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.

There are two diagrams. In diagram 1, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. There are three downward-sloping, parallel convex curves which represent altruistic preferences. The level of Zoë’s utility increases further away from the origin. In diagram 2, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. There are three parallel vertical lines which represent self-interested preferences. The level of Zoë’s utility increases further away from the vertical axis.
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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.

In this diagram, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. Coordinates are (amount for Zoë, amount for Yvonne). A downward-sloping straight line connects points (0, 200) and S (200, 0). This line is labelled feasible frontier (budget constraint). The area between the budget constraint and the axes is the feasible set. There are three parallel, vertical lines at the following amounts for Zoë: £80, £140 and £200. Zoë’s utility increases further away from the vertical axis.
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Figure 4.11 How Zoë chooses to distribute her lottery winnings depends on whether she is selfish or altruistic.

The feasible set: In this diagram, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. Coordinates are (amount for Zoë, amount for Yvonne). A downward-sloping straight line connecs points (0, 200) and (200, 0). This line is labelled feasible frontier (budget constraint). The area between the budget constraint and the axes is the feasible set.
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The feasible set

The shaded area shows the feasible ways of sharing the prize. Zoë will choose a point on the budget constraint, sharing all of the £200 between them (being below the feasible frontier would mean throwing money away).

If Zoë has altruistic preferences: In this diagram, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. Coordinates are (amount for Zoë, amount for Yvonne). A downward-sloping straight line connecs points (0, 200) and (200, 0). This line is labelled feasible frontier (budget constraint). The area between the budget constraint and the axes is the feasible set. There are three parallel, downward-sloping, convex curves. The lower-most curve intersects the feasble frontier in two points. The two upper-most curves lie above the feasible frontier at all points.
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If Zoë has altruistic preferences

Zoë ’s utility depends on Yvonne’s share of the prize as well as her own. She will choose the point on the feasible frontier that gives her the highest utility.

Zoë’s choice when she is altruistic: In this diagram, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. Coordinates are (amount for Zoë, amount for Yvonne). A downward-sloping straight line connects points (0, 200) and S (200, 0). This line is labelled feasible frontier (budget constraint). The area between the budget constraint and the axes is the feasible set. There are four parallel, downward-sloping, convex curves. The lower-most curve intersects the feasble frontier in two points. The one above this one is tangential to the feasible frontier at point A (140, 60). The two upper-most curves lie above the feasible frontier at all points.
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Zoë’s choice when she is altruistic

She achieves the highest level of utility at A, where an indifference curve just touches the feasible frontier. She keeps £140 for herself, and gives £60 to Yvonne.

If Zoë had self-interested preferences: In this diagram, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. Coordinates are (amount for Zoë, amount for Yvonne). A downward-sloping straight line connects points (0, 200) and S (200, 0). This line is labelled feasible frontier (budget constraint). The area between the budget constraint and the axes is the feasible set. There are three parallel, vertical lines at the following amounts for Zoë: £80, £140 and £200. Zoë’s utility increases further away from the vertical axis.
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If Zoë had self-interested preferences

If Zoë cared only about money for herself, her utility would be highest at point S. She would keep the whole prize and give Yvonne nothing.

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:

  1. Draw Zoë’s indifference curves assuming that she cared just as much about Yvonne’s consumption as her own.
  2. Draw Zoë’s indifference curves assuming that she derived utility only from the total of her and Yvonne’s consumption.
  3. Draw Zoë’s indifference curves assuming that she derived utility only from Yvonne’s consumption.
  4. 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).

  • Yvonne will receive £60 if Zoë is altruistic.
  • If Zoë was altruistic and kept £140 before the tax bill, she will still keep £140 after the tax bill by turning completely selfish.
  • Zoë will be on a lower indifference curve after the tax bill.
  • Had Zoë been so extremely altruistic that she only cared about Yvonne’s share, then Yvonne would have received the same income before and after the tax bill.
  • 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.

There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). Four vertical lines represent Anil’s indifference curves when completely selfish. Each line passes through one of the points just mentioned. Anil’s utility increases further away from the vertical axis. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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Figure 4.12 Anil’s best response in the pest control game when he has self-interested preferences.

The four possible allocations: There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). Four vertical lines represent Anil’s indifference curves when completely selfish. Each line passes through one of the points just mentioned. Anil’s utility increases further away from the vertical axis. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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The four possible allocations

The left-hand panel shows the allocations of income for Anil (horizontal axis) and Bala (vertical axis) in each of the four possible outcomes of the game. We have drawn Anil’s indifference curves through each point.

