SI 563 Homework 1 Solution
Sep 14, 2016
Chapter2
Exercise 2: (2 points)
Consider the following strategic situation concerning the owner of a firm (O), the manager of
the firm (M), and a potential worker (W). The owner first decides whether to hire the worker, to
refuse to hire the worker, or to let the manager make the decision. If the owner lets the manager
make the decision, then the manager must choose between hiring the worker or not hiring the
worker. If the worker is hired, then he or she chooses between working diligently and shirking.
Assume that the worker does not know whether he or she was hired by the manager or the
owner when he or she makes this decision. If the worker is not hired, then all three players get
a payoff of 0. If the worker is hired and shirks, then the owner and manager each get a payoff
of −1, whereas the worker gets 1. If the worker is hired by the owner and works diligently, then
the owner gets a payoff of 1, the manager gets 0, and the worker gets 0. If the worker is hired
by the manager and works diligently, then the owner gets 0, the manager gets 1, and the
worker gets 1. Represent this game in the extensive form (draw the game tree).
The solution is provided in the picture below:
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SI 563 Homework 1 Solution
Sep 14, 2016
Note: solutions may vary in layout, you can get full marks as long as all the essential information are
included.
Exercise 3: (2 points)
Draw the extensive form for the following game (invent your own payoff vectors, because I
give you no payoff information). There is an industry in which two firms compete as follows:
First, firm 1 decides whether to set a high price (H) or a low price (L). After seeing firm 1’s price,
firm 2 decides whether to set a high price (H) or a low price (L). If both firms selected the low
price, then the game ends with no further interaction. If either or both firms selected the high
price, then the attorney general decides whether to prosecute (P) or not (N) for anticompetitive
behavior. In this case, the attorney general does not observe which firm selected the high price
(or if both firms selected the high price).
The solution is provided in the picture below:
Chapter 3
Exercise 2: (2 points)
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SI 563 Homework 1 Solution
Sep 14, 2016
Suppose a manager and a worker interact as follows. The manager decides whether to hire
or not hire the worker. If the manager does not hire the worker, then the game ends. When
hired, the worker chooses to exert either high effort or low effort. On observing the worker’s
effort, the manager chooses to retain or fire the worker. In this game, does “not hire” describe
a strategy for the manager? Explain.
Note: You are not required to draw the game for your answer.
No, “Not Hire” is not a complete strategy. A complete strategy defines the agent’s action at
every information set, including contingencies for what to do at information sets that can never
be reached implementing the strategy. Although the manager playing “Not Hire” ends the game,
“Not Hire” has to be paired with what the manager would do in the case of “High” and “Low”
effort in order to define a complete strategy.
Note that the manager has three, not two, information sets. A lot of teams answered that the
manager has two information sets. His (or her) first is the decision to hire or not. He has two
more because he can observe the worker’s effort and make a different decision in the case
the worker exerts “High” versus “Low” effort. The manager’s strategies are:
{Hire, Retain High, Retain Low}; {Hire, Retain High, Fire Low}; {Hire, Fire High, Retain Low}
{Hire, Fire High, Fire Low}; {Don’t Hire, Retain High, Retain Low}; {Don’t Hire, Retain High,
Fire Low}; {Don’t Hire, Fire High, Retain Low}; {Don’t Hire, Fire High, Fire Low}
Exercise 3: (2 points)
Draw the normal-form matrix of each of the following extensive-form games.
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SI 563 Homework 1 Solution
Sep 14, 2016
Note: Each strategy should be complete, which is to say that it defines an action for every
information set the player faces. Break out all the different strategy so that every different
strategy gets a row/column. Even if two strategies lead to identical outcomes (they both end
the game after one step for example), they still each get their own row/column. Many teams
lost points on this question for failing to define complete strategies or failing to include all the
complete strategies for each player. There’s no certain order you which you have to list players’
strategies, so answers might look a little bit different. Convention is to put player 1 on the left
and player two on the top. That way payoffs can be entered (Player 1 payoff, Player 2 payoff).
