Game Theory with Applications in Corporate Finance
Assignment 2
Due Date: 4 March 2022 by midnight
1. (This is Problem 9U1 in the textbook) Consider again the case of the 2011 Citrus. Almost all cars depreciate over time, and so it
is with the Citrus. Every month that passes, all sellers of Citrus – regardless of type – are willing to accept $100 less than they were
the month before. Also, with every passing month, buyers are maximally willing to pay $400 less for an orange than they were the
previous month and $200 less for a lemon. Assume that the example in the text takes place in month 0. Eighty percent of the
Citruses are oranges, and this proportion never changes.
(a) Fill out three versions of the following table for month 1, month 2, and month 3.
Willingness to accept of sellers
Willingness to pay of buyers
Oranges
Lemons
(b) Graph the willingness to accept of the sellers of oranges over the next 12 months. On the same figure, graph the price that
buyers are willing to pay for a Citrus of unknown type (given that the proportion of oranges is 0.8). (Hint: Make the vertical axis
range fro 10 000 to 14 000.)
(c) Is there a market for oranges in month 3? Why or why not?
(d) In what month does the market for oranges collapse?
(e) If owners of lemons never experienced depreciation, i.e., they were never willing to accept anything less than $3000, would
this affect the timing of the collapse of the market for oranges? Why or why not? In what month does the market for oranges
collapse in this case?
(f) If buyers experienced no depreciation for a lemon, i.e., they weree always willing to pay up to $6000 for a lemon, would this
affect the timing of the collapse of the market for oranges? Why or why not? In what month does the market for oranges collapse
in this case?
2. (This is Problem 9U2 in the textbook). An economy has two types of jobs, Good and Bad, and and two types of workers,
Qualified and Unqualified. The population consists of 60% Qualified and 40% Unqualified. In a Bad job either type of worker
produces 10 units of outputs. In a Good job, a Qualified worker produces 100 units of output and an Unqualified worker produces
0. There is enough demand for worker that for each type of job, companies must pay what they expect the appointee to produce.
Companies must hire each worker without observing his type and pay him before knowing his actual output. But Qualified
worker can signal their qualification by getting educated. For a Qualified worker, the cost of getting educated to level n is
n2
2
,
2
whereas for an Unqualified worker, it is n . These costs are measured in the same unit as output, and n must be an integer.
(a) What is the minimum level of n that will achieve separation of types?
(b) Now suppose that the signal is made unavailable. Which kind of job will be filled by which types of workers and at what wage?
Who will gain and who will lose from this change?
3. (This is Problem 9U9 in the textbook) Consider Spence’s job-market signalling model with the following specifications. There
are two types of workers, 1 and 2. The productivities of the two types, as functions of the level of education, are
W1 [E] = E and W2 [E] = 1.5 E.
The cost of education of the two types, as functions of the level of educations, are
2
C1 [ E ] =
E2
2
and C2 [E] =
E2
3
.
Each worker’s utility is equal to his or her income minus the cost of education. Companies that seek to hire these workers are
perfectly competitive in the labor market.
(a) If types are public information (observable and verifiable), find expressions for the levels of education, income, and utilities of
the two types of workers.
Now suppose that each worker’s types is his or her private information.
(b) Verify that if the contracts of part (a) are attempted in this situation of information asymmetry, then type 2 does not want to
take up the contract of type 1, but type 1 does want to take up the contract intended for type 2, so “natural” separation cannot
prevai.
(c) If we leave the contract of type 1 as in part (a), what is the range of contracts (education-wage pairs) for type 2 that can
achieve separation?
(d) Of the separating contracts, which one do you expect to prevail? Give a verbal but not a formal explanation for your answer.
(e) Who gains or loses from the information asymmetry? How much?
4. Consider a Cournot duopoly operating in a market with inverse market demand p[Q] = a – Q, where Q = q1 + q2 is the aggregate
quantity on the market. Both firms have total cost ci [qi ] = γ qi , where γ > 0 is a constant. Demand is uncertain: it is high (a = aH )
with probability θ and low (a = aL ) with probability 1 – θ. Furthermore, information is asymmetric: firm 1 knows whether demand
is high or low, but firm 2 does not. All of this is common knowledge. The two firms simultaneously choose quantities.
What is the Bayesian Nash equilibrium? In answering this question, presume that the values of the parameters aH , aL , θ are such
that all the equilibrium quantities are positive.
5. Consider the following asymmetric information model of Bertrand duopoly with differentiated products. Demand for firm i is
qi pi , pj = a – pi + bi pj . Costs are zero for both firms. The sensitivity of firm i ‘ s demand to firm j ‘ s price is either high or low. That
is bi is either bH or bL , with 0 < bL < bH . For each firm, bi = bH with probability θ and bi = bL with probability 1 – θ, independent of
the realization of bj . Each firm knows its own bi , but not its competitor’s. All of this is common knowledge.
What conditions define a symmetric pure-strategy Bayesian Nash equilibrium of this game? Solve for such an equilibrium.
6. (This is Problem 14U4 in the textbook) Cheapskates is a very minor-league professional hockey team. Its facility is large enough
to accommodate all of the 1000 fans who might want to watch its home games. It can provide two types of seats- ordinary and
luxury. There are also two sorts of fans: 60% of the fans are blue-collar fans, and the rest are white-collar fans. The costs of
providing each type of seat and the fans’ willingness to pay for each type of seat are given in the following table (measured in
dollars):
Willingness to Pay
Cost
Blue-Collar
White-Collar
Ordinary
4
12
14
Luxury
8
15
22
Seating
Each fan will buy at most one seat, depending on the consumer surplus he would get (maximum willingness to pay minus the
actual price paid) from the two kinds. If the surplus for both is negative, then he won’t buy any. If at least one kind gives him nonnegative surplus, then he will buy the kind that gives him the larger surplus. If the two kinds give him equal non-negative surplus,
then the blue-collar fan will buy the ordinary kind of seat, and the withe-collar fan will buy the luxury kind.
The team owners provide and price their seating to maximize profit., measured in thousands of dollars per game. They set prices
for each kind of seats, sell as many tickets as are demanded at these prices and then provide the numbers and types of seats of
each kind for tickets have sold.
3
(a) First, suppose that the team owners can identify the type of each individial fan who arrives at the ticket window (presumably
bythe color of his collar) and can offer him just one type of seat at a stated price on a take-it-or-leave-it basis. What is the owners’
maximum profit π * under this system?
(b) Now suppose that the owners cannot identify any individual fan, but they still know the proportion of blue-collar fans. Let the
price of an ordinary seat be X and the price of a luxury seat be Y. What are the incentive compatibility constraints that will ensure
the blue-collar fans buy the ordinary seats and the white-collar fans buy the luxury seats? Graph these constraints on an
(X, Y ) – coordinate plane.
(c) What are the fans’ participation constraints for the fans’ decisions on whether to buy tickets at all? Add these constraints to
the grap in part (b).
(d) Given the constraints in (b) and (c), what prices X and Y maximize the owners’ profit π2 under this price system? What is the
value of π2 ?