Anil’s best response if Bala chooses T: There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). Four vertical lines represent Anil’s indifference curves when completely selfish. Each line passes through one of the points just mentioned. Anil’s utility increases further away from the vertical axis. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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Anil’s best response if Bala chooses T

Which allocations are feasible for Anil depends on what Bala does. If Bala chooses T, Anil’s choice is between (I, T) and (T, T). He will choose T, because (T, T) gives him higher utility.

Anil’s best response if Bala chooses I: There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). Four vertical lines represent Anil’s indifference curves when completely selfish. Each line passes through one of the points just mentioned. Anil’s utility increases further away from the vertical axis. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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Anil’s best response if Bala chooses I

If Bala chooses I, Anil again has a choice between two allocations. He will choose T again, because (T, I) gives him higher utility than (I, I). So Anil’s dominant strategy is T.

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.

There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). There are five parallel, downward-sloping, convex curves. The second from the bottom passes through point (I, T) and the second from the top passes trhough point (I, I). Anil’s utility increases further away from the origin. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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Figure 4.13 Anil’s best response in the pest control game when he is altruistic towards Bala.

Altruistic indifference curves: There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). There are five parallel, downward-sloping, convex curves. The second from the bottom passes through point (I, T) and the second from the top passes trhough point (I, I). Anil’s utility increases further away from the origin. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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Altruistic indifference curves

If Anil cares about Bala’s income as well as own income, his indifference curves slope downward. His utility rises if his own income increases; it also rises if Bala’s income increases.

Anil’s best response if Bala chooses T: There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). There are five parallel, downward-sloping, convex curves. The second from the bottom passes through point (I, T) and the second from the top passes trhough point (I, I). Anil’s utility increases further away from the origin. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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Anil’s best response if Bala chooses T

With these indifference curves, if Bala chooses T, Anil will choose I because (I, T) gives him more utility than (T, T). (I, T) is on a higher indifference curve than (T, T). Although Anil’s own monetary pay-off is lower at (I, T), he values the additional benefit to Bala.

Anil’s best response if Bala chooses I: There are 2 diagrams. In Diagram 1, the horizontal axis shows Anil’s payoff, ranging from 0 to 5, and the vertical axis shows Bala’s payoff, ranging from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). Four points are labelled: I, T with coordinates (1, 4), T, T with coordinates (2, 2), I, I with coordinates (3, 3) and T, I with coordinates (4, 1). There are five parallel, downward-sloping, convex curves. The second from the bottom passes through point (I, T) and the second from the top passes trhough point (I, I). Anil’s utility increases further away from the origin. Diagram 2 shows Anil and Bala’s available actions, which are IPC or Toxic Tide. Payoffs are expressed as (Anil’s, Bala’s). If both choose IPC, payoffs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, payoffs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, payoffs are (4, 1). If both choose Toxic Tide, payoffs are (2, 2).
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Anil’s best response if Bala chooses I

If Bala chooses I, Anil will also choose I because (I, I) is on a higher indifference curve than (T, I). Although (I, I) gives him lower income, he prefers it because it doesn’t inflict damage on Bala. So I is Anil’s dominant strategy.

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).

  • Experimental evidence supports the assumption of purely self-interested agents.
  • Non-self-interested behaviour can be motivated by self-interest.
  • One explanation for non-self-interested behaviour is that the stakes are sufficiently low.
  • The decision to act in a self-interested way is context-dependent.
  • 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:

  • When Anil is purely self-interested, using Toxic Tide is his dominant strategy.
  • When Anil is altruistic, using Toxic Tide is his dominant strategy.
  • When Anil is purely self-interested, (T, T) is the dominant strategy equilibrium even though it is on a lower indifference curve for him than (T, I).
  • If Anil is altruistic, and Bala’s preferences are the same as Anil’s, (I, I) can be attained as the dominant strategy equilibrium.
  • (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.

In this diagram, the horizontal axis shows the amount for Zoë (in pounds), and ranges between 0 and 240. The vertical axis shows the amount for Yvonne (in pounds) and ranges between 0 and 240. Coordinates are (amount for Zoë, amount for Yvonne). A downward-sloping straight line connects points (0, 200) and S (200, 0). This line is labelled feasible frontier (budget constraint). The area between the budget constraint and the axes is the feasible set. There are four parallel, downward-sloping, convex curves. The lower-most curve intersects the feasble frontier in two points. The one above this one is tangential to the feasible frontier at point A (140, 60). The two upper-most curves lie above the feasible frontier at all points.
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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:

  1. How would she choose to split the lottery winnings between herself and Zoë? (Hint: Your answer will be a function of \(a\) and \(b\).)
  2. 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

  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.

  1. Francis Ysidro Edgeworth. 2003. Mathematical Psychics and Further Papers on Political Economy. Oxford: Oxford University Press. 

  2. H. L. Mencken. 2006. A Little Book in C Major. New York, NY: Kessinger Publishing.