# Public Goods Experiment

This material is replicated on a number of sites
as part of the
SERC Pedagogic Service Project

Initial Publication Date: June 25, 2010

## Summary

In this experiment, students must decide how to divide their "endowment" of a good between private consumption and a public good. The private consumption provides a benefit (earnings) to only the individual and the public good provides a benefit to each person in the group, even those who do not contribute to the public good. This hand-run experiment is suitable for use in small classes in the range of 5 to 40 students. Some variations on this basic experiment are also described.

## Learning Goals

This experiment allows students to discover for themselves the tension between self-interest and group-interest in the public good problem. Some students quickly recognize the incentive to free-ride, while others more readily focus on the group-gains associated with contributions. The hand-run version includes time for group-discussion after several rounds of the game have been completed. This allows students to teach one another about the competing interests between groups and individuals, and also demonstrates the difficulty of maintaining high levels of contributions to public goods without an enforcement mechanism.

Concepts Covered:

1. The two defining characteristics of a public good.

2. The tension between individual and group interests in the voluntary provision of public goods.

3. Why the voluntary provision of public goods may lead to inefficient levels of provision.

4. How the public benefit from a public good affects contributions (theoretically and empirically).

Concepts Covered:

1. The two defining characteristics of a public good.

2. The tension between individual and group interests in the voluntary provision of public goods.

3. Why the voluntary provision of public goods may lead to inefficient levels of provision.

4. How the public benefit from a public good affects contributions (theoretically and empirically).

## Context for Use

Considerations:

1. This experiment works well in classes with up to 40 students. If you have a larger class you can try the large-class version described on EconPort.

2. This experiment works best if you conduct the experiment first, followed by class discussion and lecture.

Time Required: This experiment can be conducted in one 50-minute class. The experiment takes up to 20 minutes, including reading instructions. The lecture material provides up to 30 minutes of lecture.

Materials:

1. One deck of cards for every 13 students. Each deck should be ordered by number, with all four suits of a number together (i.e., 2 of hearts, 2 of diamonds, 2 of clubs, 2 of spades, then 3 of hearts, 3 of diamonds, 3 of clubs, 3 of spades, etc.).

2. Printed instructions (which include a record-sheet) for your students. [LINK TO PDF]

Time Required: This experiment can be conducted in one 50-minute class. The experiment takes up to 20 minutes, including reading instructions. The lecture material provides up to 30 minutes of lecture.

1. This experiment works well in classes with up to 40 students. If you have a larger class you can try the large-class version described on EconPort.

2. This experiment works best if you conduct the experiment first, followed by class discussion and lecture.

Time Required: This experiment can be conducted in one 50-minute class. The experiment takes up to 20 minutes, including reading instructions. The lecture material provides up to 30 minutes of lecture.

Materials:

1. One deck of cards for every 13 students. Each deck should be ordered by number, with all four suits of a number together (i.e., 2 of hearts, 2 of diamonds, 2 of clubs, 2 of spades, then 3 of hearts, 3 of diamonds, 3 of clubs, 3 of spades, etc.).

2. Printed instructions (which include a record-sheet) for your students. [LINK TO PDF]

Time Required: This experiment can be conducted in one 50-minute class. The experiment takes up to 20 minutes, including reading instructions. The lecture material provides up to 30 minutes of lecture.

## Description and Teaching Materials

(These instructions are written assuming that the teacher reads them to students in class, but can easily be modified for reading outside of class prior to the lecture period. They can also be given in writing to students, but many professors simply read them to the students and highlight the key rules on the board. In this case, the students can create their own record sheet on their own paper. Cards can be handed out while instructions are being read.)

[link to PDF file of instructions for printing, and then reproduce instructions below]

This is a simple card game. Each of you will be given 4 cards; two of these cards are red (hearts or diamonds) and two of these cards are black (clubs or spades). All of your cards will be the same number. The exercise will consist of a number of rounds (you have 15 rounds listed on your record sheet - we will probably complete fewer than 15 rounds). When a round begins I will come to each of you in order, and you will play TWO of your four cards by placing these two cards face down on top of the stack in my hand. Your earnings in dollars are determined by what you do with your red cards.

In the first several rounds: you will earn $4 for each red card that you keep and $0 for each black card that you keep. Red cards that are placed on the stack will affect everyone's earnings in the following manner. I will count up the number of red cards, and everyone will earn this number of dollars. Black cards placed on the stack have no effect on the count.

