Unit 4 Strategic interactions and social dilemmas

4.5 Evaluating outcomes: The Pareto criterion

Whatever the economic interaction—shopping, negotiating wages, or managing fish stocks —we want to be able both to describe the outcome, and to evaluate it: is it better or worse than alternative outcomes? The first involves facts; the second involves values.

allocation
In an economic interaction, an allocation is a particular distribution of goods or other things of value to all participants.
Pareto criterion
The Pareto criterion is a way of comparing two allocations, A and B. It states that A is is an improvement on B if at least one person would be strictly better off with A than B (in other words, would strictly prefer A to B) and nobody would be worse off. We say that A Pareto dominates B.

We call the outcome of an economic interaction an allocation. In a game, an allocation is a particular distribution among the players of goods or other things that they value. For example, if two firms compete to sell goods in a market, the allocation might consist of the profits for each firm.

Suppose that we want to compare two possible allocations, A and B, that may result from an economic interaction. Can we say which is better? If we find that everyone concerned would prefer A, most people would agree that A is a better allocation than B. This is what the Pareto criterion, named after the Italian economist and sociologist Vilfredo Pareto, would tell us.

The Pareto criterion

According to the Pareto criterion, allocation A is better than allocation B if at least one party would be strictly better off with A than B, and nobody would be worse off.

Pareto efficient, Pareto efficiency
An allocation is Pareto efficient if there is no feasible alternative allocation in which at least one person would be better off, and nobody worse off.

We say that A Pareto-dominates B, or that A would be a Pareto improvement over B.

When we say an allocation makes someone ‘better off’ we mean that they prefer it, which does not necessarily mean they get more money.

Great Economists Vilfredo Pareto

Portrait of Vilfredo Pareto

Vilfredo Pareto (1848–1923), an Italian economist and sociologist, earned a degree in engineering for his research on the concept of equilibrium in physics. He is mostly remembered for his criterion for ranking allocations, and the related concept of Pareto efficiency. He wanted economics and sociology to be fact-based sciences, similar to the physical sciences he had studied when he was younger.

His empirical investigations led him to another important contribution, on allocations of wealth. The distribution of wealth did not seem to resemble the familiar bell curve, with a few rich, a few poor, and a large middle-income class. Instead he proposed what came to be called Pareto’s law, according to which, across the ages and differing types of economies, there would be very few rich people and a lot of poor people.

His 80–20 rule—derived from Pareto’s law—asserted that the richest 20% of a population typically held 80% of the wealth. If he’d lived in the US in 2020, he would have observed that 90% of the wealth was held by the richest 20%, suggesting that his law might not be as universal as he’d thought.

In Pareto’s view, the economic game was played for high stakes, with big winners and losers. He urged economists to study conflicts over the division of goods, and he thought the time and resources devoted to these conflicts were part of what economics should be about.1 In his most famous book, the Manual of Political Economy (1906), he wrote that: ‘The efforts of men are utilized in two different ways: they are directed to the production or transformation of economic goods, or else to the appropriation of goods produced by others.’

Exercise 4.5 Pareto’s law

Pareto’s law states that the richest 20% of a population typically holds 80% of the wealth. Verify this law for yourself by downloading the global wealth data.

  1. Choose five countries and one particular year (ranging from 1995 to 2020, inclusive), and calculate the percentage of wealth held by the richest 20% (columns K and L of the spreadsheet).
  2. Comment on how well your chosen countries fit Pareto’s law.
Pareto dominate, Pareto dominant
Allocation A Pareto dominates allocation B if it is better according to the Pareto criterion. That is, at least one person would be strictly better off with A than B, and nobody would be worse off. See also: Pareto criterion.
Pareto improvement
A change that benefits at least one person without making anyone else worse off. See also: Pareto dominant, Pareto criterion.

Figure 4.6 shows the pay-off matrix in the prisoners’ dilemma game from Section 4.4. The two farmers, Anil and Bala, can use either integrated pest control, or the pesticide Toxic Tide. On the right, we have plotted the four allocations in a graph, with Anil’s pay-off on the horizontal axis and Bala’s on the vertical axis, labelling the actions I and T for convenience. Anil and Bala are self-interested, preferring allocations with a higher pay-off for themselves.

There are 2 diagrams. Diagram 1 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). In Diagram 2, 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).
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Figure 4.6 The four allocations in the pest control game.

The graph shows that the allocation at (I, I) Pareto-dominates (T, T): both players would prefer it. Figure 4.7 explains how to compare all four allocations graphically.

