Unit 4 Strategic interactions and social dilemmas

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

If you and your friend find $100 lying in the street, there are potential benefits for both of you. How those are shared will depend on your preferences, and on social norms.

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). See also: reservation option.
ultimatum game
A game in which the first player proposes a division of a ‘pie’ with the second player, who may either accept, in which case they each get the division proposed by the first person, or reject the offer, in which case both players receive nothing.

The same question arises in many economic interactions, whether social (sharing the gains from a community project) or private (determining the price of a second-hand car). There are potential economic rents for both parties. How much each party receives depends on preferences; it also depends on the rules of the game—the process that determines the outcome.

The ultimatum game

How people share mutual benefits has been studied experimentally using a two-person one-shot game known as the ultimatum game, or take-it-or-leave-it game. It has been used around the world with participants including students, farmers, warehouse workers, and hunter-gatherers. By observing their choices, we investigate the participants’ preferences and motives, such as pure self-interest, altruism, inequality aversion, or reciprocity.

The participants are invited to play a game in which they can win some money, depending on how they and other participants play. Real money is at stake, to ensure that their decisions reflect their actions in real life.

The experiment is explained to the players. They are randomly matched in pairs; one is randomly assigned as the Proposer, and the other 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 can either accept or reject the offer.
  • If the offer is rejected, both players get nothing.
  • Otherwise, the Responder receives y and the Proposer gets 100 – y.

For example, suppose the Proposer offers to give $35, and keep $65. It is a take-it-or-leave-it offer: the Responder either accepts $35 or gets nothing. But if the offer is rejected, the Proposer also gets nothing.

If the Responder accepts the Proposer’s offer, both players receive a rent (a slice of the pie); their next best alternative is to get nothing (the pie is thrown away).

simultaneous game
A game in which the players choose their strategies simultaneously, for example, the prisoners’ dilemma. See also: sequential game.
sequential game
A game in which players do not all choose their strategies at the same time, and players who choose later can see the strategies already chosen by the other players. An example is the ultimatum game. See also: simultaneous game.

In contrast to simultaneous games like the prisoners’ dilemma, in which players choose their actions at the same time, the ultimatum game is a sequential game. One player, the Proposer, chooses an action first, followed by the Responder. To think about how they will decide, we first consider a simpler case.

Simplified example

Suppose the Proposer has only two choices: a ‘fair offer’ ($50), or an ‘unfair offer’ ($20). Then we could represent the game using a pay-off matrix. But to capture the sequential structure as well as the pay-offs, it is more helpful to use a game tree as in Figure 4.16, with the pay-offs in the last row.

This diagram shows the game tree for the ultimatum game. The proposer can choose to make an offer of an equal split or an unfair offer of 20 to the responder and 80 to the proposer. The responder can then choose to accept or reject the offer. If the responder accepts, the pie is split between them as proposed. If the responder rejects, they both get nothing.

Figure 4.16 Game tree for a simplified ultimatum game.

This is a strategic interaction: the Proposer moves first, but will need to think carefully about the likely response of the Responder.

So to try to predict the outcome, first consider the Responder’s strategy. Put yourself in the place of the Responder. Would you accept (50, 50)? Would you accept (80, 20)? Or might you reject it, sacrificing your pay-off to punish the Proposer for offering so little?

Now switch roles. If you were the Proposer, what would you offer?

Social preferences in the ultimatum game

Now return to the general case, 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 would expect the outcome of the ultimatum game to depend on the Proposer’s and Responder’s preferences, including their attitude to reciprocity, and prevailing social norms.

A Responder with purely self-interested preferences would 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 would anticipate that the Responder would accept any offer and offer the minimum possible amount—one dollar—knowing it would be accepted. This outcome would be Pareto efficient, although unfair.

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 would be accepted.
homo economicus
Latin for ‘economic man’, used to describe an economic actor who is assumed to make decisions entirely in pursuit of their own of self-interest.

But in a community where many people cared about fairness, the Proposer might prefer to offer $50. Even a selfish Proposer might expect the Responder to reject a low offer, and therefore offer not much less. The best strategy in this case would be to make the lowest offer that the Responder would be likely to accept. In the extension to this section, we calculate the Responder’s minimum acceptable offer for the case of a 50-50 fairness norm and reciprocal preferences.

Section 4.12 describes the results of experiments using the ultimatum game. These are not consistent with the behaviour of Homo economicus—most Proposers do not make the lowest possible offer. As in experiments using the public good game (Section 4.8), social norms and social preferences seem to matter.

Exercise 4.12 A sequential prisoners’ dilemma

Return to the prisoners’ dilemma pest control game that Anil and Bala played in Figure 4.4b, but now suppose that the game is played sequentially, like the ultimatum game. One player (chosen randomly) chooses a strategy first (the first mover), and then the second moves (the second mover).

  1. Suppose you are the first mover and you know that the second mover has strong reciprocal preferences, meaning the second mover will act kindly towards someone who upholds social norms not to pollute and will act unkindly to someone who violates the norm. What would you do?
  2. Suppose the reciprocal person is now the first mover interacting with the person they know to be entirely self-interested. What do you think would be the outcome of the game?

Extension 4.11 When will an offer in the ultimatum game be accepted?

This extension explains how to calculate, using simple algebra, the Responder’s minimum acceptable offer in an example of the ultimatum game in which the social norm is a 50-50 split, and the Responder has reciprocal preferences.

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 would get from refusing the offer and getting no money, but ensuring that the Proposer also gets nothing.

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

We will model the case when the social norm would be a 50-50 split, and the Responder cares about the norm being upheld. When the proposal, \(y\), is $50 or above, (\(y \geq 50\)), the Responder feels positively disposed towards the Proposer and would accept it; rejecting it would hurt both herself and the Proposer (which she has no wish to do when they conform to the social norm, or are even more generous).

But if the offer is below $50, then she feels that the social norm is not being respected, and she may want to punish the Proposer for this breach. If she does reject the offer, this will come at a cost to her, because rejection means that both receive nothing.

To model the Responder’s reciprocity motive, suppose that her anger at an offer below the social norm depends on the size of the breach. Specifically, if the Proposer offers \(y \geq 50\), her satisfaction from rejecting the offer would be:

\[R(50 - y)\]

This means that she would 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 she cares a lot about whether the Proposer is acting generously and fairly or not; if \(R = 0\), she does not care about the Proposer’s motives at all.

On the other hand, if she accepts the offer, she will receive the amount of money, \(y\). So she should reject the offer if \(y\) is lower than the satisfaction from rejecting \(y\):

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

We can rearrange this inequality to write it as:

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

So the Responder’s minimum acceptable offer is \(\frac{50R}{1 + R}\).

For example, if \(R = 1\), her minimum acceptable offer is $25. This represents the case of 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 offers have to be. If \(R = 4\), she will reject any offers below $40.

Exercise E4.2 Acceptable offers

  1. How might the minimum acceptable offer depend on the method by which the Proposer acquired the $100 (for example, did she find it on the street, win it in the lottery, or receive it as an inheritance)?
  2. Suppose that the fairness norm in this society is 50-50. Can you imagine anyone offering more than 50% in such a society? If so, why?