Unit 4 Strategic interactions and social dilemmas

4.13 Coordination games and conflicts of interest

In all the games discussed so far, there is a single Nash equilibrium that gives us a prediction of the outcome. We should expect all players to do the best they can, given what others are doing, and we can usually reason that they are likely to choose their Nash equilibrium strategies—particularly if they have dominant strategies.

But what happens if there is more than one Nash equilibrium?

We have already mentioned one situation in which there are two possible equilibria: driving on the right or on the left. If others drive on the right, your best response is to drive on the right, too. If they drive on the left, your best response is to drive on the left. Everyone driving on the right is a Nash equilibrium, which is what happens in the US. And so is everyone driving on the left, as happens in Japan.

Even in simple economic problems, there may be more than one Nash equilibrium. Suppose that when Bala and Anil choose their crops, their pay-offs are as shown in Figure 4.21. As in Figure 4.2b, Anil’s land is better for growing cassava, and Bala’s for rice. But now if the two farmers produce the same crop, there is such a large fall in price that it is better for each to specialize, even in the crop they are less suited to grow. Follow the steps in Figure 4.21 to find the two equilibria.

This diagram shows Anil and Bala’s available actions, which are growing rice or growing cassava. Payoffs are expressed as (Anil’s, Bala’s). If both grow rice, payoffs are (2, 3). If Anil grows rice and Bala grows cassava, payoffs are (4, 4). If Anil grows cassava and Bala grows rice, payoffs are (6, 6). If both grow cassava, payoffs are (3, 2). If Bala is going to choose Rice, Anil’s best response is to choose Cassava. If Bala is going to choose Cassava, Anil’s best response is to choose Rice. If Anil chooses Rice, Bala’s best response is to choose Cassava, and if Anil chooses Cassava Bala should choose Rice.
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Figure 4.21 A third rice–cassava game: more than one Nash equilibrium.

Anil’s best response to Rice: This diagram shows Anil and Bala’s available actions, which are growing rice or growing cassava. Payoffs are expressed as (Anil’s, Bala’s). If both grow rice, payoffs are (2, 3). If Anil grows rice and Bala grows cassava, payoffs are (4, 4). If Anil grows cassava and Bala grows rice, payoffs are (6, 6). If both grow cassava, payoffs are (3, 2). If Bala is going to choose Rice, Anil’s best response is to choose Cassava.
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Anil’s best response to Rice

If Bala is going to choose Rice, Anil’s best response is to choose Cassava. We place a dot in the bottom left-hand cell.

Anil’s best response to Cassava: This diagram shows Anil and Bala’s available actions, which are growing rice or growing cassava. Payoffs are expressed as (Anil’s, Bala’s). If both grow rice, payoffs are (2, 3). If Anil grows rice and Bala grows cassava, payoffs are (4, 4). If Anil grows cassava and Bala grows rice, payoffs are (6, 6). If both grow cassava, payoffs are (3, 2). If Bala is going to choose Rice, Anil’s best response is to choose Cassava. If Bala is going to choose Cassava, Anil’s best response is to choose Rice.
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Anil’s best response to Cassava

If Bala is going to choose Cassava, Anil’s best response is to choose Rice. Place a dot in the top right-hand cell. Anil does not have a dominant strategy.

Bala’s best responses: This diagram shows Anil and Bala’s available actions, which are growing rice or growing cassava. Payoffs are expressed as (Anil’s, Bala’s). If both grow rice, payoffs are (2, 3). If Anil grows rice and Bala grows cassava, payoffs are (4, 4). If Anil grows cassava and Bala grows rice, payoffs are (6, 6). If both grow cassava, payoffs are (3, 2). If Anil chooses Rice, Bala’s best response is to choose Cassava, and if Anil chooses Cassava Bala should choose Rice.
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Bala’s best responses

If Anil chooses Rice, Bala’s best response is to choose Cassava, and if Anil chooses Cassava, Bala should choose Rice. The circles show Bala’s best responses. He doesn’t have a dominant strategy either.

There are two Nash equilibria: This diagram shows Anil and Bala’s available actions, which are growing rice or growing cassava. Payoffs are expressed as (Anil’s, Bala’s). If both grow rice, payoffs are (2, 3). If Anil grows rice and Bala grows cassava, payoffs are (4, 4). If Anil grows cassava and Bala grows rice, payoffs are (6, 6). If both grow cassava, payoffs are (3, 2). If Bala is going to choose Rice, Anil’s best response is to choose Cassava. If Bala is going to choose Cassava, Anil’s best response is to choose Rice. If Anil chooses Rice, Bala’s best response is to choose Cassava, and if Anil chooses Cassava Bala should choose Rice.
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There are two Nash equilibria

If Anil chooses Cassava and Bala chooses Rice, both of them are playing best responses (a dot and a circle coincide). So (Cassava, Rice) is a Nash equilibrium. But so is (Rice, Cassava).

