4.2 Social interactions: Game theory

On which side of the road should you drive? If you live in Japan, the UK, or Indonesia, you drive on the left. If you live in South Korea, France, or the US, you drive on the right. If you grew up in Sweden, you drove on the left until 5 p.m. on 3 September 1967, and at 5.01 p.m., you started driving on the right. The government sets a rule, and we follow it.

But suppose we just left the choice to drivers to select one side of the road or the other. If everyone else was already driving on the right, self-interest (avoiding a collision) would be a major motivating factor in leading people to drive on the right as well.

Devising policies to promote people’s wellbeing requires an understanding of the difference between situations in which self-interest can promote general wellbeing, and cases in which it leads to undesirable results. To analyse this, we will introduce game theory, a way of modelling how people interact.

In some economic models there is just one decision-maker—like the model in Unit 3 of a worker (Karim) deciding on his working hours. Karim faces a set of feasible options determined by his budget constraint, and chooses the best possible outcome for himself—which does not depend on what anyone else decides to do. He is not engaged in a social interaction.

social interactions: Situations in which the actions taken by each person affect other people’s outcomes as well as their own.

Social interactions are situations in which there are two or more people, and the actions taken by each person affect both their own outcome and other people’s outcomes. For example, one person’s choice of how much to heat their home will affect everyone’s experience of global climate change.

We use four terms:

strategic interaction: A social interaction in which the participants are aware of the ways in which their actions affect others (and the ways in which the actions of others affect them).
strategy: An action (or action plan) that a person may choose, while being aware that the outcomes for themselves and others depend on their own strategy and the strategies chosen by others.
game: A model of strategic interaction that describes the players, the feasible strategies, the order of play, the information that the players have, and their pay-offs. See also: game theory.
game theory: A branch of mathematics that studies strategic interactions, meaning situations in which each actor knows that the benefits they receive depend on the actions taken by all. See also: game.

When people are engaged in a social interaction and are aware of the ways that their actions affect others, and vice versa, we call this a strategic interaction.
A strategy is defined as an action (or action plan) that a person may choose while being aware of the mutual dependence of the outcomes on their own and others’ actions.
Models of strategic interactions are described as games.
Game theory is a set of models of strategic interactions. It is widely used in economics and elsewhere in the social sciences.

Read the article ‘Game Theory in Economics and Beyond’ to learn about how game theory is used in other disciplines, including political science, biology, philosophy, and computer science.

To understand how game theory can clarify strategic interactions, imagine two farmers, who we will call Anil and Bala. They face a problem: should they grow rice or cassava? We assume that they have the ability to grow both types of crop, but can only grow one type at a time.

Anil’s land is equally suitable for growing rice and cassava. Bala’s land is likewise good for producing rice, but less suitable for cassava. They both sell whatever crop they produce in a nearby village market. On market day, if they bring less rice to the market, the price will be higher. Likewise, the price of cassava depends on how much cassava they have grown.

The farmers choose what to grow independently, which means they do not meet together to discuss a course of action. This assumption may seem odd in a model of just two farmers, but understanding what happens when players act independently will give us insight into problems in which many people—billions, in the case of climate change—interact.

Game

A description of a social interaction, which specifies:

the players: who is interacting with whom
the feasible strategies: which actions are open to the players
the order of play: when the players choose their actions
the information: what each player knows when making their decision
the pay-offs: what the outcomes will be for each of the possible combinations of actions.

Figure 4.1 describes the farmers’ interaction, which is what we call a game.

Anil’s choices are the rows of the table and Bala’s are the columns. We call Anil the ‘row player’ and Bala the ‘column player’. When an interaction is represented in a table like Figure 4.1, each entry describes the outcome of a hypothetical situation. For example, the upper-left cell should be interpreted as:

‘Suppose (for whatever reason) Anil planted rice and Bala planted rice, too. What would the outcome be?’

There are four possible hypothetical situations. Figure 4.1 describes what would happen in each case.

This diagram shows Anil and Bala’s available actions, which are growing rice or growing cassava. If Anil and Bala grow rice, there is a glut of rice, which sells at a low price, and a shortage of cassava. Moreover, Anil is better able to produce cassava than rice. If Anil grows rice and Bala grows cassava, there is no market glut, and rice and cassava sell at high prices. However, Anil and Bala produce the crop for which they are less suited. If Anil grows cassava and Bala grows rice, there is no market glut, and rice and cassava sell at high prices. Moreover, Anil and Bala produce the crop for which they are better suited. If Anil and Bala grow cassava, there is a glut of cassava, which sells at a low price, and a shortage of rice. Moreover, Bala is better able to produce rice than cassava.

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https://www.core-econ.org/microeconomics/04-strategic-interactions-02-game-theory.html#figure-4-1

Figure 4.1 Planting rice or cassava: social interactions between Anil and Bala.

To simplify the model, we assume the following:

simultaneous game: A game in which the players choose their strategies simultaneously, for example, the prisoners’ dilemma. See also: sequential game.
pay-off: The pay-off for an individual player in a game is the benefit that the player receives as a result of the joint actions of all the players.

There are no other people involved or affected in any way.
The selection of which crop to grow is the only decision that Anil and Bala need to make.
They interact just once. (This is called a ‘one-shot game’.)
It is a simultaneous game: the players make their decisions simultaneously, not knowing what the other person has decided to do.
They know exactly what would happen in each possible outcome.

Figure 4.2a shows the pay-offs for Anil and Bala in each of the four hypothetical situations—the incomes they would receive if the hypothetical row and column actions were taken. Remember that their incomes come from selling the crops they produce, so they depend on market prices, and the prices depend on the total amount of each crop brought to market.

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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https://www.core-econ.org/microeconomics/04-strategic-interactions-02-game-theory.html#figure-4-2a

Figure 4.2a The pay-offs from crop choice.

Because the price falls when the market is flooded with one crop, the total pay-offs to the two farmers are lower in the cases when both produce the same crop than in the cases when they produce different crops.
Since Bala’s land is less suitable for cassava, the highest pay-offs are obtained when Bala specializes in rice and Anil specializes in cassava.

Having established the rules of the game—who can do what, when they can do it, and how each player’s actions determine their pay-off—we can go on to consider what the players will choose to do, and what the outcome of the game might be.

Question 4.1 Choose the correct answer(s)

In a simultaneous one-shot game:

A player observes what others do before deciding how to act.
A player can change their action after observing what other players have chosen.
Players coordinate among themselves to choose the actions that lead to a mutually beneficial outcome.
A player chooses an action taking into account the possible actions that other players can take.

A simultaneous game means that players all make a decision on their action simultaneously, without knowing what the other player(s) have chosen.
In a one-shot game, each person only takes action once, without knowing what the other player(s) have chosen, and cannot change their action afterwards.
The players take actions non-cooperatively, driven by self-interest. The outcome may not necessarily be the best possible outcome for each player.
An essential element of strategic games is that each player takes into account the possible actions of other players, when the actual choices made are unknown.

Unit 4 Strategic interactions and social dilemmas