978-0393919684 Chapter 4 Lecture Note

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CHAPTER 4

Simultaneous-Move Games: Discrete Strategies

Teaching Suggestions

This chapter develops a succession of examples to illustrate the concept of Nash

equilibrium when each player chooses from a finite number of discrete pure strategies. It also

addresses the various issues that arise when looking for equilibria in such games, including

dominance solvability, multiple equilibria, and nonexistence of equilibrium. You will probably

want to look for examples that are different, but related to, the ones in the book for use in class

lectures. And as usual, you may find it worthwhile to have your students play illustrative

games among themselves, or as a class, in order to motivate some of the concepts and ideas

that you will present. Two of the games that we suggest below (Games 1 and 2) can be played

before starting the material from the chapter. The discussion that follows such games can be

used as a natural springboard into the methods of illustrating and then solving

simultaneous-move games.

The concepts of dominant and dominated strategies, also illustrated with the first two

in-class games, are usually easily grasped. A few real-world examples of dominant strate gies

can be presented to the class. Dixit and Nalebuff, in Thinking Strategically, use an example

from baseball; with two out and a three-ball, two-strike count on the batter, all base runners

who are subject to being retired by a forced out should run with the next pitch. Another

example comes from the television show Singled Out. When the leading contestant is one

question away from winning the game, any contestant who answers after the leader (and who

wants to win the game) should give an answer different from the leader (see Greg Trandel,

“Using a TV Game Show to Explain the Concept of a Dominant Strategy,” Journal of

Economic Education, 1999).

It is nice to work up to the general concept of Nash equilibrium, and even to the

existence of multiple equilibria, using increasingly large game tables. Some variants of Table

4.1 can be used for this purpose. Start with the original matrix:

Column

L M R

Row

T 3, 1 2, 3 10, 2

H 4, 5 3, 0 6, 4

L 2, 2 5, 4 12, 3

B 5, 6 4, 5 8, 7

In this form, the game is dominance solvable and has a unique Nash equilibrium: (L,

M). By changing the 6 in the (H, R) cell to a 9, you create a game in which there is no domi-

nance; cell-by-cell searching for a Nash equilibrium yields the same equilibrium as before at

(L, M). With that change in place you can do two things: (1) Change the 7 in the (B, R) cell to a

6 and get a game with two Nash equilibria, one at (L, M) and the other at (B, L); or (2) leave

the 7 in the (B, R) cell but change the 5 in the (L, M) cell to a 3 and get a game with no

equilibrium in pure strategies. This is a fun example for students to work on in groups during

class or as an assignment between lectures.

Here are some suggestions for alternative illustrating or motivating games:

Prisoners’ Dilemma

An oil-pricing game based on the Organization of Petroleum Exporting Countries

(OPEC; see Game 2 in Chapter 10 of this Instructor’s Manual).

Dominance Solvability

To illustrate a game in which only one player has a dominant strategy, you can use the

example of the United States and the Soviet Union from Exercise S4 of Chapter 9 or the

color-coordination example from Game 2 below.

Assurance-Type Games

You can use classic games like the Stag Hunt, referring back to Rousseau, and bring a

bit of history and culture into the subject; students from the humanities will like this.

Battle of the Sexes

Here a standard sexist example would be one in which a man and woman try to choose

a movie to see together (the man prefers violent films like Independence Day or Black Hawk

Down, and the woman prefers romantic or weepie films like The English Patient or, indeed, A

Beautiful Mind). Gibbons suggests an example of two firms that would both gain if they could

agree on a product standard, although each has a preferred standard (Game Theory for Applied

Economists [Princeton, N.J.: Princeton University Press, 1992]).

