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CHAPTER 5
Simultaneous-Move Games: Continuous Strategies, Discussion, and Evidence
Teaching Suggestions
The basic concepts of simultaneous moves, best responses, and Nash equilibrium were
motivated in Chapter 4 in the context of discrete strategies. The new issues arising when
strategies are continuous variables are of a slightly more technical kind and include drawing
best-response curves, finding best-response equations, and solving these simultaneously to find
Nash equilibria. Our suggestions and games in this chapter are accordingly of a slightly more
technical nature.
For most teachers who come to game theory from economics, the analysis of games
with continuous strategies is familiar terrain. Any example of a duopoly, considering firm price
setting, quantity setting, advertising, or R&D behavior, will serve well to convey the ideas of
Nash equilibrium and rationalizability. For others, or for those who prefer to limit the
analytical complexity of their courses, Sections 1 and 3.B can be omitted without significant
loss of continuity, as most games described in later chapters are discrete, not continuous. Many
of you will probably want to cover a little on rationalizability (Section 3.A), and almost all will
include or even expand on the discussion and criticisms of Nash equilibrium (Sections 2 and 4).
If you teach more-sophisticated students, you may find that they see the use of
continuous strategies as perfectly natural and an obvious extension of the discrete games
described most often in the text. For other students, the leap from a table in which you identify,
say, Row’s strategy that is her best response to each of Column’s strategies to a situation in
which that identification process leads to a continuous function can be hard to grasp. You can
convince students of the usefulness of best-response curves without undue emphasis on the
mathematics if you concentrate on the intuition behind the algebra. Explaining what a
best-response function does and that it can be derived as the solution to an optimization
problem may be sufficient for many purposes; you can then supply the functions and work
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
through the mechanics of graphing them. Once you have them on the board, most students will
be able to follow the interpretation and will see how (and where) the Nash equilibrium arises.
Some students may have trouble following the deriva tions for, or understanding the
significance of, a best-response function. It may help these students to go through a derivation
in which the best-response rule is built up from scratch. Here is a simple example.
If you used the Guessing Half of the Average game from Chapters 1 and 2 of this
manual or plan to use it here, it serves the purpose very well. When playing the game, you
probably restricted the choices to integers between 0 and 100 for ease of computation. But the
analysis proceeds more conveniently by treating them as continuous variables. First consider a
two-player version. Denote their choices by X and Y. For any given Y, the choice of X that
comes closest to half of the average is found by setting
which yields X = Y/3. This is the X player’s best-response function. Similarly, the Y player’s
best-response function is Y = X/3. These can be drawn in an (X, Y) graph to show that they
intersect only at the origin. Or algebraically, the two together give X = (X/3)/3 = X/9, which has
the unique solution X = 0. Then Y = 0 also.
X and Y are bounded above by 100. So you can carry out the same kind of sequential
reasoning as in the Cournot duopoly of Figure 5.7 in the text to show that X = 0 and Y = 0 are
the only rationalizable strategies for the two players.
When there are more players, the calculation goes as follows. With n players, denote
the choices by X1 , X2 , X3 , . . . Xn . Then Player 1’s best-response function is
The other players’ best-response functions are similar. Then the first of these equations,
compared across players, shows that X1 = X2 = X3 = … = Xn in the Nash equilibrium. Call the
common value X. Then X satisfies X = X/2 or X = 0.
You can give your own general discussion of the Nash equilibrium concept in lecture
format, or you can conduct a class discussion on it. For the latter, you can let it develop out of
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
games currently being played, or remind the class of the whole list of games they have played
and studied so far and ask them to bring together all the concepts they have learned so far.
Either way, there are some pertinent questions to be addressed.
1. In what kind of games does the Nash equilibrium concept provide a useful prediction
of the outcome? The general experience is that it works in games with unique Nash equilibria,
when the players have experience playing these or similar games with similarly experienced
others. The experience of your class may differ from this in some respects. There is also the
question of whether the Nash equilibrium prediction is an adequate approximation, as in the
game of guessing half of the average after a few trial runs.
2. When Nash equilibrium does not work, why does it fail? Possible reasons:
(i) The players’ objectives or payoff functions are different from what the game theorist
assumed when analyzing the game.
(ii) The calculations involved in solving for the Nash equilibrium were too complex or
unfamiliar.
(iii) There were multiple Nash equilibria and no adequate device was available for
selecting one of them; for example, the players did not have enough common background or
beliefs to locate a focal point. You will need to guide or channel the discussion to bring out
these points. If your class is sufficiently sophisticated, you may be able to tell them about
alternative or more general solution concepts. You can explain rationalizability in greater
detail, or even talk about quantal-response equilibrium. However, we suspect that in most
classes using this book, these concepts will be too difficult to treat in any depth.
Game Playing in Class
If you did not play the bargaining game from Chapter 4 of the manual (Game 3—
Divide a Dollar), you can play it here, allowing continuous strategy choices (although for
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
eventual settlements you will have to round them up or down as appropriate). Another similar
game was suggested to us by Roy Ruffin of the University of Houston, and goes as follows:
GAME 1—Cournot’s Cake
This is a divide-a-dollar game with a twist to the settlement rule. Two players play the
game, and simultaneously choose the fractions X and Y, respectively, that they would like to
receive. The amount that each actually gets is the smaller of her choice and the surplus (if any)
of the dollar that remains after summing their two choices. Thus the X player gets min(X, 1 – X
– Y) and the Y player gets min(Y, 1 – X – Y).
For any given Y < 1, the X player’s best response as a function of X increases up to
X = (1 – Y)/2 and decreases thereafter. So X = (1 – Y)/2 is the X player’s best-response function.
Similarly, the Y player’s best-response function is Y = (1 – X)/2. Graphing or solving these
algebraically gives the Nash equilibrium: X = Y = 1/3.
These are the amounts that would be produced by Cournot duopolists with zero costs
and facing the inverse demand function P = 1 – X – Y, giving the price P as a function of the
quantities X and Y, hence the name for the game. Having played (or analyzed) the game, you
can bring out this parallel, and give an interesting example of how the common mathematics of
game theory underlies two seemingly very different situations—duopoly and bargaining.
Computer software packages for playing simultaneous-move games (such as oligopoly)
are now widely available, and you may be able to use one of them in your class. Two examples
are
1. University of Virginia: veconlab.econ.virginia.edu/admin.htm.
2. University of Arizona: www.econport.org:8080/econport/request?
page=web_experiments_software_nfg.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
