Problems

Problems
Tutorial 1: Topics in Microeconomics
Kirori Mal College, University of Delhi
1. Assume the cost of commuting to work for an individual with wage w is wf(n) by car and wx +
t by metro where n is the number of cars on the road, and x and t are constants. Let there be a
total of N commuters. Assume that f(0) = 0 and f’, f” > 0.
(a) Assume everyone makes the same wage. What will be the equilibrium number of drivers?
How will the equilibrium number change with the wage?
(b) What is the socially efficient number of drivers? How does this number change with wage?
2. Suppose that identical duopoly firms have constant marginal costs of Rs. 10 per unit. Firm 1 faces
a demand function q1 = 100 − 2p1 + p2, where q1 is Firm 1’s output, p1 is Firm 1’s price, and p2
is Firm 2’s price. Similarly, the demand Firm 2 faces is q2 = 100 − 2p2 + p1. Solve for the Nash
(Bertrand) equilibrium.
3. Consider the following social problem. A pedestrian is hit by a car and lies injured on the road.
There are n people in the vicinity of the accident. The injured pedestrian requires immediate
medical attention, which will be forthcoming if at least one of the n people calls for help. Simultaneously
and independently, each of the n bystanders decides whether or not to call for help (by
dialing 100 on a cell phone or pay phone). Each bystander obtains v units of utility if someone
(anyone) calls for help. Those who call for help pay a personal cost of c. That is, if person i calls
for help, then he obtains the payoff v − c; if person i does not call but at least one other person
calls, then person i gets v; finally, if none of the n people calls for help, then person i obtains zero.
Assume v > c. Find the symmetric mixed strategy Nash equilibrium of this n-player normal-form
game. Compute the probability that at least one person calls for help in equilibrium. Note how
this depends on n.
4. The famous British spy 001 has to choose one of four routes, a, b, c, or d (listed in order of speed in
good conditions) to ski down a mountain. Fast routes are more likely to be struck by an avalanche.
At the same time, the notorious rival spy 002 has to choose whether to use(y) or not to use(x) his
valuable explosive device to cause an avalanche. The payoffs of this game are represented here:
002→ x y
001↓
a 12,0 0,6
b 11,1 1,5
c 10,2 4,2
d 9,3 6,0
Is there any route which 001 will definitely not take? Find a Nash equilibrium in which one player
plays a pure strategy si and the other player plays a mixed strategy σj. Find a different mixedstrategy
equilibrium in which the same pure strategy si is assigned zero probability. Are there any
other equilibria?
5. Consider a game with n players. Simultaneously and independently, the players choose between
X and Y. That is, the strategy space for each player i is Si ={X, Y }. The payoff of each player
who selects X is 2mx − m 2 x + 3, where mx is the number of players who choose X. The payoff of
each player who selects Y is 4 − my, where my is the number of players who choose Y. Note that
mx + my = n.
(a) For the case of n = 2, represent this game in the normal form and find the pure-strategy Nash
equilibria(if any).
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(b) For the case of n = 3, find the pure-strategy Nash equilibria(if any). Determine whether this
game has symmetric mixed-strategy Nash equilibrium.
6. Consider the game in the following figure.
2→ L M R
1↓
U x,x x,0 x,0
C 0,x 2,0 0,2
D 0,x 0,2 2,0
Compute the pure-strategy and mixed-strategy Nash equilibria for this game, and note how they
depend on x. In particular, what is the difference between x > 1 and x < 1?
7. Army A has a single plane with which it can strike one of three possible targets. Army B has one
antiaircraft gun that can be assigned to one of the targets. The value of target k is vk, with v1
> v2 > v3 > 0. Army A can destroy a target only if the target is undefended and A attacks it.
Army A wishes to maximize the expected value of the damage and army B wishes to minimize
it. Formulate the situation as a (strictly competitive) strategic game and find its mixed strategy
Nash equilibria.
8. Farber(1980) proposes the following model of final-offer arbitration. There are three players: a
management (i = 1), a union (i = 2), and an arbitrator (i = 3). The arbitrator must choose
a settlement t ∈ ℜ from the two offers, s1 ∈ ℜ and s2 ∈ ℜ, made by the management and the
union respectively. The arbitrator has exogenously given preferences v0 = −(t − s0) 2 . That is, he
would like to be as close to his ”bliss point,” s0, as possible. The management and the union
don’t know the arbitrator’s bliss point; they know only that it is drawn from the distribution P
with continuous positive density p on [s 0 , s0]. The management and the union choose their offers
simultaneously. Their objective functions are u1 = −t and u2 = +t, respectively. Derive and
interpret the first-order conditions for a Nash equilibrium. Show that the two offers are equally
likely to be chosen by the arbitrator.
9. Consider a first-price sealed-bid auction of an object with two bidders. Each bidder i’s valuation
of the object is vi, which is known to both bidders. The auction rules are that each player submits
a bid in a sealed envelope. The envelopes are then opened, and the bidder who has submitted the
highest bid gets the object and pays the auctioneer the amount of his bid. If the bidders submit
the same bid, each gets the object with probability 1
2 . Bids must be in dollar multiples (assume
that valuations are also).
(a) Are any strategies strictly dominated?
(b) Are any strategies weakly dominated?
(c) Is there a Nash equilibrium? What is it? Is it unique?
10. Cosider a normal form game between three major car producers, C, F, and G. Each producer can
produce either large cars, or small cars but not both. That is, the action set of each producer i,
i = C, F, G is Ai = {SM, LG}. We denote by a i the action chosen by player i, a i ∈ A i , and by
πi aC , aF , aG the profit of firm i. Assume that the profit function of each player i is defined by
⎧
π i ⎪⎨
=
⎪⎩
Answer the following questions:
γ if a j = LG, for all j = C, F, G;
γ if a j = SM, for all j = C, F, G;
α if a i = LG, and a j = SM for all j = i;
α if a i = SM, and a j = LG for all j = i;
β if a i = a j = LG, and a j = SM, j = k = i;
β if a i = a j = SM, and a j = LG, j = k = i.
(a) Does there exist a pure-strategy Nash equilibrium when α > β > γ > 0?
(b) Does there exist a pure-strategy Nash equilibrium when α > γ > β > 0?
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Notes
Problem 1 is taken from OCW, MIT: Microeconomic Theory - 1 (Fall 2009 Practice Exam)
Problem 2 is taken from Perloff: Microeconomics: Theory and Applications with Calculus
Problems 3, 4, 5, and 6 are taken from Watson: Strategy
Problem 7 is taken from Osborne and Rubinstein: A Course in Game Theory
Problem 8 is taken from Fudenberg and Tirole: Game Theory
Problem 9 is taken from Mas-colell, Whinston and Green: Microeconomic Theory
Problem 10 is taken from Shy: Industrial Organization
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