PART ONE—Introduction and General Principles
CHAPTER 1: Basic Ideas and Examples
and CHAPTER 2: How to Think about Strategic Games
Teaching Suggestions
These chapters try to arouse the students’ interest and prepare the groundwork for the analysis to
come. Chapter 1 uses several stories or examples to illustrate different kinds of strategic games, and
Chapter 2 begins to connect them to a framework of concepts and terminology. Some teachers may find it
better to mingle the two—pause after each story to define the concepts and terms relevant to it and then
do the same with the next story. Others may prefer to present the stories from Chapter 1 as a way of
stimulating student interest and capturing student attention, following their presentation with a summary
of the terminology and the classifications found in Chapter 2.
Much of Chapter 2 may be most useful to students as a reference source—somewhere for them to
turn to remind themselves about the difference between decisions and games, the game-theoretic meaning
of rationality, or what distinguishes cooperative games from noncooperative games. You will certainly
want to define those terms and concepts that you will focus on during the semester, and em phasize the
connections between the stories and the technical jargon used in analyzing them. It is probably not
necessary at this stage to go into great detail about each level of classification, each term, and each
assumption. A brief summary or discussion in the context of various stories will allow you to move
forward; you can always return to specific concepts when they become crucial to your analysis later in the
term.
Stories to Motivate the Subject and the Main Ideas
Some will find that it is not a good idea to retell our stories from Chapter 1. Students can easily
read them in advance, and depending on the culture of the institution at which you teach, many may do
so. Others will find that you can retell our stories on the first day of class; while a few students may have
read into the book already, the majority may be waiting for a syllabus (or even to buy the book). In either
case, you can use our stories as the basis for variation or discussion. You can provide your own variations
on the stories or ask students to come up with stories of their own.
Whether you tell our stories or your own, or have students think of their own, you can start the
discussion of a story by posing questions such as “Was this a game with strategic interaction or only a
decision problem?” “If a game, who were the strategically active players?” “What were the strategies
available to the players—not merely what they did but what else they could have done?” “In light of their
strategies, can we make sense of why they did what they did?” And so on. This process can quickly build
up to a framework for understanding strategies as complete plans of action, rollback, each player’s
simultaneously thinking of what everyone else is doing, or even equilibrium. It is also a good way to
make the transition from the stories to the concepts of Chapter 2.
Most students in an elementary course will not have an extensive background in economics,
politics, or business studies. Therefore, motivating them by using examples from these disciplines to
introduce the ideas of strategies and games may not work. We have chosen the stories in Chapter 1 to
relate to the lives of the students—relations with parents, siblings, and friends; sports; and so on. If your
class has some specific background, you should of course use it for sources of stories or cases. Thus,
economics or business teachers may be able to use the OPEC cartel to motivate the prisoners’ dilemma,
repeated play, and different strategic situations of large and small players, or a very simple version of a
Keynesian low-level equilibrium trap (no firm invests and creates jobs, because none thinks that it can
sell the output profitably, because incomes are low, because firms aren’t investing) to motivate the idea of
lock-in equilibria in games with positive feedbacks. In courses more specifically targeted to political
science students, teachers may be able to use campaign advertising or special-interest lobbying stories to
introduce the prisoners’ dilemma.
In later chapters, of course, we do develop examples from economics, politics, and so on, in each
case explaining the discipline-specific contexts to the extent necessary.
Game Playing in Class
Playing a few well-designed games in class, and watching others play them, brings to life the
concepts of strategy, backward induction, and Nash equilibrium far better than any amount of formal
statement or problem-set drill. There are several games that are appropriate for use on the first or second
day of class. These games are simple but can be used to convey important points about basic tools and
concepts, including rollback analysis, multiple equilibria, focal points, and so on. Indeed, we like to start
the course with two classroom games, before teaching or even mentioning any of these concepts at all.
Games 1 and 2 go well together as do 5 and 6; other pairings can be devised as appropriate to your own
course. We generally choose one sequential-move game and one simultaneous-move game from the list
below. The concepts relevant to each emerge naturally during the discussion of each game.
The list provided here is sufficiently long that you will not be able to use all of the games on the
first day of class. You will probably find that some of the games we discuss here are better suited for use
later in the semester when you are covering the material relevant to the game. We have noted those games
that are repeated later in the Instructor’s Manual and the chapter in which they reappear.
