Unit 4 Strategic interactions and social dilemmas

4.3 Best responses in the rice–cassava game: Nash equilibrium

best response
In game theory, a player’s best response is the strategy that will bring about the player’s most-preferred outcome, given the strategies adopted by the other players.

Game theory describes social interactions; it can also help to predict what will happen. For this, we need the concept of best response: the strategy that will lead to a player’s most preferred outcome, given the strategies that the other players select.

In Figure 4.2b, we represent the pay-offs for Anil and Bala in the rice–cassava game using a standard format called a pay-off matrix. A matrix is just any rectangular array of numbers. The first (bottom-left) number in each box is the reward for the row player (whose name begins with A as a reminder that their pay-off is first). The second number is the column player’s pay-off. Each pay-off depends on how much of the crop the player can grow, but also on demand: that is, how the market price varies with the amount available.

Think about best responses in this game. Suppose you are Anil, and you consider the hypothetical case in which Bala has chosen to grow rice. Which response yields you the higher pay-off? You would grow cassava (receiving a pay-off of 6, compared with only 4 if you grew rice instead).

Work through the steps in Figure 4.2b to find the best responses in each hypothetical situation, using a handy method for keeping track of best responses by placing dots and circles in the pay-off matrix.

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 (4, 4). If Anil grows rice and Bala grows cassava, pay-offs are (6, 3). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 2). Growing cassava is Anil’s best response to Bala growing rice. Growing rice is Anil’s best response to Bala growing cassava. Growing rice is Bala’s best response to Anil growing rice. Growing rice os Bala’s best response to Anil growing cassava.
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Figure 4.2b Finding best responses in the rice–cassava game.

Finding best responses: 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 (4, 4). If Anil grows rice and Bala grows cassava, pay-offs are (6, 3). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 2).
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Finding best responses

Begin with the row player (Anil) and ask: ‘What would be his best response if the column player (Bala) decided to play Rice?’

Anil’s best response if Bala grows rice: 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 (4, 4). If Anil grows rice and Bala grows cassava, pay-offs are (6, 3). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 2). Growing cassava is Anil’s best response to Bala growing rice.
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Anil’s best response if Bala grows rice

If Bala chooses Rice, Anil’s best response is Cassava—that gives him 6, rather than 4. Place a dot in the bottom left-hand cell. A dot in a cell means that this is the row player’s best response.

Anil’s best response if Bala grows cassava: 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 (4, 4). If Anil grows rice and Bala grows cassava, pay-offs are (6, 3). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 2). Growing cassava is Anil’s best response to Bala growing rice. Growing rice is Anil’s best response to Bala growing cassava.
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Anil’s best response if Bala grows cassava

If Bala chooses Cassava, Anil’s best response is Rice—giving him 6, rather than 5. Place a dot in the top right-hand cell.

Now find the column player’s best responses: 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 (4, 4). If Anil grows rice and Bala grows cassava, pay-offs are (6, 3). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 2). Growing cassava is Anil’s best response to Bala growing rice. Growing rice is Anil’s best response to Bala growing cassava. Growing rice is Bala’s best response to Anil growing rice.
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Now find the column player’s best responses

If Anil chooses Rice, Bala’s best response is to choose Rice too (4 rather than 3). Circles represent the column player’s best responses. Place a circle in the upper left-hand cell.

Bala’s best response is the same if Anil chooses cassava: 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 (4, 4). If Anil grows rice and Bala grows cassava, pay-offs are (6, 3). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 2). Growing cassava is Anil’s best response to Bala growing rice. Growing rice is Anil’s best response to Bala growing cassava. Growing rice is Bala’s best response to Anil growing rice. Growing rice is Bala’s best response to Anil growing cassava.
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Bala’s best response is the same if Anil chooses cassava

Lastly, if Anil chooses Cassava, Bala’s best response is Rice again (he gets 6 rather than 2). Place a circle in the lower left-hand cell.

