Self Test Questions
Theory
1. What is 'two-person zero-sum game' ?
2. Define the following:
- Saddle point
- Pure strategy.
- Pay-off matrix.
- Optimal strategies
3. Explain briefly the importance of the principle of dominance.
4. What are the advantages & limitations of game theory?
Practical
1. Consider the game whose pay-off matrix is given below. Find its
solution.
| Player B | ||||
|---|---|---|---|---|
| Player A | I | II | III | |
| I | -4 | -6 | 3 | |
| II | -3 | -3 | 6 | |
| III | 2 | -3 | 4 | |
2. Two companies A and B are competing for the same product. Their different strategies are given in the following pay-off matrix:
| Company B | ||||
|---|---|---|---|---|
| Company A | I | II | III | |
| I | 2 | -2 | 3 | |
| II | -3 | 5 | -1 | |
Determine the best strategies and find the value of the game.
3. Solve the following games:
(a)| Player B | ||||||
|---|---|---|---|---|---|---|
| Player A | I | II | III | IV | V | |
| I | 4 | 0 | 1 | 7 | -1 | |
| II | 0 | -3 | -5 | -7 | 5 | |
| III | 3 | 2 | 3 | 4 | 3 | |
| IV | -6 | 1 | -1 | 0 | 5 | |
| V | 0 | 0 | 6 | 0 | 0 | |
(b
| Player B | ||||
|---|---|---|---|---|
| Player A | I | II | III | |
| I | -2 | 15 | -2 | |
| II | -5 | -6 | -4 | |
| III | -5 | 20 | -8 | |
(c)
| Player B | ||||
|---|---|---|---|---|
| Player A | I | II | III | |
| I | 2 | -1 | 3 | |
| II | 2 | -1 | 2 | |
| III | -1 | 0 | 0 | |
| IV | 2 | 0 | 4 | |
(d)
| Player B | |||||
|---|---|---|---|---|---|
| Player A | I | II | III | IV | |
| I | -5 | 3 | 1 | 20 | |
| II | 5 | 5 | 4 | 6 | |
| III | -4 | -2 | 0 | -5 | |
(e)
| Player B | |||||
|---|---|---|---|---|---|
| Player A | I | II | III | IV | |
| I | 3 | -5 | 0 | 6 | |
| II | -4 | -2 | 1 | 2 | |
| III | 5 | 4 | 2 | 3 | |
4. Use the dominance principle to solve the following game:
| 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 2 | 1 | 1 |
| 4 | 3 | 1 | 3 | 2 | 2 |
| 4 | 3 | 7 | -5 | 1 | 2 |
| 4 | 3 | 4 | -1 | 2 | 2 |
| 4 | 3 | 3 | -2 | 2 | 2 |
5. Solve the following games:
(a)
| Player B | ||||
|---|---|---|---|---|
| Player A | I | II | III | |
| I | -5 | -1 | -1 | |
| II | 4 | 0 | 2 | |
| III | -5 | 2 | 0 | |
(b)
| Player B | ||||
|---|---|---|---|---|
| Player A | 6 | 8 | 3 | 13 |
| 4 | 1 | 5 | 3 | |
| 8 | 10 | 4 | 12 | |
| 3 | 6 | 7 | 12 | |
6. Solve the following game algebraically
| Player B | ||||
|---|---|---|---|---|
| Player A | I | II | III | |
| I | 4 | 2 | 4 | |
| II | 2 | 4 | 0 | |
| III | 4 | 0 | 8 | |
7. Reduce each of the following games by using the rule of dominance and then solve the reduced game by any of the method you have studied:
(a)| B1 | B2 | B3 | |
|---|---|---|---|
| A1 | 3 | 8 | 5 |
| A2 | 6 | 2 | 7 |
| A3 | 4 | 5 | 6 |
(b)
| B1 | B2 | B3 | B4 | B5 | |
|---|---|---|---|---|---|
| A1 | 8 | 7 | 6 | -1 | 2 |
| A2 | 12 | 10 | 12 | 0 | 4 |
| A3 | 14 | 6 | 8 | 14 | 16 |
8. Two players A & B, without showing each other put a coin on a table with head or tail up. If the coins show the same side (both head or tail), the player A takes both the coins, otherwise B gets them. Construct the matrix of the game and solve it.
9. In a game of matching coins with two players, suppose A wins one unit of the value when there are two heads; wins nothing when there are two tails and loses 1/2 units of value when there is one head and one tail. Determine the pay off matrix, the optimal strategies for both the players.