(e) The owners are considering whether to set prices so that only the white-collar fans will buy tickets. What is theor profit πw if
they decide to cater to only white-collar fans?
(f) Comparing π2 and πw , determine the pricing policy that the owners will set. How does their profit achieved from this policy
compare with the case of full information, where they earn π * ?
(g) What is the cost of coping with the information asymmetry in part f? Who bears this cost? Why?
Games with Incomplete Information
1. A Simple Game with Incomplete Information
John is in love with Mary, but he does not know whether Mary likes him or not. From his own experience with Mary, he thinks there is a chance of 25% that mary likes him, and a chance of 75% that Mary
does not like him. John is thinking of inviting Mary to dinner. The utility of John depends on (i) his
strategy – invite or do not invite – (ii) the responese of Mary when asked – accept or refuse the invitation
– and (iii) the type of Mary (whether she likes him or not). As for Mary, her utility also depends on (i) the
strategy chosen by John – invite or do not invite – (ii) her response when asked – accept or not accept and (iii) her type (whether she likes him or not).
The following two matrices represent the payoffs for John and Mary, as functions of the type of Mary
and the combination of strategies chosen by the two individuals.
Mary 1 (0.25)
(Mary likes John)
Invite
John
Do not invite
Accept
(10, 7)
Refuse
(2, 2)
(3, 4)
(3, 4)
Mary 2 (0.75)
(Mary does not like John)
Accept
(8, 1)
Refuse
(2, 4)
(3, 4)
(3, 4)
In the above matrices, Mary 1 represents the type and Mary 2 represents the type
.
We would like to find the answers to the following questions:
(i) Should John invite Mary to dinner?
(ii) Should Mary accept the offer if she likes him?
(iii) Should Mary accept the offer if she does not like him?
The game just described is called as game with incomplete information because John does not know
the type of Mary, i.e., he does not know whether he is facing a girl who likes him or a girl who does not
like him. In the two payoff matrices, Mary 1 represents the type Mary likes John, and Mary 2 represents
the type Mary does not like John.
1.1. The Solution of the Game: the Bayesian Nash Equilibrium
2
Games with Incomplete Information_01_02_2022.nb
Obviously, Mary knows whether she likes John or not, i.e., she knows her own type. If she likes him,
then she knows that the first matrix represents the payoff matrix for the game between John and her.
On the other hand, if Mary does not like John, then from her perspective, it is the second matrix that
represents the game between John and her.
John, on the other hand, because he does not know the type of Mary, thinks that the first matrix applies
with probability 0.25, and the second matrix applies with probability 0.75. In his decision making, he
has to take into account both of these possibilities.
Strictly speaking, the game being analyzed can be considered as a game that involves three players:
John, Mary 1, and Mary 2. Mary 1 knwos that she is playing the game represented by the first payoff
matrix, while Mary 2 knows that she plays the game represented by the second payoff matrix. As for
John, he does not know exactly which game he is playing. He thinks that he play the first game with
probability 0.25 and the second game with probability 0.75.
For Mary 1, the strategy gives her a payoff of 7 if John invites her to dinner, and a payoff of 4
if John does not invite her to dinner. On the other hand, if Mary 1 plays the strategy , then
she obtains a payoff of 2 if John invites her to dinner and a payoff of 4 if John does not invite her to
dinner. Thus is the dominant strategy for Mary 1, and she will play .
For Mary 2, the strategy gives her a payoff of 1 if John invites her to dinner, and a payoff of 4
if John does not invite her to dinner. On the other hand, if Mary 2 plays the strategy , then
she obtains a payoff of 4 if John invites to dinner and a payoff of 4 if John does not invite her to dinner.
Thus is the weakly dominant strategy for Mary 1, and she will play .
For John, according to the first-order rationality principle, he knows that Mar 1 will play ,
and Mary 2 will play . If John plays , then there is 25% chance that he is playing
against Mary 1 in which case she will acceplt the offer, giving him a payoff of 10. On the other hand,
because there is a 75% chance that it is Mary 2 that he is asking to dinner, and he knows that Mary 2 will
refuse, with the ensuing consequence that he will obtain a payoff of 2. Thus playing > gives
John an expected payoff of
(1) 0.25 × 10 + 0.75 × 2 = 4.0
On the other hand, if John plays , Mary 1 will play , which is the dominant
strategy for her, and the payoff for John is then 3. For Mary 2, she plays , and John’s payoff
is 3. Thus, playing gives John a payoff of 3.
Because yields John a higher payoff (4) than (3), John will invite Mary to
dinner. As for Mary, she accepts the offer if she likes John, but turns down the offer if she does not like
John. This equilibrium is called a Bayesian Nash equilibrium of the game with incomplete information.
Under the Bayesian Nash equilibrium, the expected payoff for John is 4. For Mary 1, her payoff is 7. For
Games with Incomplete Information_01_02_2022.nb
3
1.2. The Value of Perfect Information
Suppose that the little sister of Mary knows whether Mary likes John or not, and is willing to provide
John with this information at a price. What is that price?
If John accepts such a proposition, and if Mary likes him, then this information will be given to him by
the little sister. In this case, and the probability of this event is 0.25, he will invite mary out to dinner
because is the dominant strategy of Mary 1. This event gives him a payoff of 10. On the
other hand, if Mary does noit like him, then with this information – which occurs with probability 0.75 he will not invite Mary out to dinner because he knows that is weakly dominant for Mary 2.
The payoff John obtains under this event is then equal to 3. Thus, the expected payoff for John if he
accepts the proposition of Mary’s little sister is
(2) 10 × 0.25 + 3 × 0.75 = 4.75.
The difference between (2) and (1), namely
(3) 4.75 – 4 = 0.75
is the gain in expected payoff that John obtains if he accepts the proposition of Mary’s sister, and this
gain is the maximum amount John is willing to pay for the information. The payoff differential 0.75
represents the value of perfect information on the type of Mary.
2. A More Complicated Games with Incomplete Information: A Two-Person
Game in which Each Player has Two Types
In this game, there are two players: player I and player II. Player I has two types a and b, while player II
has two types c and d. A player knows his type, but not the type of the other player. Thus, in reality
there are 4 possible games:
(i) The game between Ia and IIc,
(ii) The game between Ia and IId,
(iii) The game between Ib and IIc,
(iv) The game between Ib and IId.
In game (i) player I knows that his type is a, but does not know whther he plays against IIc or IId. On the
other hand, in this game, player II knows that his type is c, but does not know whether he is paying
against Ia or Ib. The interpretation of games (ii), (iii), and (iv) are similar.
The game evolves through time as follows. Nature moves first, and makes a random choice among the
following set of possible type profiles:
{(Ia, IIc), (Ia, IId), (Ib, IIc), (Ib, IId)}.
