Problem 1 [38pt]
Consider the following road network:
(
2 min
if y ≤ 0.5
1.5 + y min if y > 0.5
y
H
B
h
1−y
(
1 min
if 1 − y ≤ 0.7
0.3 + (1 − y) min if 1 − y > 0.7
C
x
A
1−x
2 min
L
l
3 min
c min s
D
Numbers on the edges show the travel time (in minutes) on each road; x, 1 − x, y, 1 − y are
fractions of all drivers on roads L, l, H and h, respectively. All drivers start at A and finish at
B.
The difference from the assignment problem in (a)–(f) is that we will assume that there are
only two drivers (hence, when one driver takes a road, then 0.5 fraction of all drivers take this
road).
(a) [4pt] Identify the pure strategies of each player. Compute the payoffs to each of the
players from six strategy profiles. If you compute more than six payoffs, the first six will be
checked by a marker. You can answer this question by drawing a table with strategies and
payoffs. Note that c is not given a numeric value; do not substitute any value for c. Do not
eliminate any strictly dominated strategies, but assume that no driver drives away from B or
uses the same road twice (that is, do not consider such strategies as LslL . . . ).
For (b)–(e), we set c = 1.
(b) [3pt] Find best response(s) for one of the players for each strategy you have identified
in (a).
(c) [3pt] Identify all strictly dominated strategies and prove that they are. Explain why
other strategies are not strictly dominated.
(d) [3pt] Identify all strictly dominant strategies and prove that they are. Explain why
other strategies are not strictly dominant.
(e) [4pt] Find all pure strategy Nash equilibria of this game. Explain why they are Nash
equilibria and why there are no other pure strategy Nash equilibria.
(f) [4pt] Suppose now that c = 0.3. Find all pure and mixed strategy Nash equilibria of
that game.
We now return to the model with infinitely many drivers (the road network is exactly the
same). Suppose that driving time on s is different for different drivers and given by the
function c(i) = i, where i ∈ [0, 1] is the index of a driver. Hence, for any x, the fraction x
of all drivers have the driving time less than x and the fraction (1 − x) of drivers have the
driving time more than x. The minimum driving time, for driver i = 0, is 0 and the maximum
driving time, for driver i = 1, is 1.
(g) [8pt] Find a pure-strategy Nash equilibrium of this game.
We now return to the problem with two drivers. Suppose that the one driver moves first
(we will call her the first driver) and the other driver moves next (the second driver). Suppose
further that, when deciding upon his strategy, the second driver sees whether the first driver
1
took road L or l, but does not know what happens when the first driver reaches points C or D,
respectively. Note that the second driver decides upon a strategy (that is, a complete route)
when he starts driving and does not change the route en route.
(h) [5pt] Represent this game as a sequential game (draw a tree). As in (a), you need to
identify only six payoffs.
(i) [4pt] Find one subgame perfect Nash equilibrium (SPNE) and explain why it is an SPNE.
Problem 2 [11pt]
Suppose that agent’s preferences are represented by a linear utility function
u(m, t) = m + kt,
where t ∈ R+ is the amount of toilet paper that the agent consumes, m ∈ R+ is the amount of
other goods she consumes and k > 0 is a parameter (a given value and not a good consumed
by the agent). Goods t and m are perfectly divisible throughout the problem.
Let the price of m be 1 and the price of toilet paper be pt > 0.
(a) [3pt] Write a Marshallian demand function for this agent for any value of k > 0.
(b) [3pt] Write a Hicksian demand function for this agent for any value of k > 0.
Suppose in the remainder of the problem that the price of a roll of toilet paper has the
following schedule:
(
$0.4 if t ≤ 2;
pt =
$2
if t > 2.
(c) [5pt] Let k = 1. What is the consumer’s optimal bundle for any income Y > 0?
Problem 3 [13pt]
Consider a two-person household with arbitrary preferences 1 and 2 , which satisfy all the
axioms we imposed on individual preferences. Note that, unlike the assignment problem, we
do not specify any utility function for these individuals.
