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 1110234.

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

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

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

Q3:

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<π‘ƒοŠ¨οŽ•ο…οŠͺοŽ•ο….
  • BThere is a column 𝑗 with 𝑃β‰₯π‘ƒοŠ¨οŽ•ο…οŠͺοŽ•ο….
  • CThere is a column 𝑗 with 𝑃>π‘ƒοŠ¨οŽ•ο…οŠͺοŽ•ο….
  • DThere is a column 𝑗 with π‘ƒβ‰€π‘ƒοŠ©οŽ•ο…οŠͺοŽ•ο….

Q4:

Suppose the second row does not dominate the first row for the payoff matrix ο”π‘Žπ‘π‘π‘‘ο . If 𝑐>π‘Ž, what can you conclude?

  • A 𝑑 = 𝑏
  • B 𝑑 > 𝑏
  • C 𝑑 < 𝑏
  • D 𝑑 β‰₯ 𝑏
  • E 𝑑 ≀ 𝑏

Q5:

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 greater than the element below it: 𝑃>π‘ƒοŠ§οŽ•ο…οŠ¨οŽ•ο… for each 𝑗.
  • BEvery entry in the first row is greater than or equal to the element below it: 𝑃β‰₯π‘ƒοŠ§οŽ•ο…οŠ¨οŽ•ο… for each 𝑗.
  • CEvery entry in the first row is less than or equal to the element below it: π‘ƒβ‰€π‘ƒοŠ§οŽ•ο…οŠ¨οŽ•ο… for each 𝑗.
  • DEvery entry in the first row is less than the element below it: 𝑃<π‘ƒοŠ§οŽ•ο…οŠ¨οŽ•ο… for each 𝑗.

Q6:

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

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

Q7:

Use dominance to reduce the payoff matrix ⎑⎒⎒⎣2βˆ’49110βˆ’1βˆ’1βˆ’311βˆ’1⎀βŽ₯βŽ₯⎦.

  • A  2 βˆ’ 4 9 βˆ’ 1 βˆ’ 1 βˆ’ 3 
  • B  2 βˆ’ 4 1 1 
  • C  βˆ’ 4 9 1 0 
  • D  2 βˆ’ 4 9 1 1 0 
  • E  2 βˆ’ 4 9 1 1 0 βˆ’ 1 βˆ’ 1 3 

Q8:

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

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

Q9:

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 or equal to the element to its left: 𝑃β‰₯π‘ƒοƒοŽ•οŠ¨οƒοŽ•οŠ§ for each 𝑖.
  • BEvery entry in the second column is greater than the element to its left: 𝑃>π‘ƒοƒοŽ•οŠ¨οƒοŽ•οŠ§ for each 𝑖.
  • CEvery entry in the second column is less than or equal to the element to its left: π‘ƒβ‰€π‘ƒοƒοŽ•οŠ¨οƒοŽ•οŠ§ for each 𝑖.
  • DEvery entry in the second column is less than the element to its left: 𝑃<π‘ƒοƒοŽ•οŠ¨οƒοŽ•οŠ§ for each 𝑖.

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 𝑃=ο”βˆ’4910 cannot be reduced.

What is the largest row minimum?

  • A 𝑃 = 0  οŽ• 
  • B | 𝑃 | = βˆ’ 9
  • C 𝑃 = 1  οŽ• 
  • D 𝑃 = 9  οŽ• 
  • E 𝑃 = βˆ’ 4  οŽ• 

What is the smallest column maximum?

  • A 𝑃 = 1  οŽ• 
  • B 𝑃 = 0  οŽ• 
  • C 𝑃 = 9  οŽ• 
  • D 𝑃 = βˆ’ 4  οŽ• 
  • E | 𝑃 | = βˆ’ 9

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 4Γ—4 payoff matrix?

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

Q13:

A game has the payoff matrix 𝑃=0βˆ’1110βˆ’1βˆ’110. Suppose player 𝑅 (row) uses strategy 𝐴=(0.50.50).

What does this strategy mean if 100 games are played?

  • APlayer 𝑅 will use π‘ŸοŠ§ approximately 100 times, and π‘ŸοŠ¨ approximately 0 times.
  • BPlayer 𝑅 will use π‘ŸοŠ§ approximately 50 times, and π‘ŸοŠ¨ approximately 50 times.
  • CPlayer 𝑅 will use π‘ŸοŠ§ approximately 0 times, and π‘ŸοŠ© approximately 100 times.
  • DPlayer 𝑅 will use π‘ŸοŠ¨ approximately 50 times, and π‘ŸοŠ© approximately 50 times.
  • EPlayer 𝑅 will use π‘ŸοŠ§ approximately 5 times, and π‘ŸοŠ¨ approximately 5 times.

Consider the product 𝐴𝑃=(0.5βˆ’0.50).

