The following is an interpretation of the rivalry between the United States and the Soviet
Union for geopolitical influence in the 1970s and 1980s. Each side has the choice of two
strategies: Aggressive and Restrained. The following payoff matrix prevails:
|U SA||Soviet Union|
For each player, 4 is best and 1 is worst.
a.Does either country have a dominant strategy?
b.Suppose the countries move simultaneously. Find the Nash equilibrium.
c.Next consider three different and alternative ways in which the game could be
played with sequential moves:
|The U.S. moves first and the Soviet Union moves second
The Soviet Union moves first and the U.S. moves second
The Soviet Union moves first, the U.S. moves second, but the Soviet Union
has a further move where they can change their first move.
For each case, draw the game tree and find the sub-game perfect equilibrium.
d.What are the key strategic issues (e.g. commitment, credibility etc.) for the two
2. Two manufacturers currently are competing for sales in two different but equally profitable
product lines. In both cases the sales volume for manufacturer 2 is three times as large as that
for manufacturer 1. Because of a recent technological breakthrough, both manufacturers will
be making a major improvement in both products. However, they are uncertain as to what
development and marketing strategy to follow.
If both product improvements are developed simultaneously, either manufacturer can have
them ready for sale in 12 months. Another alternative is to have a “crash program” to
develop only one product first to try to get it marketed ahead of the competition. By doing
this, manufacturer 2 could have one product ready for sale in 9 months, whereas manufacturer
1 would require 10 months (because of previous commitments for its production facilities). For
either manufacturer, the second product could then be ready for sale in an additional 9
For either product line, if both manufacturers market their improved models simultaneously, it
is estimated that manufacturer1would increase its share of the total future sales of this product
by 8 percent of the total (from 25 to 33 percent). Similarly, manufacturer 1 would increase its
share by 20, 30, and 40 percent of the total if it marketed the product sooner than
manufacturer 2 by 2, 6, and 8 months, respectively. On the other hand, manufacturer1would
lose 4, 10, 12, and 14 percent of the total if manufacturer 2 marketed it sooner by 1, 3, 7,
and 10 months, respectively.
Formulate this problem as a two-person, zero-sum game, and then determine which strategy
the respective manufacturers should use according to the minimax criterion.
3. Two politicians soon will be starting their campaigns against each other for a certain political
office. Each must now select the main issue she will emphasize as the theme of her campaign.
Each has three advantageous issues from which to choose, but the relative effectiveness of
each one would depend upon the issue chosen by the opponent. In particular, the estimated
increase in the vote for politician 1 (expressed as a percentage of the total vote) resulting
from each combination of issues is as follows:
|Issue for Politician 2|
However, because considerable staff work is required to research and formulate the issue
chosen, each politician must make her own choice before learning the opponent’s choice.
Which issue should she choose?
For each of the situations described here, formulate this problem as a two-person, zero-sum
game, and then determine which issue should be chosen by each politician according to the
a. The current preferences of the voters are very uncertain, so each additional percent of
votes won by one of the politicians has the same value to her. Use the minimax
b. A reliable poll has found that the percentage of the voters currently preferring
politician 1 (before the issues have been raised) lies between 45 and 50 percent.
(Assume a uniform distribution over this range.) Use the concept of dominated
strategies, beginning with the strategies for politician 1.
c. Suppose that the percentage described in part (b) actually were 45 percent. Should
politician 1 use the minimax criterion? Explain. Which issue would you recommend?