Sometimes these two persons need to make joint decisions. They decide very simply: for
any two alternatives, (x1 , y1 ) and (x2 , y2 ), they vote. If they both pick the same alternative,
say (x1 , y1 ), we say that household prefers (x1 , y1 ). If one member of the household prefers
(x1 , y1 ) and the other is indifferent, then we say the household prefers (x1 , y1 ). If they pick
different alternatives (say person 1 picks (x1 , y1 ) and person 2 picks (x2 , y2 )), then we say that
household is indifferent between (x1 , y1 ) and (x2 , y2 ). Similarly, if they are both indifferent,
we say that the household is indifferent. The resulting choice between two bundles defines the
preferences of the household, H .
In lectures, we defined transitivity using “no-worse-than” () preference relation. If you
prefer, you can use the following alternative definition of transitivity: preferences are transitive
if, whenever (x1 , y1 ) (x2 , y2 ) and (x2 , y2 ) (x3 , y3 ), then (x1 , y1 ) (x3 , y3 ).
Prove that the preferences of the household are transitive or give a counter-example showing
that preferences are not transitive.
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Problem 4 [26pt]
You should provide explanations for all your answers. While you are welcome to use Bayes’
formula, you are not required to do so.
Consider an information cascade problem with asymmetric signals. There are two states of
the world, Y and N , with prior belief that P (Y ) = P (N ) = 0.5. Ann’s utility when she takes
action ΥA in state Y or action ηA in state N is 1; it is 0 otherwise.
(a) [3pt] Suppose Ann has not received any signal about the true state of the world and
must base her action only on the information given above. What is Ann’s optimal action?
Explain.
Suppose Ann receives signals yA and nA with probabilities p(yA |Y ) = 0.25, p(nA |Y ) = 0.75,
p(yA |N ) = 0, p(nA |N ) = 1.
(b) [3pt] Suppose Ann receives signal yA . What is Ann’s optimal action?
(c) [3pt] Suppose Ann receives signal nA . What is Ann’s optimal action?
Suppose Bob’s signals yB , nB are generated according to the following probabilities: P (yB |Y ) =
P (nB |N ) = 0.9, P (nB |Y ) = P (yB |N ) = 0.1
(d) [3pt] Suppose Bob observes Ann’s action and receives signal yB . What is Bob’s optimal
action? (You need to provide an answer for each possible action of Ann)
(e) [3pt] Suppose Bob observes Ann’s action and receives signal nB . What is Bob’s optimal
action? (You need to provide an answer for each possible action of Ann)
(f) [6pt] Suppose that Carol, before receiving her own signal, wants to calculate her belief
about states Y and N . Calculate Carol’s belief for each feasible sequence of actions by Ann
and Bob; do not simplify your expressions.
Suppose Carol’s signals yC , nC are generated according to the following probabilities: P (yC |Y ) =
P (nC |N ) = 0.9, P (nC |Y ) = P (yC |N ) = 0.1. Suppose that it is important for Carol to match
the action with the state only when the state is N , but not when the state is Y . That is,
Carol’s utility is equal to 1 when she takes action ηC in state N ; in all other cases, her utility
is 0.
(g) [5pt] Identify Carol’s optimal actions for every feasible sequence of decisions by Ann and
Bob.
Problem 5 [12pt]
Consider two utility functions:
u1 (x, y) = x − α|x − y|, α > 0
1
x
− , β > 0.
u2 (x, y) = x − β
x+y 2
Construct two specific bundles, (x1 , y1 ), (x2 , y2 ), which would allow a researcher to distinguish whether an individual has preferences 1 or 2, or show that it is impossible.
Notes: You are only allowed to offer the individuals the choice between two bundles; you
are not allowed to design a more sophisticated experiment. If you think that the preferences
can be distinguished, you must provide specific bundles (e.g., numbers). Please economise on
your algebra; it is not the basis for your assessment.
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