In this diagram, the horizontal axis displays Anil’s payoff, and ranges from 0 to 5. The vertical axis displays Bala’s payoff and ranges from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). If Anil uses IPC and Bala uses Toxic Tide, represented by point (I, T), payoffs are (1, 4). If both use Toxic Tide, represented by point (T, T), payoffs are (2, 2). If both use IPC, represented by point (I, I), payoffs are (3, 3). If Anil uses Toxic Tide and Bala uses IPC, represented by point (T, I), payoffs are (4, 1). All outcomes above and to the right of point (T, I) are better for both Anil and Bala than point (T, I). All outcomes above and to the right of point (T, T) are better for both Anil and Bala than point (T, T). All outcomes above and to the right of point (I, I) are better for both Anil and Bala than point (I, I). All outcomes above and to the right of point (I, T) are better for both Anil and Bala than point (I, T).
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Figure 4.7 Applying the Pareto criterion to the pest control game.

Comparing (T, T) and (I, I): In this diagram, the horizontal axis displays Anil’s payoff, and ranges from 0 to 5. The vertical axis displays Bala’s payoff and ranges from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). If Anil uses IPC and Bala uses Toxic Tide, represented by point (I, T), payoffs are (1, 4). If both use Toxic Tide, represented by point (T, T), payoffs are (2, 2). If both use IPC, represented by point (I, I), payoffs are (3, 3). If Anil uses Toxic Tide and Bala uses IPC, represented by point (T, I), payoffs are (4, 1). All outcomes above and to the right of point (I, I) are better for both Anil and Bala than point (I, I). All outcomes above and to the right of point (T, T) are better for both Anil and Bala than point (T, T).
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Comparing (T, T) and (I, I)

The allocation at (I, I) lies in the rectangle north-east of (T, T), so an outcome where both players use IPC Pareto-dominates one where both use Toxic Tide: they both prefer (I, I).

Compare (T, T) and (T, I): In this diagram, the horizontal axis displays Anil’s payoff, and ranges from 0 to 5. The vertical axis displays Bala’s payoff and ranges from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). If Anil uses IPC and Bala uses Toxic Tide, represented by point (I, T), payoffs are (1, 4). If both use Toxic Tide, represented by point (T, T), payoffs are (2, 2). If both use IPC, represented by point (I, I), payoffs are (3, 3). If Anil uses Toxic Tide and Bala uses IPC, represented by point (T, I), payoffs are (4, 1). All outcomes above and to the right of point (T, I) are better for both Anil and Bala than point (T, I). All outcomes above and to the right of point (T, T) are better for both Anil and Bala than point (T, T).
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Compare (T, T) and (T, I)

If Anil uses Toxic Tide and Bala IPC, then Anil is better off but Bala is worse off than when both use Toxic Tide. The Pareto criterion cannot say which of these allocations is better.

No allocation Pareto-dominates (I, I): In this diagram, the horizontal axis displays Anil’s payoff, and ranges from 0 to 5. The vertical axis displays Bala’s payoff and ranges from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). If Anil uses IPC and Bala uses Toxic Tide, represented by point (I, T), payoffs are (1, 4). If both use Toxic Tide, represented by point (T, T), payoffs are (2, 2). If both use IPC, represented by point (I, I), payoffs are (3, 3). If Anil uses Toxic Tide and Bala uses IPC, represented by point (T, I), payoffs are (4, 1). All outcomes above and to the right of point (T, I) are better for both Anil and Bala than point (T, I). All outcomes above and to the right of point (T, T) are better for both Anil and Bala than point (T, T). All outcomes above and to the right of point (I, I) are better for both Anil and Bala than point (I, I).
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No allocation Pareto-dominates (I, I)

None of the other allocations lie to the north-east of (I, I), so it is not Pareto-dominated.

What can we say about (I, T) and (T, I)?: In this diagram, the horizontal axis displays Anil’s payoff, and ranges from 0 to 5. The vertical axis displays Bala’s payoff and ranges from 0 to 5. Coordinates are (Anil’s payoff, Bala’s payoff). If Anil uses IPC and Bala uses Toxic Tide, represented by point (I, T), payoffs are (1, 4). If both use Toxic Tide, represented by point (T, T), payoffs are (2, 2). If both use IPC, represented by point (I, I), payoffs are (3, 3). If Anil uses Toxic Tide and Bala uses IPC, represented by point (T, I), payoffs are (4, 1). All outcomes above and to the right of point (T, I) are better for both Anil and Bala than point (T, I). All outcomes above and to the right of point (T, T) are better for both Anil and Bala than point (T, T). All outcomes above and to the right of point (I, I) are better for both Anil and Bala than point (I, I). All outcomes above and to the right of point (I, T) are better for both Anil and Bala than point (I, T).
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What can we say about (I, T) and (T, I)?

Neither of these allocations are Pareto-dominated, but they do not dominate any other allocations either.

The Pareto criterion is cautious. It makes no value judgement beyond the least controversial one that an allocation is better if it makes at least one person better off and no one worse off. So it does not rank (I, I) and (T, I)—although you may be able to think of other reasons for ranking (I, I) above (T, I).

An allocation that is not Pareto-dominated by any other allocation is called Pareto efficient. The game in Figure 4.7 has three Pareto-efficient allocations: (I, T), (I, I), and (T, I). Allocation (T, T) is not Pareto efficient because it is dominated by (I, I).

