Solutions for Test 3
First Version
Question 1: (7 points) What are the expected payoffs in this game, if both players Ann and Beth would play optimally?
At the beginning, will Ann choose "up" or "down"?
When Ann would have to decide between "hot" and "cold", which one would she choose?
When she would have to decide between "north" and "south", how would she decide?
How would Beth decide between the options "left" and "right",
between the options "west" and "east", and between the options "fast" and "slow"
if she would have to?
As can be seen in the analysis, the expected payoff values for both are 4,
and Ann will choose "down", "hot", and "north", and Beth will choose "left", "east", and "slow".
Question 2:
(3 points)
How many pure strategies does Ann have in this game? How many pure strategies does Beth have?
Describe one pure strategy for Ann and one for Beth. Describe one information set for Ann and one for Beth.
Ann has 3 information sets: The one where she walks or stays,
the one where she decides between Lugano and Milano,
and the one where she chooses between green, blue, or red.
Since she has 2 options in the first, 2 option in the second, and 3 options in the third,
she has 2·2·3=12 pure strategies.
They are (walk,Lugano,green), (walk,Lugano,blue), (walk,Lugano,red),
(walk,Milano,green), (walk,Milano,blue), (walk,Milano,red),
(stay,Lugano,green), (stay,Lugano,blue), (stay,Lugano,red),
(stay,Milano,green), (stay,Milano,blue), (stay,Milano,red).
Beth has two information sets, the one where she calls or doesn't, and the one where she chooses
between high, middle or low. Since she has 2 options in the first case and 3 options in the other,
she has 2·3=6 pure strategies. They are (call,high), (call,middle), (call,low),
(don't,high), (don't,middle), (don't,low).
Second Version
Question 1: (7 points) What are the expected payoffs in this game, if both players Ann and Beth would play optimally?
At the beginning, will Ann choose "up" or "down"?
When Ann would have to decide between "hot" and "cold", which one would she choose?
When she would have to decide between "north" and "south", how would she decide?
How would Beth decide between the options "left" and "right",
between the options "west" and "east", and between the options "fast" and "slow"
if she would have to?
As can be seen in the analysis, the expected payoff value is 4 for Ann
and 4.75 for Beth (with a neutral Ann).
Ann will choose "up" or "down", "cold", and "north", and Beth will choose "right", "west", and "fast".
Question 2:
(3 points)
How many pure strategies does Ann have in this game? How many pure strategies does Beth have?
Describe one pure strategy for Ann and one for Beth. Describe one information set for Ann and one for Beth.
Ann has 3 information sets: The one where she chooses between high, middle, or low,
the one where she decides between today or tomorrow,
and the one where she chooses between green, blue, or red.
Since she has 3 options in the first, 2 option in the second, and 3 options in the third,
she has 3·2·3=18 pure strategies.
They are (high,today,green), (high,today,blue), (high,today,red),
(high,tomorrow,green), (high,tomorrow,blue), (high,tomorrow,red),
(middle,today,green), (middle,today,blue), (middle,today,red),
(middle,tomorrow,green), (middle,tomorrow,blue), (middle,tomorrow,red),
(low,today,green), (low,today,blue), (low,today,red),
(low,tomorrow,green), (low,tomorrow,blue), (low,tomorrow,red).
Beth has two information sets, the one where chooses between hot and cold, and the one where she chooses
between Lugano and Milano. Since she has 2 options in the first case and 2 options in the other,
she has 2·2=4 pure strategies. They are (hot,Lugano), (hot,Milano), (cold,Lugano), (cold,Milano).