5.5 The ultimatum game: Dividing a pie (or leaving it on the table)

One way to understand the different preferences people might have is to use an experiment like the ultimatum game.

ultimatum game: A game in which the first player proposes a division of a “pie” with the second player, who may either accept the offer, in which case each player gets the division proposed by the first person, or reject the offer, in which case both players receive nothing.

The ultimatum game is based on a thought experiment: If you and another person find a $100 bill on the street, splitting the money has potential benefits for both of you. But how you share the benefits, and the rules for doing so, are affected by your preferences, by who has power in the interaction, and by institutions. In this section we outline how the game works, and in the next section we explore how real people actually play the game.

The ultimatum game is a two-person, one-shot, take-it-or-leave-it game. It has been used around the world with students, farmers, warehouse workers, and hunter-gatherers.

Participants in the ultimatum game can get an amount of money, depending on how they and other participants play. Real money is at stake to ensure that their decisions reflect real life. The experiment is explained to the players, who are randomly matched in pairs. One is randomly assigned as the Proposer, and the other as the Responder. They do not know each other, but they know that all players were recruited in the same way. Players remain anonymous.

The point of the experiment is to observe how the players will decide to share an amount of money—say $100—which we call the “pie.”

The rules of the game

The Proposer is provisionally given $100.
The Proposer decides how much money, y, to offer to the Responder; y can be anything from $0 to $100.
The Responder chooses one of two actions: Accept or Reject the offer.
If the Responder Rejects the offer, both players get nothing.
If the Responder Accepts, then the Responder receives y and the Proposer gets $100 – y.

Suppose the Proposer offers $40 while keeping $60. This is a take-it-or-leave-it offer: The Responder either accepts $40 or gets nothing. If the Responder Accepts, then both players receive a rent (a slice of the pie) because their next-best alternative is to get nothing (not playing the game). But remember, a crucial rule of the game is that if the Responder Rejects, both get nothing.

economic rent: Economic rent is the difference between the net benefit (monetary or otherwise) that an individual receives from a chosen action, and the net benefit from the next-best alternative (or reservation option).

Tying the experiment to ideas we introduced in Chapter 1, any accepted positive offer leaves both players better off than no deal. In economics, the benefit someone receives above their next-best alternative is called economic rent. Here, the Responder’s outside option is zero—the game pays nothing if they reject—so any accepted offer is pure economic rent for the Responder. Similarly, whatever the Proposer keeps is their economic rent from the interaction. The total pie to be divided is, in this sense, the combined potential economic rent for both players.

Understanding this, the choice the Responder makes depends on their preferences: Do they care only about the payoffs they receive, or do they care about fairness and/or have other social preferences?

A Responder who cares only about their own payoffs

For example, when the Proposer offers $20 to a Responder who dislikes being treated unfairly (in their eyes), that Responder might play Reject and get $0 rather than get what they consider to be an unfair amount of $20. If the Proposer offers $50, which the Responder perceives as fair, then the Responder will Accept that offer.

Even if the Proposer does not care about fairness, if they know that they are playing with someone who cares about fairness, then they will not make a very low offer because it will be rejected. The Proposer needs to anticipate what a Responder who cares about fairness will accept.

Suppose the Proposer believes that a fair-minded Responder will reject an offer of $20 but accept a higher offer. Even if the Proposer cares only about their own payoffs, they will try to work out how much is just enough to get the Responder who cares about fairness to accept their offer. For some fair-minded Responders, any offer lower than $50 would be unacceptable, and they will reject it. They want an exact 50-50 split. Other Responders might think that, although it’s not ideal for them, a 60-40 split where the Proposer offers them $40 is acceptable. For yet other Responders, a 70-30 split may be considered far from ideal (they would make a 50-50 offer if they were the Proposer), but they consider it acceptable. The challenge for the Proposer is to figure out what kind of Responder they’re paired with and what they consider acceptable.

This illustration shows four plates, each holding a slice of pie representing a different offer in the ultimatum game, alongside the corresponding Gini coefficient. For an offer of $50, the plate shows half a pie remaining, corresponding to a Gini coefficient of 0. For an offer of $33, the plate shows about a third of a pie remaining, corresponding to a Gini coefficient of 0.34. For an offer of $8.50, the plate shows a small sliver of pie remaining, corresponding to a Gini coefficient of 0.83. For an offer of $0, the plate is empty except for crumbs, corresponding to a Gini coefficient of 1.

