Unit 4 Strategic interactions and social dilemmas

4.12 Fair farmers, self-interested students? Experimental results of the ultimatum game

Responders do sometimes reject offers of more than half of the pie. In experiments in Papua New Guinea, offers of more than half of the pie were commonly rejected by Responders who preferred to receive nothing than to participate in a very unequal outcome even if it was in the Responder’s favour, or to incur the social debt of having received a large gift that might be difficult to reciprocate. The Responders were inequality averse, even if the inequality in question benefited them.1

An experiment using the ultimatum game was run with a group of farmers in Kenya, and a group of students in the US. Proposers could offer 0, 10, 20, 30, 40, or 50% of the pie to Responders. Figure 4.17 illustrates the choices made by Responders. The height of each bar indicates the fraction who were willing to accept the offer indicated on the horizontal axis. For example, offers of 10% of the pie were accepted by 58% of the students who received them, but only 2% of the farmers. Offers of half of the pie were accepted by all Responders in both countries, as you would expect.

In this bar chart, the horizontal axis shows the fraction of the pie offered by the Proposer to the Responder, ranging from 0 to 50%, and the vertical axis shows the fraction of responders who would accept the offer, ranging from 0 to 100%. There are two sets of bars, one for Kenyan farmers and one for US students. For both groups, the fraction of responders who would accept the offer increases with the fraction of the pie offered by the proposer. When Proposers offer 0% of the pie, none of the Responders accept the offer, and when proposers offer 50% of the pie, all Responders accept the offer. For shares of 10%, less than 5% of farmers accept whereas around 60% of students accept. Similarly, for shares of 20%, around 10% of farmers accept whereas almost 70% of students accept. For shares of 30%, around 50% of farmers accept whereas around 80% of students accept. For shares of 40%, around 90% of farmers and students accept the offer.
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Figure 4.17 Offers accepted by Responders.

Adapted from Joseph Henrich, Richard McElreath, Abigail Barr, Jean Ensminger, Clark Barrett, Alexander Bolyanatz, Juan Camilo Cárdenas, Michael Gurven, Edwins Gwako, Natalie Henrich, Carolyn Lesorogol, Frank Marlowe, David Tracer, and John Ziker. 2006. ‘Costly Punishment Across Human Societies’. Science 312 (5781): pp. 1767–1770.

Remember that a self-interested Responder would accept any offer: that is always as good or better for themselves than rejection. So it is clear that many Responders were not motivated purely by self-interest. No one in either group accepted an offer of zero: they preferred to ensure that the Proposer got nothing as well. Kenyan farmers, in particular, were very unlikely to accept low offers, and almost half rejected offers of 30%. It appears that in both groups, Responders were influenced by considerations of inequality aversion, reciprocity, or social norms. However, the results suggest that attitudes to fairness, and willingness to lose money to punish unfair behaviour, differed between the groups.

Question 4.12 Choose the correct answer(s)

Consider an ultimatum game where the Proposer offers a proportion of $100 to the Responder, who can either accept or reject the offer. If the Responder accepts, both the Proposer and the Responder keep the agreed share, while if the Responder rejects, then both receive nothing. Figure 4.17 shows the results of a study that compares the responses of US university students and Kenyan farmers.

From this information, we can conclude the following:

  • In the general population, Kenyans are more likely to reject low offers than Americans.
  • Just over 50% of Kenyan farmers accepted the offer of the Proposer keeping 30%.
  • Both groups of Responders are indifferent between accepting and rejecting an offer of receiving nothing.
  • The Kenyan farmers in the study place higher importance on fairness than the US students.
  • The Kenyan farmers in the experiment are more likely to reject low offers than the US students. This does not imply that all Kenyans are more likely to reject low offers than all Americans.
  • The graph shows the share offered to the Responder, so just over 50% of Kenyan farmers accepted the offer of the Responder receiving 30%.
  • In both groups of Responders, 100% rejected the offer of receiving nothing.
  • The fact that Kenyan farmers were more likely to reject unfair offers, and hence forgo any income, indicates that they value fairness more.

Now let’s consider the behaviour of Proposers. The full height of each bar in Figure 4.18a indicates the fraction of the Kenyan and US Proposers who made the offer shown on the horizontal axis.

