Unit 4 Strategic interactions and social dilemmas

4.4 Dominant strategy equilibrium and the prisoners’ dilemma

In the rice–cassava game in Figure 4.2b, Bala’s best response is to choose Rice, whatever his opponent chooses to do. When this happens, we say that the player has a dominant strategy. Bala’s dominant strategy is Rice.

Figure 4.3 shows the pay-offs in another version of the rice–cassava game. Here we have assumed that not only is Bala’s land better for growing rice; also, Anil’s land is better for cassava. Use the dot-and-circle method to work out the best responses. You should find that in this game, both players have dominant strategies.

This diagram shows Anil and Bala’s available actions, which are growing rice or growing cassava. Pay-offs are expressed as (Anil’s, Bala’s). If both grow rice, pay-offs are (3, 5). If Anil grows rice and Bala grows cassava, pay-offs are (4, 4). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 3).

Figure 4.3 Another rice–cassava game: dominant strategies for both players.

dominant strategy equilibrium
A dominant strategy equilibrium is a Nash equilibrium in which the strategies of all players are dominant stategies.

You should also find that the Nash equilibrium is the same as before: Anil chooses Cassava and Bala chooses Rice. But because Cassava is a dominant strategy for Anil and Rice is a dominant strategy for Bala, we can be especially confident in predicting this outcome. We say that (Cassava, Rice) is not only a Nash equilibrium but also a dominant strategy equilibrium.

The dominant strategy equilibrium is beneficial for both players. Just like the game in the previous section, this is an invisible hand game. But in the next example, something very different happens.

The pest control game

Imagine that Anil and Bala are now facing another problem. Each is deciding how to deal with pest insects that destroy the crops they cultivate in their adjacent fields. Each has two feasible strategies:

  • The first is to use an inexpensive chemical called Toxic Tide. It kills every insect for miles around. Toxic Tide also leaks into the water supply that they both use.
  • The alternative is to use integrated pest control (IPC), in which beneficial insects are introduced to the farm. The beneficial insects eat the pest insects.

If just one of them chooses Toxic Tide, the damage is quite limited. If they both choose it, water contamination becomes a serious problem, and they need to buy a costly filtering system. Figure 4.4a describes their interaction.

This diagram shows Anil and Bala’s available actions, which are IPC or Toxic Tide. If Anil and Bala both choose IPC, beneficial insects spread over both fields, eliminating pests, and there is no water contamination. If Anil chooses IPC and Bala chooses Toxic Tide, Bala’s chemicals spread to Anil’s field and kill his beneficial insects, resulting in limited water contamination. If Anil chooses Toxic Tide and Bala chooses IPC, Anil’s chemicals spread to Bala’s field and kill his beneficial insects, resulting in limited water contamination. If Anil and Bala both choose Toxic Tide, all pests are eliminated but there is heavy water contamination, requiring a costly filtration system.

Figure 4.4a Social interactions in the pest control game.

Both Anil and Bala are aware of these outcomes. They know that their pay-off (the income from selling their crops, minus the costs of their pest control method and of any water filtration required), will depend not only on their own decision, but also on the other’s choice. This is a strategic interaction. The pay-offs are shown in Figure 4.4b.

This diagram shows Anil and Bala’s available actions. Pay-offs are expressed as (Anil’s, Bala’s). If both choose IPC, pay-offs are (3, 3). If Anil chooses IPC and Bala chooses Toxic Tide, pay-offs are (1, 4). If Anil chooses Toxic Tide and Bala chooses IPC, pay-offs are (4, 1). If both choose Toxic Tide, pay-offs are (2, 2).

Figure 4.4b Pay-off matrix for the pest control game.

How will they play the game? Again, we can use the dot-and-circle method to figure this out (draw the dots and circles in the matrix for yourself).

Anil’s best responses are:

  • if Bala chooses IPC: Toxic Tide (cheap eradication of pests, little water contamination)
  • if Bala chooses Toxic Tide: Toxic Tide (IPC costs more and cannot work since Bala’s chemicals will kill beneficial pests).

So Toxic Tide is Anil’s dominant strategy. You can check, likewise, that Toxic Tide is a dominant strategy for Bala. So we predict that both will use it. Both players using insecticide is the dominant strategy equilibrium of the game.

The predicted outcome is not good for either player: they each receive pay-offs of 2, but they would have been better off if both had used IPC, with pay-offs of 3 each.

prisoners’ dilemma
A prisoners’ dilemma is a game that has a dominant strategy equilibrium, but also has an alternative outcome that gives a higher pay-off to all players. So the Nash equilibrium is not Pareto efficient.

The pest control game, with its undesirable outcome, is a particular example of a game called the prisoners’ dilemma.

The prisoners’ dilemma

The name of this game comes from a story about two prisoners (we call them Thelma and Louise) whose strategies are either to Accuse (implicate) the other in a crime that the prisoners may have committed together, or Deny that the other prisoner was involved.

If both Thelma and Louise Deny, they will receive a 1-year sentence for a less serious crime.

If one person Accuses the other, while the other Denies, the accuser will be freed immediately (a sentence of 0 years), whereas the other person gets a long jail sentence (10 years).

