## Classroom Expernomics

### A Simple Game Theory Experiment for Teaching Oligopoly

##### Daniel Seiver Department of Economics Miami University Oxford, OH

For a number of years, I have been using a simple and brief classroom experiment to illustrate the power of game theory in explaining the behavior of oligopolists. The whole presentation takes about fifteen minutes of class time, and it has worked well in the Principles of Microeconomics course.

I begin with the simplest case in game theory, a 2x2 payoff matrix with 2 players (see Figure 1). This Prisoner's Dilemma is used to show how a payoff matrix is read, and also introduces the students to the minimax strategy (without ever using that term) After putting Figure 1 on the board, I provide some motivation for each of the payoff boxes: if both remain silent, they can only be convicted on a minor charge (upper left); if one "squeals" on the other, the squealer gets off with a suspended sentence, and they throw the book at the other one (upper right, lower left). If both confess, they each get a tenner (lower right). While it is clearly best overall for both to remain silent, I first take the viewpoint of player A, and point out that player A should consider whether to confess or not if player B confesses. It is clear to all that if player B confesses, player A is better off confessing, saving 10 years of prison time. The first revelation comes when I examine what player A should do if B doesn't confess: it still pays to confess, as A saves 6 months of prison time! (You may want to point out here that A might not want to be around when B gets out.) Since this is a symmetric matrix, it is easy to show that player B can also determine that confessing is the best strategy.

At this point I move directly to the pricing strategy of two oligopolists. I prepare in advance two paper copies of a 3x3 profit matrix (see Figure 2). I pick three students in the front center of the class to be Team A, and give them 3 minutes to pick a price from the three possible prices in the matrix. I also pick three students in a back corner of the class to be Team B, and given them the same charge. At this point I write the 3x3 matrix on the board, and encourage the rest of the class to "play along at home" and try to guess what prices will be chosen by Teams A and B. I also prepare in advance a piece of paper with "\$8, \$8" written on it. For maximum effect, I put this Nash solution in a sealed envelope and give it to another student to hold during the three minutes. At the end of the three minutes, I ask each team to write down their price, and then ask Team A to announce their price, and then Team B likewise. In nine of the last ten years, both have announced \$8, at which point I call for the envelope and tear it open to reveal the correctly predicted \$8 prices. In the one exceptional year, Team A picked \$9, but as soon as Team B announced \$8, Team A asked to change to \$8, which I graciously allowed them to do before calling for the envelope.

At this point, I review the profit matrix and each team's optimal strategy, and show that there was nothing magical in my prediction. Figure 2 is of course just a 3x3 version of the Prisoner's Dilemma. I then point out that the combined profits of the \$8-\$8 equilibrium is the lowest of all nine elements of the matrix, while the \$10-\$10 combined profits are the highest. So if the two firms could just "agree" to charge \$10, they would both make bigger profits. This leads naturally to a discussion of price-fixing. I still like to tell the "phases of the moon" story here, since the executives involved used to cheat on the price-fixing agreement, which is easily illustrated with Figure 2. This is also a good time to remind your future business leaders that a number of these convicted executives did time "in the joint."

My experience has convinced me that this classroom experiment is a quick and effective way to illustrate the basic principle of oligopolistic interdependence, the urge to collude, and efforts (often failed) to avoid "price wars." In addition, the very best students may want to pursue "game theory" further. (I recommend starting with Axelrod's paperback The Evolution of Cooperation.) My interpretation of examination results over the years also suggests that students on average perform better on oligopoly questions, although I have not done any controlled experiments to support this armchair empiricism.

Figure 1

```						Player A
Don't Confess		Confess
Don't Confess	6 mos., 6 mos.		0, 20 yrs
Player B
Confess		20 yrs, 0 		10 yrs, 10 yrs.

Note: Player A's payoffs are listed first.
```

Figure 2

```			 	        Team A
\$10		  \$9		  \$8
\$10	16, 16		19, 12		22, 9
Team B	        \$9	13, 18		15, 15		18, 12
\$8	10, 20		12, 16		14, 13
```

### Predation in the Classroom

##### Andrew Kleit Department of Economics Louisiana State University Baton Rouge, LA

Predation is one of the oldest concepts in the industrial organization literature. Yet until relatively recently, there was no firm theoretical basis for predation. Consider a finite game of full information where an incumbent (the potential predator) faces a sequential series of potential entrants. In such games, it is usually more profitable for the incumbent to "accommodate" entry rather than "preying" on it, at least in the last period of the game. Knowing this, the last potential entrant will cho ose to enter, realizing that its entry will be accommodated. Given this, and the process of "backward induction," one is able to generate the "Chain Store Paradox" and conclude that accommodate entry will occur in every period, much to the incumbent's regret.

