibrium offer x

Start at the bottomWhat will Responder accept?

Offer of 1, 2, … 10, but not 0

Ultimatum game: Standard Prediction

Proposer

0 10

1

Responder

9, 1 0, 0

Accept Reject

If proposer offers 1 will responder accept?

Yes, 1 is better than zero

Does proposer have an incentive to deviate from this strategy? Could he do better by offering 2?

No, while responder earnings rise to 2, the proposer does not need to make this offer to induce acceptance and with earnings of 8 the proposer is worse off.

Ultimatum game: Standard Prediction

Proposer

0 10

1

Responder

9, 1 0, 0

Accept Reject

X = 1 is a Nash equilibrium.

It is not fair to the responder, but shows the advantage of being the one who calls the equilibrium.

Prediction of Standard Economic Theory

A “rational” proposer would offer $1 and keep $9 for him/herself.

A “rational” responder would accept $1, reasoning that $1 is better than nothing, and not crash the deal.

Findings to the contrary suggest that people do no have purely self-regarding preferences

Class Experiment

I will pay two students, chosen at random, 1/0 of their cumulative earnings after 12 rounds of Treatments 1 and 2 run in Veconlab.

Treatment #1 – Dictator Game run by hand. $10 (10 x $1) given to a student chosen at random who is then invited to share some with neighbor.

Treatment #2 – Dictator Game run in Veconlab .

Treatment #3 – Ultimatum Game run in Veconlab .

Standard Empirical Findings

Results contradict predictions of standard game theory.

In Industrial cultures, irrespective of monetary sum, modal offers are around 50% of total amount.

Low offers (20%) have about a 50% chance of being rejected.

First Explanation – Low Stakes

Unequal allocations are rejected only because the absolute amount of the offer is low. If the amount to be split were ten million dollars a 90:10, the split would probably be accepted rather than spurning a million dollar offer.

Cameron and Hoffman et al. (1994) – the higher 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 at this level, with 30 USD offers being turned down in Indonesia, as in the US, even though this equates to two week's wages in Indonesia.

Second Explanation

Bolton (1991) - Utility includes social comparison

U(xR, xR/xP)

xR = absolute earnings received by responderxR/xP = ratio of responder’s earning to proposer’s earning (1 if both get zero)

Example: Responder rejects $2 out of $10 offer

U($2, 0.25) < U($0, 1)

+ +

Problem with Second Explanation?

Social comparison theory does not distinguish between: a) distaste for unequal allocations b) willingness to punish someone who has behaved “unfairly”

Blount (1996): Subjects are more likely to accept small (uneven) offers it they come from a random device than from a person

Conclusion: People are punishing unfairness, not rejecting inequality

Sanfey, et al. Science (2003).

Third Explanation: Intentions Matter

Mathew Rabin (1993)

Intentional acts of meanness

Punish

Unintentional acts of meanness

Tolerate

Fairness equilibrium - both parties will sacrifice to reward (punish) other player’s cooperative (uncooperative) act

Push-Pull Game: Prisoner’s DilemmaColumn Player

Row

Player

Cooperate Defect

Defect

3

5 2

0

3 5

0 2

Cooperate

Nash Equilibrium

Fairness Equilibrium (both parties punish uncooperative behavior of others)

Fairness Equilibrium (both parties willing to sacrifice to reward other player’s cooperative act)

Implications

If intentions matter we can explain…

1) Difference in results from ultimatum game when played with computer

2) Simultaneous,…

Positive altruism (helping friends with gifts or trust)

Negative altruism (punishing enemies at a cost to oneself)

Camerer and Thaler (1996) Rabin has it right - Manners Matter

Findings: People share in dictator games, but sharing shrinks when a) the relationship with the other player is made less personal or 2) when proposer “earned” the right to the $10.

Conclusion: “Manors” require you to share a windfall with a friend, do not require that you give up a hard-earned bonus to a stranger.”

Findings: Responders reject low (but greater than $1) offers.

Conclusion: Responders in ultimatum game are willing to turn down rude offers at a cost to themselves.

More Evidence for Rabin’s “Manners Matter”

Finding: When Responders must compete with one another to take an offer from single Proposer, minimal acceptable offers fell to 10% (Roth et al. 1991).

