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GAME 4—The Tire Story (The Pure Coordination Game)
Another game that we have successfully played in the first lecture is based on the “We
can’t take the exam; we had a flat tire” story from Section 2 of Chapter 1. Even if the students
have read ahead, the discussion in the text makes it clear that there is no obvious focal answer to
the question, “Which tire?”
Bring along a stack of index cards and, when you are ready to play this game, hand one
card to each student. After relating the story, ask each student to pretend that she is one of those
taking the exam and must answer the tire question on the card. Collect the cards and tabulate the
answers on the board. Start a discussion about why different students chose different tires; focus
on the difficulties of obtaining a focal equilibrium when players have different backgrounds or
concerns. You can also relate the discussion back to the material in the text regarding the
necessity of being prepared to face a strategically savvy opponent at any time.
GAME 5—Single-Offer Bargaining (The Ultimatum Game)
Two players, A and B, are chosen. Player A offers a split of a dollar (whole dimes only).
If B agrees, both get paid the agreed coins and the game is over. If B refuses, it is B’s turn, but
now the sum is only 80 cents. If A accepts B’s offer, the two get paid the agreed coins. If A
refuses, the game is over and neither gets anything.
Do this five times in succession with different pairs and the second-round totals falling
successively to 70, 60, 50, and 40 cents. Keep a record of the successive outcomes.
Again hold a brief discussion. The aim is to get students to start thinking about rollback
and subgame perfectness and, if students understand these strategies but still don’t play them,
why they don’t. Also, consider how the discrepancy changes with the second-round fraction.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
GAME 6—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 he wants—1 cent or $1 or some value in between. If the amounts
requested for a given pair of students (or for two randomly selected student responses) add up to
$1 exactly, 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. Thus, it is also relevant
to (and discussed again in) Chapter 4. But you may prefer to conduct an 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, with the tire game described in
Game 4 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 (Chapter 4). 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 7—Auctioning a Penny Jar (Winner’s Curse)
Show a jar of pennies; pass it around so each student can have a closer look and form an
estimate of the contents. Show students a stack of 100 pennies to give them a better idea of what
the jar might contain. While the jar is going around, explain the rules. Everyone submits a “sealed
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
bid”; hand out blank cards and ask students to write their names and bids on the cards and return
them. (This is also a good way for you to get to remember their names during the first meeting of
the class or the section.) The winner will pay his bid and get money (paper and silver, not
pennies) equal to that in the jar. Ties for a positive top bid split both prize and payment equally.
When you explain the rules, emphasize that the winner must pay his bid on the spot in cash.
After you have collected and sorted the cards, write the whole distribution of bids on the
board. Our experience is that if the jar contains approximately $5.00, the bids average to $3.50
(including a few zeros). Thus, the estimates are on the average conservative. But the winner
usually bids about $6.00. Hold a brief discussion with the goal of getting across the idea of the
winner’s curse.
The emphasis of this game is a concept relating to auctions that are not covered in the
text until Chapter 16. It is a simple enough game to play early in the semester if you want to
increase interest in the topics or hook additional students. One could certainly save this game
until ready to cover auctions.
GAME 8—All-Pay Auction of $10
Everyone plays. Show the students a $10 bill, and announce that it is the prize; the known
value of the prize guarantees that there is no winner’s curse. Hand out cards. Ask each student to
write her name and a bid (in whole quarters). Collect the cards. The highest positive bid wins
$10; if two or more tie with the highest positive bids, they share the $10 equally. All players pay
the instructor what they bid, win or lose.
Be sure to emphasize before bids are submitted that “This is for real money; you must
pay your bid in cash on the spot. You can make sure of not losing money by writing $0.00. But of
course if almost everyone does that, then someone can win with 25 cents and walk away with a
tidy profit of $9.75.”
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Once you have collected the cards, write the distribution of bids on the board. Hold a
brief discussion about the distribution and the value of the optimal bid. This game usually leads to
gross overbidding; a profit of $50 in a class or section of 20 students is not uncommon. If that
happens, you will have to find ways of returning the profit to the class; we have done this by
having a party if the sum is large enough or by bringing cookies to the next meeting if the sum is
small. Of course, do not announce this plan in advance.
This game is also treated in Chapter 16. If you play the game on the first day, you can
lead up to at least some of the points made there, even though the analysis at this early stage
cannot go anywhere close to that level. If you prefer to follow this game with a more in-depth
discussion and, perhaps, the derivation of the formula for the optimal bid, then you want to wait
and play it when you get to Chapter 16.
Movie Excerpts
Many movies contain scenes that illustrate some aspect of strategic interaction. You can
screen these scenes in class as an introduction to that topic and let a discussion lead on to
theoretical analysis of it. Most movies or excerpts from films are best shown later in the course,
in conjunction with the particular theoretical ideas being developed. But one short scene worth
showing at the outset, because many of your students will have seen the movie, is the poison
scene from The Princess Bride. (The “poison death scene” is available in various versions on
YouTube.) In it the hero (Westley) challenges one of the villains (Vizzini) to a duel of wits.
Westley will poison one of two wine cups without Vizzini observing his action and set one in
front of each of them. Vizzini will choose a cup for himself; Westley then must drink from the
other cup. Vizzini goes through a whole sequence of arguments as to why Westley would or
would not choose one cup or the other. Finally, he believes he knows which cup is safe and drinks
from it. Westley drinks from the other. Just as Vizzini is laughing and advising Westley to “never
go against a Sicilian when death is on the line,” Vizzini drops dead.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
You will want to pause the video at this point to have a brief discussion. The students will
quickly see that each of Vizzini’s arguments is inherently self-contradictory. If Westley thinks
through to the same point that leads Vizzini to believe that a particular cup will contain the
poison, then he should instead put the poison in the other cup. Any systematic action can be
thought through and defeated by the other player. Therefore, the only correct strategy is to be
unsystematic or random. This is a good way to motivate the idea of mixed strategies.