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Figure 5.6 Ultimatum game offers with corresponding slices of a pie for the same proportion of the offer and the corresponding Gini coefficients when the pie can be shared between two people. We assume that the Responder accepts the offer (even the offer of $0).

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As Figure 5.6 shows, the size of the slice of the pie that each player gets in the ultimatum game corresponds to a particular level of the Gini coefficient (explained in Section 5.2). With a 50-50 offer, there is no inequality, and the Gini coefficient equals zero. With smaller offers, the Gini coefficient increases. When someone cares about fairness, they may care about a measure of inequality like the Gini coefficient as a measurement of the fairness of outcomes in society.

Extension 5.5a Using a game tree to understand the ultimatum game

Simplified example

game tree: A game tree shows the players, the actions, the sequence of moves, and the payoffs resulting from each possible pathway through the game that results from the actions the players take.
interdependence principle, principle of interdependence: The outcomes people obtain in economic interactions depend on the actions that they and others take in response to each other and on what they believe about the future.

Suppose the Proposer has only two choices: a “fair offer” ($50) or an “unfair offer” ($20). We can represent this game using the game tree in Figure E5.7, which shows the payoffs at the bottom of the tree. Game trees allow us to see how people with interdependent outcomes make choices in sequential games. Work through the steps to see how the game tree is constructed.

This diagram is a game tree illustrating the two choices a proposer has- offering an 80-20 split and a 50-50 split. The responder can accept or reject both splits. If he accepts the 80-20 split, the proposer gets 80 and he gets 20. If he rejects the offer, neither of them get anything. If he accepts the 50-50 split, the proposer gets 50 and he gets 50. Again, If he rejects the offer, neither of them get anything.

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Figure E5.7 Game tree for a simplified ultimatum game. The Proposer’s payoff is listed before the comma in blue, and the Responder’s payoff is listed after the comma in red.

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The Proposer can offer to split (80,20) or (50,50)

The Proposer has an endowment of $100 given by the experimenter. In this game tree, we limit the Proposer’s offers to an (80,20) split where the Proposer keeps $80 and offers the Responder $20 and a (50,50) split where the Proposer keeps $50 and offers $50 to the Responder.

$The Responder’s choices when the Proposer offers a split of (80,20): This bar chart shows the choices of students at Emory University. The horizontal axis displays the fraction of the pie offered by the Proposer to the Responder, from 0% to 50%. The vertical axis displays the share of Responders who would accept the offer, from 0% to 100%. No students accepted an offer of 0%. Over 50% of students accepted an offer of 10%. All students accepted an offer of 50%, the highest amount they could be offered.$

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The Responder’s choices when the Proposer offers a split of (80,20)

When the Proposer offers a split of (80,20), the Responder gets $20 if they Accept the offer, and the Proposer keeps $80. If the Responder Rejects the offer, then both players get $0.

$The Responder’s choices when the Proposer offers a split of (50,50): This bar chart shows the choices of Kenyan farmers. The horizontal axis displays the fraction of the pie offered by the Proposer to the Responder, from 0% to 50%. The vertical axis displays the share of Responders who would accept the offer, from 0% to 100%. No farmers accepted an offer of 0%. Almost no farmers accepted an offer of 10%. All farmers accepted an offer of 50%, the highest amount they could be offered.$

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The Responder’s choices when the Proposer offers a split of (50,50)

When the Proposer offers a split of (50,50), the Responder gets $50 if they Accept the offer, and the Proposer keeps $50. If the Responder Rejects the offer, then both players get $0.

$The complete game tree: This bar chart compares the choices of students at Emory University and Kenyan farmers. The horizontal axis displays the fraction of the pie offered by the Proposer to the Responder, from 0% to 50%. The vertical axis displays the share of Responders who would accept the offer, from 0% to 100%. Students are more likely to accept lower offers than farmers are: over 50% of students accepted offers of 10% or 20%, while far fewer farmers accepted at those levels. The two groups converge only at offers of 40% and 50%, where acceptance is high or complete for both.$

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The complete game tree

Here we see all the actions of both players and the terminal nodes of the game with each set of payoffs. The way the players solve the game depends on the Responder’s preferences and the Proposer’s preferences and beliefs about the Responder.