In this bar chart, the horizontal axis shows the fraction of the pie offered by the Proposer to the Responder, ranging from 0 to 50%, and the vertical axis shows the fraction of Proposers making the offer indicated, ranging from 0 to 60%. There are two sets of bars, one for Kenyan farmers and one for US students. 5% of farmers and 20% of students made offers of 0%. 10% of farmers and 10% of students made offers of 10%. 10% of farmers and 20% of students made offers of 20%. Nearly 15% of farmers and 35% of students made offers of 30%. 50% of farmers and 5% of students made offers of 40%, almost all of which were accepted. 10% of farmers and 5% of students make offers of 50%, all of which are accepted.
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Figure 4.18a Offers made by Proposers.

Adapted from Joseph Henrich, Richard McElreath, Abigail Barr, Jean Ensminger, Clark Barrett, Alexander Bolyanatz, Juan Camilo Cárdenas, Michael Gurven, Edwins Gwako, Natalie Henrich, Carolyn Lesorogol, Frank Marlowe, David Tracer, and John Ziker. 2006. ‘Costly Punishment Across Human Societies’. Science 312 (5781): pp. 1767–1770.

Figure 4.18a seems to suggest that the farmers made more generous offers. 60% of them offered 40% or more. Only 11% of the students made such generous offers.

But were the farmers really being generous? Their offers will depend on what they expect the Responders to do. From Figure 4.17, we know that 89% of farmers accepted an offer of 40% of the pie, but lower offers were much less likely to be accepted. If this is what the Proposer-farmers expected, then making a low offer is risky. Even if they were not at all altruistic, they may have decided to offer 40%.

Figure 4.18b again shows the offers made, but also in darker shading the proportion that were rejected (from the data in Figure 4.17). Proposers could use their knowledge of the preferences and norms of their own community to estimate the likelihood of Responders rejecting different offers. If they did this successfully, their expectations would be similar to actual rejection decisions.

In this bar chart, the horizontal axis shows the fraction of the pie offered by the Proposer to the Responder, ranging from 0 to 50%, and the vertical axis shows the fraction of Proposers making the offer indicated, ranging from 0 to 60%. There are two sets of bars, one for Kenyan farmers and one for US students. 5% of farmers and 20% of students made offers of 0%, all of which are rejected. 10% of farmers and 10% of students made offers of 10%. All the farmers’ offers were rejected, whereas slightly less than half of the students’ offers were rejected. 10% of farmers and 20% of students made offers of 20%. Almost all the farmers’ offers were rejected, whereas one-third of the students’ offers were rejected. Nearly 15% of farmers and 35% of students made offers of 30%. Almost half of the farmers’ offers were rejected, whereas only one-seventh of students’ offers were rejected. 50% of farmers and 5% of students made offers of 40%, almost all of which were accepted. 10% of farmers and 5% of students make offers of 50%, all of which are accepted.
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Figure 4.18b Actual offers, showing the proportion that Proposers could expect to be rejected.

Adapted from Joseph Henrich, Richard McElreath, Abigail Barr, Jean Ensminger, Clark Barrett, Alexander Bolyanatz, Juan Camilo Cárdenas, Michael Gurven, Edwins Gwako, Natalie Henrich, Carolyn Lesorogol, Frank Marlowe, David Tracer, and John Ziker. 2006. ‘Costly Punishment Across Human Societies’. Science 312 (5781): pp. 1767–1770.

Consider the Kenyan farmers’ offers first.

  • Very few offered zero (4% of them as shown in the far left-hand bar) and all of those offers would have been rejected (the entire bar is dark).
  • 10% offered half the pie (right-hand bar). This ensured an acceptance rate of 100% (the entire bar is light).
  • Offers of 30% were almost as likely to be rejected as accepted (the dark and light parts of the bar are almost the same size).

Now imagine you are a Kenyan farmer who only cares about your own pay-off. Offering the Responder nothing is out of the question because that will ensure that you get nothing when they reject your offer. Offering half will get you half for sure—because the Responder will surely accept.

But perhaps it would be better to offer less than half, risking rejection, but gaining more if you are lucky?

When pay-offs are uncertain, we can compare them by calculating the expected pay-off of each one. In this game, the expected pay-off from an offer is the pay-off you receive if it is accepted, multiplied by the probability of acceptance. It gives you a measure of what you would receive, on average, if you had the opportunity to make the offer many times.

In Figure 4.19, we have calculated the Proposers’ expected pay-offs from different offers, using the probabilities of acceptance by the farmers. (You can calculate the expected pay-offs from other possible offers yourself, using Figure 4.18b to find the probabilities.)

Offer Probability of acceptance Pay-off if accepted Expected pay-off
50% of pie 1 50% = 1 × 50 = 50%
40% of pie 0.96 60% = 0.96 × 60 = 58%
30% of pie 0.52 70% = 0.52 × 70 = 36%

Figure 4.19 The farmers’ expected pay-offs from some possible offers.