Now change the game into one of asymmetric information. Assume that a small percentage of incumbent firms are "hard" competitors, and that entrants cannot tell which incumbents are hard, at least without observing their actions. These hard competitors actually prefer predating and losing money to accommodating entry and making money. Given the existence of hard competitors, "soft" profit-maximizing competitors will desire to mimic hard competitors and prey on any entry that occurs early in the game. For an incumbent to do otherwise would be to advertise to all potential entrants that it is soft. This would in turn invite entry, and deny the incumbent the opportunity to make monopoly profits. Thus, even soft firms will predate (at least early in the relevant game) so as not to generate a reputation for being soft. (See, for example, Milgrom and Roberts (1982).)

Now look at the situation from the point of view of an entrant early in the game. It knows that while the probability of it actually entering against a hard competitor is small, and by itself not enough reason to deter entry, no matter who it enters against it will suffer the pain of predation. Given this, no entry will occur until late in the game when soft incumbents have little or no reason to protect their reputations.

This combination of strategic conclusions: 1) that entry makes sense in the one period game; 2) that in a multi-period game even soft incumbents will predate early in the game; and 3) that, given 2), potential entrants will not enter early in the game, is a significant challenge for most undergraduates. To help the students in my senior level industrial organization class understand this, I run the experiment printed below, which is a modification of Jung, Kagel, and Levin (1994). The experiment takes about 75 minutes to perform.

In their article Jung, Kagel, and Levin (at 74) indicate that it is important that all concepts be expressed in neutral terms. To achieve this, I take the following steps. First, I run the experiment before I teach the concept of predation in class. Second, the experiment title has no connection to the purpose of the experiment. Third, instead of entrants and incumbents, players are called "starters" and "finishers." Fourth, instead of "soft" and "hard" finishers (incumbents), finishers are either "blue" or "green." Fifth, starters move either "up" or "down," instead of staying out of or entering the market. Finally, instead of accommodating or preying, finishers move either "left" or "right."

In addition, it is important to prevent student play from being affected by the reputation of particular students. To achieve this, experiment pairings are kept anonymous and randomized through use of an "indicator" system. Students report to me that this system leaves them unable to determine whom they are playing against.

In my experience, in the first round almost all entrants choose to enter in the first period. Soft incumbents almost always accommodate, while hard incumbents predate. The ensuing three periods then generate entry against accommodating soft incumbents ( who by accommodating in the first period have already identified themselves) and no entry against hard incumbents. By the second round, about half of the students realize that to accommodate entry is to identify yourself to the world as soft and to invite entry. They therefore predate in the early periods of that round, discouraging later entry. By the third round, most players have figured through the three relevant steps.

To make sure almost all students understand the concepts, it would probably be necessary to run a fourth round of the experiment. Unfortunately, the limits on class time preclude this. Nevertheless, I find this experiment serves as an important tool to encourage students to think through the strategic implications of predation theory.

##### References

Jung, Yun Joo, John Kagel,and Dan Levin, "On the Existence of Predatory Pricing: An Experimental Study of Reputation and Entry Deterrence in the Chain-Store Game," RAND Journal of Economics 25(1): 72-93 (1994).

Milgrom, Paul, and John Roberts, "Predation, Reputation, and Entry Deterrence," Journal of Economic Theory, 27: 280-312 (1982).

#### The Jelly Bean Experiment

There are two types of players: starters and finishers. Starters have two types of moves, UP and DOWN. Finishers have two types of moves, LEFT and RIGHT. There are two types of finishers, BLUE and GREEN. Only two of the finishers are green, the rest are blue. Play is sequential, starters move first and are followed by finishers.

If a starter chooses UP, she always receives a payoff of 0 points. If she choose DOWN and the relevant finisher chooses LEFT, she receives 10 points. If she chooses DOWN and the finisher chooses RIGHT, she receives -5 points.

For BLUE finishers, if a starter chooses UP, the relevant finisher always receives 25 points. If the starter chooses DOWN, and the finisher chooses LEFT, the finisher receives 10 points. If the starter chooses DOWN, and the finisher chooses RIGHT, the finisher receives -2.5 points.

For GREEN finishers, if a starter chooses UP, the relevant finisher always receives 25 points. If the starter chooses DOWN, and the finisher chooses LEFT, the finisher receives -2.5 points. If the starter chooses DOWN, and the finisher chooses RIGHT, the finisher receives 10 points.

The payoff tables are as follows:

Payoff Table: Blue Finisher

```				Blue Finisher
Left		Right
Up		0, 25		0, 25
Starter
Down		10, 10		-5, -2.5
```

Payoff Table: Green Finisher

```				Green Finisher
Left		Right
Up		0, 25		0, 25
Starter
Down		10, -2.5	-5, 10
```

The experiment is conducted as follows. At the beginning of each round students are divided into starters and finishers. Round one has four periods, while rounds two and three have six periods each. Two numbers are randomly selected to be assigned to green finishers. Both starters and finishers are randomly assigned numbers. Please do not tell anyone which number or type you are.