Another game with the battle of the sexes payoff structure is based on C. P. Snow’s

famous essay, The Two Cultures and the Scientific Revolution (New York: Cambridge

University Press, 1959). Two groups of faculty, one from the sciences and one from the

humanities, have suggested a use for an antiquated lecture hall that is about to be renovated:

the scientists want to use it as a laboratory; the humanists hope to see a small theater. The

university president has said that she will allocate funds for renovation but only in the presence

of unified faculty opinion on the final product. Neither group wants to let the possibility of

renovation die and would like to see something done with the space. The payoff structure is

then as follows, a standard battle of the sexes:

Humanities faculty

Lab Theater

Science

Faculty

Lab 2, 1 0, 0

Theater 0, 0 1, 2

Chicken

You can use a labor-management negotiation game in which each side must choose

between being soft or tough. Or you can use the example of two children who find a new set of

64 crayons, and each can choose to fight for the right to most of the crayons, or to share.

Sharing gets each 32; a fighter gets 48 if the other player will share (and take 16), but if both

fight, they destroy all the crayons and each gets 0.

But perhaps the best new example of Chicken you can use, and make it a more general

vehicle for clarifying the concept of Nash equilibrium, comes from the scene in A Beautiful

Mind, where Nash is supposed to have acquired the inspiration for his equilibrium concept. In

this scene, Nash is in a bar with two male friends. A blonde and her three brunette friends walk

in. All three men would like to win the blonde’s favor. But if they all approach her, each will

stand at best a one-third chance; the movie suggests that she would reject all three. The men

will have to turn to the brunettes, but then the brunettes will reject them also, because “no one

likes to be second choice.” The movie-Nash says that the solution is for them all to ignore the

blonde and go for the brunettes. One of the other two men thinks this is just a ploy on Nash’s

part to get the others to go for the brunettes so he can be the blonde’s sole suitor. And if you

think about the situation game theoretically, the movie-Nash is wrong and the friend is right.

Consider a two-male version of this game. Ordering their payoffs in the various outcomes from

1 to 4 (where 4 is best), the payoff matrix is

Male 2

Blonde Brunette

Male 1

Blonde 1, 1 4, 2

Brunette 2, 4 3, 3

It is not a Nash equilibrium for both of them to go for brunettes: given the strategies of the

other, either of them gains by deviating and going for the blonde. There are two Nash equilibria

in pure strategies, one where Male 1 goes for the blonde and Male 2 for a brunette, and the

other the other way around. There are also mixed-strategy Nash equilibria, so you can use the

example in Chapter 7 as a vehicle for illustrating mixed-strategy equilibria in non-zero-sum

games. This example is used as the basis for Exercise S13 in this chapter.

See Anderson and Engers, “A Beautiful Blonde: A Nash Coordination Game” for a

more thorough analysis of this game. They show that the game has multiple equilibria, but the

only outcome that cannot be a Nash equilibrium is the supposedly brilliant solution found by

the movie-Nash! The paper is available online at http://ideas.repec.org/p/vir/virpap/359.html.

Nonexistence

As an alternative to the Tennis-Point Game to illustrate the nonexistence of pure

strategy equilibrium in zero-sum games, you can use any variant of a cops-robbers game. The

classic version is the Sherlock Holmes-Moriarty story that is told in John von Neumann and

Oskar Morganstern’s Theory of Games and Economic Behavior (Princeton, N.J.: Princeton

University Press, 1944), pp. 176–178. We will elaborate on this in the context of

mixed-strategy equilibrium in Chapter 7 of this manual. Or you can use any topical story to

motivate nonexistence. The situation in which for many months when the U.S. Army was

looking for Saddam Hussein, he had to choose between hiding in Baghdad or Tikrit and the

U.S. Army had to choose where to concentrate its search for him, lends itself to analysis.

Equilibrium Selection

Once you have covered the idea of multiple equilibria and presented the different

possible ways that a single equilibrium might emerge, either due to focal points or coordination

or mixing, you may want to play one of the other in-class games described below. Games 3 and

4 address the issue of tacit coordination and give students the opportunity to see that, while in

some games there are outcomes that are obviously focal in the eyes of all players, there are

many situations in which individuals will believe that there is an obvious focal point only to

discover later that their opponents thought otherwise. We have successfully used these games

during our discussions of simultaneous-move games, as well as immediately before the

discussion of strategic moves in Chapter 9. Using them then allows a discussion of the

importance of pregame communication for arriving at outcomes favorable to you; this naturally

leads into the idea of both unconditional commitments and conditional-response rules.