The following matrix represents the probabilities of the four possible type profiles:
4
Games with Incomplete Information_01_02_2022.nb
Player II
IIc
IId
1
Ia
100
0
Player I
9
Ib
90
100
100
According to the above matrix, Nature chooses the type profile (Ia, IIc) with probability
1
100
and the type
profile (Ia, IId) with probability 0. In the same manner, Nature chooses the type profile (Ib, IIc) with
9
90
and the type profile (Ib, IId) with probability 100
.
probability 100
If the random choice made by Nature is (Ia, IIc), Nature reveals secretly to player I that his type is Ia et
secretly to player II that his type is IIc. However, Ia does not know whether he is playing against IIc or
IId. In the same manner, IIc does not know whether he is playing against Ia or Ib.
Each player has 2 strategies, say 1 and 2. The following 4 matrices represent the payoffs for the two
players as functions of their types and their strategies:
IIc
Ia
1
2
1
(8,-8)
(0,0)
Ib
1
2
(-8,8)
(-12,12)
IId
2
(-8,8)
(-4,4)
1
(-4,4)
(0,0)
2
(8,-8)
(12,-12)
(12,-12)
(16,-16)
(4,-4)
(0,0)
(-4,4)
(-8,8)
2.1. The Solution of the Game: The Bayesian Nash Equilibrium
Games with Incomplete Information_01_02_2022.nb
5
Applying the Bayes’ formula, we obtain
1
p[c, a] = Prob {IIc Ia} =
+0
100
p[d, a] = Prob {IId Ia} =
= 1,
100
1
0
1
100
+0
= 0,
9
p[c, b] = Prob {IIc Ib} =
100
9
+
100
=
90
100
90
p[d, b] = Prob {IId Ib} =
100
9
100
+
90
=
100
1
p[a, c] = Prob {Ia IIc} =
100
1
100
+
9
1
,
10
=
9
,
10
100
100
1
100
p[a, d] = Prob {Ia IId} =
+
0
90
9
10
,
11
=
9
p[b, c] = Prob {Ib IIc} =
1
,
11
100
= 0,
100
90
p[b, d] = Prob {Ib IId} =
100
90
0+ 100
= 1.
We shall carry out the computations with the help of Mathematics. First, we enter the data on the
payoff matrices. Because in the four payoff matrices, the gain of one player is the opposite of the gain
of the other player, we only need to enter the payoff for one player, say player I.
The following matrix gives the gain for Ia when he plays against Ic.
In[1]:=
Out[1]=
In[2]:=
m[a, c] = {{8, – 8}, {0, – 4}}
{{8, – 8}, {0, – 4}}
MatrixForm[m [a, c]]
Out[2]//MatrixForm=
8 -8
0 -4
In[3]:=
m[a, c] // MatrixForm
Out[3]//MatrixForm=
8 -8
0 -4
The payoff matrix when Ia plays against IId
In[4]:=
Out[4]=
In[5]:=
m[a, d] = {{- 4, 8}, {0, 12}}
{{- 4, 8}, {0, 12}}
m[a, d] // MatrixForm
Out[5]//MatrixForm=
-4 8
0 12
6
Games with Incomplete Information_01_02_2022.nb
The payoff matrix when Ib plays against IIc
In[6]:=
m[b, c] = {{- 8, 12}, {- 12, 16}}
Out[6]=
{{- 8, 12}, {- 12, 16}}
In[7]:=
m[b, c] // MatrixForm
Out[7]//MatrixForm=
– 8 12
– 12 16
The payoff matrix when Ib plays against IId
In[8]:=
Out[8]=
In[9]:=
m[b, d] = {{4, – 4}, {0, – 8}}
{{4, – 4}, {0, – 8}}
MatrixForm[m[b, d]]
Out[9]//MatrixForm=
4 -4
0 -8
The conditional probailities that represent the beliefs of one player concerning the type of the other
player.
In[10]:=
Out[10]=
In[11]:=
Out[11]=
In[12]:=
p[c, a] = 1
1
p[d, a] = 0
0
p[c, b] =
1
11
1
Out[12]=
In[13]:=
11
p[d, b] =
10
11
10
Out[13]=
In[14]:=
11
p[a, c] =
1
10
1
Out[14]=
In[15]:=
10
p[b, c] =
9
Out[15]=
10
9
10
Games with Incomplete Information_01_02_2022.nb
In[16]:=
Out[16]=
In[17]:=
Out[17]=
7
p[a, d] = 0
0
p[b, d] = 1
1
The game between the two players I and II in which each of them has 2 possible types can be considered as a game with 4 players: Ia, Ib, IIc, IId, where each type of each player can be considerd as a
distinct player. However, the computations of the expected payoff for each type of each player require
some particular attention. For example, in the game between (Ia, IIc), player Ia knows that Ib is not
involved and that player Ia does not know against whom he is playing: IIc or IId. For player Ia, player Ib
does not appear in the game he is playing. Similarly, player IIc knows that player IId is not in the game,
but player Ia or player Ib might be his adversary with probability p[a, c] and p[b, c], respectively.
A combination of strategies for the four players Ia, Ib, IIc, IId can be represented by a list of positive
integers: {i, j, k, ℓ}, i = 1, 2, j = 1, 2, k = 1, 2, ℓ = 1, 2, where i is the strategy chosen by player Ia; j is the
strategy chosen by player Ib; k is the strategy chosen by player IIc; and ℓ is the strategy chosen by player
IId.
For example, the list {1, 1, 1, 1} represents the combination of the following strategies:
(i) player Ia chooses strategy 1;
(ii) player Ib chooses strategy 1;
(iii) player IIc chooses strategy 1;
(iv) player IId chooses strategy 1.
As another example, the list {2, 1, 1, 2} represents the combination of the following strategies:
(i) player Ia chooses strategy 2;
(ii) player Ib chooses strategy 1;
(iii) player IIc chooses strategy 1;
(iv) player IId chooses strategy 2.
Under the list {1, 1, 1, 1}, player Ia chooses strategy 1. Player Ia plays against IIc with probability p[c, a]
and against player IId with probability p[d, a]. Because under the list {1, 1, 1, 1}, player IIc chooses
strategy 1, the payoff obtained by player Ia is 8, and this event has probability p[c, a]. Also, because
under the list {1, 1, 1, 1}, player IId chooses strategy 1, the payoff obtained by player Ia is -4, and this
event has probability p[d, a]. Thus, the expected payoff for player Ia under the combination of strategy
{1, 1, 1, 1} is
(1) 8 p[c, a] + (-4) p[d, a] = 8 × 1 + (-4)× 0 = 8.
Under the list {1, 1, 1, 1}, player Ib chooses strategy 1. Player Ib plays against IIc with probability p[c, b]
and against player IId with probability p[d, b]. Because under the list {1, 1, 1, 1}, player IIc chooses
strategy 1, the payoff obtained by player Ib is -8, and this event has probability p[c, b]. Also, because
8
Games with Incomplete Information_01_02_2022.nb
under the list {1, 1, 1, 1}, player IId chooses strategy 1, the payoff obtained by player Ib is 4, and this
event has probability p[d, b]. Thus, the expected payoff for player Ib under the combination of strategy
{1, 1, 1, 1} is
1
10
+ 4× =
(2) -8 p[c, b] + 4 p[d, b] = -8 × 11
11
32
.