When the cards are counted, I will not reveal who made which decisions, but I will return your cards to you at the end of the round (by returning to each of you in reverse order and giving you the top two cards from the stack in my hand). To summarize, your earnings for the period will be calculated as:

Earnings = $4 x (number of red cards you kept) + $1 x (total number of red cards I collect)

After several rounds, I will announce a change in the earnings for each red card you keep. Red cards placed on the stack will always earn $1 for each person. Use the space below to record your decisions, your (hypothetical) earnings, and your cumulative earnings.

[IMAGE OF RECORDSHEET HERE -- from bottom of student instructions pdf]

Additional Instructions for the Professor

1. You should conduct several rounds with the earnings specified in the students' instructions ($4 for each red-card kept and $1 for each red-card turned in), until the number of red-cards contributed starts to decline. Typically there is a marked decrease in contributions after three or four rounds. After this, change the value of a red-card kept to $2, while keeping the value of a red-card turned in fixed at $1. This does not change the Nash equilibrium, but because the opportunity cost of keeping a card declines, one typically observes an increase in contributions as a result of this change. This increase is particularly dramatic since it occurs after contributions had been declining for two or three rounds in the first treatment. After several rounds with this payment structure, announce that you will give students a chance to talk for several minutes. They can talk about anything they like, but tell them that you will not enforce any agreements that they make and that they will not be given another chance to talk with one another before the end of the experiment. Give students up to 5 minutes for discussion, then conduct several more rounds.

2. If you are running short of time, reduce the number of rounds conducted under the second treatment ($2 for each red card kept) and conduct only one or two rounds after the discussion period. It's best not to skip the discussion period since this is when many students start to figure out the conflict between what is best for the individual and what is best for the group. However, we suggest that you not allow students to talk to one another at any other time during the experiment.

3. Each student should start a round with two red cards and two black cards, therefore it is essential that you return cards you give a student the same two cards that he or she played. An easy way to do this is to go back to each student in the reverse order that you used to collect the cards, then give the student the top two cards from the stack in your hand. This will work as long as (a) each student played two cards as instructed, and (b) you keep the cards in order when you count the number of red cards turned in. Since students are originally given four cards with the same number (i.e., 4 of hearts, diamonds, clubs, and spades), if you make a mistake in returning cards it can be quickly corrected.

4. 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. Remind them that they earned $4 for each red card kept; so the column labeled "earnings for cards kept" will be $0 if they kept no red cards (they turned in both), $4 if they kept 1 red card, and $8 if they kept both red cards. Then announce the earnings that EVERYONE will receive from the red cards turned in (for example, if 18 red cards were turned in, announce that everyone should write down $18 in the column labeled "$1 x (total # of red cards in stack)"). Be sure to remind them that each person will receive this amount, even those who turned in no red cards. Then tell them to fill in their total earnings in the period: earnings from cards kept + earnings from cards in the stack.

5. After each round, you should write on the board the total number of red cards that were turned in. Be sure to indicate the value of the red card kept for each decision, since you will refer to this and the number of cards contributed during the class discussion.

Student instructions for public goods experiment (Acrobat (PDF) 56kB Oct12 09)

[link to PDF file of instructions for printing, and then reproduce instructions below]

This is a simple card game. Each of you will be given 4 cards; two of these cards are red (hearts or diamonds) and two of these cards are black (clubs or spades). All of your cards will be the same number. The exercise will consist of a number of rounds (you have 15 rounds listed on your record sheet - we will probably complete fewer than 15 rounds). When a round begins I will come to each of you in order, and you will play TWO of your four cards by placing these two cards face down on top of the stack in my hand. Your earnings in dollars are determined by what you do with your red cards.

In the first several rounds: you will earn $4 for each red card that you keep and $0 for each black card that you keep. Red cards that are placed on the stack will affect everyone's earnings in the following manner. I will count up the number of red cards, and everyone will earn this number of dollars. Black cards placed on the stack have no effect on the count.

When the cards are counted, I will not reveal who made which decisions, but I will return your cards to you at the end of the round (by returning to each of you in reverse order and giving you the top two cards from the stack in my hand). To summarize, your earnings for the period will be calculated as:

Earnings = $4 x (number of red cards you kept) + $1 x (total number of red cards I collect)

After several rounds, I will announce a change in the earnings for each red card you keep. Red cards placed on the stack will always earn $1 for each person. Use the space below to record your decisions, your (hypothetical) earnings, and your cumulative earnings.