But this is not the main point of the story. Resume the video. The princess is surprised to
find that Westley had put the poison in the cup he placed closer to himself. “They were both
poisoned,” he replies, “I have been building up immunity to Iocaine for years.” Thus, the game
being played was really one of asymmetric information; Vizzini did not know Westley’s payoffs
and did not think the strategy of poisoning both cups was open to him. So you can get this idea in
at the outset of the course and tell the students that it will be discussed in greater depth in Chapter
8.
Here is some general advice for screening movie excerpts.
DO:
1. Take suggestions from students about other movies or games you can use; their ideas
may be more appealing to other students than yours. Always ask and discuss in advance
just what strategic issue the excerpt illustrates.
2. Come prepared with a note indicating the exact time where your excerpt begins on a
DVD or with the correct YouTube URL. The effectiveness of any clip can be lost if
students have to wait for you to find it.
DON’T:
1. Don’t assume that students know the general plots of the movies from which you
show excerpts. Prepare a brief explanation of the situation and the film’s characters as it
pertains to your excerpt, and give it just before you start the DVD.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
2. Don’t show a whole movie or a long clip when only a small point pertains to strategy.
The students will get distracted by the other incidental aspects.
News and Current Affairs
You will find that there is no shortage of motivation and illustration for game theory. Just
keep an eye on recent news events. These will give you several events of topical interest to tie in
with your course. More than likely, you will find examples of various concepts throughout the
semester that help you make even clearer your points about multiperson prisoners’ dilemmas,
credibility of strategic moves, the importance of patience in bargaining, and so on. Negotiations
between the government of Israel and the Palestinian authority and between the North Korean
regime and its neighboring countries and the United States provide continuing opportunities for
discussion. At a more trivial but perhaps more engaging level, so do incidents from the CBS
reality show Survivor.
One way to increase your students’ willingness or desire to apply what they are learning
to actual events is to provide a semester-long assignment requiring each of them to add to a class
collection of real-world events amenable to analysis using the theory of games. It may take them
a few weeks before they are able to do the appropriate analysis on their own, but you can add
weekly event analyses to the collection until such time as they are able to take over that role. You
do not have to restrict their examples to newsworthy events; many interesting and relevant
examples can be culled from recent films or novels. The film Waking Ned Devine, for instance,
has a wonderful example of a collective-action game and shows how individual incentives can be
different from those of the group as a whole.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Books and Web Sites
There are plenty of other sources for examples. The following books have several cases
and stories, varying greatly in their context and relevance, sometimes with an explicitly
game-theoretic analysis and sometimes without:
Steven J. Brams, The Presidential Election Game (New Haven, Conn.: Yale University Press,
1978).
Steven J. Brams, Biblical Games (Cambridge, Mass.: MIT Press, 1980).
Steven J. Brams, Superpower Games (New Haven, Conn.: Yale University Press, 1985).
Steven J. Brams and Alan D. Taylor, Fair Division (London: Cambridge University Press,
1996).
Adam Brandenburger and Barry Nalebuff, Co-opetition (New York: Doubleday, 1996).
Michael Chwe, Jane Austen: Game Theorist (Princeton, N.J.: Princeton University Press,
2013).
Avinash Dixit and Barry Nalebuff, Thinking Strategically (New York: Norton, 1991).
Avinash Dixit and Barry Nalebuff, The Art of Strategy (New York: Norton, 2008).
Paul Fisher, Rock, Paper, Scissors: Game Theory in Everyday Life (New York: Basic Books,
2008).
Erving Goffman, The Presentation of Self in Everyday Life (New York: Doubleday, 1959).
John Kay, Why Firms Succeed (London: Oxford University Press, 1995).
Paul Kennedy, ed., Grand Strategies in War and Peace (New Haven, Conn.: Yale University
Press, 1991).
John McMillan, Games, Strategies and Managers (London: Oxford University Press, 1996).
Howard Raiffa, The Art and Science of Negotiation (Cambridge, Mass.: Harvard University
Press, 1982).
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Richard Thaler, The Winner’s Curse: Paradoxes and Anomalies of Economic Life (Princeton,
N.J.: Princeton University Press, 1992).
In addition, there are several Web sites dealing with game theory from the perspectives of
research, teaching, and history of thought. Here are a few URLs:
Marko Grobelnik, Robert A. Miller, and Vesna Prasnikar have some downloadable software
(for matrix games and extensive-form games) for running classroom experiments at
www.comlabgames.com/strategicplay.
For teaching and research resources and many additional links, see
David Levine, UCLA, for a focus on economic theory: levine.sscnet.ucla.edu.
Alvin Roth, Stanford University, for a focus on laboratory experiments:
web.stanford.edu/~alroth/alroth.html.
For ideas on in-class experiments, some of which are appropriate for use with topics covered in
this text, see
www.marietta.edu/~delemeeg/expernom.html
gametheory.tau.ac.il
veconlab.econ.virginia.edu/admin.htm
The latter two sites above allow your students to play games online and collect data from their
responses for your use in-class discussion. Instructor registration is required.
There is also a chronology of the subject by Paul Walker of the University of Canterbury,
New Zealand: www.econ.canterbury.ac.nz/personal_pages/paul_walker/gt/hist.htm.
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