At the beginning of each period, each starter is assigned her matching finisher randomly by use of an "indicator number" written on the board. Indicator numbers work as follows. Let A be the number of the starter, B be the indicator number, and C be the number of matched pairs in the game. Given A, B, and C, player A plays against finisher

```A + B		if A + B <(=) C; A + B C if A + B> C.
```

Thus, if there are 8 matched pairs in the game and the indicator number is 3, starter 4 is matched against finisher 3 + 4 = 7. If the indicator number is 6, starter 3 plays against finisher 3 + 6 - 8 = 1. Starters should write down their matched finishe r as soon as the indicator number is put on the board.

The starter then writes on her own index card her number, the number of the finisher, circled, and the chosen strategy, either "up" or "down." The starter should then write her chosen strategy down on her scorecard. The index cards are collected, and the chosen strategies are written on the board next to the number of the relevant finisher. (Please note that starter strategies are not revealed by number.)

All finishers are then asked to write their number on an index card. (Finishers are asked to determine their strategies before their matches' strategies are written on the board.) Those finishers whose relevant starters have chosen "down" should then write their relevant response strategies, "left" or "right." Finisher responses are then posted on the board, ending the period. At the end of each period players are asked to write down on their scoresheets their strategies, the strategies chosen by their "matched" players (the starters or finishers they are paired against), the relevant payoff, and the cumulative payoff for the round.

At the end of each round, please add up your total points for the round. To start a new round, starters become finishers and finishers become starters. The sequence described above is then repeated.

It is important that players not reveal their strategies to one another, or reveal what number they are playing. Please try not to discuss what happens in the experiment, except to clarify issues about the rules.

Starter Scoresheet: Round 1

```Name:
Starter #:
Round #:
```

Your payoffs are as follows: if you choose UP, you always receive a payoff of 0 points; if you choose DOWN and the relevant finisher chooses LEFT, you receive 10 points; if you choose DOWN and the finisher chooses RIGHT, you receive (minus) -5 points.

```Period	 Your Match's    Your     Your Match's   Your    Total Payoffs
Number   	Strategy    Strategy	Payoff    This Round
1
2
3
4
```

Starter Scoresheet: Rounds 2 and 3

```Name:
Starter #:
Round #:
```

Your payoffs are as follows: if you choose UP, you always receive a payoff of 0 points; if you choose DOWN and the relevant finisher chooses LEFT, you receive 10 points; if you choose DOWN and the finisher chooses RIGHT, you receive (minus) -5 points.

```Period	   Your Match's   Your    Your Match's  Your  Total Payoffs
Number	 Strategy  Strategy    Payoff  This Round
1
2
3
4
```

BLUE Finisher Scoresheet: Round 1

```Name:
Starter #:
Round #:
```

You are a BLUE finisher. Your payoffs are as follows: if the relevant starter chooses UP, you always receive a payoff of 25 points; if the starter chooses DOWN and you choose LEFT, you receive 10 points; if the starter chooses DOWN and you choose RIGHT, you receive (minus) -2.5 points.

```Period	    Your Match's   Your Strategy    Your Payoff    Total Payoffs
Strategy                                      This Round
1
2
3
4
```

BLUE Finisher Scoresheet: Rounds 2 and 3

```Name:
Starter #:
Round #:
```

You are a BLUE finisher. Your payoffs are as follows: if the relevant starter chooses UP, you always receive a payoff of 25 points; if the starter chooses DOWN and you choose LEFT, you receive 10 points; if the starter chooses DOWN and you choose RIGHT, you receive (minus) -2.5 points.

```
Strategy                                      This Round
1
2
3
4
```

GREEN Finisher Scoresheet: Round 1

```Name:
Starter #:
Round #:
```

You are a GREEN finisher. Your payoffs are as follows: if the relevant starter chooses UP, you always receive a payoff of 25 points; if the starter chooses DOWN and you choose LEFT, you receive (minus) -2.5 points; if the starter chooses DOWN and you choose RIGHT, you receive 10 points.

```Period	    Your Match's   Your Strategy    Your Payoff    Total Payoffs
Strategy                                      This Round

1
2
3
4
```

GREEN Finisher Scoresheet: Rounds 2 and 3

```Name:
Starter #:
Round #:
```

You are a GREEN finisher. Your payoffs are as follows: if the relevant starter chooses UP, you always receive a payoff of 25 points; if the starter chooses DOWN and you choose LEFT, you receive (minus) -2.5 points; if the starter chooses DOWN and you choose RIGHT, you receive 10 points.

```Period	    Your Match's  Your Strategy    Your Payoff    Total Payoffs
Strategy                                      This Round
1
2
3
4
```

Instructor's Scoresheet

```This is Round #:
Green finishers are:

Period:		      Period:
Finisher #	Starter   Finisher 	Starter    Finisher
Play       Play         Play        Play
1
2
3
4
5
6
7
8
9
10
11

```