[IMAGE OF RECORDSHEET HERE -- from bottom of student instructions pdf]

Additional Instructions for the Professor

1. You should conduct several rounds with the earnings specified in the students' instructions ($4 for each red-card kept and $1 for each red-card turned in), until the number of red-cards contributed starts to decline. Typically there is a marked decrease in contributions after three or four rounds. After this, change the value of a red-card kept to $2, while keeping the value of a red-card turned in fixed at $1. This does not change the Nash equilibrium, but because the opportunity cost of keeping a card declines, one typically observes an increase in contributions as a result of this change. This increase is particularly dramatic since it occurs after contributions had been declining for two or three rounds in the first treatment. After several rounds with this payment structure, announce that you will give students a chance to talk for several minutes. They can talk about anything they like, but tell them that you will not enforce any agreements that they make and that they will not be given another chance to talk with one another before the end of the experiment. Give students up to 5 minutes for discussion, then conduct several more rounds.

2. If you are running short of time, reduce the number of rounds conducted under the second treatment ($2 for each red card kept) and conduct only one or two rounds after the discussion period. It's best not to skip the discussion period since this is when many students start to figure out the conflict between what is best for the individual and what is best for the group. However, we suggest that you not allow students to talk to one another at any other time during the experiment.

3. Each student should start a round with two red cards and two black cards, therefore it is essential that you return cards you give a student the same two cards that he or she played. An easy way to do this is to go back to each student in the reverse order that you used to collect the cards, then give the student the top two cards from the stack in your hand. This will work as long as (a) each student played two cards as instructed, and (b) you keep the cards in order when you count the number of red cards turned in. Since students are originally given four cards with the same number (i.e., 4 of hearts, diamonds, clubs, and spades), if you make a mistake in returning cards it can be quickly corrected.

4. 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. Remind them that they earned $4 for each red card kept; so the column labeled "earnings for cards kept" will be $0 if they kept no red cards (they turned in both), $4 if they kept 1 red card, and $8 if they kept both red cards. Then announce the earnings that EVERYONE will receive from the red cards turned in (for example, if 18 red cards were turned in, announce that everyone should write down $18 in the column labeled "$1 x (total # of red cards in stack)"). Be sure to remind them that each person will receive this amount, even those who turned in no red cards. Then tell them to fill in their total earnings in the period: earnings from cards kept + earnings from cards in the stack.

5. After each round, you should write on the board the total number of red cards that were turned in. Be sure to indicate the value of the red card kept for each decision, since you will refer to this and the number of cards contributed during the class discussion.

Student instructions for public goods experiment (Acrobat (PDF) 56kB Oct12 09)

## Teaching Notes and Tips

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. First ask, "Will someone who turned in a red card explain why they did so?" Typical answers include:

* I wanted to encourage other people to turn in red cards so I did so at first.

* Everyone would earn more money if people turned in red cards.

* If I kept both red cards I would only earn $8, but if everyone turned in both red cards, everyone would have earned $40. (This would be the type of answer one would get in a class with 20 people: 20 people x 2 red cards = 40 red cards turned in.)

Next, ask "These are all good reasons for turning in a red card, so will someone who did NOT turn in any red cards explain why they didn't do so?" (Sometimes no one wants to admit that they didn't turn in a red card, so if no one answers this question, you can rephrase this as a hypothetical question: "OK, will someone try to explain why someone MIGHT NOT want to turn in a red card?") Typical answers include:

* Because I can make more money if I keep my red cards.

* I earned money from the cards everyone turned in AND money from cards that I kept, so I earned more by keeping my red cards.

At this point, it's helpful to summarize the discussion so far: If we're thinking of the group, everyone can do better by turning in their red cards; but each individual can earn more money by keeping their red cards and still earning money from the red cards others turn in.

Once students understand this, you can introduce the term "public good" and its two defining characteristics: non-rivalry (one person's enjoyment of the good does not diminish others' ability to enjoy it) and non-excludability (everyone enjoys the benefits of a public good, even those who don't contribute to it).

Ask your students how turning in a red card represents making a contribution to a public good. As they answer, make sure that they understand that EVERYONE receives a dollar from the public good, whether there are 10 people in the class or 200 people in the class: the more people there are, the greater the benefits that can be enjoyed. They should also point out that everyone receives a dollar, even those that don't contribute to it.

Next, ask your students what it is about the characteristics of a public good that give people an incentive to free-ride. If they have trouble answering this question, have them relate their answer to the previous discussion about turning in red cards: everyone received $1 for each red-card that was turned in, even if they didn't contribute. More people may have contributed if receiving the benefit ($1 per card) had been limited to only those who contributed.

