Summary: In this experiment, students participate in a normal-form game with two pure-strategy equilibria that can be Pareto ranked. The experiment can be conducted outside of class on EconPort or during class with paper and pencil. This game's payoffs are setup to reflect a minimum-effort coordination game: two players each choose a level of effort, with the outcome determined by the lowest effort decision. Because effort is costly, one only wants to make a high-effort choice if all others do so also.
Motivation: After students are first introduced to the concept of equilibrium, they often struggle with the notion that a game can have two different equilibrium outcomes. Participating in the game gives students the opportunity to make decisions and thereby think more carefully about the incentives and strategic nature of the game. This game is most typically used in upper-level courses that use game-theory or to illustrate the concept of Pareto-ranking. However, it would also be useful in an introductory macroeconomics course to illustrate the Keynesian model, which asserts a difference between equilibrium income and potential income; students often are confused by the idea that more than one equilibrium outcome is possible, with some equilibria better than others.
Time Required: This experiment can be conducted in one 50-minute class. The experiment takes less than 20 minutes, and the subsequent discussion and lecture can easily be completed in less than 30 minutes.
- Determining Nash equilibrium strategies.
- Pareto optimality.
- How outcomes may be Pareto ranked.
- Institutional factors that affect outcomes.
A simple in-class coordination game experiment is described in "Coordination" by C. Monica Capra and Charles A. Holt, Southern Economic Journal, January 1999. Full details, including subject instructions, are included in their paper. This section highlights the main features of this classroom game.
- One deck of cards for every 26 students. Each deck should be ordered alternating a red card (heart or diamond) with a black card (club or spade).
- Written instructions (which include a record-sheet) for your students; this is found in the article referenced above.
- This experiment works well in almost any size class, as long as you can see all students and distinguish between red and black cards when they hold them up.
- This experiment works best if you conduct the experiment first, followed by class discussion and lecture.
- The instructions are intentionally neutral: no mention of effort (or the cost of effort) is made. The parallel between effort or other applicable contexts is made during the class discussion period.
- This experiment works fine with hypothetical earnings, or you can randomly select one student to pay a fraction of earnings.
The professor distributes two cards to each student: one red (heart or diamond) and one black (club or spade). Students play one card in each round, first by (privately) holding the card to their chest and when instructed revealing this card by holding it up facing the instructor at the front of the room. Playing a red card indicates making a low-effort decision and playing a black card indicates a high-effort decision. The high-effort outcome is obtained only if all players choose high effort (play a black card). However, making high-effort is costly: if one makes a high-effort but not all others do so, one incurs the cost of the high-effort, but still attains the low-effort outcome.
In various rounds of the game, the instructor can change the size of the group (pairs, or larger groups of students) and the value of making a costly effort.
Additional Instructions for the Professor:
- For the initial rounds of the game, students will earn $1 if they play a red card (low effort) and $4 if they play a black card (high effort) and the other does also, but $0 if they play a black card and the other plays a red card. This reflects a situation in which there is no cost to choosing a low-effort level, and the earnings from low-effort are $1. However, making high-effort costs $1, and results in earnings of $5 (minus $1 effort = $4) if the high-effort outcome is attained (i.e., both make high effort); if the low-effort outcome is attained, then the person earns $1 (low-effort earnings) minus $1 (the cost of making a high-effort) = $0.
- In the first round, ask about 10 students to make a decision simultaneously (for example, the first two rows of the class). Tell them they will be paired with one other person in their row, but do not tell them who they will be paired with. You will see that all have made their decisions when they are each holding a card to their chest. Then choose two people at a time to reveal their decisions, and instruct these students to write the other's decision (and their earnings) on their record sheet. As students reveal their decisions, you should write them on the board or on a transparency. For example: (R, R), (R, B), or (B, B).
- In the second round, ask another 10 or so students to make a decision simultaneously, then repeat these procedures. Repeat these procedures until at least several groups of 10 have made decisions.
- In the third round, increase the size of the group. For example, tell students that they are in a group with everyone in their row. Earnings are as described above ($1 if one plays a red card; $4 if ALL play a black card; $0 if one plays a black card and anyone else in the group plays a red card), however it requires EVERYONE in the group to play a black card for anyone to earn $4. As before, write the results on the board as each group reveals their decisions. It is not necessary to write down each person's decision; for example, you can write "all red", "all black", "2 red", etc.
- Repeat this for several rounds (or until all students have made a decision, either in treatment 1 or 2).
- If desired, conduct additional rounds in which the value of a high-effort decision is reduced. For example, one earns $1 for playing a red card, and $2 if ALL play a black card.
- It is helpful to conduct several rounds of each treatment. For example, in the initial treatment (where students are paired) have at least two or three rows of students make decisions before going on to the next treatment. This ensures that everyone understands the procedures and the earnings from each decision.
- Students typically need some help in filling out their record sheet during the first round or two, so you should lead them through the process. After the first pair's decisions are revealed, instruct each to write down what the other chose, and then write down their earnings. Tell each person what they earned, based on what they played and what the other played. For example, if both played a red card, tell them both to write down that they earned $1. If one played a black card and the other played a red-card, tell the person who played a black card to write down $0 and the person who played a red card to write down $1. Do this for several pairs of students. Once you are done with a round, ask students if they have any questions about how to fill out their record sheet. If any do, go over each of the three possible outcomes so that everyone can see how to record their earnings.
