Unit 4 Strategic interactions and social dilemmas
4.7 Social preferences: Altruism
Before you start
To understand the model in this section, you will need to know how to model preferences and choice using indifference curves and utility. If you are not familiar with these concepts, read Sections 3.2–3.5 (especially 3.3) before beginning work on it.
In real-world examples and experiments, people often play the cooperative strategy in prisoners’ dilemma games—rather than choosing to defect, the dominant strategy for self-interested players. One possible explanation is altruism.
- preferences
- A description of the relative values a person places on each possible outcome of a choice or decision they have to make.
- utility
- A numerical indicator of the value that one places on an outcome. Outcomes with higher utility will be chosen in preference to lower valued ones when both are feasible.
- social preferences
- An individual is said to have social preferences if their individual utility depends on what happens to other people, as well as on their own pay-offs.
In Unit 3, we model economic decision-makers by specifying their preferences, using indifference curves and the concept of utility. If individuals are self-interested, the only things that affect their utility are the goods they obtain for themselves, such as their own consumption, and leisure. So far, we have assumed self-interest in our game-theoretic models, with each agent’s utility given by their own pay-off.
But people generally do care about what happens to others. When people have social preferences, their utility depends not only on what they obtain for themselves, but also on things that affect the wellbeing of other people.
Altruism is a social preference in which an individual’s utility is increased by benefits to others. Other social preferences are inequality aversion (a preference for more equal outcomes); and spite and envy—in which cases, benefits to others may reduce the individual’s utility.
Modelling altruistic preferences
In Exercise 3.3, we model a budgeting decision for Zoë, a university student in London, assuming that she cares only about goods she consumes herself. But suppose Zoë faces a different decision. She is given some tickets for the national lottery, and one of them wins a prize of £200. Will she decide to keep all the money for herself, or share some of it with her flatmate, Yvonne? Her decision depends on how much she cares about Yvonne: that is, on whether Zoë has altruistic or self-interested preferences in this situation.
This is not a game; as in Unit 3, there is a single decision-maker, and we can model the decision in the same way. Zoë’s problem is how to allocate her ‘budget’ of £200 between two ‘goods’: her own share, and if she is altruistic, Yvonne’s share. So we use indifference curves to represent Zoë’s preferences between the two goods that affect her utility.
The left-hand panel of Figure 4.10 shows Zoë’s preferences if she is altruistic. Increases in her amount of money would raise her utility, but so would increases for Yvonne. The indifference curves slope downwards, showing that she is willing to give up some of her own money to give more to Yvonne.
Figure 4.10 The shape of Zoë’s indifference curves depends on whether she is altruistic or self-interested.
The right-hand panel shows the shape of her indifference curves if she was entirely self-interested: increases in her own amount of money raise her utility, but money for Yvonne has no effect. She only cares about the good on the horizontal axis—the money she receives herself.
Altruism does not mean that Zoë cares as much about Yvonne as herself. In the left-hand panel, where they receive similar amounts of money, the curves are quite steep: at the point on the middle curve where each receives £120, Zoë would be willing to give up only £4 to give another £10 to Yvonne. If she were more altruistic, the curves would be flatter; if she were more self-interested, they would be steeper (remember that with pure self-interest they are vertical).
Figure 4.11 solves Zoë’s decision problem. Any way of distributing the prize between Zoë and Yvonne is feasible, if the total amount is less than or equal to £200. Zoë will choose the point in the feasible set that gives the highest utility, so her choice depends on whether or not she has altruistic preferences.
Figure 4.11 How Zoë chooses to distribute her lottery winnings depends on whether she is selfish or altruistic.
The feasible set
If Zoë has altruistic preferences
Zoë’s choice when she is altruistic
If Zoë had self-interested preferences
If Zoë is altruistic in this situation, she chooses point A, giving Yvonne £60 out of her prize. She is willing to bear a cost to benefit somebody else. If she was purely self-interested, she would choose S, giving Yvonne nothing. In general, whether people behave altruistically may depend on the situation they face. So Zoë might be self-interested when she decides how to allocate her student budget, but altruistic when she wins the lottery.