11
Under the list {1, 1, 1, 1}, player IIc chooses strategy 1. Player IIc plays against Ia with probability p[a, c]
and against player Ib with probability p[b, c]. Because under the list {1, 1, 1, 1}, player Ia chooses
strategy 1, the payoff obtained by player IIc is -8, and this event has probability p[a, c]. Also, because
under the list {1, 1, 1, 1}, player Ib chooses strategy 1, the payoff obtained by player IIc is 8, and this
event has probability p[b, c]. Thus, the expected payoff for player IIc under the combination of strategy
{1, 1, 1, 1} is
1
9
+ 8× =
(3) -8 p[a, c] + 8 p[b, c] = -8 × 10
10
64
10
=
32
.
5
Under the list {1, 1, 1, 1}, player IId chooses strategy 1. Player IId plays against Ia with probability
p[a, d] and against player Ib with probability p[b, d]. Because under the list {1, 1, 1, 1}, player Ia
chooses strategy 1, the payoff obtained by player IId is 4, and this event has probability p[a, d]. Also,
because under the list {1, 1, 1, 1}, player Ib chooses strategy 1, the payoff obtained by player IId is -4,
and this event has probability p[b, d]. Thus, the expected payoff for player IId under the combination of
strategy {1, 1, 1, 1} is
(4) 4 p[a, d] + -4 p[b, d] = 4 × 0 + (-4)× 1 = -4.
Putting (1)-(4) together, we obtain the following list of expected payoffs for the four players
{Ia, Ib, IIc, IId} :
, 32 , -4.
(5) {1, 1, 1, 1} 8, 32
11 5
The computations needed to obtain (5) are laborious. Because there are 4 players and each player has
two possible strategies, there are in total 16 combinations of strategies
(6) {i, j, k, ℓ}, i = 1, 2, j = 1, 2, k = 1, 2, ℓ = 1, 2.
Now we write a Mathematica program to obtain the lists of payoffs for all the possible combinations of
strategies (6).
In[18]:=
u[i_, j_, k_, ℓ_] := {p[c, a] × m[a, c]〚i, k〛 + p[d, a] × m[a, d]〚i, ℓ〛,
p[c, b] × m[b, c]〚j, k〛 + p[d, b] × m[b, d]〚j, ℓ〛,
– p[a, c] × m[a, c]〚i, k〛 – p[b, c] × m[b, c]〚i, k〛,
– p[a, d] × m[a, d]〚i, ℓ〛 – p[b, d] × m[b, d]〚j, ℓ〛}
Using the function just defined, we obtain the following list of expected payoffs for {Ia, Ib, IIc, IId} under
the combination of strategies {1, 1, 1, 1} :
In[19]:=
Out[19]=
u[1, 1, 1, 1]
8,
32
11
,
32
5
, – 4
The list of expected payoffs for {Ia, Ib, IIc, IId} under the combination of strategies {1, 2, 1, 1} :
Games with Incomplete Information_01_02_2022.nb
In[20]:=
Out[20]=
9
u[1, 2, 1, 1]
8, –
12
11
,
32
5
, 0
The list of vectors of payoffs for (Ia, Ib, IIc, IId) that correspond to all possible combinations of strategies {i, j, k, ℓ} :
In[21]:=
Out[21]=
t[0] = Table[{{i, j, k, ℓ}, u[i, j, k, ℓ]}, {i, 1, 2}, {j, 1, 2}, {k, 1, 2}, {ℓ, 1, 2}]
{1, 1, 1, 1}, 8,
32
,
32
, – 4, {1, 1, 1, 2}, 8, –
48
,
32
, 4,
11
5
11
5
52
28
, – 10, – 4, {1, 1, 2, 2}, - 8, , – 10, 4,
{1, 1, 2, 1}, - 8,
11
11
12 32
92 32
,
, 0, {1, 2, 1, 2}, 8, ,
, 8,
{1, 2, 1, 1}, 8, 11
5
11
5
16
64
, – 10, 0, {1, 2, 2, 2}, - 8, , – 10, 8,
{1, 2, 2, 1}, - 8,
11
11
32 54
48 54
,
, – 4, {2, 1, 1, 2}, 0, ,
, 4,
{2, 1, 1, 1}, 0,
11
5
11
5
52
28
, – 14, – 4, {2, 1, 2, 2}, - 4, , – 14, 4,
{2, 1, 2, 1}, - 4,
11
11
12 54
92 54
,
, 0, {2, 2, 1, 2}, 0, ,
, 8,
{2, 2, 1, 1}, 0, 11
5
11
5
16
64
, – 14, 0, {2, 2, 2, 2}, - 4, , – 14, 8
{2, 2, 2, 1}, - 4,
11
11
The list of vectors of payoffs for (Ia, Ib, IIc, IId) that correspond to all possible combinations of strate, 32 , -4 contains
gies {i, j, k, ℓ} has dimension 2 × 2 × 2 × 2. An element of this list, say {1, 1, 1, 1}, 8, 32
11 5
two lists. The first list {1, 1, 1, 1} represents the combination of strategies chosen by the four players
, 32 , -4 gives the expected payoff for the four players under the
Ia, Ib, IIc, IId, and the second list 8, 32
11 5
combination of strategies {1, 1, 1, 1}. To make t[0] more readable, we now suppress the unnecessary
pairs of parentheses.
10
Games with Incomplete Information_01_02_2022.nb
In[22]:=
Out[22]=
t[1] = Flatten[t[0], 3]
{1, 1, 1, 1}, 8,
{1, 1, 2, 1},
{1, 2, 1, 1},
{1, 2, 2, 1},
{2, 1, 1, 1},
{2, 1, 2, 1},
{2, 2, 1, 1},
{2, 2, 2, 1},
32
,
32
, – 4, {1, 1, 1, 2}, 8, –
48
,
32
, 4,
11
5
11
5
52
28
, – 10, – 4, {1, 1, 2, 2}, - 8, , – 10, 4,
- 8,
11
11
12 32
92 32
,
, 0, {1, 2, 1, 2}, 8, ,
, 8,
8, 11
5
11
5
16
64
, – 10, 0, {1, 2, 2, 2}, - 8, , – 10, 8,
- 8,
11
11
32 54
48 54
,
, – 4, {2, 1, 1, 2}, 0, ,
, 4,
0,
11
5
11
5
52
28
, – 14, – 4, {2, 1, 2, 2}, - 4, , – 14, 4,
- 4,
11
11
12 54
92 54
,
, 0, {2, 2, 1, 2}, 0, ,
, 8,
0, 11
5
11
5
16
64
, – 14, 0, {2, 2, 2, 2}, - 4, , – 14, 8
- 4,
11
11
To find the Bayesian Nash equilibrium, we go through the list t[1] one element at a time.
First, consider the list {1, 1, 1, 1}, 8,
32 32
, ,
11 5
-4. If player Ia switches from strategy 1 to strategy 2,
then the new combination of strategies that now applies is {2, 1, 1, 1}, and the list of expected payoffs
for the 4 players is
32 54
{2, 1, 1, 1}, 0, , , -4.
