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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.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