(a)
2
CE
DE
CF
DF
A
(0,0)
(1,1)
(0,0)
(1,1)
B
(2,2)
(2,2)
(3,4)
(3,4)
1
(b)
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SI 563 Homework 1 Solution
Sep 14, 2016
2
I
O
OU
(1,1)
(1,1)
OD
(1,1)
(1,1)
IU
(4,0)
(-1,-1)
ID
(3,2)
(-1,-1)
1
(c)
2
AC
AD
BC
BD
UE
(3,3)
(3,3)
(5,4)
(5,4)
UF
(3,3)
(3,3)
(5,4)
(5,4)
DE
(6,2)
(2,2)
(6,2)
(2,2)
DF
(2,6)
(2,2)
(2,6)
(2,2)
1
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SI 563 Homework 1 Solution
Sep 14, 2016
(d)
2
1
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A
B
UXW
(3,3) (5,1)
UXZ
(3,3) (5,1)
UYW
(3,3) (3,6)
UYZ
(3,3) (3,6)
DXW
(4,2) (2,2)
DXZ
(9,0) (2,2)
DYW
(4,2) (2,2)
DYZ
(9,0) (2,2)
SI 563 Homework 1 Solution
Sep 14, 2016
(e)
2
1
U
D
A
(2,1) (1,2)
B
(6,8) (4,3)
C
(2,1) (8,7)
(f)
2
1
UXP
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A
B
(3,8) (1,2)
SI 563 Homework 1 Solution
Sep 14, 2016
UXQ
(3,8) (1,2)
UYP
(8,1) (2,1)
UYQ
(8,1) (2,1)
DXP
(6,6) (5,5)
DXQ
(6,6) (0,0)
DYP
(6,6) (5,5)
DYQ
(6,6) (0,0)
Chapter 4
Exercise 1 (2 points)
Evaluate the following payoffs for the game given by the normal form pictured here.
[Remember, a mixed strategy for player 1 is 𝜎1 ∈ ∆{𝑈, 𝑀, 𝐷}, where 𝜎1 (𝑈) is the probability
that player 1 plays strategy U, and so forth. For simplicity, we write 𝜎1 as (𝜎1 (𝑈), 𝜎1 (𝑀), 𝜎1 (𝐷)),
and similarly for player 2.]
(a) 𝑢1 (𝑈, 𝐶)
(b) 𝑢2 (𝑀, 𝑅)
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SI 563 Homework 1 Solution
Sep 14, 2016
(c) 𝑢2 (𝐷, 𝐶)
1 2
(d) 𝑢1 (𝜎1 , 𝐶) 𝑓𝑜𝑟 𝜎1 = (3 , 3 , 0)
1 1 1
(e) 𝑢1 (𝜎1 , 𝑅) 𝑓𝑜𝑟 𝜎1 = ( , , )
4 2 4
(f) 𝑢1 (𝜎1 , 𝐿) 𝑓𝑜𝑟 𝜎1 = (0,1,0)
1 2
(g) 𝑢2 (𝜎1 , 𝑅) 𝑓𝑜𝑟 𝜎1 = (3 , 3 , 0)
1 1
1 1 1
(h) 𝑢1 (𝜎1 , 𝜎2 ) 𝑓𝑜𝑟 𝜎1 = (2 , 2 , 0) 𝑎𝑛𝑑 𝜎2 = (4 , 4 , 2)
(a) 𝑢1 (𝑈, 𝐶) = 0
(b) 𝑢2 (𝑀, 𝑅) = 4
(c) 𝑢2 (𝐷, 𝐶) = 6
1 2
20
(d) 𝑢1 (𝜎1 , 𝐶) 𝑓𝑜𝑟 𝜎1 = (3 , 3 , 0) =
1 1 1
(e) 𝑢1 (𝜎1 , 𝑅) 𝑓𝑜𝑟 𝜎1 = (4 , 2 , 4) =
3
21
4
(f) 𝑢1 (𝜎1 , 𝐿) 𝑓𝑜𝑟 𝜎1 = (0,1,0) = 2
1 2
(g) 𝑢2 (𝜎1 , 𝑅) 𝑓𝑜𝑟 𝜎1 = (3 , 3 , 0) =
1 1
11
3
1 1 1
(h) 𝑢1 (𝜎1 , 𝜎2 ) 𝑓𝑜𝑟 𝜎1 = ( , , 0) 𝑎𝑛𝑑 𝜎2 = ( , , ) =
2 2
4 4 2
1 1
1 1
1 1
1 1
1 1
1 1
9
× × 0 + × × 10 + × × 3 + × × 10 + × × 2 + × × 4 =
2 4
2 4
2 2
2 4
2 4
2 2
2
Note: We do not ask each step of your calculation. However, if you got a wrong answer, you
can still get partial credits if you have correct explanation.
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