You will also want to distinguish carefully for your students the differences between

zero-sum and non-zero-sum games sometime during your presentation of simultaneous-move

games. Zero-sum games are often easiest to understand, since player payoffs are in direct

conflict and winners and losers can be identified. Non-zero-sum games, in which winners and

losers are often difficult to define, may be less intuitive. Students often want the Nash

equilibrium to be an outcome in which one player gets her absolute best payoff, so that there is

an obvious winner; other times students are swayed by the idea that it is best to beat the other

player by as much as possible rather than to do as well as one can for oneself.

Game Playing in Class

GAME 1—Price Setting (Prisoners’ Dilemma)

Students can quickly grasp the concept of a dominant strategy if required to think

through a game situation in which they each have one. The easiest way to do this is to set up a

game with a prisoners’ dilemma structure and either have pairs of students play against each

other or provide a handout on which students can indicate their choice for play against a

random opponent in the class; in either case, you must assign students to be one of the two

players. (If you are not playing in pairs, you can assign all students to be the same player.)

Set up a game in which players take on roles of competing manufacturers setting high

or low prices for their products; four different values of H, C, D, or L, where H > C > D > L are

the possible payoffs (profits) depending on the combination of prices chosen. Then if both

players choose high, each gets C, while if both choose low, each gets D. When one chooses

high and the other chooses low, the one choosing high gets L and the one choosing low gets H.

If your students have not yet seen games illustrated in tables, they may find themselves

trying to find a way to diagrammatically represent the choices available to them. Most will

ultimately arrive at the obvious pricing choice, however. You can astonish your students with

the predictive powers of game theory by coming to class with your predic tion of their choices

sealed in an envelope. Once they have indicated their choices, you can open the envelope and

reveal your prediction. You can also use the results to stimulate a discussion on the role of

dominant strategies as well as on the particulars of the prisoners’ dilemma, including incentives

to cheat and the result that players are better off if both can cooperate than they are in the Nash

equilibrium of the game.

GAME 2—Pick a Color (Battle of the Bismarck Sea)

This game is based on the Battle-of-the-Bismarck-Sea game, presented in Exercise U8

of this chapter, in which only one player has a dominant strategy. It is best used before you do

any discussion of dominance in class. Students can be divided into pairs to play each other, or

you can design a handout on which students indicate their choices for play against a random

opponent in the class. The game itself involves picking one color from two possible choices,

and students must be assigned to be Player 1 or Player 2.

In the first version of this game, players choose between white and blue. Payoffs are

such that if both players choose white, then Player 1 gets 50 and Player 2 gets 50. If both

players choose blue, then Player 1 gets 25 and Player 2 gets 75. If Player 1 chooses white and

Player 2 chooses blue, then Player 1 gets 75 and Player 2 gets 25. If Player 1 chooses blue and

Player 2 chooses white, then Player 1 gets 50 and Player 2 gets 50.

In the second version, players choose between orange and black. Payoffs are such that

if both players choose orange, then Player 1 gets 75 and Player 2 gets 25. If both players

choose black, then Player 1 gets 50 and Player 2 gets 50. If Player 1 chooses orange and Player

2 chooses black, then Player 1 gets 25 and Player 2 gets 75. If Player 1 chooses black and

Player 2 chooses orange, then Player 1 gets 50 and Player 2 gets 50.

As is the case in Game 1, if the students have not yet seen game tables, they may find

it somewhat difficult to come up with a good way to think about what they should do in each

case. Then one of the topics of discussion can involve the depiction of this game in its normal

form. Further discussion can focus on the idea of dominant strategies and the intuitive results

that you should use one if you have it and expect an opponent to use one if he has one.

