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Worksheet: Payoff Matrix for Game Theory

Q1:

Use dominance to reduce the payoff matrix

  • A 2 4 1 1
  • B 2 4 9 1 1 0 1 1 3
  • C 2 4 9 1 1 0
  • D 4 9 1 0
  • E 2 4 9 1 1 3

Q2:

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

Q3:

The payoff matrix 𝑃 = 4 9 1 0 cannot be reduced.

What is the largest row minimum?

  • A 𝑃 = 9 1 , 2
  • B 𝑃 = 4 1 , 1
  • C 𝑃 = 1 2 , 1
  • D 𝑃 = 0 2 , 2
  • E | 𝑃 | = 9

What is the smallest column maximum?

  • A 𝑃 = 1 2 , 1
  • B 𝑃 = 0 2 , 2
  • C 𝑃 = 9 1 , 2
  • D 𝑃 = 4 1 , 1
  • E | 𝑃 | = 9

Does this payoff matrix have a saddle point?

  • Ayes
  • Bno

Q4:

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

Q5:

Which of the following is a strategy for a player in a game with a 4 × 4 payoff matrix?

  • A [ 0 . 8 0 . 3 0 0 . 2 ] 𝑇
  • B [ 0 . 5 0 . 3 0 0 . 2 ] 𝑇
  • C [ 0 . 5 0 . 3 0 . 1 0 . 2 ] 𝑇
  • D [ 0 . 5 0 . 3 0 0 . 2 ] 𝑇
  • E [ 0 . 6 0 0 0 . 8 ] 𝑇

Q6:

Players 𝑅 and 𝐶 play a game in which they both call “heads” or “tails” at the same time. They use the payoff matrix 2 2 1 1 , 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 𝐶 gets 2 points from player 𝑅 ; if both players call “tails,” player 𝐶 gets 1 point from player 𝑅 ; and if both players call different sides, player 𝑅 gets 1 point from player 𝐶 if he calls “heads” or 2 points if he calls “tails.”
  • BIf both players call “heads,” player 𝐶 gets 1 point from player 𝑅 ; if both players call “tails,” player 𝐶 gets 2 points from player 𝑅 ; and if both players call different sides, player 𝑅 gets 1 point from player 𝐶 if he calls “heads” or 2 points if he calls “tails.”
  • CIf both players call “heads,” player 𝐶 gets 1 point from player 𝑅 ; if both players call “tails,” player 𝐶 gets 2 points from player 𝑅 ; and if both players call different sides, player 𝑅 gets 2 points from player 𝐶 if he calls “heads” or 1 point if he calls “tails.”
  • DIf both players call “heads,” player 𝐶 gets 2 points from player 𝑅 ; if both players call “tails,” player 𝐶 gets 1 point from player 𝑅 ; and if both players call different sides, player 𝑅 gets 2 points from player 𝐶 if he calls “heads” or 1 point if he calls “tails.”
  • EIf both players call “heads,” player 𝐶 gets 2 points from player 𝑅 ; if both players call “tails,” player 𝐶 gets 2 points from player 𝑅 ; and if both players call different sides, player 𝑅 gets 1 point from player 𝐶 if he calls “heads” or 1 point if he calls “tails.”

Q7:

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 .

Q8:

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 .

Q9:

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

Q10:

Use dominance to reduce the payoff matrix 1 1 1 0 2 3 4 .

  • A [ 1 ]
  • B [ 4 ]
  • C [ 1 2 ] 𝑇
  • D [ 2 ]
  • E [ 1 0 4 ] 𝑇

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

  • A 𝑅 = [ 0 1 ] , 𝐶 = [ 1 0 0 ] 𝑇
  • B 𝑅 = [ 0 1 ] , 𝐶 = [ 0 1 1 ] 𝑇
  • C 𝑅 = [ 1 1 ] , 𝐶 = [ 1 0 0 ] 𝑇
  • D 𝑅 = [ 1 0 ] , 𝐶 = [ 1 0 0 ] 𝑇
  • E 𝑅 = [ 0 1 ] , 𝐶 = [ 1 0 1 ] 𝑇

Q11:

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 .

Q12:

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.

Q13:

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

  • A 0 1 1 0
  • B 1 1 1 1
  • C 1 0 0 1
  • D 1 1 1 1
  • E 1 1 1 1

Q14:

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

The entry 𝑀 of the payoff matrix represents how much player 𝑅 is paid by player 𝐶 if 𝑅 uses move 𝑟 and 𝐶 uses move 𝑐 .

If player 𝑅 chooses move 𝑟 and player 𝐶 chooses move 𝑐 , who wins?

  • Aneither
  • Bplayer 𝑅
  • Cplayer 𝐶

If player 𝑅 chooses move 𝑟 and player 𝐶 chooses move 𝑐 , who wins?

  • Aneither
  • Bplayer 𝑅
  • Cplayer 𝐶

If player 𝐶 thinks that player 𝑅 will choose 𝑟 , should she choose 𝑐 or 𝑐 ?

  • A 𝑐
  • B 𝑐

What move should player 𝐶 choose if she thinks that player 𝑅 will choose 𝑟 ?

  • A 𝑐
  • B 𝑐
  • C 𝑐
  • D 𝑐

Explain why player 𝐶 should never choose 𝑐 .

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

What other move will never be used by player 𝐶 ?

  • A 𝑐
  • B 𝑐
  • C 𝑐

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

  • A 2 1 2 1 1 0 2 1 0 2 2 1
  • B 2 1 2 1 1 0 2 1 0 2 2 1
  • C 2 1 2 1 1 0 2 1 0 2 2 1
  • D 2 2 1 2 1 1 1 2 2 0 0 1
  • E 2 2 1 2 1 1 1 2 2 0 0 1

Q15:

Player 𝑅 uses a strategy of the form [ 𝑥 1 𝑥 ] , where 0 𝑥 1 , for a game with payoff matrix 4 9 1 0 .

For what value of 𝑥 are [ 1 0 ] and [ 0 1 ] both optimal strategies for player 𝐶 ?

  • A 3 4
  • B 1 9
  • C 9 1 4
  • D 1 1 4
  • E 4 9

What is the expected payoff in this case?

  • A 9 1 4
  • B 1 1 4
  • C 3 4
  • D 1 9
  • E 4 9

Q16:

Suppose the second row does not dominate the first row for the payoff matrix 𝑎 𝑏 𝑐 𝑑 . If 𝑐 > 𝑎 , what can you conclude?

  • A 𝑑 = 𝑏
  • B 𝑑 > 𝑏
  • C 𝑑 𝑏
  • D 𝑑 < 𝑏
  • E 𝑑 𝑏

Q17:

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 .