Here is some general advice about playing games in class:
DO:
1. Use real money for prizes. Even small amounts raise student interest and attention. You can make
students’ scores in the game count for a small fraction of the course credit.
2. Take enough rolls of coins to class with you when playing games that involve paying out or taking in
small amounts of change; do not expect your students to have the right change. (We have also been
successful using food items, like M&Ms or jelly beans, as an alternative to actual currency.)
3. Use games with stories, not just abstract trees or matrices.
4. Follow the game immediately with a discussion that brings out the general concepts or methods of
analysis that the game was supposed to illustrate. If there isn’t enough time for a good discussion,
circulate an explanation or post it on the course Web site.
DON’T:
1. Don’t make the game so complex that the main conceptual point is smothered.
2. If the games count toward the course credit, don’t choose games where the outcome depends
significantly on uncertainty rather than skill.
GAME 1—21 Flags
This is a simple Nim-like sequential-move game that we have adopted from Episode 6 of the
Survivor Thailand television series that aired in fall 2002 (Survivor Season 5). The game is played
between two players (or teams) who alternate in taking turns to remove some number of flags from a field
of 21 available for play. Each time it is a player’s turn, he chooses to remove one, two, or three flags; the
player to remove the last flag is the winner. For easier in-class play, you can use coins instead of flags; lay
them out on the glass of the overhead projector so the whole class can easily see what is going on.
In the Survivor show, the game was played as an “immunity challenge” between two “tribes.” The
losing tribe had to vote out one of its members, weakening it for future competitions. In the specific
context, this loss had a crucial effect on the eventual outcome of the game. Thus, a million dollars hinged
on the ability to do the simple calculation. The actual players got almost all of their moves wrong, so if
you have access to the DVD version of the show, you should consider showing it to your students. Seeing
the video and then playing a similar game will be a good way for your students to learn the concepts.
The correct solution is simple. If you leave the other player (or team) with 4 flags, he must
remove 1, 2, or 3, and then you can take the rest and win. To make sure of leaving the other player with
four flags, at the turn before, you have to leave him facing 8 flags. Carrying the logic further, that means
leaving 12, 16, and 20 flags on subsequent turns. Therefore, starting with 21 flags, the first player should
remove 1, and then proceed to take 4 minus whatever the other takes at his immediately preceding turn.
Have several pairs of students play this game in front of the class. We have found that the first
pair makes choices almost at random, but the second pair does better, figuring out one or perhaps two of
the final rounds correctly. By the third or at most the fourth time, the players will have figured out the full
backward induction.
You can then hold a brief discussion. You should nudge or guide it a little toward three
conclusions. First, bring out the idea of backward induction, or the importance of solving sequential-move
games backward from the final moves. Second, identify the idea of “correct strategies” that constitute a
solution of the game; explain that you will soon give this solution a formal name, the rollback
equilibrium. Finally, get to the idea that one can learn correct strategies by actually playing a game. With
this conclusion comes the idea that if a game is played by experienced players, we might expect to
observe correct strategies and equilibrium outcomes. This will give students some confidence in the
concepts of backward induction and rollback equilibrium.
The last remark motivates a brief digression. Over the last decade, behavioral game theorists have
made a valuable contribution to the stock of interesting games that can be played in classrooms. However,
many of them come to the subject with a negative agenda, namely, to argue that everything in
conventional game theory is wrong. Our experience suggests otherwise. To be sure, it takes time and
experience merely to understand the rules of any game, and a lot of practice and experimentation to play
it well. But students learn quite fast, often faster than their teachers. Some basic difficulties with the
foundations of the subject undeniably remain and give interest to it at a research level. But it is
counterproductive to tell beginners that what they are about to learn is all wrong; it destroys their whole
motivation to learn. We find it better to convey a sense of guarded optimism about the standard Nash
theory, without pretending that it closes the subject. Of course we believe this to be the truth of the matter.
GAME 2—Guessing Half of the Average
This is the well-known simultaneous-move game of the “generalized beauty contest.” Choose 10
students in the class, and give them blank cards. Each student is to write his or her name on the card, and
a number between 0 and 100. The cards will be collected and the numbers on them averaged. The student
whose choice is closest to half of the average is the winner. These rules are, of course, explained in
advance and in public.
