978-0393919684 Chapter 10 Lecture Note Part 2

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Game Playing in Class

GAME 1—Paired Prisoners’ Dilemma

Students can be paired off and instructed to play several versions of a particular game

with a prisoners’ dilemma structure. Provide each pair with a sheet describing the payoff table for

the game and with space available to indicate the choices made by each player in each round and

in each version of the game.

First, tell students that the game will end after a known number of rounds, say 10. Have

them play through 10 rounds, keeping track of their choices. Then have them play another

10-round game, and a third if there is time. As with the dime games, you should see convergence

to the rollback equilibrium of always cheating within a few plays of the finite game; students who

try to cooperate originally will lose out to cheating rivals at the end of the game and will cheat

earlier and earlier in each successive play of the game.

Second, try a version in which the students play a round at a time without knowing when

the game will end; you call out an end to the game after some number of minutes. This version is

similar to the Serendip version available on the Web (see the URL above) except that the rival’s

strategy is not as predictable. In addition, there is uncertainty about how long the game will last

since it could be very short (unlike the Web version) or relatively long. You can do this several

times and ask students to describe how their behavior changes in future plays of the game.

Each of these games gives students the opportunity to experience the actual play of a

prisoners’ dilemma. The discussion that follows can be used to consider discrepancies between

predicted and actual behavior, changes in behavior in later plays of the same game, how players

update information about a rival’s play, and other issues related to players’ willingness to

cooperate (including cultural norms, friendships, rivalries, etc.).

GAME 2—Leadership in an Oil Production Game

Students can be broken into pairs to play this game once, with each student representing

one country; then each should switch partners and play as if she were the other country. Students

could also play this game individually, indicating which action they choose when playing as each

country; you can pair sets of answers to show outcomes. This game is based on the rational-pigs

game in John McMillan’s Games, Strategies, and Managers (London: Oxford University Press,

1996), p. 13; he uses the Saudi Arabia analogy on p. 14.

Ask students to consider a very simplified version of the situation facing members of

OPEC in the 1970s. In this simplified story, we assume that OPEC is made up of only two

countries: Saudi Arabia and Kuwait. Suppose that each country can produce a unit of oil (maybe a

million barrels) at a cost of $1. Saudi Arabia is a big country and can produce either 4 or 5 units

of oil; Kuwait is a small country and can produce either 1 or 2 units of oil.

Given output and consumption in the rest of the world, the following formula tells us the

price at which oil is sold: Price = 10 – (QS + QK), where QS is the number of units produced by

Saudi Arabia (SA), and QK is the number of units produced by Kuwait (K). Then profits are

calculated as follows: If SA produces 4 units and K produces 1 unit, the price of a unit of oil

equals 5 (from the above formula). Because the cost of producing a unit of oil equals $1, a

country earns a profit of $4 for each unit it sells. Because SA produces 4 units, its total profit

equals $16; because K produces 1 unit, its total profit equals $4. Profit per unit obviously depends

on total production, as the following table illustrates:

Joint profit (SA’s profit plus K’s profit, or OPEC profit) is maximized if total production

equals 5 units.

Finally, tell students to put themselves in this situation and worry only about their own

profits; then ask them: (1) How many units would you produce if you were Saudi Arabia? and (2)

How many units would you produce if you were Kuwait?

When discussing this game, you can first collect student input on their choices, writing on

the board the number of students who chose 4 as Saudi Arabia and 2 as Kuwait. Then you can

show how the game can be analyzed more formally. It is nice to show that the symmetric version

of the game (in which Saudi Arabia and Kuwait each choose either 2 or 3 units) is a standard

prisoners’ dilemma; the smaller output level is the cooperative strategy. (You might even want to

use this as your example of a standard prisoners’ dilemma if you play the game before you start

the material from this chapter.) Once you change the choices available to the players to be

consistent with the description of the game and change the payoffs accordingly, you will be able

to illustrate how the analysis changes. Encourage students to provide hypotheses for why Saudi

Arabia’s incentives have changed and build on their ideas to show that large players often incur a

much higher cost of cheating than do small players. Ask students to think of other examples of

dilemma situations with leadership or provide some of your own.

