# Worksheet: Introduction to Game Theory

In this worksheet, we will practice representing a game using matrices.

**Q1: **

What is the definition of a saddle point of a payoff matrix?

- Aan entry which is the smallest row maximum and the smallest column maximum at the same time
- Ban entry which is the largest row minimum and the largest column minimum at the same time
- Can entry which is the largest row minimum and the smallest column maximum at the same time

**Q2: **

Use dominance to reduce the payoff matrix .

- A
- B
- C
- D
- E

What are the optimal strategies for player (row) and player (column)?

- A ,
- B ,
- C ,
- D ,
- E ,

**Q3: **

What can you conclude if the second row of the payoff matrix does not dominate the fourth row?

- AThere is a column with .
- BThere is a column with .
- CThere is a column with .
- DThere is a column with .

**Q4: **

Suppose the second row does not dominate the first row for the payoff matrix . If , what can you conclude?

- A
- B
- C
- D
- E

**Q5: **

Suppose that the first row dominates the second row in a payoff matrix . What does this mean in terms of entries?

- AEvery entry in the first row is greater than the element below it: for each .
- BEvery entry in the first row is greater than or equal to the element below it: for each .
- CEvery entry in the first row is less than or equal to the element below it: for each .
- DEvery entry in the first row is less than the element below it: for each .

**Q6: **

A game is given by a payoff matrix. What is meant by a strategy for player (column)?

- Aa matrix of nonnegative entries that add up to 4
- Ba matrix of nonnegative entries that add up to 1
- Ca matrix of nonnegative entries that add up to 3
- Da matrix of nonnegative entries that add up to 1

**Q7: **

Use dominance to reduce the payoff matrix

- A
- B
- C
- D
- E

**Q8: **

A game is given by a payoff matrix. What is meant by a strategy for player (row)?

- Aa matrix of nonnegative entries that add up to 1
- Ba matrix of nonnegative entries that add up to 3
- Ca matrix of nonnegative entries that add up to 1
- Da matrix of nonnegative entries that add up to 4

**Q9: **

Suppose that the second column dominates the first column in a payoff matrix . What does this mean for the entries of ?

- AEvery entry in the second column is greater than or equal to the element to its left: for each .
- BEvery entry in the second column is greater than the element to its left: for each .
- CEvery entry in the second column is less than or equal to the element to its left: for each .
- DEvery entry in the second column is less than the element to its left: for each .

**Q10: **

What is a pure strategy for a player in a game with a payoff matrix?

- AIt is one with three entries that equal 1 while all the others equal 0.
- BIt is one with all entries that equal 0.
- CIt is one with one entry that equals 0 while all the others equal 1.
- DIt is one with all entries that equal 1.
- EIt is one with one entry that equals 1 while all the others equal 0.

**Q11: **

The payoff matrix cannot be reduced.

What is the largest row minimum?

- A
- B
- C
- D
- E

What is the smallest column maximum?

- A
- B
- C
- D
- E

Does this payoff matrix have a saddle point?

- Ayes
- Bno

**Q12: **

Which of the following is a strategy for a player in a game with a payoff matrix?

- A
- B
- C
- D
- E

**Q13: **

A game has the payoff matrix Suppose player (row) uses strategy .

What does this strategy mean if 100 games are played?

- APlayer will use approximately 100 times, and approximately 0 times.
- BPlayer will use approximately 50 times, and approximately 50 times.
- CPlayer will use approximately 0 times, and approximately 100 times.
- DPlayer will use approximately 50 times, and approximately 50 times.
- EPlayer will use approximately 5 times, and approximately 5 times.

Consider the product .

If player chooses move each time in 100 plays, what is the expected net payoff?

- AThe expected net payoff is 50 from player to player .
- BThe expected net payoff is 100 from player to player .
- CThe expected net payoff is 5 from player to player .
- DThe expected net payoff is 50 from player to player .
- EThe expected net payoff is 100 from player to player .

What strategy should play, and what would be their expected net payoff in 100 plays with this strategy?

- A should use the pure strategy which gives an expected net payoff of 100 from player to player .
- B should use the pure strategy which gives an expected net payoff of 100 from player to player .
- C should use the pure strategy which gives an expected net payoff of 50 from player to player
- D should use the pure strategy which gives an expected net payoff of 100 from player to player .
- E should use the pure strategy which gives an expected net payoff of 5 from player to player .

Suppose that for the next 60 plays, uses the strategy . What strategy must use to maximise their expected winnings?

- APlayer must play the mixed strategy .
- BIt doesnβt matter what strategy player chooses, their expected winnings will be the same.
- CPlayer must play either purely , purely , or any mixed strategy involving and but not .
- DPlayer must play .
- EPlayer must play .

**Q14: **

Two players, and , are playing a game. At each turn, player has three possible moves: , , and , and player has four possible moves: , , , and .

The entry of the payoff matrix represents how much player is paid by player if uses move and uses move .

If player chooses move and player chooses move , who wins?

- Aplayer
- Bplayer
- Cneither

If player chooses move and player chooses move , who wins?

- Aneither
- Bplayer
- Cplayer

If player thinks that player will choose , should she choose or ?

- A
- B

What move should player choose if she thinks that player will choose ?

- A
- B
- C
- D

Explain why player should never choose .

- Abecause whatever chooses, will be at least as good as
- Bbecause there is a chance that neither player will win
- Cbecause whatever chooses, will be at least as good as
- Dbecause whatever chooses, will be at least as good as

What other move will never be used by player ?

- A
- B
- C

What would the payoff matrix for the same game be if players and swapped roles so that player βs moves were listed along the rows instead?

- A
- B
- C
- D
- E

**Q15: **

Player uses a strategy of the form , where , for a game with payoff matrix .

For what value of are and both optimal strategies for player ?

- A
- B
- C
- D
- E

What is the expected payoff in this case?

- A
- B
- C
- D
- E