11 5
Furthermore, the expected payoff for player Ia falls from 8 to 0, and this means that player Ia will not
change strategy when players Ib, IIc, IId all choose strategy 1. Thus, strategy 1 is the best response of
player Ia when players Ib, IIc, IId all choose strategy 2.
Similarly, starting from the combination of strategies {1, 1, 1, 1}, if player Ib switches to strategy 2, then
the list of ecpected payoffs becomes
12 32
{1, 2, 1, 1}, 8, – , , 0.
11 5
The expected payoff for player Ib falls from
32
11
to
-12
11
, and this means that player Ib will not deviate,
either.
Also, starting from the combination of strategies {1, 1, 1, 1}, if player IIc switches to strategy 2, then the
list of ecpected payoffs becomes
52
{1, 1, 2, 1}, -8, , -10, -4
11
The expected payoff for player IIc falls from
32
5
to -10, and this means that player IIc will not deviate,
either.
Finally, starting from the combination of strategies {1, 1, 1, 1}, if player IId switches to strategy 2, then
Games with Incomplete Information_01_02_2022.nb
11
the list of ecpected payoffs becomes
48 32
{1, 1, 1, 2}, 8, – , , 4
11 5
The expected payoff for player IId rises from -4 to 4, and this means that player IId will deviate.
We have just shown that the combination of strategies (1, 1, 1, 1} is not a Nash equilibrium.
Next, consider the combination of strategies
48 32
{1, 1, 1, 2}, 8, – , , 4.
11 5
If player Ia chnages strategy, then the new combination of strategies that applies is
, 54
, 4
{2, 1, 1, 2}, 0, – 48
11
5
and this action leads to a fall in his expected payoff from 8 to 0. Thus, player Ia will not deviate.
If player Ib chnages strategy, then the new combination of strategies that applies is
92 32
{1, 2, 1, 2}, 8, – , , 8,
11 5
and this action leads to a fall in his expected payoff from – 48
to – 92
. Thus, player Ib will not deviate.
11
11
If player IIc chnages strategy, then the new combination of strategies that applies is
28
{1, 1, 2, 2}, -8, – , -10, 4
11
and this action leads to a fall in his expected payoff from
32
5
to -10. Thus, player IIc will not deviate.
If player IId chnages strategy, then the new combination of strategies that applies is
32 32
{1, 1, 1, 1}, 8, , , -4
11 5
and this action leads to a fall in his expected payoff from 4 to -4. Thus, player IId will not deviate.
The combination of strategies {1, 1, 1, 2} thus constitutes a Bayesian Nash equilibrium. One can verify
(with a lot of hard work) that the remaining combinations of strategies, just like {1, 1, 1, 1}, are not
Nash equilibria.
2.2. The Bayesian Nash equilibrium of the game
The game has a unique Bayesian Nash equilibrium in pure strategies, which is given by
48 32
{1, 1, 1, 2}, 8, – , , 4.
11 5
There might exist Bayesian Nash equilibria in mixed strategies. It remains to be found.
February 1, 2022
ECO4170
Games with Incomplete Information
1. The problem
John is in love with Mary, and is thinking of asking her out for dinner at a fancy restaurant. The problem
is that John does not know whether Mary likes him or not. From his interactions with her, John thinks
that Mary likes him with probability 0.25, and Mary does not like him with probability 0.75. That is, John
does not know the type of Mary, and we model this situation as a game played by three players: John,
Mary who likes John (Mary 1), and Mary who does not like John (Mary 2). Strictly speaking, either Mary
likes John or Mary does not like John. So, the two possible types of Mary can be interpreted as two
distinct players. When we use the terminology Mary, we mean either Mary 1 or Mary 2.
The payoffs for John and the two types of Mary are represented by the following two matrices:
Mary 1 (0.25)
Accept
John
Invite
Co not
invite
(10, 7)
(3, 4)
Mafry 2 (0.75)
Do not
accept
(2, 2)
(3, 4)
2. The Bayesian Nash equilibrium
Accept
(8, 1)
(3, 4)
Do not
accept
(2, 4)
(3, 4)
2
ECO4170_01 February 2022.nb
Mary 1 (0.25)
Accept
John
Invite
(10, 7)
Co not
invite
(3, 4)
Mafry 2 (0.75)
Do not
accept
Accept
(2, 2)
(3, 4)
Do not
accept
(8, 1)
(3, 4)
(2, 4)
(3, 4)
If Mary likes John, the payoff matrix for Mary is the first matrix
So, for Mary 1, she only looks at the firsdt matrix in making her choice
Observe that Accept is the dominant strategy of Mary 1. So, if Mary likes John
she will accept his invitation
Mary 1 (0.25)
Accept
John
Invite
Co not
invite
(10, 7)
(3, 4)
Mafry 2 (0.75)
Do not
accept
(2, 2)
(3, 4)
Accept
Do not
accept
(8, 1)
(3, 4)
(2, 4)
(3, 4)
If Mary does not like John, her type is Mary 2, and it is the second payoff
matrix that represents her possible payoffs. Thus, for Mary 2, her dominant
strategfy is > .
ECO4170_01 February 2022.nb
Mary 1 (0.25)
Accept
John
Invite
Co not
invite
(10, 7)
(3, 4)
3
Mafry 2 (0.75)
Do not
accept
(2, 2)
(3, 4)
Accept
Do not
accept
(8, 1)
(3, 4)
(2, 4)
(3, 4)
As for John, because he does not know whether he is playing against Mary 1
or Mary 2, he must consider both possibilities.
If John invites Mary to supper, and if it is Mary 1, he know that Mary 1 will play
her dominant strategy, which is > . The payoff for John under this
scenario is 10, and this event has probability 0.25. On the other hand, if it is
Mary 2 who is playing against him, John knows that Mary 2 will refuse. His
payoff under this scenario is 2.
Thus, the expected payoff for John if he invites Mary to supper is
0.25 × 10 + 0.75 × 2 = 4.0
If John does not invite Mary to supper, then his payoff is 3.0. Because × 4 > 3,
John will invite Mary to supper.
The Bayesian Nash equilibrium can be described as follows:
• John invites Mary to supper.
• Mary 1 (Mary who likes John) accepts the invitation.
• Mary 2 (Mary who does not like John) rejects the invitation.
• The expected payoff for John is 4.0
3. The value of perfect information
Now imagine that Mary has a little sister, who knows exactly whether Mary likes John or not. Suppose
that the little sister of Mary says to John that . How much is John willing to pay for this information?
Suppose that the little sister tells John the type of Mary
4
ECO4170_01 February 2022.nb
Mary 1
[0.25]
[0.75]
Mary 2
Because Mary 1 occurs with probability 0.25, there is a 25 × % chance that
the little sister will tell John that Mary likes him. Under this scenario, John
knows that he is playing against Mary 1, and his best choice, given this
information is to invite Mary to dinner. His payoff is then 10.
The other scenario, which occurs with probability 0.75, is that the little sister
tells John that Mary does not like him. Armed with this informatio, the optimal
decision for John is not to invite Mary to supper.
