Worksheet: Payoff Matrix for Game Theory

In this worksheet, we will practice using the game theory, which is using a matrix to represent payoffs in a game as an application of linear algebra and matrix methods.

Q1:

Players and play a game in which they both call ‘heads’ or ‘tails’ at the same time. They use the following payoff matrix, where the first row and column represent ‘heads’ and the second row and column represent ‘tails’: Describe the payoff rules for this game.

  • AIf both players call ‘heads’, player gets 2 points from player ; if both players call ‘tails’, player gets 1 point from player ; and if both players call different sides, player gets 1 point from player if he calls ‘heads’ or 2 points if he calls ‘tails’.
  • BIf both players call ‘heads’, player gets 1 point from player ; if both players call ‘tails’, player gets 2 points from player ; and if both players call different sides, player gets 1 point from player if he calls ‘heads’ or 2 points if he calls ‘tails’.
  • CIf both players call ‘heads’, player gets 1 point from player ; if both players call ‘tails’, player gets 2 points from player ; and if both players call different sides, player gets 2 points from player if he calls ‘heads’ or 1 point if he calls ‘tails’.
  • DIf both players call ‘heads’, player gets 2 points from player ; if both players call ‘tails’, player gets 1 point from player ; and if both players call different sides, player gets 2 points from player if he calls ‘heads’ or 1 point if he calls ‘tails’.
  • EIf both players call ‘heads’, player gets 2 points from player ; if both players call ‘tails’, player gets 2 points from player ; and if both players call different sides, player gets 1 point from player if he calls ‘heads’ or 1 point if he calls ‘tails’.

Q2:

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

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

Q3:

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 ,

Q4:

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

  • AThere is a column 𝑗 with 𝑃 𝑃 2 , 𝑗 4 , 𝑗 .
  • BThere is a column 𝑗 with 𝑃 > 𝑃 2 , 𝑗 4 , 𝑗 .
  • CThere is a column 𝑗 with 𝑃 𝑃 3 , 𝑗 4 , 𝑗 .
  • DThere is a column 𝑗 with 𝑃 < 𝑃 2 , 𝑗 4 , 𝑗 .

Q5:

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

  • A
  • B
  • C
  • D
  • E

Q6:

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

  • AEvery entry in the first row is less than the element below it: 𝑃 < 𝑃 1 , 𝑗 2 , 𝑗 for each 𝑗 .
  • BEvery entry in the first row is less than or equal to the element below it: 𝑃 𝑃 1 , 𝑗 2 , 𝑗 for each 𝑗 .
  • CEvery entry in the first row is greater than the element below it: 𝑃 > 𝑃 1 , 𝑗 2 , 𝑗 for each 𝑗 .
  • DEvery entry in the first row is greater than or equal to the element below it: 𝑃 𝑃 1 , 𝑗 2 , 𝑗 for each 𝑗 .

Q7:

A game is given by a 3 × 4 payoff matrix. What is meant by a strategy for player 𝐶 (column)?

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

Q8:

Use dominance to reduce the payoff matrix

  • A
  • B
  • C
  • D
  • E

Q9:

A game is given by a 3 × 4 payoff matrix. What is meant by a strategy for player 𝑅 (row)?

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

Q10:

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

  • AEvery entry in the second column is greater than the element to its left: 𝑃 > 𝑃 𝑖 , 2 𝑖 , 1 for each 𝑖 .
  • BEvery entry in the second column is greater than or equal to the element to its left: 𝑃 𝑃 𝑖 , 2 𝑖 , 1 for each 𝑖 .
  • CEvery entry in the second column is less than the element to its left: 𝑃 < 𝑃 𝑖 , 2 𝑖 , 1 for each 𝑖 .
  • DEvery entry in the second column is less than or equal to the element to its left: 𝑃 𝑃 𝑖 , 2 𝑖 , 1 for each 𝑖 .

Q11:

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

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

Q12:

Players and play a game in which they both call ‘heads’ or ‘tails’ at the same time. If they both call the same thing, player wins one point. If they call different things, player wins one point. Write the payoff matrix for this game.

  • A
  • B
  • C
  • D
  • E

Q13:

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

Q14:

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

  • A
  • B
  • C
  • D
  • E

Q15:

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 0 times, and approximately 100 times.
  • BPlayer will use approximately 100 times, and approximately 0 times.
  • CPlayer will use approximately 5 times, and approximately 5 times.
  • DPlayer will use approximately 50 times, and approximately 50 times.
  • EPlayer will use approximately 50 times, and approximately 50 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 50 from player to player .
  • CThe expected net payoff is 5 from player to player .
  • DThe expected net payoff is 100 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 either purely , purely , or any mixed strategy involving and but not .
  • BIt doesn‘t matter what strategy player chooses, their expected winnings will be the same.
  • CPlayer must play the mixed strategy .
  • DPlayer must play .
  • EPlayer must play .

Q16:

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?

  • Aneither
  • Bplayer
  • Cplayer

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 there is a chance that neither player will win
  • Bbecause whatever chooses, will be at least as good as
  • 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

Q17:

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

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