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In this lesson, we will learn how to use 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.

Q2:

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

Q3:

Use dominance to reduce the payoff matrix .

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

Q4:

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

Q5:

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

Q6:

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

Q7:

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

Q8:

Use dominance to reduce the payoff matrix

Q9:

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

Q10:

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

Q11:

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

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.

Q13:

The payoff matrix cannot be reduced.

What is the largest row minimum?

What is the smallest column maximum?

Does this payoff matrix have a saddle point?

Q14:

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

Q15:

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

What does this strategy mean if 100 games are played?

Consider the product .

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

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

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

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?

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

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

Explain why player should never choose .

What other move will never be used by player ?

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?

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 ?

What is the expected payoff in this case?

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