1. Game Theory II Solutions 1
Two companies, A and B, are bidding on a contract for a large construction project. Three bids are possible, low, medium, and high. The lowest bid placed by the two companies will be chosen, but if the bids are equal, the project will be awarded to company A which has more experience. If a company wins with the low bids, the costs will equal income and no profit is made. If a company wins with a medium bid, a profit of 5 is realized and if a company wins with a high bid, a profit of 10 is earned. The table below shows the payoff matrix for company A. If the company looses the bid to its opponent, this is counted as a loss with respect to the relative position of the two companies. The bids are sealed and the companies have no information about each other except that they both use a minimax policy.
What is the optimal bidding policy for the two companies?
Who will win the contract?
What profit will be made?

2. Game Theory II Solutions 2
Prelude from course notes:
By this reasoning, each player should minimize his maximum losses.
Player 1 should select the minimum payoff to player 1 is largest.
Player 2 should select the maximum payoff to player 1 is minimum.
Player 1 should select the minimum payoff to player 1 is largest, maximin criteria.
Player 2 should select the maximum payoff to player 1 is minimum, minimax criteria.
Ref: Hillier & Lieberman
By this reasoning, each player should minimize his maximum losses.
Player 1 should select the minimum payoff to player 1 is largest.
Player 2 should select the maximum payoff to player 1 is minimum.
By this reasoning, each player should minimize his maximum losses minimax .
Ref: Jensen, Student Guide to O.R.
Soln:
A can choose a low or Med
strategy
A will choose a low
Strategy. Why?
B will choose a low strategy
Company A will win the bid but there is little chance a profit will be made.
How do construction firms make $ in this environment?

3. Game Theory Solutions 3
At a particular point in a football game, the offensive team and the defensive team must select a strategy for the next play. Both teams have the same statistical information that predicts the expected yardage to be gained for various combinations of offensive and defensive strategies. The information is given in the following table, which shows yards gained for the offensive team for every combination. Determine the minimax policy for both teams. What is the expected gain in yardage?
Soln:
Unstable Game, there is no saddle point. We must go to a mixed strategy.

4. Game Theory Solutions 5
At a particular point in a football game, the offensive team and the defensive team must select a strategy for the next play. Both teams have the same statistical information that predicts the expected yardage to be gained for various combinations of offensive and defensive strategies. The information is given in the following table, which shows yards gained for the offensive team for every combination. Determine the minimax policy for both teams. What is the expected gain in yardage?
Soln:
Unstable Game, there is no saddle point. We must go to a mixed strategy.

5. Game Theory Solutions 4

6. Game Theory Solutions 5

7. Game Theory Solutions 5

8. Game Theory Solutions 6