The Nash equilibrium of this game is 0; it results from an iterated dominance argument. Since the
average can never exceed 100, half of the average can never exceed 50. Therefore, any choice above 50 is
dominated by 50. Then the average can never exceed 50, . . . The first time the game is played, the winner
is usually close to 25. This outcome fits Nagel’s observation (1995) that the outcome is as if the students
expect the others to choose at random, averaging 50, and then choose half of that. Next, choose a different
set of 10 students from the class (who have watched the outcome of the first group’s game). This second
group chooses much smaller numbers, and the winner is close to 10 (as if one more round of the
dominance calculation were performed) or even 5 or 6 (as if two more rounds were performed). The third
group of 10 chooses much smaller numbers, including several zeros, and the winner’s choice is as low as
3 or 4. Incidentally, we have found that learning proceeds somewhat faster by watching others play than
when the same group of 10 plays successively. Perhaps the brain does a better job of observation and
interpretation if the ego is not engaged in playing the game.
Again hold a brief discussion. The points to bring out are:
1. The logical concept of dominance, iterated elimination of dominated strategies, and the
culmination in a Nash equilibrium.
2. Getting close to the Nash equilibrium by the experience of playing the game. Whether it is a
crucial flaw of the theory that zero is rarely exactly attained or true that the theory gives a good
approximation can be a point to be debated depending on the time available.
3. The idea that if you have good reason to believe that others will not be playing their Nash
equilibrium strategies, then your optimal choice will also differ from your own Nash equilibrium
strategy.
The discussion can also touch on the question of what would happen if the object of the game
were to come closest to the average, not half of the average. That game is, of course, Keynes’s famous
metaphor for the stock market, where everyone is trying to guess what everyone else is trying to guess.
The game has multiple Nash equilibria, each sustained by its own bootstraps. Details of this are best
postponed to a later point in the course where you cover multiple equilibria more systematically, but a
quick mention in the first class provides students with an interesting economic application at the outset.
You can also stress the importance of this game in their own lives. Part or even all of their social security
is likely to be in individual accounts. When they decide how to invest this sum, they will have to think
through the question: Will the historical pattern of returns and volatility of various assets persist when
everyone makes the same decisions that I am now contemplating? This interaction between individual
choice (strategy) and aggregate outcomes (equilibrium) comes naturally to someone who is trained to
think game-theoretically, but others are often liable to forget the effect of everyone’s simultaneous
choices. In the context of saving for retirement this can be very costly.
GAME 3—Claim a Pile of Dimes (The Centipede Game)
This simple two-player game is similar to Game 1 (21 Flags) in terms of the concepts and issues
that it highlights. In this game, two players, A and B, are chosen. The instructor places a dime on the
table. Player A can say “Stop” or “Pass.” If Stop, then A gets the dime and the game is over. If Pass, then a
second dime is added and it is B’s turn to say Stop or Pass. This goes on to the maximum of a dollar (five
turns each). The players are told these rules in advance. As with the flag game, you can play this game
five times in succession with different pairs of players for each game. Keep a record of where the game
stops for each pair.
This game is discussed in the text (Chapter 3, Figure 3.7), but most students will not have read
that far ahead at this stage. Our experience is that the simple, theoretical subgame-perfect equilibrium of
immediate pickup is never observed. Most games go to 60 or 70 cents, but you do see the students
thinking further ahead. Later pairs learn from observing the outcomes of earlier pairs, but the direction of
this learning is not always the same. Sometimes they collude better; sometimes they get closer to the
subgame-perfect outcome.
After the five pairs have played, hold a brief discussion. Ask students why they made the choices
that they did. Develop the idea of rollback (or backward induction). Investigate why they did not achieve
the rollback equilibrium. Did the players fail to figure it out, or did they understand it instinctively but
have different objective functions?
This game could also be played to motivate the ideas of rollback right before they are covered
with the material in Chapter 3. If you prefer to cover simultaneous-move games first, then you might want
to save this game until after you have completed that material. However, if you are following the order of
the material in the book, rollback is likely to be the subject of your lectures within the first two weeks;
you could use this game to motivate the following week’s lectures.