6.2 Coordinating economic interactions

Devising policies to promote people’s well-being requires knowing the difference between situations in which self-interest can promote general well-being (like buying a shirt) and cases in which it leads to undesirable results (like the drying up of the Great Salt Lake).

To analyze and compare these different types of problems, we return to game theory, as introduced in Chapter 1 and Chapter 2.

Imagine two farmers, Ana and Ben, who face a problem: should they grow soy or corn? We assume they can grow both crops, but they can grow only one at a time. For simplicity, we assume that Ana and Ben are the only farmers in their local economy.

interdependence principle, principle of interdependence: The outcomes people obtain in economic interactions depend on the actions that they and others take in response to each other and on what they believe about the future.
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.
payoff table, game table: A table whose rows and columns list the actions (or strategies) available to the players; each cell gives the payoffs each player receives for that combination of actions.
simultaneous game: A game in which the players choose their strategies simultaneously: for example, the prisoners’ dilemma. See also: sequential game.

Ana’s land is equally suitable for growing soybeans and corn. Ben’s land is good for producing soy but less suitable for corn. They both sell whichever crop they produce in local agricultural markets. On market day, if they bring less soy to the market, the price of soy will be higher. Likewise, the price of corn depends on how much corn they have grown. Ana and Ben’s interaction, therefore, demonstrates the interdependence principle in action: what each person gets depends on what they do and on what other people do.

The farmers choose what to grow independently, which means they do not meet to discuss their strategy or course of action. This assumption may seem odd in an example with just two farmers, but understanding what happens when two players act independently provides insight into the problems that arise when many people interact.

Figure 6.1 illustrates the farmers’ interaction using a game table: a table with rows and columns that depicts the players, their strategies, and how each player’s payoff depends on the strategies chosen. Game tables are different from the game trees you learned previously, but they convey similar information. Game tables summarize each player’s possible actions and payoffs. Generally, we use game trees to depict sequential games and game tables to depict simultaneous games, in which players make their choices at the same time or independently commit to their choices. We call their interaction the “corn–soy game.”

Everyday economics 6.2

Gluts occur when a lot of a commodity is produced, and the price decreases. Sometimes the price is so low that it drives the producers, such as farmers, out of business. A shortage occurs when too little of a commodity is produced, and the price increases. Gluts often occur because people make plans about the future based on current information. For example, corn and soy farmers in the US have responded to increases in the prices of those crops—driven by regulations on biofuels and exporters seeking to ship soybeans to other countries—in the hope that the same price will hold next year and they can make more profit. But if everyone plants more of the same crop, the price will go down due to a glut.

Ana’s strategies are the rows of the table, and Ben’s strategies are the columns. We call Ana the row player and Ben the column player. When an interaction is represented in a table like Figure 6.1, each entry describes the outcome of a hypothetical situation. For example, the upper-left cell poses a hypothetical question and summarizes the answer.

Question: Suppose that Ana (row) plants soy and Ben (column) plants soy. What will the market outcomes be, and what will each person get?
Answer: Because they both plant the same crop, there is a glut, meaning that a lot of the crop is produced. When farmers grow a lot of one crop, the market price will be low. Because the price is low, they each make low profits. Because neither of them produces corn, there is a shortage of corn (and the price of corn will be high).

There are four possible hypothetical situations, each with specific outcomes. Figure 6.1 summarizes what will happen in each case.

This diagram shows a payoff matrix summarizing the outcomes of the corn-soy game between Ana and Ben. Ana’s choices, soy or corn, form the rows, and Ben’s choices, soy or corn, form the columns. If both choose soy, both produce soy, resulting in a glut of soy with low prices and a shortage of corn. If Ana chooses soy and Ben chooses corn, there is no market glut and prices are high for both crops, but Ben produces the crop for which his land is unsuited. If Ana chooses corn and Ben chooses soy, there is no market glut and prices are high for both crops. If both choose corn, both produce corn, resulting in a glut of corn with low prices and a shortage of soy, and Ben produces the crop for which his land is unsuited.

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-1

Figure 6.1 Planting soy or corn (the corn–soy game): a social interaction between Ana and Ben when planting soy or corn.

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-1a

The players and their strategies

We start with the strategies of each player before we think about what occurs in each cell. Make sure you understand which actions Ana (the row player) can take and which actions Ben (the column player) can take.

Outcome when both plant soy, (Soy, Soy): This diagram builds on the previous step. The cell for the outcome where both choose soy is filled in: both produce soy, resulting in a glut of soy with low prices and a shortage of corn.

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-1b

Outcome when both plant soy, (Soy, Soy)

Now consider what happens when Ana plays Soy and Ben plays Soy. In that case, there is a glut of soy (meaning it has a low price) and a shortage of corn (meaning corn has a high price).