GAME 3—Divide a Dollar

This game asks pairs of students (or each student individually if you use a handout and

match students with a random opponent after the fact) to divide a dollar between them. Each

writes the amount of the dollar she wants, from $.01 to $1 inclusive. If the amounts requested

within a given pair of students (or for two randomly selected student responses) add up to

exactly $1, then each student gets the amount requested. If the two amounts add up to anything

other than $1, each player gets nothing. You can play this with real money if you can afford it;

we have managed to play this particular game without actual cash with perfectly acceptable

results.

In actual play this is a game with discrete strategies, 100 for each player, or fewer if

the choices are restricted to be multiples of nickels or dimes or even quarters. But you may

prefer to conduct the analysis of the game treating the choices as continuous variables, in

which case the game could be placed in Chapter 5.

Discussion of the game will bring out the idea that a game can have multiple equilibria

(any two values summing to $1 can make up an equilibrium) but that sometimes one of those

multiple equilibria is focal. This game is in direct contrast, then, to the tire game described in

Chapter 1 in which there are also multiple equilibria but none are as obviously focal as 50 cents

each is here. You can consider the various types of games in which multiple equilibria arise,

including an assurance-type game, Battle of the Two Sexes, and Chicken. In some cases, there

are focal outcomes; in others, players may prefer to alternate among the different equilibria.

You can lead from here into the idea of mixed strategies without much difficulty.

GAME 4—Schelling’s Tacit Coordination Quiz

If they have played Game 3, students will see that sometimes games with multiple

equilibria will have obvious focal points. This game, which replicates a quiz designed by

Thomas Schelling (see The Strategy of Conflict [London: Oxford University Press, 1960], chap.

3), shows that tacit coordination is often not so simple. The nice thing about Schelling’s quiz is

that players often believe that some of their answers are obviously the right ones. We generally

encourage students to try to coordinate with their classmates by offering edible rewards for

every question on which the entire class manages to coordinate answers; sometimes we go

years without having to make good on this promise. The questions are

1. Write heads or tails.

2. Circle one of the numbers listed below:

7 100 13 261 99 555

3. Circle one of the 16 x’s below:

xxxx

4. You are to meet someone in your local city or town. You have not been instructed where

to meet, you have no prior understanding with the person on where to meet, and you cannot

communicate with each other. You are simply told that you will have to guess where to meet,

that the other person is being told the same thing, and that you will just have to try to make

your guesses coincide. Where do you go?

5. You are told the date but not the hour of the meeting in question 4. The two of you must

guess the exact minute of the day for the meeting. At what time will you appear?

6. Write some positive number.

7. Name an amount of money.

8. On the first ballot of an election, candidates received the following votes:

Smith 19, Robinson 29, Brown 15, Jones 28, White 9

The second ballot is about to be taken. You have no interest in the outcome, except that you

will be rewarded if someone gets a majority on the second ballot and you vote for the one who

does. Similarly, all voters are interested only in voting with the majority, and everybody knows

that it is in everybody’s interest. For whom do you vote on the second ballot?

9. You are to divide the list of the following six names into two subsets of three each.

Another player is independently doing the same. Both of you will get a prize if the two of you

divide up the list in the same way, else neither will get anything. The list is

Cornwallis, Montgomery, Napoleon, Rommel, Washington, Wellington.

Discussion of the answers to these questions can be quite interesting. Often students

are surprised that not everyone wrote heads or that 6 was not the obvious choice for a positive

number to write as the answer to question 6. You can have them suggest ideas for the different

beliefs held by different students; this will lead to thoughts on how different social, economic,

cultural, and other issues can play into games of this type. In question 9, the list consists of the

winners and losers in three famous battles, so an obvious way to divide the list is to have

winners in one group and losers in the other. But the list has three British commanders and

three non-British ones, so a Briton might choose that as the basis for division. This result helps

makes the useful point that the existence of a focal point can depend on the historical and

cultural backgrounds of the players. For your information, if all students do manage to

coordinate on answering one question, it is usually question 8.