Mutual best responses: 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 (4, 4). If Anil grows rice and Bala grows cassava, pay-offs are (6, 3). If Anil grows cassava and Bala grows rice, pay-offs are (6, 6). If both grow cassava, pay-offs are (5, 2). Growing cassava is Anil’s best response to Bala growing rice. Growing rice is Anil’s best response to Bala growing cassava. Growing rice is Bala’s best response to Anil growing rice. Growing rice is Bala’s best response to Anil growing cassava.
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Mutual best responses

The dot and circle coincide in the lower left-hand cell. If Anil chooses Cassava and Bala chooses Rice, the players are playing best responses to each other.

equilibrium
An equilibrium is a situation or model outcome that is self-perpetuating: if the outcome is reached it does not change, unless an external force disturbs it. By an ‘external force’, we mean something that is determined outside the model.

The method used in Figure 4.2b reveals that there is one pair of strategies which are best responses to each other: Anil chooses Cassava and Bala chooses Rice. We call this pair of strategies an equilibrium of the game. In general, an equilibrium is a self-perpetuating situation. In this case, Anil choosing Cassava and Bala choosing Rice is an equilibrium, because neither of them would want to change their decision after seeing what the other player chose.

Nash equilibrium
A Nash equilibrium is an economic outcome where none of the individuals involved could bring about an outcome they prefer by unilaterally changing their own action. More formally, in game theory it is defined as a set of strategies, one for each player in the game, such that each player’s strategy is a best response to the strategies chosen by everyone else. See also Game theory.
invisible hand game
A game in which there is a single Nash equilibrium that is Pareto efficient may be called an invisible hand game. See also: Nash equilibrium, Pareto efficient.

In game theory, a set of strategies in which each player plays a best response to the strategies of the other players is called a Nash equilibrium. In the Nash equilibrium of this game:

  • Anil chooses Cassava
  • Bala chooses Rice.

We can use the shorthand (Cassava, Rice) to refer to this Nash equilibrium, listing the row player’s (Anil’s) strategy first. Can we predict that Anil and Bala will play their Nash equilibrium strategies? Thinking about the decision from Anil’s point of view suggests that he might not be sure what to do, because his best response depends on Bala’s decision (he wants to make the opposite choice from Bala). However, Bala’s decision is easier: whatever Anil does, Bala gets a higher pay-off from choosing Rice. So we would expect him to choose Rice. Then, if Anil thinks about Bala’s decision, he will also expect Bala to choose Rice. This simplifies the problem for Anil: his best response to Rice is Cassava.

Reasoning in this way suggests that the Nash equilibrium does give us a plausible prediction of the outcome of the game. Usually, if we find that a game has just one Nash equilibrium, it is the most plausible outcome—although we may feel more confident about this in some games than others.

In the Nash equilibrium, both Anil and Bala specialize in one crop, so market gluts are avoided. Moreover, both crops are produced on land well suited for growing them. Simply pursuing their self-interest—that is, choosing the strategy giving them the highest pay-off—results in an outcome with the highest possible pay-off for each player.

In this example, the Nash equilibrium is also the outcome that each would have chosen if they had a way of coordinating their decisions. Although they independently pursued their self-interest, they were guided by market prices to an outcome that was in their best interests.

Figure 4.2 is an example of a type of game that is sometimes called an invisible hand game, because it reflects Adam Smith’s idea that forces that are not explicit (‘invisible’) can guide the players to the outcome that is best for both of them. An invisible hand game has the property that players acting independently in their own self-interest reach an equilibrium that is in the joint interest of the players involved.

Pursuing self-interest without regard for others is sometimes considered to be morally bad, but economic analysis has identified conditions in which it leads to socially desirable outcomes. In the next section we discuss a contrasting case where the outcome of self-interested decisions is not in the self-interest of any of the players: the prisoners’ dilemma game.

Question 4.2 Choose the correct answer(s)

Adam likes going to the cinema more than watching football. Bella, on the other hand, prefers watching football to going to the cinema. If one person chooses their favourite activity, the other person prefers to spend time together rather than spend an afternoon apart. The following table represents the enjoyment levels (pay-offs) of Bella and Adam, depending on their choice of activity (the first number is Adam’s enjoyment level while the second number is Bella’s).

