Lesson test: Mixed strategies

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Mixed strategies

1.[2p]
In a mixed equilibrium, why is a player indifferent between the pure strategies they mix over?

Because otherwise shifting probability towards the better one would raise their expected payoff

Correct

Because indifference is an extra assumption imposed to make the algebra work

The answer is: Because otherwise shifting probability towards the better one would raise their expected payoff

Because the game must be zero-sum

The answer is: Because otherwise shifting probability towards the better one would raise their expected payoff

Because both players have the same payoffs

The answer is: Because otherwise shifting probability towards the better one would raise their expected payoff

2.[2p]
Whose payoffs determine a player's own equilibrium mixing probabilities?

The other player's, through their indifference condition

Correct

Their own, through their indifference condition

The answer is: The other player's, through their indifference condition

Both players' equally

The answer is: The other player's, through their indifference condition

Neither: the probabilities are chosen freely

The answer is: The other player's, through their indifference condition

3.[2p]
In the audit game, evading pays the taxpayer -100 if audited and 50 if not, and declaring pays 0. What audit probability makes the taxpayer indifferent? Give a decimal.

CorrectNot quite: 0.333

4.[3p]
The authority earns 80 from auditing an evader, -20 from auditing an honest return, -50 from missing an evader, and 0 otherwise. What evasion probability makes the authority indifferent? Give a decimal.

CorrectNot quite: 0.133

5.[3p]
In that same game the penalty for a caught evader is tripled from 100 to 300, with no change to the authority's payoffs. What is the new equilibrium probability of evasion? Give a decimal.

CorrectNot quite: 0.133

6.[3p]
A harsher penalty in an inspection game with no change to the enforcer's payoffs

lowers the audit rate and leaves the offence rate where it was

Correct

lowers the offence rate and leaves the audit rate where it was

The answer is: lowers the audit rate and leaves the offence rate where it was

lowers both rates

The answer is: lowers the audit rate and leaves the offence rate where it was

has no effect on anything

The answer is: lowers the audit rate and leaves the offence rate where it was

7.[1p]
Nash's theorem guarantees that every finite game has at least one equilibrium once mixed strategies are allowed.

True

Correct

False

The answer is: True

8.[3p]
Player 1 wins 2 if both play Heads, 1 if both play Tails, and loses 1 otherwise; the game is zero-sum. With what probability does player 1 play Heads in equilibrium? Give a decimal.

CorrectNot quite: 0.4

9.[3p]
Match each reading of a mixed strategy to what it says.

Literal randomisation
Population frequency
Harsanyi purification

a share of a large population plays each action
the player really does use a chance device
private payoff shocks make each player pure

Show the answer

Literal randomisation: the player really does use a chance device Population frequency: a share of a large population plays each action Harsanyi purification: private payoff shocks make each player pure

1.[2p] In a mixed equilibrium, why is a player indifferent between the pure strategies they mix over?

2.[2p] Whose payoffs determine a player's own equilibrium mixing probabilities?

3.[2p] In the audit game, evading pays the taxpayer -100 if audited and 50 if not, and declaring pays 0. What audit probability makes the taxpayer indifferent? Give a decimal.

4.[3p] The authority earns 80 from auditing an evader, -20 from auditing an honest return, -50 from missing an evader, and 0 otherwise. What evasion probability makes the authority indifferent? Give a decimal.

5.[3p] In that same game the penalty for a caught evader is tripled from 100 to 300, with no change to the authority's payoffs. What is the new equilibrium probability of evasion? Give a decimal.

6.[3p] A harsher penalty in an inspection game with no change to the enforcer's payoffs

7.[1p] Nash's theorem guarantees that every finite game has at least one equilibrium once mixed strategies are allowed.

8.[3p] Player 1 wins 2 if both play Heads, 1 if both play Tails, and loses 1 otherwise; the game is zero-sum. With what probability does player 1 play Heads in equilibrium? Give a decimal.

9.[3p] Match each reading of a mixed strategy to what it says.

1.[2p]
In a mixed equilibrium, why is a player indifferent between the pure strategies they mix over?

2.[2p]
Whose payoffs determine a player's own equilibrium mixing probabilities?

3.[2p]
In the audit game, evading pays the taxpayer -100 if audited and 50 if not, and declaring pays 0. What audit probability makes the taxpayer indifferent? Give a decimal.

4.[3p]
The authority earns 80 from auditing an evader, -20 from auditing an honest return, -50 from missing an evader, and 0 otherwise. What evasion probability makes the authority indifferent? Give a decimal.

5.[3p]
In that same game the penalty for a caught evader is tripled from 100 to 300, with no change to the authority's payoffs. What is the new equilibrium probability of evasion? Give a decimal.

6.[3p]
A harsher penalty in an inspection game with no change to the enforcer's payoffs

7.[1p]
Nash's theorem guarantees that every finite game has at least one equilibrium once mixed strategies are allowed.

8.[3p]
Player 1 wins 2 if both play Heads, 1 if both play Tails, and loses 1 otherwise; the game is zero-sum. With what probability does player 1 play Heads in equilibrium? Give a decimal.

9.[3p]
Match each reading of a mixed strategy to what it says.