Exercise 4.8 Altruism and selflessness
Using the same axes as in Figure 4.11:
- Draw Zoë’s indifference curves assuming that she cared just as much about Yvonne’s consumption as her own.
- Draw Zoë’s indifference curves assuming that she derived utility only from the total of her and Yvonne’s consumption.
- Draw Zoë’s indifference curves assuming that she derived utility only from Yvonne’s consumption.
- For each of these cases, provide a real-world situation in which Zoë might have these preferences, and make sure to specify how Zoë and Yvonne derive their pay-offs.
Question 4.6 Choose the correct answer(s)
In Figure 4.11, Zoë has just won the lottery and has received £200. She is considering how much (if at all) of this sum to share with her flatmate, Yvonne. Before she manages to share her winnings, Zoë receives a tax bill for these winnings of £40. Assume Zoë’s preferences are altruistic and fixed (they are the same before and after winning the lottery). Based on this information, read the following statements and choose the correct option(s).
- Without the tax, Zoë would have given exactly £60 to Yvonne. With the total income now at £160, Zoë will choose to give less than this.
- We assume that preferences are fixed. Hence Zoë will remain altruistic and give Yvonne some of her winnings.
- The tax bill can be depicted as an inward shift of the feasible frontier. Therefore, Zoë will no longer be able to obtain the same level of utility as she did before the tax bill.
- Yvonne would have received £200 and £160, respectively, before and after the tax bill.
How altruism can change behaviour in the prisoners’ dilemma
What would happen in the pest control game if the farmers were altruistic? Would their strategies be different?
We start by modelling Anil’s preferences in the same way as Zoë’s. We already know that if he is self-interested, his dominant strategy is T (the pesticide). We can show this in a different way using his indifference curves: we find feasible allocations that maximize his utility. The left-hand panel of Figure 4.12 shows Anil’s indifference curves when he is self-interested, and the four potential allocations in the game: that is, the monetary pay-offs corresponding to the pay-off matrix on the right.
But this is a game, so Anil has to think strategically. He does not have a free choice between the four allocations: what is feasible for him depends on Bala’s choice.
Figure 4.12 Anil’s best response in the pest control game when he has self-interested preferences.
The four possible allocations
Anil’s best response if Bala chooses T
Anil’s best response if Bala chooses I
Now, suppose that Anil has altruistic preferences towards Bala, similar to Zoë’s towards her flatmate: then, his utility depends not only on his own monetary pay-off, but also on that of Bala. In Figure 4.13, we repeat the analysis for this case. Work through the steps to deduce that his dominant strategy is now I, rather than T.
Figure 4.13 Anil’s best response in the pest control game when he is altruistic towards Bala.
Altruistic indifference curves
Anil’s best response if Bala chooses T
Anil’s best response if Bala chooses I
Figure 4.13 shows that altruism can make Anil’s behaviour cooperative. With these indifference curves, IPC is his dominant strategy. But whether this happens depends on how altruistic he is. If he had cared a bit less about Bala’s income, his indifference curves would have been steeper and he might have made a different choice.
How will Anil’s altruism affect the equilibrium of the game? If Bala is similarly altruistic, he will choose IPC, too. Mutual cooperation will result in the good outcome, (I, I), being the dominant strategy equilibrium. However, if Bala is still self-interested, he will choose T as before. The dominant strategy equilibrium will be (I, T), resulting in an allocation of 4 to Bala and 1 to Anil. Bala will benefit from high profits, while Anil (willingly) bears the monetary cost of Bala’s choice. Different equilibria are possible depending on the degree of altruism felt by each player.
If people care about one another, social dilemmas are easier to resolve. Cooperative equilibria are possible in a prisoners’ dilemma. This helps us understand the historical examples—such as irrigation, or the Montreal Protocol to protect the ozone layer—in which people mutually cooperate rather than free-riding.
When economists disagree Homo economicus in question: Are people entirely selfish?
For centuries, economists (and others) have debated whether people are entirely self-interested or are sometimes happy to help others—even when it costs them something. Homo economicus (economic man) is the nickname for the selfish and calculating character that you may find in economics textbooks. Have economists been right to imagine Homo economicus as the only actor on the economic stage?
