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CHAPTER 6
Combining Sequential and Simultaneous Moves
Teaching Suggestions
This chapter introduces three new concepts and associated techniques: (1) multistage games
where a simultaneous-move game occurs at one or more of the stages; (2) the effects of changing the
rules concerning the order of moves in a game—the source and nature of first- and second-mover
advantages; and (3) the connection between Nash and subgame-perfect equilibria—an introduction to
the notion of credibility.
Numerous examples are available to illustrate the general idea of a combination of sequential
and simultaneous moves.
Sports Example
In football, the coach for the offense calls a play, and simultaneously the coach for the defense
chooses his players and formation. On the field, the players on each side can observe the formation of
the other side. Then the quarterback can change the play at the line of scrimmage, and the defense’s
captain on the field can shout for some minor adjustments. Each can also call a time-out.
Politics Example
For many legislative and executive positions in the United States, party primary elections
precede the general election. If politics is polarized between the two main parties, taking an extreme
ideological position is the surer way to winning a primary. But that enables one’s opponent in the
general election to label one an extremist and reduces one’s chances of winning the general election.
The optimal strategies in the primaries, and therefore the equilibrium of the full game, depend on a
forward-looking calculation of these consequences.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Economic Example
Two firms simultaneously choose locations; price competition follows at the second stage. In a
game of location choice alone, there would be a Hotelling equilibrium with identical central location
choices. In the two-stage game, the centralizing tendency is offset, and may be reversed, by the firms’
foresighted calculation that the closer they are to each other, the fiercer will be the price competition.
In these examples, the payoffs are likely to be situation specific. You can set up the general
structure, and then hold a discussion about the actual payoffs to be entered. This process will give the
class a valuable lesson in “setting up a game to be analyzed.”
Consider the politics example, and one of the primaries. If Candidate A chooses an extremist
position and B a moderate position, then A is considered more likely to win. (But strategic voters in the
primary can be foresighted, too, and Candidate B may actually remind them of the importance of
choosing a more electable candidate.) But with this strategy, A will have a lower probability of winning
the general election than he would as an established moderate. The payoffs to A are not obvious or
unambiguous. If he suffers a bad loss in the general election, A could disappear into obscurity
(McGovern?), or such a loss may give him an even higher status among the party faithful, and perhaps
years later he may come to be regarded as the prophet of a new political philosophy (Goldwater?). If B
wins the primary despite taking on a moderate position, he may be more likely to win the general
election and enjoy the prestige and perks of office; but if he is really an ideological extremist, the
moderate platform may constrain him in the kinds of policies he can pursue. The choices and outcomes
can vary depending on the relative importance the candidates attach to these alternatives.
In the location-price (economics) example, the nature of the payoffs—profits—is not in doubt.
But how profits vary with different choices will depend on specific characteristics of the market
including the distribution of the consumers in physical space or a space of preferences, the degree of
substitutability between the two firms’ products, and so on. If your class is mathematically
sophisticated, you can construct and solve actual models of these interactions. Some references are
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Claude D’Aspremont, J. Jaskold Gabszewicz, and Jaques-François Thisse, “On
Hotelling’s ‘Stability in Competition,’ ” Econometrica, vol. 47, no. 5 (September 1979), pp.
1145–1150.
Avner Shaked and John Sutton, “Relaxing Price Competition through Product
Differentiation,” Review of Economic Studies, vol. 49, no. 1 (January 1982), pp. 3–13.
Another collection of interesting two-stage oligopoly examples, leading to a general principle
about when a first-stage strategic commitment involves too much versus too little aggression, is Drew
Fudenberg and Jean Tirole, “The Fat Cat Effect, the Puppy Dog Ploy, and the Lean and Hungry Look,”
American Economic Review, vol. 74, no. 2 (May 1984), (Papers and Proceedings), pp. 361–368. An
incidental great merit of this paper is that students will read it attracted by the title, and then learn a lot
from it.
We have given several examples in the text to illustrate the consequences of changing the rules
of a game from sequential to simultaneous moves and vice versa, or reversing the order of moves. You
can replace these with any other games of similar basic structure—Prisoners’ Dilemma, Chicken, and
so on. If you want to emphasize economic applications, you can take the Cournot and Bertrand models
of Chapter 5 or some variants of them to show that there is a first-mover advantage in quantity-setting
duopoly, but generally a second-mover advantage in price-setting duopoly.
