ECON 2112. HOMEWORK 3
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ECON 2112. HOMEWORK 3
DUE: 11 PM, 31ST MARCH (FRIDAY)
Exercise 1. (6 marks)
Adeline and Bani are a team in a consulting irm. Once a task is assigned, Adeline irst decides whether
to work separately (S ) or work together (T). Subsequently, Adeline and Bani choose their efort levels.
We assume that efort choices are made separately but simultaneously.1 Let ea and eb denote the level of efort exerted by Adeline and Bani respectively where ei ∈ {0, 1} and i = a, b. If Adeline chooses S ,
each individual i’s payof is
ei + ej
Ui = 2 
ei .
If Adeline chooses T
Ui = vmin{ei , ej} ei ,
where i, j ∈ {a, b} but j 丰 i 2 and min{ei , ej} = ei if ei 
ej and min{ei , ej} = ej if ei 之 ej
Assume v equals the largest digit in your student ID.
(i) (2 marks) Suppose Adeline chooses S . Consider the subgame associated with that choice (i.e. the one that arise following S ).
. Write the corresponding payof matrix.
. Identify all pure strategy Nash equilibria.
. Is there a Nash equilibrium in mixed strategies where at least one player chooses both
efort levels with strictly positive probability? Explain.
(ii) (2 marks) Suppose Adeline chooses T . Consider the subgame associated with that choice (i.e. the one that arise following T) and answer the three questions stated in part (i).
(iii) (2 marks) Now consider the entire game
. Identify a pure strategy proile that is a Subgame Perfect Nash equilbrium (SPNE). Explain
why it is SPNE.
. Identify a pure strategy proile that is a Nash equilbrium (NE) but not SPNE. Explain why
it is a NE but not SPNE.
Note: The function min{a, b} refers to minimum of two numbers a and b. For example, min{3, 4} = 3; min{3, 1} = 1. In the irst case, i.e. between 3 and 4, 3 is minimum. In the second case, i.e., between 3 and 1, 1 is minimum. So minimum of a and b is a when a 
b, and minimum ofa and b is b when b 
a.
Exercise 2. (6 marks)
Little Italy has 100 consumers all of whom buy either one pizza or nothing. Each consumer has a reservation price of $15. That is, each consumer is willing to pay at most $15 for a pizza. There are two pizza stores - A and B - located next to each other. It costs $7 to make a pizza. Let pa and pb denote the price charged by stores A and B respectively. For simplicity, assume
pi ∈ {12, 15}
for both i = A, B. Hereafter we ignore the dollar sign. Thus, prices can take just one of the two values - 12 or 15. If pi < pj, all consumers buy pizza from store i. If pi = pj = p then 50 consumers buy from store A while 50 buy from store B.
(i) (1 mark) Write down the payof matrix. State the unique Nash equilibrium of the one-shot game.
(ii) (1 mark) Suppose the game is repeated twice. Is there a subgame perfect equilibrium where pa = pb = 15 in the irst period? Assume that after irst period, one-period proits are realised and each store knows whether or not the other store has charged 15 or 12. Explain.
(iii) (1.5 marks) Suppose the game is ininitely repeated. Consider the following trigger strategy by player i: Play pi = 15 in the irst stage. In the t-th stage, if the outcome of all t 
1 preceding stages has been (15,15) then play pi = 15; otherwise, play pi = 12. Assume a common discount factor of 6 for both stores A and B. Can the two stores sustain pa = pb = 15 each period if 6 =3(1) ? Explain.
(iv) (2.5 marks) Now add pi = 9 to the set of possible prices. That is, suppose pi ∈ {9, 12, 15}
How does the inclusion of this new strategy pi = 9 afect your indings. In particular, . Are there any additional pure-strategy Nash equilibrium in the one-shot game? . Does you answer to part (ii) change with inclusion of this additional strategy? . Does your answer to part (iii) change? [Assume 6 =3(1)]
2023-03-28