We cannot know if the Kenyan farmers actually made these calculations, of course. But if they did, they would have discovered that offering 40% maximized their expected pay-off. And Figure 4.18a shows that 40% is by far the most popular offer for them.

This means that the results of the experiment are consistent with the Proposers behaving self-interestedly, given what they expected of Responders. On the other hand, the Responders’ behaviour was not consistent with pure self-interest: fairness and reciprocity seemed to matter to them. But both Proposers and Responders were drawn randomly from the same population, and it is therefore likely that Proposers were influenced by similar motives. Offers of close to half the pie may have reflected aversion to inequality, rather than a desire simply to maximize their expected pay-off.

Similar calculations indicate that, among the students, the expected pay-off-maximizing offer was 30%, and this was the most common offer among them. The students’ lower offers could be because they correctly anticipated that lowball offers (even as low as 10%) would sometimes be accepted. They may have been trying to maximize their pay-offs and hoping that they could get away with making low offers.

How do the two populations differ? Although many of the farmers and the students offered the amount that would maximize their expected pay-offs, the similarity ends there. The Kenyan farmers were more likely to reject low offers. Is this a difference between Kenyans and Americans, or between farmers and students? Or is it related to local social norms, rather than nationality and occupation? These experiments do not resolve such questions. Before jumping to the conclusion that Kenyans are more averse to unfairness than Americans, note that the experiment was also run with rural Missourians in the US. They were even more likely to reject low offers than the Kenyan farmers; almost every Missourian Proposer offered half the pie.

Exercise 4.13 Social preferences in the ultimatum game

Consider the experiment described in Figure 4.17 and Figure 4.18.

  1. Explain how inequality aversion, reciprocity, and social norms could have motivated the Responders’ willingness to reject low offers, even though by doing so they would receive nothing at all.
  2. Which of the preferences that you have studied could help to explain the behaviour of Proposers in this experiment?
  3. The table below shows the percentage of the Responders who rejected the amount offered by the Proposers in the ultimatum game played by Kenyan farmers and US university students. Construct a table like Figure 4.19, but for the US students, and verify that 30% is the expected pay-off-maximizing offer.
Share of pie offered (%) 0 10 20 30 40 50
Proportion rejected Kenyan farmers 100% 100% 90% 48% 4% 0%
US students 100% 40% 35% 15% 10% 0%
  1. How do you think the Proposer and Responder would behave if you played this game with two different sets of players—your classmates and your family? Explain whether or not you expect the results to differ across these groups. If possible, play the game separately with your classmates and your family and comment on whether the results are consistent with your predictions.

Exercise 4.14 Strikes and the ultimatum game

A strike over pay or working conditions may be considered an example of an ultimatum game.

  1. To model a strike as an ultimatum game, who is the Proposer and who is the Responder?
  2. Draw a game tree to represent the situation between these two parties.
  3. Research a well-known strike or a recent strike in a country of your choice and explain how it satisfies the definition of an ultimatum game.
  4. In this section, you have been presented with experimental data on how people play the ultimatum game. How could you use this information to suggest what kind of situations might lead to a strike?

The effect of competition

Ultimatum game experiments with two players provide insight into how people may choose to share rent in economic interactions. But the outcome of a negotiation may be different if it is affected by competition. For example, a professor who wants to hire a research assistant could consider several applicants rather than just one.

Consider a different ultimatum game in which a Proposer offers a two-way split of $100 to two respondents, rather than one.

  • If only one Responder accepts, that Responder and the Proposer get the split, and the other Responder gets nothing.
  • If no one accepts, the three players get nothing.
  • If both Responders accept, one is chosen at random to receive the split.

If you were one of the Responders, what is the minimum offer you would accept? Would it be different, now that there are two Responders rather than one? Perhaps it depends on whether you think your competitor wants the reward very much, or is strongly motivated by fairness.

Figure 4.20 shows some laboratory evidence for a large group of participants playing multiple rounds. Proposers and Responders were randomly and anonymously matched in each round. The red bars show the fraction of Responders who rejected offers in the game with one Responder and the blue bars show the fraction of all Responders who rejected offers in the game with two Responders competing. The figure shows that individual Responders were less likely to reject offers when competing with another Responder.