This diagram shows Adam and Bella’s available actions, which are football or cinema. Payoffs are expressed as (Adam’s, Bella’s). If both choose football, payoffs are (3, 5). If Adam chooses football and Bella chooses cinema, payoffs are (1, 1). If Adam chooses cinema and Bella chooses football, payoffs are (4, 3). If both choose cinema, payoffs are (6, 2).
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Based on the information above, we can conclude that:

  • Adam would be better off choosing Football, regardless of whatever Bella chooses.
  • The Nash equilibrium is (Cinema, Football); in other words, Adam chooses Cinema, Bella chooses Football.
  • The Nash equilibrium yields the highest possible enjoyment levels for both players.
  • Neither player would want to deviate from the Nash equilibrium.
  • Adam would be better off choosing Cinema, regardless of whatever Bella chooses (if Bella chooses Football, he receives 4 from choosing Cinema instead of 3 from choosing Football; if Bella chooses Cinema, he receives 6 from Cinema instead of 1 from Football).
  • The Nash equilibrium is (Cinema, Football): if Adam chooses Cinema, Bella’s best response is to choose Football (pay-off of 3 vs 2); if Bella chooses Football, Adam’s best response is to choose Cinema (pay-off of 4 vs 3).
  • In the Nash equilibrium, Adam receives a pay-off of 4 and Bella receives a pay-off of 3. Adam would attain the highest happiness level (pay-off of 6) if they could agree to go to the cinema together. Similarly, Bella would be happiest (pay-off of 5) if they could both agree to watch football.
  • This statement is a property of the Nash equilibrium. Players are doing the best they can, given what all other players are doing, so each individual would not benefit from unilaterally changing his or her current action.

Great Economists John Nash

Portrait of John Nash

John Nash (1928–2015) completed his doctoral thesis at Princeton University at the age of 21. It was just 27 pages long, yet it advanced game theory (then a little-known branch of mathematics) in ways that led to a dramatic transformation of economics. He addressed the question: when people interact strategically, what would one expect them to do? His answer was what we now call a Nash equilibrium, in which each person’s strategy is to do the best they can for themselves, given the strategies of all other players.

Nash proved that an equilibrium must exist in a wide general class of games, where players could have any goals whatsoever: they could be selfish, altruistic, spiteful, or fair-minded, for instance. His proof was remarkable, because distinguished mathematicians of the twentieth century such as Emile Borel and John von Neumann had tackled the problem without getting very far. This result is useful because strategies can be very complicated, specifying a complete plan of action for any situation that could possibly arise. Although the number of distinct strategies in chess is greater than the number of atoms in the known universe, we know that chess has a Nash equilibrium, although not whether the equilibrium involves a win for white, a win for black, or a guaranteed draw.

Game theory has completely transformed almost all fields of economics, and this would have been impossible without Nash’s equilibrium concept and existence proof. But it was not his only path-breaking contribution to economics—he also made a brilliantly original contribution to the theory of bargaining, and other pioneering contributions to mathematics for which he received the prestigious Abel Prize.

Nash shared the 1994 Nobel Prize for his work. Roger Myerson, an economist who later won the prize himself, described the Nash equilibrium as ‘one of the most important contributions in the history of economic thought’.

For much of his life, Nash suffered from mental illness that required hospitalization. He experienced hallucinations caused by schizophrenia that began in 1959, though after what he described as ‘25 years of partially deluded thinking’ he returned to teaching and research at Princeton. The story of his insights and illness are told in the book (made into a film starring Russell Crowe) A Beautiful Mind.1

Exercise 4.2 Game theory in A Beautiful Mind

In the movie A Beautiful Mind, John Nash comes up with the idea of game theory through a conversation with his friends at a bar over which woman they should speak to.

  1. Watch this YouTube video of that scene in the movie and draw a pay-off matrix to represent the situation described. (Hint: You may find this article and explanation helpful.)
  2. By referring to your pay-off matrix, explain whether John Nash’s proposal to his friends (in the movie) is or is not a Nash equilibrium.
  1. Sylvia Nasar. 2011. A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash. New York: Simon & Schuster.