Therefore, if the little sister of Mary tells John the type of Mary, and John acts
according to the information provided, his expected payoff is given by
0.25 × 10 + 0.75 × 3 =
In[2]:=
Out[2]=
4.75
0.25 × 10 + 0.75 × 3
4.75
Under the Basyesian Nash equilibrium, the expected payoff for John is 4.0. With the perfect information, his expected payoff is 4.75. Thus, John is willing to pay up to 4.75 – 4.0 = 0.75 for the information
about the type of Mary, and this amount is called the value of perfect information.
The end of the lecture
February 4, 2022
ECO4170
Adverse Selection_The Market for Lemons
Consider the market for used cars in which two types of used cars are offered for sale: oranges (used
cars that run pretty well) and lemons (used cars that have problems). The owner of a used car obviously
knows the quality of his car. In contrast, a buyer oif a used car does not know the quality of a used car.
This is a case of asymmetric information: the owner of a used car knows the quality of his used car, but
a buyer does not. However, a buyer knows the proportion of lemons offered for sale on the market.
This piece of information can be obtained by reading a market report on used cars.
Suppose that the number of used cars for each type is limited, but there are many buyers of used cars.
Let f denote the fraction of lemons in the market for used cars. Suppose that the owner of an orange
values it at $12500 and the owner of a lemon values it at $3000. As for buyers of used cars, they value an
orange at $16000 and a lemon at $6000. Because 12 500 < 16 000 and 3000 < 6000, there is an incentive
for sellers and buyers of used cars of both type to engage in an exchange.
Now when buyers cannot identify the type of a used car, there is only one market for used cars, i.e., one
cannot observe two markets – one for oranges and one for lemons because if such a situation exists,
owners of lemons will bring their cars to the market for oranges and paass their used cars as oranges,
and no buyer can detect his dishonesty. Furthermore, because only one market exists, there is only one
price of used cars that prevail on the market.
Now if a buyer purchases a used car at price p, there is a chance f that theused car is an orange and a
chance 1 – f that the used car is a lemon. The expected payoff net of the price is then given by
16 000 f + (1 – f ) 6000 – p = 6000 + 10 000 f – p.
Thus, a buyer of used cars only purchase a used car without knowing whether it is an orange or alemon
if
p ≤ 6000 + 10 000 f .
Because there are alimited number of used cars of both types and a large number of buyers, not all
buyers will be able to purchase a used car, and in their efforts to buy a used cars, the buyers will compete against each other and drive the price of a used car to
p = 6000 + 10 000 f .
The curve p : f 6000 + 10 000 f , 0 ≤ f ≤ 1,
is depicted in the following figure
2
ECO4170_04 February 2022.nb
p = 6000 + 10 000 f
16 000
12 500
6000
0.65
1
Observe that the value of f such that 12 500 = 6000 + 10 000 f is f = 0.65.
For f ≥ 0.65, we have p ≥ 12 500. In this case bith oranges and lemons are offerd
for sale, and the price of a used car is higher than or equal to 12 500.
For f < 0.65, we have p = 6000 + 10 000 f < 12 500, and owners of oranges will
not offer their cars for sale on the used car market. As for owners of lemmons,
they will obtain a net surplus by offering to sell their cars on the market.
The buyers of used cars know this and will only be willing to pay at most 6000
for a used car, which is a lemon.
Summary of the analysis
p = 6000 + 10 000 f
16 000
12 500
both lemons and oranges
6000
only lemons
0.65
1
If f < 0.65, only lemons are offered for sale on the used cars market, and
the equilibrium price of a used car (which is a lemon) is 6000.
If f ≥ 0.65, both types of used cars will be offered for sale on the used cars
market at the price of p = 6000.
Equilibrium price of used cars as a function of f
p[f ] = 6000 for f < 0.65 (Only lemmons are offered for sale : Bad cars drive
good cars out of the market)
= 8000 + 10 000 f for f ≥ 0.65 (both oranges and lemons
are offeredfor
on the market)
End of the lecture
08 February 2022
ECO4170
Asymmetric Information
Screening to separate types in the labor market
Consider a labor market in which there are many employers and a limited number of job applicants.
There are two types of job applicants: A (Able) and C (Challenged). The employers are willing to pay
$160000/year for an employee of type A and $60000/year for an employee of type C.
The job applicants can find an alternative job that pays $125000/year to an employee of type A and
$30000/year to an emplyee of type C.
Suppose that the employers in the labor market under consideration want to separate the two types of
job applicants and adopt the following criterion:
• A job applicant who took n hard courses (mathematics, physics, econometrics, mathematiocal eco-
nomics, game theory) will be considered as of type A and offered a job that pays $160000/year.
• A job applicant that took fewer than n hard courses at the university will be consided as an employee
of type C amd offered a job that pays $60000/year.
The question: What is the number of hard courses that separates the two types of
job applicants?
Now the two types of job applicants differ in terms of their tolerance for hard courses: a job applicant perhaps because of his high IQ – can easily take a hard course with little pain, while a job applicant of
type C will have a hard time taking a hard course. To model the tolerance for hard courses, suppose
that the cost – measured in dollars per year – that a job applicant of type A has to incur is $3000/year.
For a job applicant of type C, the number is $15000/year.
We want to find the value of n so that a job applicant of type A will take n hard courses while a job
applicant of type C will take fewer than n hard courses.
The decision of a job applicant of type A
For a job applicant of type A, if he does not take at least n hard courses, then he will be declared an
employee of type C and offered a job that pays $60000/year. His net payoff is then
In[30]:=
Out[30]=
a[0] = 60 000 – 3000 n
60 000 – 3000 n
2
ECO4170_08_02_2022.nb
In[31]:=
Out[31]=
a[0] = a[0] /. n 0
60 000
Now it is obvious that if a job applicant chooses to take fewer than n courses, then it is best for him not
to take any hard courses at all, and obtain the payoff of $60000/year.
On the other hand, if a job applicant of type A takes at least n hard courses, then he will be declared an
employee of type A and offered a job that pays $1260000/year. His net payoff is then given by
In[32]:=
Out[32]=
a[1] = 160 000 – 3000 n
160 000 – 3000 n
Thus, a job applicant of type A will take at least n hard courses at the university if the following condition is satisfied
In[33]:=
Out[33]=
In[34]:=
Out[34]=
In[35]:=
Out[35]=
c[1] = a[1] ≥ a[0] // FullSimplify
3 n ≤ 100
Reduce[c[1], n]
n≤
100
3
% // N
n ≤ 33.3333
Therefore, if n ≤ 33.3333, a job applicant will take at leat n hard courses at the university. The condition
n ≤ 33.3333 is called the incentive compatibility condonstraint for a jobv applicant of type A./
The decision of a job applicant of type C
For a job applicant of type C, if he does not take at least n hard courses at the university, then he will be
offered a job that pays $60000/year. As in the case of as job applicant of type A, a job applicant who
decides to take fewer than hard courses will not take any hard courses at all, and his net payoff is
In[36]:=
60 000
Out[36]=
60 000
On the other hand, if a job applicant of type C takes at least n hard courses at the university, then his
net payoff is given by (because he will be declared an employee of type A according to the criterion and
offered a job that pays $160000/year)
In[37]:=
160 000 – 15 000 n
Out[37]=
160 000 – 15 000 n
Thus, a job applicant of type C will not take any hard courses if the following incentive compatibality
constraint is satisfied:
ECO4170_08_02_2022.nb
In[38]:=
60 000 ≥ 160 000 – 15 000 n
Out[38]=
60 000 ≥ 160 000 – 15 000 n
In[39]:=
Out[39]=
In[40]:=
Out[40]=
In[41]:=
Out[41]=
3
% // FullSimplify
3 n ≥ 20
Reduce[%, n]
n≥
20
3
% // N
n ≥ 6.66667
Therefore, if 6.67 ≤ n ≤ 33.33, a job applicant of type A will take at least n hard
courses while a job applicant of type C will not take any hard courses, The
number n = 7 will then separate the two types of job applicant. It is also Pareto
efficient because it gives the job applicants the highest net payoff will
maintaining the zero profit condition of the employers and the constant net
payoff for the job applicants of type C.
