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 | |

Restrained | Aggressive | |

Restrained | (4, 3) | (1,4) |

Aggressive | (3,1) | (2, 2) |

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:

i. ii. iii. |
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

countries?

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.

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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

months.

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 | |||

Issue for Politician 1 |
1 | 2 | 3 |

1 | 7 | -1 | 3 |

2 | 1 | 0 | 2 |

3 | -5 | -3 | -1 |

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

specified criterion.

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

criterion.

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.

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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?

Why?