978-0393919684 Chapter 7 Lecture Note Part 2

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Classical Game Theory

You can give a more rigorous treatment of bimatrix games, building on some of the

material in the online appendix to this chapter. The best reference for this is still Luce and

Raiffa’s Games and Decisions (New York: Wiley, 1957).

Rationalizability

You can take the concept of domination by a mixed strategy, presented in the online

appendices, and apply it to a few examples of non-zero-sum games. The original papers are still

worth consulting: (1) B. Douglas Bernheim, “Rationalizable Strategic Behavior,” Econometrica,

vol. 52, no. 4 (July 1984), pp. 1007–1028; (2) David Pearce, “Rationalizable Strategic Behavior

and the Problem of Perfection,” Econometrica, vol. 52, no. 4 (July 1984), pp. 1029–1050; and (3)

a textbook treatment can be found in Andreu Mas Colell, Michael D. Whinston, and Jerry R.

Green, Microeconomic Theory (New York: Oxford University Press, 1995), pp. 242–245, 252,

257, and Exercise 8.C.4.

This chapter also provides a good place for a general concluding discussion about the

merits and defects of the Nash equilibrium concept. You can recall previous findings from

classroom games or other material about when the concept works, when and why it does not

work, and so on. We hope you will guide the discussion so as to leave the class with the guarded

optimism that is our own bottom line. This optimistic view consists of several important and

interrelated points:

1. The Nash equilibrium is the best available starting point for the analysis of games.

2. When a game has multiple Nash equilibria, several conceptual avenues for selecting one of

them as the predicted outcome are available—focal points, subgame perfectness (leading to the

idea of credible strategic moves), and as will be seen in later chapters that introduce the concept

of information in games, perfect Bayesian equilibrium.

3. When a game has no Nash equilibrium in pure strategies, the first thing to look for is

mixing. The next thing to try should be rationalizability; in fact, rationalizability is the purest

logical implication of common knowledge of rationality.

4. If the Nash equilibrium fails to give a good prediction, the most likely explanation is that

you have misspecified the game. Either you have misunderstood the players’ true payoffs, or the

game is in reality repeated but you have treated it as if it is a one-shot game, or there is some

information asymmetry you have ignored, and so on.

5. When playing a complex or unfamiliar game, the players experiment and make mistakes;

various behavioral approaches come into their own in such settings.

Game Playing in Class

GAME 1—Rock–Scissors–Paper

This game entails playing three different versions of the children’s game

Rock–Scissors–Paper. In the most basic version, two people simultaneously choose either rock,

scissors, or paper, usually by putting their hands into the shape of one of the three choices. The

game is scored as follows: A person choosing scissors beats a person choosing paper (because

scissors cut paper). A person choosing paper beats a person choosing rock (because paper covers

rock). A person choosing rock beats a person choosing scissors (because rock breaks scissors). If

two players choose the same object, they tie. Technically, players end up mixing over three

possible pure strategies. You can modify this in-class game so that it can be used to teach about 2

x 2 games—use it as is but focus your discussion on the points relevant to 2 x 2 games—or play it

immediately before covering the analysis of larger games in Section 5.

Instructions for students: In this experiment, each individual game (and we will play

multiple games) is worth 10 points. Because it is a zero-sum game, the winner gets 10 points and

the loser gives up 10 points. The following matrix shows the possible outcomes in the game:

In all three of the games described below, you are the row player and your professor is

the column player. For each situation, we will simulate 60 repetitions of a game. Because actually

signaling with our hands 60 times would be too time-consuming, you are instead asked to

describe your behavior by writing down how many times (on average) out of 60 games you

would play rock, how many times you would play scissors, and how many times you would play

paper.