Outcome when Ana plants soy and Ben plants corn, (Soy, Corn): This diagram builds on the previous steps. The cell for the outcome where Ana chooses soy and Ben chooses corn is added: there is no market glut, prices are high for both crops, but Ben produces a crop for which his land is less suited.

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-1c

Outcome when Ana plants soy and Ben plants corn, (Soy, Corn)

When Ana plays Soy and Ben plays Corn, there is no market glut and no shortage of either crop. Both crops fetch high prices. Ben produces corn, the crop for which his land is unsuited—his land is better suited for producing soy (as explained in the text).

Outcome when both plant corn, (Corn, Corn): This diagram builds on the previous steps. The cell for the outcome where both choose corn is added: both produce corn, resulting in a glut of corn with low prices and a shortage of soy, and Ben produces the crop for which his land is less suited.

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-1d

Outcome when both plant corn, (Corn, Corn)

When Ana plays Corn and Ben plays Corn, there is a glut of corn and a shortage of soy. Soy prices are low and corn prices are high (if any corn is produced). Ben produces corn, the crop for which his land is unsuited—his land is better suited for producing soy.

Outcome when Ana plants corn and Ben plants soy, (Corn, Soy): This diagram builds on the previous steps to show the complete matrix. The cell for the outcome where Ana chooses corn and Ben chooses soy is added: there is no market glut and prices are high for both crops, with each farmer producing the crop for which their land is well suited.

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-1e

Outcome when Ana plants corn and Ben plants soy, (Corn, Soy)

When Ana plays Corn and Ben plays Soy, there is no glut of either crop. Both crops fetch high prices. Ben produces corn, the crop for which his land is better suited.

To simplify the model and isolate the strategic logic more clearly, we make the following assumptions:

There are no other people involved or affected in any way.
The choice of which crop to grow is the only decision that Ana and Ben need to make.
The players interact just once. (In other words, they play a one-shot game.)
The game is a simultaneous game. That is, the players make their decisions simultaneously, without knowing what the other player will do.
The players know exactly what will happen in each possible outcome.

Figure 6.2 shows the payoffs for Ana and Ben in each of the four hypothetical situations—specifically, the income (in thousands of dollars per month) they will receive if the hypothetical row and column actions are taken. Remember that their incomes come from selling the crops they produce, so they depend on market prices, which in turn depend on the total amount of each crop brought to market.

There are two diagrams. Diagram 1 is a payoff matrix showing Ana and Ben each choosing between soy and corn, with Ana’s choices as rows and Ben’s as columns. Each cell reports the players’ payoffs in thousands of dollars: if both choose soy, both receive 40; if Ana chooses soy and Ben chooses corn, Ana receives 60 and Ben receives 30; if Ana chooses corn and Ben chooses soy, both receive 60; if both choose corn, Ana receives 50 and Ben receives 20. Diagram 2 presents the same payoffs using diagonally split cells, with each player’s payoff shown in their own triangle within each cell.

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-2-a-and-b

Figure 6.2 (A) and (B) Coordinating economic interactions: the corn–soy game. (A) the payoffs from crop choice and (B) the payoffs to the players in the corn–soy game depicted with each player’s color highlighted and their payoff in the corresponding triangle in each cell. The player’s payoff is their monthly income in thousands of dollars.

The payoffs for (Soy, Soy): The top-left cell is filled: when both choose soy, both receive payoffs of (40, 40).

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-2a

The payoffs for (Soy, Soy)

When both And and Ben produce Soy, there is a glut of soy and the price is low, resulting in the payoffs of (40,40). Ben produces the crop his land is better suited to, but his payoff is not as high as it would be if there were no glut of soy.

Adding the payoffs for (Soy, Corn): The top-right cell is added: when Ana chooses soy and Ben chooses corn, payoffs are (60, 30).

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-2b

Adding the payoffs for (Soy, Corn)

When Ana produces Soy and Ben produces Corn, there is no glut of either crop. Prices are favorable for both players, resulting in the payoffs of (60,30). Ben produces the crop (corn) for which his land is unsuited, so his payoff is not as great as it would be if he were producing soy and there is a glut.

Adding the payoffs for (Corn, Corn): The bottom-right cell is added: when both choose corn, payoffs are (50, 20).

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-2c

Adding the payoffs for (Corn, Corn)

When both produce corn, there is a glut of corn, and so prices are low. Because Ben’s land is not suited to the production of corn, his production is less than Ana’s and Ana gets a higher payoff than Ben does. (Her land is equally suited to corn and soy production.) The resulting payoffs are (50,20).

Adding the payoffs for (Corn, Soy): The bottom-left cell is added: when Ana chooses corn and Ben chooses soy, payoffs are (60, 60).