In some coordination games, the players want to coordinate on the same action as each other; alternatively, as in this example, they want to make the opposite choice.

coordination game
A game in which there are two Nash equilibria, one of which may be Pareto superior to the other. Also known as: assurance game.

Whatever their neighbour does, Anil and Bala each prefer to do the opposite. (Cassava, Rice) and (Rice, Cassava) are both Nash equilibria. This is a coordination game: each player would like to ensure that their action coordinates with their opponent’s action.

Which equilibrium would we expect to observe in this game?

It is clear that the Nash equilibrium (Cassava, Rice), where they specialize in the crop they produce best, is preferred to the other Nash equilibrium, (Rice, Cassava), by both farmers.

Could we say, then, that we would expect Anil and Bala to engage in the best division of labour between the two crops? Not necessarily. Remember, we are assuming that they take their decisions independently, without communicating. Imagine that Bala’s father had been especially good at growing cassava (unlike his son) and so the land remained dedicated to cassava even though it was better suited to rice. In response to this, Anil knows that Rice is his best response to Bala’s Cassava: he decides to grow rice. Bala would have no incentive to switch to what he is good at: growing rice.

The example makes an important point. If there is more than one Nash equilibrium, and if people choose their actions independently, then the players can get ‘stuck’ in an equilibrium in which all players are worse off than they would be at the other equilibrium. We would not call the game in Figure 4.21 an invisible hand game—the players may not reach the outcome that is best for both of them.

Coordination games with conflicts of interest

A conflict of interest occurs in a coordination game if players in the game would prefer different Nash equilibria.

To understand this point, consider the case of Astrid and Bettina, two software engineers who are working on a project for which they will be paid. Their first decision is whether the code should be written in Java or C++ (either programming language is equally suitable, and the project can be written partly in one language and partly in the other). They each have to choose one program or the other, but Astrid wants to write in Java because she is better at writing Java code. While this is a joint project with Bettina, her pay will be partly based on how many lines of code are written by her. Unfortunately, Bettina prefers C++ for the same reason.

The two strategies are Java and C++. The interaction and pay-offs (thousands of dollars on completion of the project) are shown in Figure 4.22.

There are two diagrams. Diagram 1 shows Astrid and Bettina’s available actions, which are the Java or C++ programming languages. If both choose Java, they work in the same language but Astrid benefits more because she is better at Java coding. If Astrid chooses Java and Bettina chooses C++, each is working in the language they are better at but working in different languages is less productive than if both work in the same language. If Astrid chooses C++ but Bettina chooses Java, each is working in the language they are less good at, so neither works fast, and working in different languages is less productive. If both choose C++, both work in the same language but Bettina benefits more because she is better at C++ coding. Diagram 2 shows Astrid and Bettina’s available actions, which are the Java or C++ programming languages. Payoffs are expressed as (Astrid’s, Bettina’s). If both use Java, payoffs are (4, 3). If Astrid chooses Java and Bettina chooses C++, payoffs are (2, 2). If Astrid chooses C++ and Bettina chooses Java, payoffs are (0, 0). If both choose C++, payoffs are (3, 6).
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Figure 4.22 Interactions and pay-offs in the choice of programming language.

From Figure 4.22, you can work out three things:

  • They both do better if they work in the same language.
  • Astrid does better if that language is Java, while the reverse is true for Bettina.
  • Their total pay-off is higher if they choose C++.

How would we predict the outcome of this game?

If you use the dot-and-circle method, you will find that each player’s best responses are to choose the same language as the other player. So there are two Nash equilibria. In one, both choose Java. In the other, both choose C++.

In this situation, we might expect the two programmers to discuss their decisions, so that they can coordinate on the same language. But which Nash equilibrium would they choose? Astrid obviously prefers that they both play Java while Bettina prefers that they both play C++. This is another example of a negotiation with a conflict of interest, although unlike the example in Section 4.10, they do not appear to have a 50-50 option. But the total pay-off from the project is higher if both choose C++. If they could agree that both would use C++, perhaps they could also agree to split the proceeds in a way that would make both of them content with the outcome.

In Exercise 4.15, you can compare the outcome of such a negotiation with what might happen under other conditions affecting their decisions.

Exercise 4.15 Conflict between Astrid and Bettina

What is the likely result of the game in Figure 4.22 if:

  1. Astrid can choose which language she will use first, and commit to it (just as the Proposer in the ultimatum game commits to an offer, before the Responder responds)?
  2. the two can make an agreement, including which language they use, and how much cash can be transferred from one to the other?
  3. they have been working together for many years, and in the past they used Java on joint projects?

Exercise 4.16 Conflict in business

In the 1990s, Microsoft battled Netscape over market share for their web browsers, called Internet Explorer and Navigator. In the 2000s, Google and Yahoo fought over which company’s search engine would be more popular. In the entertainment industry, a battle called the ‘format wars’ played out between Blu-ray and HD-DVD.

Use one of these examples to analyse whether there are multiple equilibria and, if so, why one equilibrium might emerge in preference to the others.