COMPUTER GAME—Zenda

We set this game as Exercise U4; you can instead play it out in class.

This game was invented by James Andreoni and Hal Varian; see their article, “Pre-Play

Contracting in the Prisoners’ Dilemma,” Proceedings of the National Academy of Sciences, vol.

96, no. 19 (September 14, 1999), pp. 10933–10938. The paper also contains some code in C.

Zenda is a prisoners’ dilemma, but this is concealed behind a facade of playing cards and pull and

push mouse clicks in such a way that students do not easily figure this out. (They could, from the

name and word association, but few are sufficiently widely read or addicts of the right kind of

movies.) Nevertheless, the game is best played during an early week of the semester, before you

have treated the dilemma in class.

Make sure you have an even number of students. The program matches them randomly in

pairs. Each student sees two cards for herself in the bottom half of her screen, and two cards for

the player with whom she is matched in the top half of her screen. For each student, there is a low

card called her pull card, and a high card called her push card. She can use her mouse to click on

one of these. If she clicks on the low (pull) card, she gets from a central kitty a number of coins

(points) equal to the value of that card. If she clicks on the high (push) card, her opponent gets

from the same central kitty a number of coins (points) equal to the value of that card. The

objective is to get as many coins for yourself as possible. The two matched in a pair make their

choices simultaneously. They do not see each other’s choice until both have clicked, when the

actual transfer of coins takes place. Then new random pairings are formed, and the procedure is

repeated. Depending on the time available, you can typically play up to 10 rounds of this.

(Usually most students figure out after two or three rounds that pull is their dominant strategy.)

The values of the low and high cards a player has over her 10 rounds should be alternated

in such a way as to allow each to get the same aggregate payoff if they play the correct strategies.

This evenness is important if the exercise counts toward the course grade.

Then a second phase of the game begins. Here each player has the opportunity to bribe

the other into playing push; it shows how the prisoners’ dilemma can be overcome if there is

some mechanism by which the players can make credible promises. Again, randomly matched

pairs are formed, and in each pair each player sees her and her opponent’s cards. First, each

chooses how many coins she promises to pay her opponent if (and only if) the opponent plays

push. These bribes come from the player’s own kitty (winnings from the first phase) and not from

the central kitty. The bribes are put in an escrow box. Once both have set the bribes, each can see

the bribe offered by the other. Then they play the actual game of clicking on the cards. When both

have clicked, each gets the points from the central kitty depending on the push or pull choices as

before. If your opponent plays push, she gets the bribe you offered from your escrow box; if your

opponent plays pull, your bribe is returned to you from your escrow box. (The fact that the

program resolves this disposition of the bribes makes the promise credible.) The bribe game is

also played a number of times (typically 10 rounds) with fresh random matching of pairs for each

round. Students quite quickly find the optimal bribing strategy.

You can try different variants (treatments) of the game: allow players to talk to one

another or forbid talking, keep one pairing for several rounds to see if tacit cooperation develops,

and so on. We append for your information the instructions given at the time of playing the game,

and a report and analysis is circulated later.

INSTRUCTIONS GIVEN TO ZENDA PLAYERS

Phase 1

In Zenda, you will play a simple card game with another player. Players are assigned to each

other randomly, so that you will normally play a different person each time you play Zenda.

There are two phases to Zenda. We will first play Phase 1, which is called push-pull.

When you play push-pull, you will see on your screen two cards for you, two cards for the other

player, and a central pot with a pile of chips between the two of you.

Your cards are labeled “push” and “pull.” You can choose to play a card by clicking on it

or the button under it with the mouse. When you choose a card it will become highlighted. You

can only choose to play one card in a given round.