This game is sequential, like the interaction between Bunker and the Worker. In Figure E5.7, the Proposer moves first and the Responder moves second.

the principle of doing the best you can: Doing the best you can means that, from the set of actions available to them, people will choose the action that they believe will result in the outcome that they value the most, taking into account what they believe the other player will do in response to their choice.

To predict the outcome and what people do when they do the best they can, first consider what the Responder does when they do the best they can. Put yourself in the place of the Responder. Would you accept a 50-50 split? Would you accept an 80-20 split (the Proposer offers you $20 and keeps $80)? Or would you reject an 80-20 split, sacrificing your payoff to punish the Proposer for offering less than $50? Now switch roles. If you were the Proposer, what would you offer the Responder?

The choice the Responder makes depends on their preferences: Do they care only about the payoffs they receive, or do they have social preferences?

A Responder who cares only about their own payoffs

The fallback option for the Responder is to play Reject and receive a payoff of 0. If a Responder cares only about their own payoff, then any payoff that is greater than zero is better than their fallback offer. When a Responder cares only about their own payoffs and is not concerned with equality or fairness, then they do the best they can by achieving any possible positive payoff. As noted earlier, we call a person who cares only about their own payoff a Homo economicus or “economic man.”

In terms of Figure E5.7, a Homo economicus will accept an offer of $20 when the Proposer offers $20 and accept an offer of $50 when the Proposer offers $50. Knowing that the Responder is a Homo economicus, the Proposer will choose to keep $80 and therefore make an offer of $20, which they know the Responder as a Homo economicus will accept.

Some people have social preferences, caring about inequality between players, reciprocity, and/or social norms. How will a person with these preferences play the ultimatum game? A person who has social preferences will do the best they can by comparing outcomes and choosing the outcome that they consider to be consistent with the goals they care about given their social preferences. Specifically, someone concerned about inequality may act to rectify that inequality by rejecting an offer that results in a very unequal outcome, thereby producing an equal outcome (both players get nothing). For example, when the Proposer offers $20 to a Responder who dislikes inequality, that Responder might play Reject and get $0 rather than get what they consider to be an unequal offer of $20. If the Proposer offers $50, then the Responder who cares about inequality will Accept that offer, which they perceive as fair.

If the Proposer knows that they are playing with someone who cares about inequality, then they will offer $50 knowing that a fair-minded person will reject the offer of $20 and both players will receive $0 if the Proposer makes that offer. Comparing the possible outcomes of $0 and $50, the Proposer makes an offer of $50.

Extension 5.5b The general ultimatum game and minimum acceptable offers

An outline of the general case

Return to the general case of the ultimatum game in which the Proposer can offer any amount between $0 and $100. If you were the Responder, what is the minimum amount you would be willing to accept? If you were the Proposer, what would you offer?

We expect the outcome of the ultimatum game to depend on the Proposer’s and Responder’s preferences, including their attitude to fairness and inequality or to social norms more broadly.

A Responder with purely self-interested preferences will accept any positive offer, because something, no matter how small, is always better than nothing. Therefore, in a world composed only of self-interested individuals, the Proposer will anticipate that the Responder will accept any offer and offer the minimum possible amount—one dollar—knowing that the offer will be accepted. This outcome is efficient because all the resources are used (no money is destroyed by the offer being rejected), but it will be unfair because the Proposer will have a much larger share of income than the Responder.

In contrast, in a community where many people care about fairness as determined by the level of inequality in the society, the Proposer might prefer to offer $50. Even a selfish Proposer might expect the Responder to reject a low offer of much less than $50. The Proposer’s best strategy in this case is to make the lowest offer that the Responder is likely to accept.

When will an offer in the ultimatum game be accepted?

minimum acceptable offer: In the ultimatum game, the smallest offer by the Proposer that will not be rejected by the Responder. More generally in bargaining situations, it is the least favourable offer that will be accepted.

In the ultimatum game, the minimum acceptable offer is the offer at which the pleasure of getting the money is equal to the satisfaction the Responder gets from refusing the offer and getting no money, while ensuring that the Proposer also gets nothing.

