2. Each of n ≥ 2, i = 1, ..., n can make contributions s i ∈ [0, w] (w ...

Topics in Microeconomics 2009-10 Kirori Mal College, University of Delhi
Name: ___________________________
Date: ___________________
Test
Directions: You are required to answer all 2 short questions in part I (20 points each) and to answer
any 2 of the 3 questions in Part II (30 points each). For questions with parts, all parts have equal
weight. Please submit complete and carefully written answers.
Part I
Part II
1. There are three voters 1, 2, 3 who have to vote for either candidate A or candidate B. The
candidate who gets at least two votes wins the election. If A wins, all voters get a payoff of 1;
if B is elected, all of them get 0.
a) Which of the following strategy profiles constitute a Nash equilibrium? (Tick all the correct
options, there can be more than one)
(i) Voter 1 votes for B while 2 and 3 vote for A.
(ii) Voter 2 votes for A while the other two vote for B.
(iii) All three voters vote for A.
(iv) All three voters vote for B.
Solution: i), iii) and iv)
b) Do any of the Nash equilibria involve players playing weakly dominated strategies?
Solution: i) and iv)
2. Each of n ≥ 2, i = 1, ..., n can make contributions si ∈ [0, w] (w > 0) to the production of
some public good. Their payoff functions are given by πi (s1, .., sn) = n(max{s1, .., sn}) −
si. Find all pure strategy Nash equilibria in the game.
Solution: All pure strategy Nash equilibrium involves exactly one person contributing w
and others contributing nothing. Formally,
Set of Nash Equilibria = {(s1, .., sn)| max{s1, .., sn} = w and ∑1≤ i ≤ n si = w}
3. Two travelers returning home from a remote island discover that the identical antiques
they bought have been smashed in transit. The airline manager proposes the following
scheme to elicit the value of the articles.
The two travelers independently submit compensation claims between Rs. 2 and Rs. 100
(claims must be submitted in integer amounts). The airline will reimburse each traveler
at the minimum of the two claims. In addition, if the claims differ, a reward of Rs. 2 will
be paid to the person making the smaller claim and a penalty of Rs. 2 deducted from the
reimbursement of the larger claimant.
a) Model this situation as a strategic game.
Players: {1, 2}
Actions: For each i ∈ {1, 2}, compensation claim individual i can make is ci ∈ {2, 3, …… ,
100}

Topics in Microeconomics 2009-10 Kirori Mal College, University of Delhi
Preferences: For each i ∈ {1, 2},
πi(c1, c2) = min{c1, c2} if c1 = c2
min{c1, c2}+2 if ci < max{c1, c2}
min{c1, c2} - 2 if ci > min{c1, c2}
b) Find Nash Equilibria of the game.
Solution: Unique Nash equilibrium is (2, 2)
4. Two individuals, Mukesh and Anil are bargaining over Rs 100. Mukesh moves first and
proposes a split of (x; 100 - x) (i.e. Mukesh gets x and Anil 100 - x). Anil responds to Mukesh
offer by saying either Yes or No. If he says Yes, then the split (x; 100 - x) is implemented; if
he says No, both of them get zero.
a) What are the strategies of Anil and Mukesh in this game?
Solution: Strategy set of Mukesh = {x|0 ≤ x ≤ 100} =[0, 100]
Strategy set of Anil = {f|f :[0, 100]→{Yes, No}}
Strategy of Anil is a function from all possible offers that Mukesh can make to a set that
contains two alternatives i.e. either accept or reject an offer.
b) Can the split (50; 50) be a Nash equilibrium outcome in this game? If Yes, provide the
equilibrium strategy profile. If No, explain why?
Solution: Yes
Strategy profile:
sMukesh = 50 i.e. Mukesh offers 50 to himself
sAnil(x) = Yes if 0 ≤ x ≤ 50
= No if x > 50 i.e. Anil will accept all offers that give Mukesh atmost 50 and reject all
other offers.
c) What is the subgame-perfect equilibrium in this game?
Solution:
Strategy profile:
sMukesh = 100 i.e. Mukesh offers 100 to himself
sAnil(x) = Yes if 0 ≤ x ≤ 100 i.e. Anil will accept all offers.
d) Suppose that the bargaining described above carries on for another round. Once again if
Anil says Yes to Mukesh's offer of (x; 100-x), the game ends. However if he says No to
Mukesh's offer he gets to make a counteroffer (instead of the game ending). There is now
only Rs 80 left to bargain over. Suppose Anil offers (y; 80 - y) (i.e. Mukesh gets y and Anil 80
- y). Mukesh now responds to this offer by saying Yes or No. If he says Yes, then the split (y;
80 - y) is implemented. Otherwise the game ends with Mukesh and Anil getting payoffs of 50
and zero respectively. What is the subgame-perfect Nash equilibrium outcome of this game?
Solution: Outcome will be the following: Mukesh will offer 70 i.e. the split (70, 30) and offer
will be accepted.
5. Consider a market in which there are two firms, both producing the same good. Firm i’s cost
of producing qi units of the good is Ci(qi) = 9 for qi > 0 and Ci(0) = 0 for each i ∈ {1, 2}; the
price at which output is sold when the total output is Q is Pd(Q) = max{16 – Q, 0}, where Q
= q1 + q2. Each firm’s strategic variable is output and the firms make their decisions

Topics in Microeconomics 2009-10 Kirori Mal College, University of Delhi
sequentially: one firm chooses its output, then the other firm does so, knowing the output
chosen by the first firm.
a) Model this situation as an extensive game.
Players: {1, 2}
Terminal Histories: {(q1, q2)| q1, q2 ≥ 0}
Player function: P(∅) = 1, P(q1) = 2∀ q1 ∈ [0, ∞)
Preferences: For each i ∈ {1, 2}, πi(q1, q2) = qiPd(q1+q2) – Ci(qi) = qimax{16 – q1 – q2, 0} – 9
b) Find the subgame perfect equilibrium and outcome of Stackelberg’s duopoly game.
Equilibrium:
Strategy of firm 1 → q1 = 10
Strategy of firm 2 → q2 (q1) = (16 – q1)/2 for q1 < 10,
= 0 for q1 ≥ 10
Outcome: (10, 0)