After you have covered this much, I find it helpful to start framing the discussion in terms of economic terminology. As students what the marginal cost is of turning in a red card in the game. They may give you a specific answer ($4 in the first rounds of the experiment, $2 in later rounds of the experiment) or a more general answer (how much you earned if you kept the red card). Make sure that you eventually get the specific answer, and write it on the board: marginal cost of contributing = $4. Next, ask for the marginal benefit to the INDIVIDUAL (not the group) of turning in a red card. Someone should point out that it is $1 - the amount the individual contributing the card receives if he or she turns it in. Write this on the board: Individual's marginal benefit of contributing = $1. Finally, ask for the marginal benefit to the group of turning in a red card. Students will need to know the number of people in the class to answer this. If there are 30 people in your class, this will be $1 x 30 = $30 earned for each red card that is turned in. Write this on the board also: Group's marginal benefit of contributing = $30.

Remind students (or have them tell you) the economic rule: that you should produce where MB = MC. In this case, they are not equal, so students need to think about how to handle the inequality. First, address this from the perspective of the individual. If the MB of a contribution is $1 and the MC = $4, what should the individual do? Most students will realize that this means that they should not contribute to the public good since the cost exceeds the benefit. Next, address this from the perspective of the group: MB = $30 and MC = $4, so from the group's standpoint, all should contribute to the public good. Point out that this is the basic tension in the public goods problem: what is best for the group is not best for the individual; so if individual's act in their own self-interest the public good may not be provided.

Some students will have trouble understanding this, so it may help to use a specific example from the classroom game. Have students take a minute to calculate earnings for each of the following outcomes:

* Earnings for each person if no one contributes to the public good. Answer: Money is only earned from cards kept, since nothing is turned in. So earnings are $4 x 2 red cards kept = $8.

* Earnings for each person if everyone turns in both red cards (no cards are kept by anyone). Answer (assume for this example there are 30 people in the class): Money is only earned from cards turned in, since none are kept. Since each person has 2 cards, 30 x 2 = 60 cards are turned in, which earn $1 each to every person. So each person earns 60 x $1 = $60.

* Earnings for a person if he or she keeps both red cards but everyone else turns in both red cards. Answer (assume for this example there are 30 people in the class). In this case, the person earns $4 x 2 = $8 for the two red cards kept. If the other 29 people turn in both red cards, there are 29x2 = 58 red cards turned in. So everyone earns 58 x $1 = $58 from red cards turned in. So the person who keeps both red cards earns $8 + $58 = $66 (and the others in the class only earn $58).

When students see these numbers they have a concrete example of why everyone is better off if all turn in their red cards, but also why an individual has an incentive to keep both red cards.

Summarize this by pointing out to students that economic theory, which typically assumes that individuals act to maximize their own earnings, would predict that no one would turn in any red cards in our classroom exercise. You can now turn to what happened in the classroom game to see what happened. In almost all classes, some red cards are turned in, with the number turned in decreasing over time. You can point out that economic theory was a little too pessimistic - some red cards were provided; but also, voluntary contributions fell far-short of the optimal contribution (from the group's perspective). In other words, the public goods problem does exist, even in the class.

At this point, you can turn the discussion to remedies for the public goods problem and why the government provides many public goods: even if some level of these goods may be provided through voluntary means, one might want the government to step in and provide them (funded by tax revenues) to ensure that an adequate level of provision is achieved. For example, one would not want to leave the level of national defense to the amount that could be provided through voluntary contributions. If desired, you might also point out that some things that are typically thought of as "public goods" because they are provided by the government may not fit the textbook definition of a public good. For example, it is possible to exclude people from national parks by putting up fences and making people pay to enter the park. (It is also the case that parks may not be non-rival if too many people visit them; if you go to a wilderness area your enjoyment may be diminished if there are too many others there also.) Another good example if fire-protection. People tend to think of this as a public good, but in fact, fire-protection services could be excludable. Many communities (historically and even in current-times) offer fire protection on a subscription-basis: only those who pay a subscription fee are protected in the event of a fire.

Additional Suggestions for Upper-Level Courses

The lecture material provided above is most suitable for an introductory level course. If you are teaching upper-level courses (for example, intermediate micro, public finance, or game-theory) you can supplement it with additional material outlined below.