Lastly, if both Thelma and Louise choose Accuse (meaning each implicates the other), they both get a jail sentence. This sentence is reduced from 10 years to 5 years because of their cooperation with the police. The pay-offs of the game are shown in Figure 4.5. (The pay-offs are the years in prison—so Louise and Thelma prefer lower numbers.)

This diagram shows Thelma and Louise’s available actions, which are Deny or Accuse. Pay-offs are expressed as (Thelma’s, Louise’s). If both choose Deny, pay-offs are (1, 1). If Thelma chooses Deny and Louise chooses Accuse, pay-offs are (10, 0). If Thelma chooses Accuse and Louise chooses Deny, pay-offs are (0, 10). If both choose Accuse, pay-offs are (5, 5).

Figure 4.5 Prisoners’ dilemma (pay-offs are years in prison).

In a prisoners’ dilemma, the dominant strategy equilibrium outcome (in this example, both Accuse) is worse for both than the opposite outcome (both Deny).

Our story about Thelma and Louise is hypothetical, but the prisoners’ dilemma applies to many real problems. In economic examples, the mutually beneficial strategy (Deny) is generally termed Cooperate, while the dominant strategy (Accuse) is called Defect. Note that ‘Cooperate’ is simply a name for the beneficial strategy; it does not mean they have somehow reached an agreement on how to play. Even if players can discuss their strategies beforehand, the rules of the game are that they make independent decisions.

external effect, externality
An external effect occurs when a person’s action confers a benefit or imposes a cost on others and this cost or benefit is not taken into account by the individual taking the action. External effects are also called externalities.

Economic interactions between self-interested players can result in desirable outcomes (as in the rice–cassava games) or undesirable ones (prisoners’ dilemmas, like the pest control game). Prisoners’ dilemma games model situations in which players reach undesirable outcomes, because they take no account of the costs and benefits of their actions for others: when this happens, we say that decisions have external effects. If Anil chose IPC, Bala would choose Toxic Tide because it is better for himself, ignoring the external costs his decision imposes on Anil. Later sections and units will explore how undesirable results of external effects may be avoided if the players have different preferences, or are influenced by social norms of behaviour, or can enter into binding agreements (contracts). Or policymakers may be able to help by changing the rules of the game, or the pay-offs.

Exercise 4.3 Playing the prisoners’ dilemma

In the UK TV quiz show called Golden Balls, two players are given the chance to split a cash prize or steal it from each other, but are allowed to communicate before making their final choice. Watch the clip of how one pair played this game, and answer the following questions.

  1. Draw a pay-off matrix to represent the game described in the video (players, feasible actions, pay-offs under each possible outcome; ignoring the communication round).
  2. With reference to the pay-off matrix you drew and the final outcome of this interaction, explain the reasoning behind the first player’s (Nick’s) proposed plan.
  3. How do you think the two players’ behaviour in the communication round and their actual choice would differ if the players were friends rather than strangers?

Question 4.3 Choose the correct answer(s)

Dimitrios and Ameera work for an international investment bank as foreign exchange traders. They are being questioned by the police on their suspected involvement in a series of market manipulation trades. The table below shows the cost of each strategy (in terms of the length in years of jail sentences they will receive), depending on whether they accuse each other or deny the crime. The first number is the pay-off to Dimitrios, while the second number is the pay-off to Ameera (the negative numbers signify losses). Assume that the game is a simultaneous one-shot game.

This diagram shows Dimitrios and Ameera’s available actions, which are Deny or Accuse. Payoffs are expressed as (Dimitrios’, Ameera’s). If both choose Deny, payoffs are (-2, -2). If Dimitrios chooses Deny and Ameera chooses Accuse, payoffs are (-15, 0). If Dimitrios chooses Accuse and Ameera chooses Deny, payoffs are (0, -15). If both choose Accuse, payoffs are (-8, -8).

Based on this information, we can conclude that:

  • Both traders will hold out and deny their involvement.
  • Both traders will accuse each other, even though they will end up being in jail for eight years.
  • Ameera will accuse, whatever she expects Dimitrios to do.
  • There is a small possibility that both traders will get away with two years each.
  • Accuse is a dominant strategy for both Dimitrios and Ameera, so each of them will choose Accuse regardless of what the other player does.
  • For both Dimitrios and Ameera, Accuse is a dominant strategy. Therefore, the outcome in which they both Accuse and end up with 8-year sentences is a dominant strategy equilibrium.
  • Accusing is Ameera’s best response regardless of what Dimitrios does, so she will always Accuse. It is a dominant strategy.
  • This outcome can only happen if both Dimitrios and Ameera Deny. Accusing is the dominant strategy for both of them, so this would never happen.

Exercise 4.4 Political advertising

In many democracies, political candidates are allowed to use some forms of mass media, such as public or commercial television, to advertise their campaign and attack their opponent(s). Many people consider such political advertising (campaign advertisements) to be a classic example of a prisoners’ dilemma.

  1. Using examples from a recent political campaign from a country of your choice, explain whether you think this is the case.
  2. Write down an example pay-off matrix to illustrate this case.