Pareto efficiency

An allocation that is not Pareto-dominated by any other allocation is described as Pareto efficient. If an allocation is Pareto efficient, then there is no alternative allocation in which at least one party would be better off and nobody worse off.

The concept of Pareto efficiency is widely used in economics. ‘Efficiency’ sounds like a good thing, but—since the Pareto criterion is cautious about ranking allocations—there can be many Pareto-efficient allocations. And it is often the case that we would think them undesirable for other reasons. For example, Anil playing I and Bala playing T is Pareto efficient (moving to any other allocation would make at least one player worse off), although we (and Anil) may think (I, T) is unfair: Anil’s pay-off is 1, while Bala gets 4.

If food is allocated so that some people are more than satisfied while others are starving, we might say in everyday language: ‘This is not a sensible way to provide nutrition. It is clearly inefficient.’ But Pareto efficiency means something different. A very unequal distribution of food is Pareto efficient as long as all the food is eaten by someone who enjoys it even a little, because it could not be redistributed to those who are starving without taking away food from others.

So we need to be careful with the concept of Pareto efficiency for two reasons. Firstly, it may not tell us much about what is best:

  • There is often more than one Pareto-efficient allocation: in the pest control game, there are three.
  • The Pareto criterion does not tell us which Pareto-efficient allocation is best: it does not rank (I, I), (I, T), and (T, I).
  • Even by the Pareto criterion, a Pareto-efficient allocation is not always better than a Pareto-inefficient one: we know that (T, I) is Pareto efficient, and (T, T) is not. But if you compare the two, (T, I) does not Pareto-dominate (T, T).
fairness
A way to evaluate an allocation based on one’s conception of justice.

Secondly, there may be other criteria that matter as much, or more. In particular, we may want to assess both fairness and Pareto efficiency when we evaluate outcomes.

Evaluating the outcomes of games

If we bear in mind their limitations, the Pareto criterion and Pareto efficiency can be useful tools for evaluating the outcomes of games, alongside other criteria.

Example 1: The rice–cassava games

The games in Figure 4.2 and Figure 4.3 are invisible hand games: the Nash equilibrium (Cassava, Rice), which gives pay-offs of 6 to both Anil and Bala, is Pareto efficient. This is the only Pareto-efficient allocation, and furthermore it Pareto-dominates all the other allocations.

The equilibrium allocation is fair: both players get the same pay-off.

So in these games, most people would agree that the Nash equilibrium gives the most desirable outcome.

Example 2: The pest control game

The pest control game above is an example of the prisoners’ dilemma. The dilemma for the players is that there is a dominant strategy equilibrium, (T, T), which is not Pareto efficient. There is an alternative, the ‘cooperative’ outcome (I, I), that Pareto-dominates the equilibrium allocation: both players prefer it. But if they follow their dominant strategies, they will not achieve it.

(I, I) has other potentially desirable properties too: it is fair (pay-offs are equal), and it maximizes the sum of the players’ pay-offs. If participants in an economic interaction could negotiate an agreement beforehand, they could choose the strategies that maximize the sum of the pay-offs and agree how to share them.

But in a prisoners’ dilemma, the cooperative allocation does not Pareto-dominate all the others—that is why the players don’t choose the cooperative strategy.

Question 4.4 Choose the correct answer(s)

Read the following statements about the outcome of an economic interaction and choose the correct option(s).

  • If the allocation is Pareto efficient, then you cannot make anyone better off without making someone else worse off.
  • If an allocation is Pareto efficient, it is also fair.
  • There cannot be more than one Pareto-efficient outcome.
  • If there are multiple Pareto-efficient outcomes, and an allocation is Pareto-dominated by one Pareto-efficient outcome, then it will also be Pareto-dominated by the other Pareto-efficient outcomes.
  • This statement is the definition of Pareto efficiency.
  • Pareto-efficient allocations can be very unfair (for example, one person has all the income and everyone else has zero income), in which case it is likely that at least one participant would not be happy with the outcome.
  • There can be more than one Pareto-efficient outcome. Three of the four allocations in the pest control game were Pareto efficient.
  • Pareto dominant is a statement about pairs of allocations, so if an allocation is Pareto-dominated by one Pareto-efficient outcome, it may not be Pareto-dominated by all other Pareto-efficient outcomes. For example, in the pest control game, (T, T) was Pareto-dominated by (I, I) but is not Pareto-dominated by (T, I).

Exercise 4.6 Pareto efficient and Pareto-dominated allocations

Make as many statements as you can about the four allocations in Figure 4.7, using the following formats:

  • Allocation ____ is not Pareto-dominated by allocation ____.
  • Allocation ____ does not Pareto-dominate allocation ____.
  1. Vilfredo Pareto. (1906) 2014. Manual of Political Economy: A Variorum Translation and Critical Edition. Oxford, New York: Oxford University Press.