ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
*In response to complaints about grammar from language mavens, Winston responded with a new slogan: “What do you want, good grammar or good taste?”
3.2 Solving a Game when Players Are Rational 61
cigarettes sold. Some studies by economists show that advertising doesn’t have much of an impact on the number of smokers and instead just shifts the existing set of smokers among the different brands. But then there is other ev- idence that advertising dissuades smokers from stopping and lures nonsmok- ers (in particular, youth) into trying smoking. To keep our model simple, let us assume that advertising doesn’t affect the total number of packs sold and just shifts smokers among the different brands.
Suppose the annual demand for cigarettes is 1,000,000,000 (one billion) packs and that the market share of a company depends on how much it spends on advertising relative to what its rival spends. Let ADVPM denote the advertising expenditures of Philip Morris (PM) and ADVRJR denote the
C O
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SY O
F TH
E A
D V
ER TI
SI N
G A
R C H
IV ES
advertising expenditures of R. J. Reynolds (RJR). Assume that the market share of PM equals
This quotient says that PM’s share of all packs sold equals its share of adver- tising. Such a model is overly simplistic, but all that we really need to assume is that sales are higher when more is spent on advertising. The total number of packs sold by PM is then
1,000,000,000 �
and by a similar argument, the corresponding number for RJR is
1,000,000,000 �
If each pack sold generates a profit of 10 cents (remember, we’re back in 1971), then the profit that PM gets from spending ADVPM dollars is
0.1 � 1,000,000,000 �
or
100,000,000 �
Analogously, RJR’s profit is
100,000,000 �
Our objective is to say something about how much these companies advertise. To keep things simple, assume that there are just three levels of advertising: $5 million, $10 million, and $15 million. In that case, the pay- off matrix is as shown in FIGURE 3.4 (where strategies and payoffs are in millions
a ADVRJR ADVPM � ADVRJR
b � ADVRJR. a ADVPM
ADVPM � ADVRJR b � ADVPM.
a ADVPM ADVPM � ADVRJR
b � ADVPM,
a ADVRJR ADVPM � ADVRJR
b . a ADVPM
ADVPM � ADVRJR b ,
ADVPM ADVPM � ADVRJR
.
62 CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
FIGURE 3.4 The Cigarette Advertising Game
45,45 28,57
57,28 40,40Philip Morris
R. J. Reynolds
Spend 5 Spend 10 Spend 15
60,20 45,30
20,60
30,45
35,35
Spend 5
Spend 10
Spend 15
Now suppose the ban on TV and radio adver- tising is put into effect and it has the impact of making it infeasible for the cigarette companies to spend $15 million on advertising. That is, the most that a company can spend using the re- maining advertising venues is $10 million. In the context of this simple game, each company’s strategy set is then constrained to comprise the choices of spending $5 million and spending $10 million, as shown in FIGURE 3.5.
The solution to this game is that both companies spend moderately on ad- vertising, since spending $10 million strictly dominates spending 5 million. And what has this intrusive government policy done to their profits? They have increased! Each company’s profit rises from $35 million to $40 million.
In the original game, each company had a dominant strategy of spending $15 million. This heavy advertising tended to cancel out, so each ended up with 50% of the market. If they both could have restrained their spending to $10 million, they would each still have had half of the market—thus leaving them with the same gross profits—and would have spent less on advertising, which translates into higher net profit.
By reducing the options for advertising, the TV and radio ban served to re- strain competition, reduce advertising expenditures, and raise company profits.
of dollars). For example, if PM spends $5 million and RJR spends $15 million, then PM’s market share is 5/(5 � 15), or 25%. PM then sells 250 million packs (0.25 multiplied by 1 billion) and, at 10 cents per pack, makes a gross profit of $25 million. Once we net out the cost of advertising, PM’s profit (or payoff) is $20 million.
TABLE 3.2 shows that a strategy of spending $15 million strictly dominates spending either $5 million or $10 million. For example, if RJR spends $5 mil- lion, then PM earns $60 million from spending $15 million (and gets 75% of the market), while PM earns $57 million with an advertising budget of $10 million and only $45 million by matching RJR’s paltry expenditure of $5 mil- lion. Similarly, a budget of $15 million for PM outperforms the other two op- tions when RJR spends $10 million and when it spends $15 million. Thus, PM prefers the heavy advertising campaign regardless of what RJR chooses, so heavy advertising is a dominant strategy for PM. Because the same can be shown for RJR, the prediction is that both cigarette companies inundate our television sets with attractive men and women spewing forth smoke.