Consolidation, extension, and discussion, Game Theory

For the section on dynamic games of competition, you can begin by asking if anyone in the class has played competi- tive tennis (club or collegiate or better); there is usually one. If not, a dedicated viewer of major tournaments on TV is a good substitute. Ask such a person which point in a game is the most important; the answer is invariably "break point," where the server is 30- 40 or Ad Down. Then ask which game in a set is the most important; the answer is usually the seventh or the eighth. Ask why, and the answer is often, "The coach told me" or "The TV commentators say so." Some- times they may say, "Lose that point and you've lost the game" or "Lose that game and you've lost the set." The former is trivially true, whereas the latter is not true, and you can recount (or ask them to recall) many counterexamples. Offer to explain the idea of the importance of a point or a set in a more systematic way. When you have done so, and also brought out the parallel with R&D or related competitions in business, for example the current rivalry between Microsoft and Netscape, not only will the specific point have been gotten across more clearly but the confidence of the class in the whole analytical apparatus will have increased.

For those of you who want to focus on solving for mixed- strategy equilibria in larger games than the two-by-two games which were the focus of Chapter 5, this is your chance. You can construct your own examples or add strategies to the two-by-two games you covered earlier. This is a good time to play the rock-scissors-paper game. You can use that game to motivate many of the issues relevant to mixing in larger games.

If you can bring a laptop equipped with Gambit to the class, you can solve even more complex games on the spot. For example, you can take the soccer penalty kick game, ask the students for suggestions on how the strategies or the payoffs might differ from the numbers in the text, and display the resulting equilibrium immediately. If you can hook up the computer so that the monitor screen can be directly shown on the class screen, the effect is more dramatic.

We expect you will teach the more mathematical mate- rial in this chapter only if your class has enough familiarity with algebra and the general process of mathematical reasoning. Such a class should not need any storytelling or game playing.

If your class is sufficiently sophisticated in logical rea- soning, then you can spend more time on the concept of rationalizability. Start with the example of Figure 7.1 in the text, and show how for each player, every strategy can be justified on the basis of a complete set of logically consis- tent beliefs about the other player's beliefs about your beliefs about what one might choose; that is, every strategy is rationalizable. For example, Row can justify choosing A on the basis of the belief that Column will play B, which in turn can be justified because Row believes that Column believes that Row will play C, which in turn can be justified because Row believes that Column believes that Row believes that Column will play A, and so on. Then move on to a more complex matrix, which comes from Douglas Bernheim's original paper on rationalizability.

 

COLUMN

C1

C2

C3

C4

 

 

 

ROW

R1

0, 7

2, 5

7, 0

0, 1

 

R2

5, 2

3, 3

5, 2

0, 1

 

R3

7, 0

2, 5

0, 7

0, 1

 

R4

0, 0

0, -2

0, 0

10, -1

 

 
For Row, the strategies R1, R2, and R3 are all rationalizable on the basis of a chain of beliefs constructed by arguments similar to that given above. But R4 is not rationalizable; for that strategy, Row would have to believe that Column would play C4, but that strategy is dominated for Column by an equal-probability mixture of C1 and C3. Incidentally, this also conveys the idea that a strategy may be dominated by a mixed strategy that combines some of the other pure strategies, something that we have not discussed in the book.

If  you  are  teaching  the  class  at  this  level,  you  must have used the example of a quantity-setting duopoly with downward-sloping best-response functions. Now you can discuss rationalizability in its context. Here the logic of suc- cessive rounds of beliefs leads to the elimination of more and more ranges of strategies. An example of the thought process proceeds thus: My output must be nonnegative, therefore I can be sure that the other firm will never produce more than its monopoly output (its best response to my 0), therefore it will not believe that I will produce any less than my best response to its monopoly output, and so on. This eventually leaves only the (Cournot) Nash equilibrium as rationalizable.

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