Conclusion: Individual Responders don’t have the ability to single-handedly punish unfair offers. Since the Proposer is not suggesting an offer in these games – he is taking the best offer from competing Responders – the prospect of unfair offers is removed. These findings are consistent with idea that manners about fairness norms matter.

Is a sense of fairness learned?

Murnighan and Saxon (1994)

Kindergartners accept minimal offers (e.g., one M&M out of a pile) about 70% of the time

3rd and 6th graders accept minimal offers about 40% of the time

learned behavior

Is a sense of fairness genetic?

Wallace, Cesarini, Lichtenstein and Johannesson, “Heritability of Ultimatum Game Responder Behavior” Proceedings of the National Academy of Sciences, October 2007.

Abstract: Experimental evidence suggests that many people are willing to deviate from materially maximizing strategies to punish unfair behavior. Even though little is known about the origins of such fairness preferences, it has been suggested that they have deep evolutionary roots and that they are crucial for maintaining and understanding cooperation among non-kin. Here we report the results of an ultimatum game, played for real monetary stakes, using twins recruited from the population-based Swedish Twin Registry as our subject pool. Employing standard structural equation modeling techniques, we estimate that >40% of the variation in subjects' rejection behavior is explained by additive genetic effects. Our estimates also suggest a very modest role for common environment as a source of phenotypic variation. Based on these findings, we argue that any attempt to explain observed ultimatum bargaining game behavior that ignores this genetic influence is incomplete.

Fig. 2. Scatter plot of ultimatum game acceptance thresholds for twin pairs. (A) Scatter plot for MZ twin pairs. The acceptance thresholds are highly correlated. (B) Scatter plot for DZ twin pairs. There was no significant correlation in acceptance thresholds.

Identical Twins (share all genes)

Fraternal Twins (share 50% of genes)

Sanfey, et al. Science (2000)

A fundamental adaptive mechanism by which we assert and maintain a social reputation. Unfair treatment causes people to sacrifice monetary gain to punish partners for slight.

What is the evolutionary explanation for this behavior?

Sanfey, et al. Science (2003)

Subjects whose brains were scanned by MRI while receiving an unfair offer in an ultimatum game ($1 or $2 out of $10 available) showed greater activity in the bilateral anterior insula of the brain.

Overall, work emphasizes importance of emotional influences on human decision making.

The anterior cingulate (ACC), a region of the brain that detects cognitive conflict, also showed greater activity during presentation of unfair offers. This area mediates conflict between earning money and feeling bad.

Neuroeconomics

What about Chimpanzees?

Abstract: Traditional models of economic decision-making assume that people are self-interested rational maximizers. Empirical research has demonstrated, however, that people will take into account the interests of others and are sensitive to norms of cooperation and fairness. In one of the most robust tests of this finding, the ultimatum game, individuals will reject a proposed division of a monetary windfall, at a cost to themselves, if they perceive it as unfair. Here we show that in an ultimatum game, humans' closest living relatives, chimpanzees (Pan troglodytes), are rational maximizers and are not sensitive to fairness. These results support the hypothesis that other-regarding preferences and aversion to inequitable outcomes, which play key roles in human social organization, distinguish us from our closest living relatives.

Jensen, Call and Tomasello, “Chimpanzees Are Rational Maximizers in an Ultimatum Game,” Science Magazine, 2007.

Fig. 1. Illustration of the testing environment. The proposer, who makes the first choice, sits to the responder's left. The apparatus, which has two sliding trays connected by a single rope, is outside of the cages. (A) By first sliding a Plexiglas panel (not shown) to access one rope end and by then pulling it, the proposer draws one of the baited trays halfway toward the two subjects. (B) The responder can then pull the attached rod, now within reach, to bring the proposed food tray to the cage mesh so that (C) both subjects can eat from their respective food dishes (clearly separated by a translucent divider)

Camerer and Thaler Conclusion

In repeated encounters, it is rational to treat others fairly and punish those who behave unfairly, because long-run concerns outweigh the short-run costs

How do we explain behavior in one-shot games (e.g., tips in restaurants)

People cannot curb their repeated –game impulse

5. IMPERFECT INFORMATION