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-2d

Adding the payoffs for (Corn, Soy)

When Ana produces corn and Ben produces soy, Ben is producing the crop that his land is better suited to produce. Because they each produce a different crop, there is no glut and the prices are favorable to both producers, resulting in the payoffs of (60,60).

The completed payoff table: The full 2×2 payoff matrix is shown with all outcomes: (Soy, Soy) = (40,40), (Soy, Corn) = (60,30), (Corn, Soy) = (60,60), and (Corn, Corn) = (50,20).

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https://books.core-econ.org/uoe-101/06-02.html#figure-6-2e

The completed payoff table

We can now depict the game in a payoff table with each player receiving the payoff indicated. For the row player (Ana), payoffs are listed first in the pale-blue triangle in each cell. For the column player (Ben), payoffs are listed in the pink triangle in each cell. These payoffs result from the outcomes that we expect when each player plays their strategy, and there is a glut when they produce the same crop, or there is no glut when they produce different crops.

Everyday Economics 6.3

Think about choosing a major at your college or university. If many students choose the same major, there may be a “glut” of graduates competing for similar jobs, which drives down salaries in that field. If you expect many of your classmates to choose a business major, it might pay to consider engineering or nursing instead. You are playing a coordination game with thousands of other students you will never meet, mediated by the price signal of wages. What signals do you use when choosing your courses or major? Are those signals like the market prices Ana and Ben respond to? If so, how?

Because the price falls when the market is flooded with a single crop (a glut), the total payoffs to Ana and Ben are lower when both produce the same crop than when they produce different crops.
Because Ben’s land is less suitable for corn, the highest payoffs are obtained when Ben specializes in soy and Ana specializes in corn.

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

Exercise 6.2 Python or C++? A game of specialization and strategy

Ana and Ben are coders who each work on contracts independently. Ana is better at Python than C++; Ben is better at C++ than Python. A client offers them a project that may require both Python and C++ programming. They can only choose one language each.

Draw a game table showing the four possible combinations of language choices (Python or C++) that Ana and Ben could each make. In each cell, write a brief description explaining the expected outcome for each player (you do not need to use specific numbers). Consider how a “glut” in one language (both choosing the same language) will affect each player’s payoff. You do not need to use specific numbers, but if you find it helpful, assign illustrative payoffs to each cell so that the ranking of outcomes reflects two ideas: (i) the project is best when both languages are used, and (ii) each coder is more productive in their stronger language. Then check your table against the worked solution.
If the client needs only one language—either Python or C++—how will the nature of the interaction between Ana and Ben change? Would this still be a strategic interaction? Explain.

Question 6.1

In a simultaneous one-shot game, players …

A 2×2 payoff matrix for Ana (rows) and Ben (columns), where each player chooses between Soy and Corn. Payoffs are shown in each cell, with Ana’s payoff in the bottom-left triangle and Ben’s payoff in the top-right triangle. If both choose Soy, the payoffs are (2, 4). If Ana chooses Soy and Ben chooses Corn, the payoffs are (5, 7). If Ana chooses Corn and Ben chooses Soy, the payoffs are (5, 6). If both choose Corn, the payoffs are (3, 1).

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Observe what others do before deciding how to act.
Can change their action after observing what other players have chosen.
Can coordinate with other players before making a choice.
Take into account the other players’ possible actions.

In a simultaneous game, players choose at the same time, so they cannot observe what others do before deciding. The chapter notes players “make their choices at the same time or independently commit to their choices.”
Because choices are simultaneous, a player cannot wait to see another’s action and then change their own; that describes a sequential game (a game tree like in Chapters 1 and 2), not a simultaneous one.
The farmers “do not meet to discuss their strategy”; players choose independently and so cannot coordinate beforehand.
Even without observing others’ actions, a player anticipates the other players’ possible actions and chooses a best response to what they expect. This reflects the interdependence principle: what each person gets depends on what they do and on what others do.

Question 6.2

We re-use the figure in Question 6.1 to answer this question.

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What is the outcome if Ana chooses to grow corn and Ben chooses to grow soy?

Ana gets 5 and Ben gets 7.
Ana gets 3 and Ben gets 1.
Ana gets 5 and Ben gets 6.
Ana gets 2 and Ben gets 4.

Ana 5, Ben 7 is the cell where Ana grows soy and Ben grows corn—the two crops have been swapped. Re-read the row (Ana) and column (Ben) labels.
Ana 3, Ben 1 is the cell where both grow corn. Check which crop Ben is growing in this question.
In the Corn (Ana) / Soy (Ben) cell, Ana’s payoff is 5 and Ben’s is 6. Locating the cell from the row label (Ana = Corn) and column label (Ben = Soy) gives this outcome.
Ana 2, Ben 4 is the cell where both grow soy. Here Ana grows corn, not soy.