# 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 doesn‘t 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 ’s 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