For example, if a Responder’s minimum acceptable offer is $35 (of the total pie of $100), they will be indifferent between accepting an offer of $35 and getting no money but obtaining $35 of satisfaction from rejecting the offer. But they will accept higher offers (which bring more money and less satisfaction from rejection) and reject lower offers (which bring less money and greater satisfaction from rejection).

We will model the case when the social norm is a 50-50 split, and the Responder cares about the norm being upheld. When the proposal, $y$, is $50 or above $(y ≥ 50)$, the Responder feels positively disposed toward the Proposer and will accept the offer. Rejecting it will hurt both themself and the Proposer (which they has no wish to do when both players conform to the social norm, or are even more generous).

But if the offer is below $50, then the Responder feels that the social norm is not being respected, and they may want to punish the Proposer. Rejecting the offer will come at a cost to them, because rejection means that both players receive nothing.

To model the Responder’s reciprocity motive, suppose that there anger at an offer below the social norm depends on the size of the breach. Specifically, if the Proposer offers $y ≤ 50$, their satisfaction from rejecting the offer will be $R(50 – y)$.

This equation means that the Responder will get no satisfaction from rejecting an offer of $50 (the social norm). The lower the offer, the higher is the rejection-satisfaction. $R$ is a number that measures the strength of the reciprocity motive: if $R$ is large, then the Responder cares a lot about whether the Proposer is acting generously and fairly or not; if $R = 0$, then the Responder does not care about the Proposer’s motives at all.

On the other hand, if the Responder accepts the offer, they will receive the amount of money $y$. So they should reject the offer if $y$ is lower than the satisfaction from rejecting $y$:

\[\text{Reject offer } y \text{ if } y < R(50 − y).\]

We can rearrange this inequality to write it as:

\[\begin{align*} y &< 50R - Ry \\ y + Ry &< 50R \\ y(1+R) &< 50R \\ \text{Reject offer } y \text{ if } y &< \frac{50R}{1+R} \end{align*}\]

Therefore the Responder’s minimum acceptable offer is $\frac{50R}{1+R}$. For example, if $R = 1$, there minimum acceptable offer is $25. This case represents a Responder whose rejection-satisfaction is exactly equal to the amount by which the offer falls below $50. The more the Responder cares about reciprocity, the higher the Proposer’s offer has to be. If $R = 4$, the Responder will reject any offers below $40.

Exercise E5.3 Acceptable offers

How might the minimum acceptable offer depend on the method by which the Proposer acquired the $100 (for example, did he find it on the street, win it in the lottery, or receive it as an inheritance)?
Suppose that the fairness norm in this society is 50-50. Can you imagine anyone offering more than 50% in such a society? Why or why not?

Consider the theoretical predictions of the ultimatum game as described in this section. Explain how inequality aversion, reciprocity, and social norms could motivate Responders’ willingness to reject low offers, even though they receive nothing at all when they choose to reject such offers.

Question 5.7

What distinguishes someone with social preferences from the economic man (Homo economicus)? Choose all that apply.

Social preferences involve caring only about monetary outcomes.
Social preferences mean a person’s desired outcome depends on what happens to others as well as themselves.
Economic man (Homo economicus) never cooperates with other people.
Social preferences include considerations such as altruism and inequality aversion.

Someone with social preferences may have some self regard and care about how much they receive, but they also care about others or about social norms.
Social preferences mean that a person’s desired outcome depends on what happens to others as well as themselves.
Economic man (Homo economicus) may cooperate when it is in their self-interest to do so, for example, when keeping a trade agreement is more profitable than breaking it. Non-cooperation is not a defining characteristic of Homo economicus.
Social preferences include considerations such as altruism and inequality aversion, as well as reciprocity and consideration for social norms.

Question 5.8

Which of the following outcomes can occur in the ultimatum game when the Responder is fair-minded? Choose all that apply.

The Responder accepts a 50-50 split.
The Proposer offers a 90-10 split, and the Responder accepts it.
The Responder rejects an 80-20 split because they view it as unfair.
The Responder accepts any offer larger than zero.

A fair-minded Responder will accept an equal offer.
A fair-minded Responder is likely to reject highly unequal splits.
The Responder may reject an 80-20 split to punish unfairness.
Accepting any positive offer describes a self-interested Responder.