In the book where he first used the phrase ‘invisible hand’, Adam Smith also stated: ‘How selfish soever man may be supposed, there are evidently some principles in his nature which interest him in the fortunes of others, and render their happiness necessary to him, though he derives nothing from it except the pleasure of seeing it.’ (The Theory of Moral Sentiments, 1759)
But much subsequent economic analysis has ignored the concern for others. In 1881, Francis Edgeworth, a founder of modern economics, claimed in his book Mathematical Psychics: ‘The first principle of economics is that every agent is actuated only by self-interest.’1
Yet everyone has experienced, and some have performed, acts of bravery or kindness towards others when there was little chance of a reward. Should the unselfishness evident in these acts be part of how economists reason about behaviour?
Some say ‘no’: many seemingly generous acts can be understood as attempts to gain a favourable reputation that will benefit the actor in the future. Maybe helping others is just self-interest with a long time horizon. This is what the essayist H. L. Mencken thought: ‘conscience is the inner voice which warns that somebody may be looking’.2
Since the 1990s, economists have tried to resolve the debate empirically, performing hundreds of experiments all over the world using economic games, to observe the behaviour of individuals (from hunter-gatherers to CEOs) as they make real choices.
In these experiments, self-interested Homo economicus is often in the minority. Later sections in this unit will provide evidence of behaviour that is consistent with values such as altruism or aversion to inequality—even when the amounts at stake are as high as many days’ wages.
Is the debate resolved? Many economists now think so. A model assuming self-interest may be sufficient to capture the decisions of shoppers, or firms in pursuit of profit. But it’s less appropriate in other settings, such as how we pay taxes, or why we work hard for our employer.
Question 4.7 Choose the correct answer(s)
Read the following statements about the ideas and evidence about self-interested behaviour in the ‘When economists disagree’ box, and choose the correct option(s).
- Behaviour consistent with non-self-interested motivations (such as altruism) has been observed in experiments, so self-interest cannot be the only motivation.
- One explanation for generous acts is that the individual attempts to gain a favourable reputation that will benefit them in the future.
- Social scientists have observed non-self-interested behaviour even when the stakes are as high as many days’ wages.
- Self-interest can explain some decisions such as supermarket shopping or profit-maximization, but may not be the only motive in other contexts, such as paying taxes.
Question 4.8 Choose the correct answer(s)
Figures 4.12 and 4.13 show Anil’s preferences when he is completely selfish, and also when he is somewhat altruistic, when he and Bala participate in the prisoners’ dilemma game.
Based on the graphs, we can say that:
- (T, I) is on a ‘higher’ vertical indifference curve than (I, I) (that is, it is further to the right) and (T, T) is on a higher vertical indifference curve than (I, T). So using Toxic Tide is a dominant strategy for Anil when he is purely self-interested.
- When Anil is altruistic, (I, I) is on a higher indifference curve than (T, I), and (I, T) is on a higher indifference curve than (T, T). So using IPC is Anil’s dominant strategy.
- Toxic Tide is a dominant strategy for both players, so (T, T) is a dominant strategy equilibrium. Anil would prefer (T, I), but Bala will never choose IPC.
- IPC is a dominant strategy for Anil when he is altruistic. If Bala has the same preferences, IPC will be a dominant strategy for him too, so (I, I) could be the dominant strategy equilibrium.
Extension 4.7 Maximizing utility when preferences are altruistic
We analyse Zoë’s decision problem when she has altruistic preferences, by applying the calculus methods for constrained choice developed in the extensions to Sections 3.2–3.5. You should be familiar with these before reading this extension. We obtain the mathematical solution for the utility function that we analysed graphically in the main part of this section, and then look at the case of Cobb–Douglas utility in the exercise at the end.
Zoë has won £200 in the national lottery, and is considering sharing it with her flatmate, Yvonne. She has altruistic preferences: while she is pleased to receive the money, she also cares about Yvonne, who has not been as lucky. We modelled Zoë’s decision diagrammatically in Figure 4.11, reproduced as Figure E4.1. The indifference curves represent her preferences for two ‘goods’—money for herself, and money for Yvonne—and her feasible frontier shows all the possible ways of sharing all of the prize money. She chooses point A, where the feasible frontier reaches the highest possible indifference curve.