In this bar chart, the horizontal axis shows the fraction of the pie offered by the Proposer when there is one Responder or when there are two Responders, ranging from 0 to 50%, and the vertical axis shows the fraction of offers rejected, ranging from 0 to 100%. There are two sets of bars, corresponding to the situation of one Responder or the situation of two Responders respectively. In both situations, the fraction of offers rejected decreases when the Proposer offers a larger share of the pie. For any given fraction offered by the Proposer, the fraction of offers rejected is lower when there are two Responders compared to when there is only one Responder.
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Figure 4.20 Proportion of Responders who rejected offers in the ultimatum game, according to offer size and the number of Responders.

Adapted from Figure 6 in Urs Fischbacher, Christina M. Fong, and Ernst Fehr. 2009. ‘Fairness, Errors and the Power of Competition’. Journal of Economic Behavior & Organization 72 (1): pp. 527–45.

The Responders’ behaviour when there is competition seems more similar to what we would expect of self-interested individuals concerned mostly about their own monetary pay-offs.

To explain this phenomenon to yourself, think about what happens when a Responder rejects a low offer. This means they get a zero pay-off, but the Proposer may still get a positive pay-off if the other Responder accepts. Rejecting no longer has the same impact on the Proposer. It is a less useful instrument for punishing a Proposer who is not following a social norm of fairness. This is particularly the case if Responders have different preferences or states of need: a Responder who cares about fairness cannot rely on the other Responder to reject low offers.

Consequently, even people with preferences for fairness will accept low offers to avoid the worst of both worlds. Of course, Proposers also know this, so they will make lower offers. And their position is further strengthened because they only need one of the Responders to accept. In this experiment, the average accepted offer was 43% of the pie in the one-Responder game, but only 26% with two Responders. As in the public good game, where adding an option to punish free-riders increased contributions, a change in the rules of the game can make a big difference to the outcome.

Extension 4.12 Calculating expected pay-offs

In the main part of this section we have calculated the expected pay-off from a range of different offers for the Proposer in the ultimatum game. Expected pay-offs are important in any economic model in which the outcome of a decision is uncertain. Here we look at another numerical example that illustrates a general method for calculating them.

We have discussed many examples of economic actors making decisions in which they choose the outcome that maximizes their pay-off—perhaps in terms of utility, or profit. We often model their choices under the assumption that they know what the consequences of each choice will be. But in some situations, choices have to be made between actions with uncertain outcomes. For example, when you buy a household appliance, such as a washing machine, you may be offered—at additional cost—a warranty to cover the cost of repairs for the next two years. Should you take it? You don’t know whether your appliance will break down in that time. Your decision will depend on your estimated probability of breakdown, and how much you expect repairs to cost. To decide, you could calculate and compare your expected pay-offs from the two options: buying the warranty, or not doing so.

We sometimes model strategic interactions, like the ones in this unit, as having known—certain—pay-offs: we assumed Anil and Bala knew the exact costs and benefits of the potential outcomes of their pest control choices. But pay-offs to individuals in strategic interactions may be uncertain, particularly if players do not know the preferences of their opponent. The ultimatum game in this section is an example. Proposers may be aware of how preferences vary across their community. But they do not know the preferences of the particular Responder they are matched with in the experiment, so they cannot predict their response with certainty. They therefore need to use what they know about the proportions of Responders who are likely to reject each of the possible offers, and compare the expected pay-offs.

The following simple example illustrates more about how expected pay-offs are calculated. Imagine that you receive a gift from a generous friend who is a keen gambler. The size of the gift is uncertain: it depends on the outcome of a coin toss. If you win, your friend will give you $20; otherwise you get nothing.

In this example, there is no choice to be made. We are just going to work out your expected pay-off from receiving this gift—in other words, its expected value to you.

How much money do you expect to receive? If it is a fair coin, you have a 50% chance of $20, and a 50% chance of nothing. Then we say that the expected pay-off is: \(0.5 \times 20 = \$10\). This is what you would get, on average, if you received many such gifts.

If the gift were changed to $20 if you win and otherwise $10, it is clearly worth more. To measure its value now, we calculate the average pay-off over the two possible outcomes:

\[\begin{align} \text{expected pay-off } &= 0.5 \times 20 + 0.5 \times 10 \\ &= \$15 \end{align}\]

In general, when pay-offs are uncertain, we calculate expected pay-offs by multiplying each possible pay-off by its probability, and adding them up.

  1. Joseph Henrich, Robert Boyd, Samuel Bowles, Colin Camerer, and Herbert Gintis (eds). 2004. Foundations of Human Sociality: Economic Experiments and Ethnographic Evidence from Fifteen Small-Scale Societies. Oxford: Oxford University Press.