978-0393919684 Chapter 11 Lecture Note Part 2

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Game Playing in Class

GAME 1—Check Marks

Prepare a handout or cards and ask students to place either a blank line or a check mark

on the paper. Tell them that if fewer than half of the students in the class answer with a check

mark, 5 points (if you count experiment points toward the grade or if this is included on a quiz)

will be added to the score of each person who makes the check mark. But if more than half of the

class answers with a check mark, then 5 points will be taken away from the score of each person

who makes the check mark. (You can achieve the same incentive structure in a variety of ways.)

Collect the cards or quizzes and collate the answers to the question.

Consideration of the results from this game can lead to an interesting discussion. The

game has asymmetric Nash equilibria in pure strategies and a symmetric Nash equilibrium in

mixed strategies. Which one (if any) would you or the students expect to emerge? What kind of

coordination would be needed to achieve a particular asymmetric pure-strategy outcome? What

ethical or redistributive issues would arise if the students did act in coordination? Should students

who did badly on previous quizzes be allowed to get the extra points here, or should the

allocation be made by a lottery that gives an equal chance to everyone? If a plan of joint action is

devised, would anyone try to cheat? And so on. Exercise U2 at the end of the chapter addresses

these same issues.

GAME 2—Investment Game

This game starts with each person receiving an imaginary $5 from the instructor. We have

also been successful using small candies as a substitute for currency in this game; students can be

given the rights to 10 jellybeans instead, for example. Students are each asked to divide their

“funds” between two different investment opportunities: Asset A and Asset B. Asset A is a riskless

investment with a zero rate of return; any money invested in this asset is returned intact at the end

of the game. Asset B works in the following way: each student chooses to invest some amount

between $0 and $5 (or 0 and 10 jellybeans, inclusive) in Asset B. These amounts will be added

together, and the sum will become the Asset B investment pool. The size of this investment pool

will be increased by 50%. The funds in the investment pool will then be divided equally among

all the participants in the game. (In other words, an individual’s share of the Asset B investment

pool depends only on the total amount invested in the pool and not on how much the individual

personally invested in Asset B.)

You might want to give your students an example of how the game could proceed.

Suppose that there are 25 people who participate in the game, that Sarah Student invests $2.00 in

Asset A and $3.00 in Asset B, and that the total amount invested in Asset B (from all 25 people) is

$60.00. In this case, the Asset B pool is increased to $90.00 after the investment decisions are

made, and $90.00 is then divided equally among the 25 participants. Each player collects $3.60

from the Asset B investment pool. Sarah Student ends the game with her $2.00 from Asset A and

$3.60 from the Asset B pool for a total of $5.60.

Ask students to divide the $5 between the two assets in any way they choose. After

thinking about this game for a few moments, ask them to indicate how much of the $5 they wish

to invest in each asset and have them indicate this on a card or handout.

This game is a multiperson prisoners’ dilemma game or a collective-action game with a

payoff structure resembling a prisoners’ dilemma. Those who invest in Asset A defect while those

who invest in Asset B cooperate. You can calculate the amount of free riding that takes place in

the game by taking the proportion of the dollars invested in Asset A to the total number of dollars

available to be invested and discuss the implications of the outcomes observed.

You can also use a slightly simpler version of this game in which students can put their

entire $5 either into Asset A or into Asset B. This case is easier to analyze, and you can draw a

simple diagram showing payoffs from cooperation (this payoff is [7.5  number investing in

B]/number in class) and defection (this payoff is 5 + [7.5  number investing in B]/number in

class). Defection is a dominant strategy for each player, but the class as a whole would be better

off if all cooperated.

There are many versions of this public goods game that appear in the literature. For some

ideas about ways to vary this game see www.marietta.edu/~delemeeg/expernom/s93.html.

GAME 3—Fishing Game

Here are directions for students to be written on a handout. Imagine that you own a fleet

of boats that are used to catch fish in a particular lake. There is one other fleet of fishing boats

that also operates in this lake. On any given single day, the amount of fish caught by your fleet

depends both on the number of boats you sent out and on the number the other fleet sent out. The

following table shows the number of pounds of fish caught by each fleet in a day, depending on

how many boats each fleet sent out. (Your catch is shown first.) Suppose that a pound of fish can

be sold for $1 and that it costs $10 to send out a fishing boat for a day.

Other fleet

0 boats 1 boat 2 boats 3 boats 4 boats 5 boats

Your

fleet

0 boats 0, 0 0, 32 0, 60 0, 84 0, 104 0, 120

1 boat 32, 0 30, 30 28, 56 26, 78 24, 96 22, 110

2 boats 60, 0 56, 28 52, 52 48, 72 44, 88 40, 100

3 boats 84, 0 78, 26 72, 48 66, 66 60, 80 54, 90

4 boats 104, 0 96, 24 88, 44 80, 60 72, 72 64, 80

5 boats 120, 0 110, 22 100, 40 90, 54 80, 64 72, 72

If you want to send out the number of boats that will maximize your own profit (bearing

in mind that the owner of the other fleet will also be picking a number of boats to maximize his

profit), how many of your boats would you send out?

Now, suppose that you can pick the number of boats that both your fleet and the other

fleet send out (but, to be fair, you have to allow the other fleet to send out exactly the same

number of boats that you send out). Again assuming that you are interested in maximizing your

profit, how many of your boats would you send out in this case?

You may also want to show students the following table indicating how many fish are

caught for a given number of boats on the lake and the profits gained for the different

combinations of boats sent out by each fleet:

2 Total boats Fish caught

1 32

2 60

3 84

4 104

5 120

6 132

7 140

8 144

9 144

10 144

The original matrix was produced by allocating the number of fish caught proportionally among

all the boats sent out.

The following matrix shows the profit earned by each firm as a function of the number of

boats sent out by each fleet. It was constructed from the original matrix by subtracting the $10

cost of sending out each fishing boat; students often do this calculation for themselves as they

play the game if they are not provided with the following matrix:

Other fleet

0 boats 1 boat 2 boats 3 boats 4 boats 5 boats

Your

fleet

0 boats 0, 0 0, 22 0, 40 0, 54 0, 64 0, 70

1 boat 22, 0 20, 20 18, 36 16, 48 14, 56 12, 60

2 boats 40, 0 36, 18 32, 32 28, 42 24, 48 20, 50

3 boats 54, 0 48, 16 42, 28 36, 36 30, 40 24, 40

4 boats 64, 0 56, 14 48, 24 40, 30 32, 32 24, 30

5 boats 70, 0 60, 12 50, 20 40, 24 30, 24 22, 22

This game shows that a good individual outcome may be inversely related to a good

outcome for the other player so that some type of coordination is necessary to get a jointly

optimal outcome. When each player pursues a course of action designed only to better his payoff,

the final outcome may be bad for him. This game has an externality; putting more of your boats

on the lake hurts the other fleet’s potential catch (and potential profit). From a societal perspective

(total profit), an outcome with two fleets and three boats each is better (132 fish and a total of 72

profit) than a single fleet sending out all five of its boats (120 fish and $70 profit).

To analyze the game from an individual’s perspective, you could derive best-response

curves showing each fleet’s best number of boats (the number of boats that gives each fleet the

most profit) given the number of boats sent out by the other fleet. To analyze the game from a

social perspective, you can graph total profits (to both fleets) as a function of the number of boats

sent out (from 1 to 10). You will want to ask the class about solutions to the externality problem

here and get them to think about how the externality might be internalized in this case.