Figure E4.1 How Zoë chooses to distribute her lottery winnings when she is altruistic.
This is a constrained choice problem very similar to Karim’s problem in Section 3.5. In both cases, the decision-maker’s objective is to maximize their utility, which depends on two goods when their choice is constrained: having more of one good means having less of the other. We now use the mathematical technique explained in Extension 3.5 to analyse Zoë’s problem.
Zoë’s utility function (the one we used to draw the indifference curves in Figure E4.1) is:
\[u(z,y)=y(z-20)^2\]where \(y\) is the amount of money she gives to Yvonne, and \(z\) is the amount she keeps for herself.
Her budget constraint—that is, the equation of the feasible frontier—is \(y+z=200\). She wants to maximize her utility, given this constraint.
Zoë’s constrained choice problem
Choose \(z\) and \(y\) to maximize \(u(z,y)\) subject to the constraint \(z+y=200\).
Again, we can solve it either by substituting the constraint into the objective function, \(u\), or by using the condition that the slopes of the indifference curve and budget constraint are the same at the point where utility is maximized: that is, the first-order condition MRS = MRT. We will use the second method here.
The marginal rate of substitution is the absolute value of the slope of the indifference curve, \(-\frac{dy}{dz}\). We can calculate it using the formula in Extension 3.3:
\[\text{MRS} = \frac{\text{Zoë's marginal utility of keeping money for herself}}{\text{Her marginal utility of giving money to Yvonne}} = \frac{\partial u / \partial z}{\partial u / \partial y}\]The marginal utilities are found by partial differentiation:
\[\frac{\partial u}{\partial z}=2y(z−20) \text{ and } \frac{\partial u}{\partial y}=(z−20)^2\]Applying the formula:
\[MRS = \frac{2y}{z−20}\]The marginal rate of transformation is the absolute value of the slope of the feasible frontier, \(y+z=200\). Writing this as \(y = 200 - z\), the slope is –1, so the MRT is 1. In other words, she can transform her money into money for Yvonne at the rate of one for one. Then from the first-order condition MRS = MRT we get:
\[\frac{2y}{z-20}=1 \Rightarrow z=20+2y\]This means that Zoë would always choose to give more to herself than to Yvonne, whatever the value of the prize.
To find the outcome (the actual values of \(y\) and \(z\)) we also use the condition that it must lie on the feasible frontier \(y+z=200\).
These two equations for \(y\) and \(z\) can be solved by using the first to substitute for \(z\) in the second:
\[y+20+2y=200 \Rightarrow y=60 \text{ and hence } z=140\]This is point A in Figure E4.1; Zoë keeps £140 and gives £60 to Yvonne.
Exercise E4.1 Interpreting altruistic preferences in the Cobb–Douglas case
Suppose Yvonne has the utility function, \(u(y, z) = y^a z^b\), where \(y\) equals the amount Yvonne has and \(z\) equals the amount Zoë has. If Yvonne wins £200 in the lottery:
- How would she choose to split the lottery winnings between herself and Zoë? (Hint: Your answer will be a function of \(a\) and \(b\).)
- Calculate the amount that Yvonne gives to Zoe for the values of \(a\) and \(b\) shown in the table. Explain (using the table and by referring to the utility function) how the amount that Yvonne gives to Zoe changes with \(a\) and \(b\).
| a | b | Amount given to Zoe (£) |
|---|---|---|
| 0.1 | 0.9 | |
| 0.2 | 0.8 | |
| 0.3 | 0.7 | |
| 0.4 | 0.6 | |
| 0.5 | 0.5 | |
| 0.6 | 0.4 | |
| 0.7 | 0.3 | |
| 0.8 | 0.2 | |
| 0.9 | 0.1 |
- Repeat Question 1, but keep the amount won in the lottery as the variable \(y\) instead of £200. How does the share that Yvonne gives Zoe and the share that Yvonne keeps for herself change with \(y\)?
Read more: Sections 15.1, 17.1, 17.3 of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.