Students should try to solve for the Nash equilibrium of a one-shot game with the payoffs used in the initial rounds of the classroom experiment ($4 for each red card kept and $1 for every red card turned in by anyone). They should realize that keeping both red cards is a dominant strategy Nash equilibrium in a one-shot game. Next, students can think about the equilibrium of this game if it were repeated for a fixed (and known) number of rounds. For example, if students knew that they would make a contribution decision in each of 5 rounds. In this way, the concept of backward induction can be introduced.

After this, students should solve for the Nash equilibrium in a one-shot game in which the value of a red-card kept is reduced to $2. In this case, there is still a dominant strategy to keep both red cards.

Typically, the number of red cards turned in is significantly higher when the value of a red card kept is reduced from $4 to $2 (at least in the initial rounds). This can spur discussion of why behavior changed when the theoretic prediction is that there should be no red cards turned in as long as the value of a red-card kept exceeds $1 (the value to the individual of turning one in).

Finally, the discussion can turn to "trigger-strategies" and what can cause them to work or fail. Typically during the experiment-discussion period, people suggest trigger-strategies (without using this terminology). In almost all discussion periods, one or more students suggest that everyone turn in both red cards, but that if it is apparent (from the number turned in) that someone didn't that no one will turn in any red cards after this. Not all groups come to an agreement to try this strategy, but many do; in almost all cases one or more people do not turn in both red cards and the strategy fails. In fact, it is often the case that the person who suggested the trigger strategy is one of the people who does not turn in both red cards.

* I wanted to encourage other people to turn in red cards so I did so at first.

* Everyone would earn more money if people turned in red cards.

* If I kept both red cards I would only earn $8, but if everyone turned in both red cards, everyone would have earned $40. (This would be the type of answer one would get in a class with 20 people: 20 people x 2 red cards = 40 red cards turned in.)

Next, ask "These are all good reasons for turning in a red card, so will someone who did NOT turn in any red cards explain why they didn't do so?" (Sometimes no one wants to admit that they didn't turn in a red card, so if no one answers this question, you can rephrase this as a hypothetical question: "OK, will someone try to explain why someone MIGHT NOT want to turn in a red card?") Typical answers include:

* Because I can make more money if I keep my red cards.

* I earned money from the cards everyone turned in AND money from cards that I kept, so I earned more by keeping my red cards.

At this point, it's helpful to summarize the discussion so far: If we're thinking of the group, everyone can do better by turning in their red cards; but each individual can earn more money by keeping their red cards and still earning money from the red cards others turn in.

Once students understand this, you can introduce the term "public good" and its two defining characteristics: non-rivalry (one person's enjoyment of the good does not diminish others' ability to enjoy it) and non-excludability (everyone enjoys the benefits of a public good, even those who don't contribute to it).

Ask your students how turning in a red card represents making a contribution to a public good. As they answer, make sure that they understand that EVERYONE receives a dollar from the public good, whether there are 10 people in the class or 200 people in the class: the more people there are, the greater the benefits that can be enjoyed. They should also point out that everyone receives a dollar, even those that don't contribute to it.

Next, ask your students what it is about the characteristics of a public good that give people an incentive to free-ride. If they have trouble answering this question, have them relate their answer to the previous discussion about turning in red cards: everyone received $1 for each red-card that was turned in, even if they didn't contribute. More people may have contributed if receiving the benefit ($1 per card) had been limited to only those who contributed.

After you have covered this much, I find it helpful to start framing the discussion in terms of economic terminology. As students what the marginal cost is of turning in a red card in the game. They may give you a specific answer ($4 in the first rounds of the experiment, $2 in later rounds of the experiment) or a more general answer (how much you earned if you kept the red card). Make sure that you eventually get the specific answer, and write it on the board: marginal cost of contributing = $4. Next, ask for the marginal benefit to the INDIVIDUAL (not the group) of turning in a red card. Someone should point out that it is $1 - the amount the individual contributing the card receives if he or she turns it in. Write this on the board: Individual's marginal benefit of contributing = $1. Finally, ask for the marginal benefit to the group of turning in a red card. Students will need to know the number of people in the class to answer this. If there are 30 people in your class, this will be $1 x 30 = $30 earned for each red card that is turned in. Write this on the board also: Group's marginal benefit of contributing = $30.

Remind students (or have them tell you) the economic rule: that you should produce where MB = MC. In this case, they are not equal, so students need to think about how to handle the inequality. First, address this from the perspective of the individual. If the MB of a contribution is $1 and the MC = $4, what should the individual do? Most students will realize that this means that they should not contribute to the public good since the cost exceeds the benefit. Next, address this from the perspective of the group: MB = $30 and MC = $4, so from the group's standpoint, all should contribute to the public good. Point out that this is the basic tension in the public goods problem: what is best for the group is not best for the individual; so if individual's act in their own self-interest the public good may not be provided.