The participation constraint
If a job applicant of type A accepts the alternative job, then his net patoff is
$125000/year. On the other hand, if he takes at leat n -= 7 hard courses, then his
net payoff is
160 000 – 3000 × 7 = $139000 year > $125000 year.
Thus, if n = 7, a job applicant of type A will take n hard courses and obtain a job
that pays $160000/year, and the condition 139 000 > 125 000 is known as the
participation constraint for a job applicant of type C.
Similarly, for a job applicant of type C, he will not take any hard courses at the
university and will be offered a job that pays $60000/year, the annual salary that
is higher than $30000/year, the salary offered by the alternative job. Hence the
participation constraint for a job applicant of type C is also satisfied.
Conclusion: n = 7 is the number of hard courses that separates the two types of
job applicants.
The end of the lecture
11 February 2022
ECO4170
Asymmetric Information: A Highway Procurement Problem
A state government is carrying out a highway procurement project. To simplify the problem, assume
that there is a single construction firm that bids for the project.
The decision variable for the state government is n, the number of lanes of the proposed new highway.
The quality of the land on which the new highway is to be built is either good or poor. The state government does not know the quality of the land, and believes that it is good with probability 23 and poor
with probability 13 . The construction firm, because of its expertise, knows exactly the quality of the soil.
It is more costly to build a highway on land with poor quality, and the state government has to pay the
construction firm its opportunity cost. Suppose that the cost of building one lane on good quality soil is
$3B and the cost of building one lane on poor quality is $5B.
The gross social welfare obtained from a highway of n lanes is given by
In[1]:=
Out[1]=
u = 15 n 15 n –
1
2
n2
n2
2
1. The Base Scenario
Suppose that the state government has perfect information about the quality of the soil. If the quality
of the soil is good, it solves the following maximization problem:
maxn 15 n – 12 n2 – 3 n.
In[2]:=
Out[2]=
Maximize15 n –
1
2
n2 – 3 n, n
{72, {n 12}}
Thus, if the quality of the land is good, the state government will propose a 12-lane highway and pays
the construction firm 12 × 3 = $36 Billion. The social welfare net of construction cost is 72.
K
r
Aα
1
-1+α
If the quality of the land is poor, and the state government knows this, it solves the following maximization problem:
2
ECO4170_11_02_2022.nb
maxn 15 n – 12 n2 – 5 n.
In[3]:=
Out[3]=
Maximize15 n –
1
2
n2 – 5 n, n
{50, {n 10}}
Thus, if the state government knows that the quality of the soil is poor, it will propose a 10-lane highway and pays the construction firm $50B. The net social welfare is 50.
2. The Naive Contracts
Suppose that the state government in its naive manner offers two contracts
(nL , rl ) = (12, 36), (nH , rH ) = (10, 50), hoping that the construction firm will choose the former contract if
the soil quality is good and the latter contract if the soil quality is poor. Will the expectations of the
state government be realized? The answer is no, as ahown by the following reasoning.
Suppose that the soil quality is good. If the construction firm chooses the contract (nL , rL ) = (12, 36), it
profit is
In[4]:=
Out[4]=
36 – 3 × 12
0
On the other hand if the construction firm chooses the contract (nH , rH ) = (10, 50), it is paid $50B for
building a 10-lane highway at a cost of $3B/lane because the soil quality is good. The profit it earns by
choosing the contract (nH , rH ) = (10, 50) is given by
In[5]:=
Out[5]=
50 – 3 × 10
20
which is positive. Hence the naive approach does not inbduce the construction firm to behave in a
socially optimal manner.
3. The Optimal Contract under Asymmetric Information
Suppose that the state government offers the construction firm two possible contracts: (nL , rL ), (nH , rH ).
Under the former contract, the construction firm is required to build an nL – lane highway and is paid
rL . Under the latter contract, the construction firm is required to build an nH – lane highway and is paid
rH . Also, suppose that the two contracts are structured so that the construction firm will be induced to
choose the contract (nL , rL ) if the quality of the soil is good and the contract (nH , rH ) if the soil quality is
poor.
Now if the soil quality is good, the construction firm will choose the contract (nL , rL ) if the profit made
form this contract is at least equal to the profit made from choosing the contract (nH , rH ). In this case,
the following incentive compatibility constraint must be satisfied:
In[6]:=
Out[6]=
g[1] = r[L] – 3 n[L] ≥ r[H] – 3 n[H]
– 3 n[L] + r[L] ≥ – 3 n[H] + r[H]
ECO4170_11_02_2022.nb
3
On the other hand, if the soil quality is poor, the following incentive compatibility constraint must hold
in order for the construction firm to choose the contract (nH , rH ) :
In[7]:=
Out[7]=
g[2] = r[H] – 5 n[H] ≥ r[L] – 5 n[L]
– 5 n[H] + r[H] ≥ – 5 n[L] + r[L]
When the construction firm to choose the contract (nH , rH ), the following participation constraint must
be satisfied:
In[8]:=
Out[8]=
g[3] = r[H] – 5 n[H] ≥ 0
– 5 n[H] + r[H] ≥ 0
When the construction firm to choose the contract (nL , rL ), the following participation constraint must
be satisfied:
In[9]:=
Out[9]=
g[4] = r[L] – 3 n[L] ≥ 0
– 3 n[L] + r[L] ≥ 0
When the four constraints g[1], g[2], g[3], g[4] are satisfied, the construction firm will choose the contract (nL , rL ) if the soil quality is good, and this event occurs with probability 23 .
When the four constraints g[1], g[2], g[3], g[4] are satisfied, the construction firm will choose the contract (nH , rH ) if the soil quality is poor, and this event occurs with probability 13 .