Similarly, changing a game from sequential play to simultaneous play can mean that new
equilibria arise—either multiple equilibria or equilibria in mixed strategies. Use the sequential-game
examples you used to convey the material from Chapter 3 to show that there might be additional
equilibria in the simultaneous-move versions of the game. This works for the Tennis-Point game if you
teach it as a sequential game or for the three-person voting example from Ordeshook (cited in Chapter
3 of this manual). The Mall-Location game, familiar to those who used the first edition of Games of
Strategy, is another option here. That game is now described in Exercise S9 of Chapter 3 and analyzed
in Exercises S10 and S11 of this chapter.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
The most important new component of the analysis in Chapter 6 is the representation of
sequential-move games in strategic form and the solution of such games from that form. The second-
(and third-) mover’s strategies are more complex in sequential games, and the payoff table must have
adequate rows (or columns or pages) to accommodate all of the possible contingent strategies available
to players. Again, the tennis-point or voting examples can be used to illustrate this idea. One nice
exercise is to assert the existence of a new number of Nash equilibria in the strategic form (like those
found for the game of Figure 6.14; see the solution below). Then show how only one of these qualifies
as a subgame-perfect equilibrium of the sequential-move game in its extensive or tree form. (In
exceptional cases of ties, there may be more.) Finally, if each player moves only once, show how the
subgame-perfect equilibrium can be found in the normal- or strategic-form payoff matrix by successive
elimination of weakly dominated strategies done in an order of players that is the opposite of the order
in which they make their moves. Interpret the other Nash equilibria as involving some statement about
a future move (threat or promise) that is not credible. Students often have difficulty grasping the idea
that the eliminated equilibria are unreasonable because of the strategies associated with them rather
than because of the (often) lower payoffs going to the players, so you will want to reinforce this idea as
often as possible. The Boeing-Airbus example from Exercise S4 in Chapter 3 is the simplest example
that can serve this purpose; it is easy to explain and to understand why Boeing’s threat to fight if
Airbus enters is not credible. The concept of credibility will be developed in more detail and used
extensively in Chapter 9, so it is useful to introduce it here.
Game Playing in Class
GAME 1—Color Coordination (with Delay)
This game should be played twice, once without the delay tactic and once with it, to show the
difference between outcomes in the simultaneous and sequential versions. As usual, the game can be
played by pairs of students, although it can also be played by all students simultaneously with those on
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
the left-hand side of the room playing against those on the right-hand side. (“Right-hand” and
“left-hand” are defined from the students’ point of view. Tell the students not to write down anything
(except their names) until they hear all of the instructions. As with the Tacit Coordination game
described in Chapter 4, you might want to provide some inducement for coordination here; chocolate
usually works well.
Ask the students to choose partners from the other side of the room or have them imagine that
each is playing with one person who is sitting on the other side of the room. Each student will
eventually be asked to write either pink or purple. If both students in the real or imaginary pair write
pink, the person on the right-hand side of the room gets 50 points and the person on the left-hand side
of the room gets 40 points. If both write purple, the person on the left-hand side of the room gets 50
points and the person on the right-hand side of the room gets 40 points. If the answers don't match,
neither player gets anything.
To play without the delay tactic, simply ask the students to choose a color and write the choice.
Then play again, immediately, but explain that you will flip a coin first. If it comes up heads, those on
the right-hand side of the room get to write their answers first; otherwise those on the left-hand side of
the room write first.
Once you have collected answers from the students, you can discuss the implications of the
game. Clearly, it is much more difficult to coordinate without the benefit of the delay tactic, and there
are two equilibria in pure strategies in the simultaneous-move game. The delay tactic makes the game
sequential and creates a first-mover advantage; outcomes from this version often come much closer to
complete coordination. As usual, you can ask students to try to come up with real-world situations that
replicate some of the conditions of the game, or you can provide some examples.
GAME 2—Zenda
This game is explained in Exercise U4 of Chapter 10. It is a two-stage game in which the
second stage is a prisoners’ dilemma and, in the first stage, the players can choose and place in an
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
escrow account sums that are rewards or bribes to their opponents for choosing the cooperative action
in stage 2. The escrow account gives credibility to the promise of the rewards. We have found that
students learn the correct amount of rewards and achieve cooperative outcomes very quickly. Therefore
the game will also serve as a useful lesson in resolving prisoners’ dilemmas.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