- Be sure to keep track of decisions, since this will form the basis of the lecture and class discussion. In introductory classes, the total number of red and black cards played is probably sufficient. However, in upper-level courses (where there is more focus on equilibrium and out-of-equilibrium outcomes) the decisions made by each person in a group is also relevant. For example, in a principles course, knowing that 8 people chose red and 8 people chose black is enough. But in upper-level courses you will want to know whether the pattern was (R, B), (R, B), (R, B), (R, B), (R, B), (R, B), (R, B), (R, B) OR (R, R), (R, R), (R, R), (R, R), (B, B), (B, B), (B, B), (B, B).
- If you are running short on time, it is most helpful to spend most of the time using two-person groups, since this forms the basis of most of the class discussion. However, you should save time to conduct one or two rounds with a large group (for example, the entire class in one group) at the end.
Once students have participated in the experiment, you should get them involved in the class discussion as much as possible. Participating in the experiment will have given them an opportunity to really think about the public goods problem (though this term has not yet been introduced) and therefore the role of the discussion period should be to help them summarize what they have learned and put it into economic terms. To facilitate this, the "lecture" is presented as a series of questions that you can ask your students, with suggestions for summarizing their answers and leading them through the material associated with the public goods problem.
One of the challenges of talking about coordination games is that most students' first impression of the game is that it is a prisoner's dilemma. The primary goal of the discussion and accompanying lecture is to illustrate to students that there can be more than one equilibrium in this game, and that one equilibrium "pareto dominates" the other. Most of the lecture should focus on the two-person game (conducted in the first few rounds of the experiment), however the effects of changes in group size can be addressed at the end of the lecture.
The best way to get students involved in the discussion is to start by asking them about their experiences in the game. For example, start by asking "Will someone who played a black card explain why you did so?" Most students will point out that it made sense to do so since if both played their black card it was possible for both to earn more money than if both played a red card. Next, ask "Given the benefit of both playing a black card, will someone who played a red card explain why you did so?" If no one answers this, you can pose the question in hypothetical terms: "OK, in this case, will someone try to explain why someone else who played a red card might have done so?" Typically, someone will point out that playing a black card is risky - you only earn $4 if the other person plays a black card and you might earn nothing; there's a chance that others in the class might be confused or just worried about what others will do; on the other hand playing a red card is safe - you get $1 no matter what the other person does.
Once people have gone through this reasoning, you can point out that the strategy in this game can be complicated - your best move depends on what you think the other person will do. If you think the other person will play a black card, you should play your black card also. But if you think the other person will play a red card, then you should play your red card also. This might still be too abstract for some students, so you may want to write the possible earnings on the board:
- If you play a black card and the other does also, you will earn $4. (And if you switched to red, you would only earn $1.)
- If you play a black card and the other plays a red card, you will earn $0. (And if you switched to red, you would earn $1.)
So: you should only play a black card if you think the other person will play a black card.
- If you play a red card, and the other does also, you will earn $1. (And if you switched to black, you would earn $0.)
- If you play a red card and the other plays a black card, you will earn $1. (But if you switched to black, you would earn $4.)
So: you should only play a red card if you think the other person will also play a red card.
In an introductory course, remind students that an equilibrium is a state of balance - a situation where nothing will change. Students should look at the four outcomes listed on the board and determine in which one(s) there is no incentive to change. They should realize that there are two possible outcomes that satisfy this: when both play red or both play black. In an upper-level course, you can write the earnings in a payoff matrix and solve for equilibrium more formally.
Once students understand that there are two equilibrium outcomes, it is more apparent that one is "better" than the other. When both play black, both earn more than when both play red. At this point, you can introduce the concept of Pareto optimality: at the (black, black) outcome, it is not possible to make one player better off without hurting the other player. In fact, in this case, BOTH lose if one switches to another outcome.
This naturally leads to the question of why one would ever wind up at the bad (red, red) outcome. If students have trouble explaining this, you can remind them of some of their answers when you asked them about why someone might ever play a red card. Often at this point the discussion naturally turns to the rounds that were conducted with a larger group size: with two people it is easier to trust the other person to play a black card. But with a bigger group, it is more likely that at least one person will play a red card, which reduces earnings for everyone.
It is also helpful if you can relate this material to naturally-occurring coordination games. Group projects can be a good example: if a group project requires the active participation of all, a student may only want to work hard if he or she believes that all others will also. This is also true of other group production activities: if producing a good requires everyone to do their part, one only wants to work as hard as the slowest-moving worker.
In a macroeconomics class, the coordination problem can be related to the Keynesian model, which distinguishes between equilibrium income and potential income. At the heart of this model is the idea that there are many potential equilibrium levels of income, with some better than others. Even though all would prefer to be at a good equilibrium, if the economy gets stuck at a bad outcome, it may be difficult to get out of it. Ask students to imagine a situation in which they played the game and the other person played a red card. What would they likely play in the next round? Most will say that they would play a red card. Ask what it would take at that point for them to switch to a black card. Most will admit that they would have to have some kind of firm assurance that the other would play a black card before being willing to try it again.
You can also point out to students that some economists point to coordination failures as a CAUSE of recessions: investment in plant and equipment is an important source of economic growth. If one firm undertakes this type of investment but others don't, it won't be enough for the economy to grow and the firm may lose money on this investment. However, if all (or most) firms undertake this type of investment, the economy will grow, leading to increased return on the investment.