If player 𝐢 chooses move π‘οŠ§ each time in 100 plays, what is the expected net payoff?

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

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

  • A 𝐢 should use the pure strategy (010) which gives an expected net payoff of 100 from player 𝑅 to player 𝐢.
  • B 𝐢 should use the pure strategy (001) which gives an expected net payoff of 100 from player 𝑅 to player 𝐢.
  • C 𝐢 should use the pure strategy (010) which gives an expected net payoff of 50 from player 𝑅 to player 𝐢
  • D 𝐢 should use the pure strategy (001) which gives an expected net payoff of 100 from player 𝑅 to player 𝐢.
  • E 𝐢 should use the pure strategy (010) which gives an expected net payoff of 5 from player 𝑅 to player 𝐢.

Suppose that for the next 60 plays, 𝑅 uses the strategy ο€Ό131216. What strategy must 𝐢 use to maximise their expected winnings?

  • APlayer 𝐢 must play the mixed strategy ο€Ό121414.
  • BIt doesnβ€˜t matter what strategy player 𝐢 chooses, their expected winnings will be the same.
  • CPlayer 𝐢 must play either purely π‘οŠ¨, purely π‘οŠ©, or any mixed strategy involving π‘οŠ¨ and π‘οŠ© but not π‘οŠ§.
  • DPlayer 𝐢 must play π‘οŠ§.
  • EPlayer 𝐢 must play π‘οŠ¨.

Q14:

Two players, 𝑅 and 𝐢, are playing a game. At each turn, player 𝑅 has three possible moves: π‘ŸοŠ§, π‘ŸοŠ¨, and π‘ŸοŠ©, and player 𝐢 has four possible moves: π‘οŠ§, π‘οŠ¨, π‘οŠ©, and 𝑐οŠͺ.

The entry π‘€οƒοŽ•ο… of the payoff matrix 𝑀=21βˆ’22βˆ’11βˆ’12βˆ’2000 represents how much player 𝑅 is paid by player 𝐢 if 𝑅 uses move π‘Ÿοƒ and 𝐢 uses move 𝑐.

If player 𝑅 chooses move π‘ŸοŠ© and player 𝐢 chooses move π‘οŠ§, who wins?

  • Aplayer 𝑅
  • Bplayer 𝐢
  • Cneither

If player 𝑅 chooses move π‘ŸοŠ© and player 𝐢 chooses move π‘οŠ©, who wins?

  • Aneither
  • Bplayer 𝐢
  • Cplayer 𝑅

If player 𝐢 thinks that player 𝑅 will choose π‘ŸοŠ¨, should she choose π‘οŠ§ or π‘οŠ¨?

  • A 𝑐 
  • B 𝑐 

What move should player 𝐢 choose if she thinks that player 𝑅 will choose π‘ŸοŠ§?

  • A 𝑐 
  • B 𝑐 οŠͺ
  • C 𝑐 
  • D 𝑐 

Explain why player 𝐢 should never choose π‘οŠ¨.

  • Abecause whatever 𝑅 chooses, 𝑐οŠͺ will be at least as good as π‘οŠ¨
  • Bbecause there is a chance that neither player will win
  • Cbecause whatever 𝑅 chooses, π‘οŠ© will be at least as good as π‘οŠ¨
  • Dbecause whatever 𝑅 chooses, π‘οŠ§ will be at least as good as π‘οŠ¨

What other move will never be used by player 𝐢?

  • A 𝑐 οŠͺ
  • B 𝑐 
  • C 𝑐 

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

  • A ⎑ ⎒ ⎒ ⎣ βˆ’ 2 1 2 βˆ’ 1 βˆ’ 1 0 2 1 0 βˆ’ 2 βˆ’ 2 1 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ βˆ’ 2 βˆ’ 1 2 βˆ’ 1 1 0 2 βˆ’ 1 0 βˆ’ 2 2 1 ⎀ βŽ₯ βŽ₯ ⎦
  • C  2 βˆ’ 2 1 2 βˆ’ 1 βˆ’ 1 1 2 βˆ’ 2 0 0 1 
  • D ⎑ ⎒ ⎒ ⎣ 2 βˆ’ 1 βˆ’ 2 1 1 0 βˆ’ 2 βˆ’ 1 0 2 2 βˆ’ 1 ⎀ βŽ₯ βŽ₯ ⎦
  • E  βˆ’ 2 2 βˆ’ 1 βˆ’ 2 1 1 βˆ’ 1 βˆ’ 2 2 0 0 βˆ’ 1 

Q15:

Player 𝑅 uses a strategy of the form [π‘₯1βˆ’π‘₯], where 0≀π‘₯≀1, for a game with payoff matrix ο”βˆ’4910.

For what value of π‘₯ are [10] and [01] both optimal strategies for player 𝐢?

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

What is the expected payoff in this case?

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

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