Your professor will choose a particular strategy to use in each of three situations; her

strategy in Situation 1 differs from her strategy in Situation 2, and so on. In each situation, your

professor uses the same strategy against every student. You will be told your professor’s exact

strategies for Situation 1 and Situation 3 when we analyze the outcomes of the games; your

professor’s exact strategy for Situation 2 is explained below. Your task in each of the three games

is to choose the strategy that you think (or guess) is most likely to maximize your total payoff.

Situation 1

This is the regular version of the game, using the payoffs above. You are asked to write

down how many times (on average) out of 60 you would play rock, how many times you would

play scissors, and how many times you would play paper. In a similar way, your professor has

written down numbers that describe her behavior. You can assume that your professor plays her

equilibrium strategy here.

Professor

Rock Scissors Paper

Student

Rock 0 10 –10

Scissors –10 0 10

Paper 10 –10 0

Situation 2

In this version of the game, the payoffs remain as above, but your professor commits (for

some reason) to picking rock 24 times out of 60 (40%), picking scissors 18 times out of 60 (30%),

and picking paper 18 times out of 60 (30%). (The order in which she will make these picks is

unknown.) No matter what anybody writes down, your professor’s behavior will follow this rule.

Knowing this, again try to maximize your total payoff.

Situation 3

In this version of the game, the row player (you) has an advantage. If you pick rock and

your professor picks scissors, you win 20 points from your professor. The payoff matrix is now:

Player

Rock Scissors Paper

Player 1

Rock 0 20 –10

Scissors –10 0 10

Paper 10 –10 0

This time, your professor will use a strategy that she thinks will be beneficial for her (but

she does not know for certain that it will be beneficial). Again, try to maximize your total payoff.

When discussing the results of these games, you will be able to note that everyone’s

payoff in Situation 1 is identical, regardless of your professor’s choice. This is a direct result of

the fact that equilibrium mixtures keep a rival indifferent; any pure strategy or combination of

pure strategies yields the same payoff against someone playing her equilibrium mixture. In

Situation 2, the professor does not play her equilibrium mix. Some students (but perhaps not

many) will figure out that this is not the equilibrium mix and that it can therefore be exploited.

You can show on the board during the discussion that the student using pure rock can obtain an

expected payoff of 0 per game, the student using pure scissors can obtain an expected payoff of

–1 per game (so –60 over 60 plays of the game), and the student using pure paper can obtain an

expected payoff of 1 per game and 60 over 60 plays of the game. Given the disequilibrium

mixture, the student can take advantage of the professor’s “error” by playing rock 100% of the

time.

Finally, in Situation 3, you can consider a number of different strategies. If you again use

your equilibrium mix, then the payoff to you is 5/6 (on average) per game, for a total of 50 over

the 60 games. We have also used a number of nonequilibrium mixes on occasion; the most

interesting one to analyze is the one that avoids the use of scissors entirely, putting equal weight

on rock and paper. This helps to make the point about the counterintuitive outcome—that the

student wants to avoid rock in the new equilibrium mix—even more stark. Against a professor

mixing only rock and paper in equal proportions, the student does best to use all paper.

This game can be used to show (1) how to calculate equilibrium mixes for larger games;

(2) that a rival’s equilibrium mix keeps you indifferent among your pure strategies or any

mixture, and so makes your choice of strategy irrelevant; (3) that exploitation is possible if a rival

is using a nonequilibrium mix—one of your pure strategies will be dominant; and (4) the

counterintuitive outcome that, in equilibrium, you decrease your use of a strategy for which one

payoff has increased.

Other Games

For non-zero-sum examples, you can set up various simple 2 x 2 on-zero-sum games,

especially Chicken, with mixed strategies explicitly allowed. You can show how the

mixed-strategy choices and outcomes compare with the Nash equilibrium predictions; it is

especially interesting to show how the mixtures change when the underlying payoffs are changed.

You are almost sure to find that players change their mixture probabilities in response to changes

in their own payoffs, which is contrary to the prediction from the opponent’s indifference

principle. This can then feed into a discussion of the reasonableness of the concept.