978-0393919684 Chapter 5 Lecture Note

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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

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 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

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.