Your choice will not be final until you click on the Set Choice button!

At the same time you are deciding which card to play, the other player is making this

decision too. Your payoff depends on the choice you make and the choice the other player makes.

If you choose the pull card, then you will pull the number of chips on that card from the

central pot to your winnings. If you choose the push card, then you will push the number of chips

on that card from the pot to the other player.

We will play 10 rounds of Phase 1, generally with different matchings in the different

rounds. The objective is to maximize your own winnings; your score for this phase will equal the

total number of chips you have at the end.

Phase 2

Now you are in Phase 2 of Zenda. In this phase you have a new option called Pay to Push.

On your screen you will see a new button labeled Set Bribe and a slider. In each round

you can offer to pay the other player a side payment if she chooses to play her push card.

You set the amount of this bribe by moving the slider with the mouse. When you are

satisfied with your decision about how much you are willing to pay the other player, you click on

your Set Bribe button.

Your choice will not be final until you have clicked Set Bribe.

No player will see how much her opponent is willing to pay until both players have

clicked on their Set Bribe button.

Your payoff will be as before, but if the other player chooses to push, the bribe you

offered will be subtracted from your winnings (and added to those of the other player). If the

other player chooses to pull, then you keep your side payment.

The other player makes the same sort of decision. The chips that you see in the box

immediately in front of your cards are the amount the other player is willing to pay you if you

choose to push. The chips that you see in the box right above this are the payment you are willing

to make to an opponent should she choose to push.

We will play 10 rounds of Phase 2, again with generally different pairings every time.

Your final score for Zenda will be the total number of chips you have in your box at the end of the

game.

REPORT AND ANALYSIS OF ZENDA (SENT TO STUDENTS)

In its basic structure, Zenda is a prisoners’ dilemma. Many of you figured this out after a

couple of rounds of play. To any who are literate or old movie buffs and figured it out from the

name, special congratulations.

Suppose the Row player has the cards 2 and 7, and the column player 4 and 8. The payoff

matrix is

Column

Pull Push

Row

Pull 2, 4 10, 0

Push 0, 11 8, 7

Pull is the dominant strategy for both. But both do better if both play Push.

PHASE 1

With a finite number of rounds, even against the same opponent, the game should unravel by

backward induction, so both should play Pull every time. When opponents change from one

round to the next and the opponent in any one round is unknown, building cooperation should be

even harder.

But some cooperation does emerge. Players experiment a little, and when both of a pair

play Push, they are encouraged to try Push again. Some people in some groups try to build up a

“culture” of cooperation by exhorting everyone to Push. (Of course, some of these people are

themselves persistent cheaters! They will probably become top political leaders in 20 years or so.)

Generally, there is very little cooperation in the first couple of rounds. Then people realize its

value, and the middle rounds have more cooperation. Sometimes as many as half the people in the

group are playing Push at this point. Then cooperation declines, and in the last couple of rounds it

collapses.

The minimum score for Phase 1 is zero (if you Push and your opponent Pulls every time);

the maximum from a 10-round phase with alternating roles is when you Pull and your opponent

pushes every time: (5  11) + (5  10) = 105. If you both play Push all the time, the score is (5 

8) + (5  7) = 75. No one came close to this. If everyone cheats all the time, everyone gets (5 

2) + (5  4) = 30. You could fall even lower if you pushed while your opponents were pulling.

PHASE 2

Here each round has a little sequential structure: (1) setting the bribe, and (2) choosing the

card. Use backward induction. Begin with the choice of the card once the bribes are set.

Suppose you are offered a bribe x, your pull card is y, you offer your opponent the bribe

z, and her push card is w. Your payoff matrix is

Other

Pull Push

You

Pull y y + w – z

Push x x + w – z

At this stage bribe is the dominant strategy for each player. The maximum score for a

10-round Phase 2 with alternating roles is (5  6) + (5  9) = 75. Many students and groups did

figure this phase out after one or two rounds.