Some students will have trouble understanding this, so it may help to use a specific example from the classroom game. Have students take a minute to calculate earnings for each of the following outcomes:

* Earnings for each person if no one contributes to the public good. Answer: Money is only earned from cards kept, since nothing is turned in. So earnings are $4 x 2 red cards kept = $8.

* Earnings for each person if everyone turns in both red cards (no cards are kept by anyone). Answer (assume for this example there are 30 people in the class): Money is only earned from cards turned in, since none are kept. Since each person has 2 cards, 30 x 2 = 60 cards are turned in, which earn $1 each to every person. So each person earns 60 x $1 = $60.

* Earnings for a person if he or she keeps both red cards but everyone else turns in both red cards. Answer (assume for this example there are 30 people in the class). In this case, the person earns $4 x 2 = $8 for the two red cards kept. If the other 29 people turn in both red cards, there are 29x2 = 58 red cards turned in. So everyone earns 58 x $1 = $58 from red cards turned in. So the person who keeps both red cards earns $8 + $58 = $66 (and the others in the class only earn $58).

When students see these numbers they have a concrete example of why everyone is better off if all turn in their red cards, but also why an individual has an incentive to keep both red cards.

Summarize this by pointing out to students that economic theory, which typically assumes that individuals act to maximize their own earnings, would predict that no one would turn in any red cards in our classroom exercise. You can now turn to what happened in the classroom game to see what happened. In almost all classes, some red cards are turned in, with the number turned in decreasing over time. You can point out that economic theory was a little too pessimistic - some red cards were provided; but also, voluntary contributions fell far-short of the optimal contribution (from the group's perspective). In other words, the public goods problem does exist, even in the class.

At this point, you can turn the discussion to remedies for the public goods problem and why the government provides many public goods: even if some level of these goods may be provided through voluntary means, one might want the government to step in and provide them (funded by tax revenues) to ensure that an adequate level of provision is achieved. For example, one would not want to leave the level of national defense to the amount that could be provided through voluntary contributions. If desired, you might also point out that some things that are typically thought of as "public goods" because they are provided by the government may not fit the textbook definition of a public good. For example, it is possible to exclude people from national parks by putting up fences and making people pay to enter the park. (It is also the case that parks may not be non-rival if too many people visit them; if you go to a wilderness area your enjoyment may be diminished if there are too many others there also.) Another good example if fire-protection. People tend to think of this as a public good, but in fact, fire-protection services could be excludable. Many communities (historically and even in current-times) offer fire protection on a subscription-basis: only those who pay a subscription fee are protected in the event of a fire.

Additional Suggestions for Upper-Level Courses

The lecture material provided above is most suitable for an introductory level course. If you are teaching upper-level courses (for example, intermediate micro, public finance, or game-theory) you can supplement it with additional material outlined below.

Students should try to solve for the Nash equilibrium of a one-shot game with the payoffs used in the initial rounds of the classroom experiment ($4 for each red card kept and $1 for every red card turned in by anyone). They should realize that keeping both red cards is a dominant strategy Nash equilibrium in a one-shot game. Next, students can think about the equilibrium of this game if it were repeated for a fixed (and known) number of rounds. For example, if students knew that they would make a contribution decision in each of 5 rounds. In this way, the concept of backward induction can be introduced.

After this, students should solve for the Nash equilibrium in a one-shot game in which the value of a red-card kept is reduced to $2. In this case, there is still a dominant strategy to keep both red cards.

Typically, the number of red cards turned in is significantly higher when the value of a red card kept is reduced from $4 to $2 (at least in the initial rounds). This can spur discussion of why behavior changed when the theoretic prediction is that there should be no red cards turned in as long as the value of a red-card kept exceeds $1 (the value to the individual of turning one in).

Finally, the discussion can turn to "trigger-strategies" and what can cause them to work or fail. Typically during the experiment-discussion period, people suggest trigger-strategies (without using this terminology). In almost all discussion periods, one or more students suggest that everyone turn in both red cards, but that if it is apparent (from the number turned in) that someone didn't that no one will turn in any red cards after this. Not all groups come to an agreement to try this strategy, but many do; in almost all cases one or more people do not turn in both red cards and the strategy fails. In fact, it is often the case that the person who suggested the trigger strategy is one of the people who does not turn in both red cards.