Thus, if the state government offers two contracts (nL , rL ) and (nH , rH ) that satisfy the four constraints
g[1], g[2], g[3], g[4], the expected net social welfare is given by
In[10]:=
g[0] =
1
Out[10]=
In[11]:=
3
In[12]:=
Out[12]=
In[13]:=
Out[13]=
In[14]:=
Out[14]=
3
1
15 n[L] –
15 n[H] –
n[H]2
2
2
n[L]2 – r[L] +
– r[H] +
2
3
1
3
15 n[H] –
15 n[L] –
n[L]2
2
1
2
n[H]2 – r[H]
– r[L]
g[0] /. {n[L] nL , r[L] rL , n[H] nH , r[H] rH }
1
Out[11]=
2
3
15 nH –
n2H
2
– rH +
2
3
15 nL –
n2L
2
– rL
g[1] /. {n[L] nL , r[L] rL , n[H] nH , r[H] rH }
– 3 nL + rL ≥ – 3 nH + rH
g[2] /. {n[L] nL , r[L] rL , n[H] nH , r[H] rH }
– 5 nH + rH ≥ – 5 nL + rL
g[3] /. {n[L] nL , r[L] rL , n[H] nH , r[H] rH }
– 5 nH + rH ≥ 0
4
ECO4170_11_02_2022.nb
In[15]:=
Out[15]=
g[4] /. {n[L] nL , r[L] rL , n[H] nH , r[H] rH }
– 3 nL + rL ≥ 0
The optimal contracts under asymmetric information can be stated formally as follows:
max((nL ,rL ),(nH ,rH )) 13 15 nH –
n2H
2
– rH +
2
3
15 nL –
n2L
2
– rL
subject to
-3 nL + rL ≥ -3 nH + rH ,
-5 nH + rH ≥ -5 nL + rL ,
-5 nH + rH ≥ 0,
-3 nL + rL ≥ 0.
Here is how the solution of the preceding constrained optimization problem is obtained by
Mathematica:
In[16]:=
Out[16]=
Maximize[{g[0], g[1], g[2], g[3], g[4]}, {n[H], r[H], n[L], r[L]}]
{54, {n[H] 6, r[H] 30, n[L] 12, r[L] 48}}
Under asymmetric information, the state government offers two contracts:
(nL , rL ) = (12, 48), (nH , rH ) = (6, 30).
The construction firm will choose the first contract if the soil quality is good and the second contract if
the soil quality is poor. The expected net social welfare is 54.
Low cost
High cost
(nL , rL )
(nH , rH )
Perfect Information
12 36
10 50
Asymmetric Information
12 48
6
30
Observe that for the low cost case, the contract requires the same number of lanes for both the perfect
information and the asymmetric information scenario, but the payment is higher under the asymmetric
information scenario. As for the high cost scenario, the number of lanes and the payment are both
lower in the case of asymmetric information than in the case of perfect information to discourage the
construction firm to choose the contract intended for the high cost case when the soil quality is good.
The End of the Lecture
15 February 2022
ECO4170
Asymmetric Information: Price Discrimination to Screen
Consumers
An airline serves a route between two cities. There are two types of passengers: business travellers and
tourists. Business travellers account for 30% of the clientele and tourists 70%. The following table
presents the data obtained by the airline
Types of service
Airline’s cost
Economy class
First class
100
150
Reservation price
Tourist Business traveller
140
225
175
300
Potential profit
Tourist Business traveller
40
125
25
150
The reference scenario
Suppose that the airline can identifier the type of a flyer when he comes to buy a ticket and that the
airline is free to sell the flyer any type of ticket at any price, and such a practice is not prohibited by the
law. Presumably, the airline can identify the type – tourist or business – of a flyer by observing the way
he is dressed: a business traveller wears a suit while the clothes worn by a tourist are casual.
Now when a tourist comes to buy a tricket, the ticket agent can sell him either an economy class ticket
or a first-class ticket. If the ticket agent decides to sell the flyer an economy-class ticket, he can sell it at
the reservation price of the flyer, and this action brings the airline the following profit
In[17]:=
Out[17]=
140 – 100
40
On the other hand, if the ticket agent decides to sell the flyer a first-class ticket, the price he can charge
the flyer is the latter’s willingness to pay 0f 175, and this action brings the following profit
In[18]:=
Out[18]=
175 – 150
25
Because 25125, it is better for the airline to sell a first class ticket to a business traveller.
Thus, when the airline can identify the type of a flyer and is allowed to practice price discrimination,
then it will sell an economy-class ticket to a tourist at the price of 140 and a first-class ticket to a business traveller at the price of 300. The profits made by the airline are
In[21]:=
Out[21]=
70 (140 – 100) + 30 (300 – 150)
7300
Pricing under asymmetric information
Now consider the realistic case in which the airline cannot identify the type of a flyer. First, consider the
naive pricing strategy: Offer two types of tickets, economy-class tickets at price 140 and first-class
tickets at price 300, hoping that a tourist will buy an economy-clas ticket and a business traveller a firstclass ticket.
For a tourist, if he buys an economy-class ticket, his surplus is
In[22]:=
Out[22]=
140 – 140
0
On the other hand, if a tourist buys a first-class ticket, his surplus is
In[23]:=
Out[23]=
175 – 300
– 125
Thus, the naive pricing strategy induces a tourist to buy an economy-class ticket. What about a business traveller?
If a business traveller buys a first-class ticket at the price 300, his surplus is
In[24]:=
Out[24]=
300 – 300
0
On the other hand, if the business traveller buys an economy-class ticket, his utility is 225 and he pays
140 for a surplus of
ECO4170_15_02_2022.nb
In[25]:=
Out[25]=
3
225 – 140
85
Thus, if the airline carries out the naive pricing strategy, all the flyers – tourists and business travellers will opt for the economy-class tickets, and the profits made by the airline are
In[26]:=
Out[26]=
100 (140 – 100)
4000
Because 4000 is much less than 7300, the naive pricing strategy is a disastrous strategy.
Now suppose that the airline adopts a pricing strategy that can separate types. Let p[0] denote the
price of an economy-class ticket and p[1] the price of a first-class ticket. If p[0] is intended to capture
the tourist type, the airline certainly can set
In[27]:=
Out[27]=
p[0] = 140
140
If a business traveller chooses to buy an economy-class ticket, he pays p[0] to obtain a surplus of
In[28]:=
Out[28]=
225 – p[0]
85
On the other hand, if the business traveller buys a first-class ticket at price p[1], his surplus is
In[29]:=
300 – p[1]
Out[29]=
300 – p[1]
Therefore, a business traveller will buy a first-class ticket if
In[30]:=
300 – p[1] ≥ 85
Out[30]=
300 – p[1] ≥ 85
In[31]:=
Out[31]=
% // FullSimplify
p[1] ≤ 215
If the airline adopts the pricing strategy p[0]=140,p[1]=215, such a strategy will separate types: a tourist
will opt for an economy-class ticket while a business traveller will purchase a first-class ticket. The
prive p[1] = 215 is the maximum price of a ticket that the airline can charge a business traveller while
separating the two types of passengers. The profits earned by the airline are then given by
In[32]:=
Out[32]=
30 (215 – 150) + 70 (140 – 100)
4750
The profits – for the case of asymmetric information – earned by the pricing strategy that separates
types are lower than the profits under perfect information (7300), but higher than the profits obtained
from the strategy that separates types (4000.
4
ECO4170_15_02_2022.nb
End of the lecture
In[33]:=
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