18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY
Profit –maximizing insurance contract with adverse selection ;
Imagine a profit maximizing risk – neutral insurance company with monopoly power.
An individual which has uncertain amount of money to spend.
There is a probability of πi the individual have endowment Y2 and a probability of
( 1- πi ) that she has endowment Y1 we know that Y1 > Y2 .
i=L,H which stands for low and high risk. πH > πL and the individual is strictly riskaverse and knows that she is high or low risk. Insurance company is risk neutral.
The individual’s utility function defined on the amount of money he consumes .
The individual is risk-averse and he is wiling to give up some of his endowment ( P )
in high endowment state of nature in return for more ( B – P ) in the low
endowment state of nature.
The insurance contract specifies two numbers y1 and y2 where y1 = Y1 – P , where
P is insurance premium and y2 = Y2 + B – P , where B is a benefit paid by insurance
company when endowment is only Y2 and P is the premium paid to the company.
yi = the total resource left to the consumer in state of i ( i = 1,2 ) . The amount
which is guaranteed by insurance company by a contract . The risk of having Y1 in
good year or Y2 in bad year will be transferred to insurance company by the
contract.
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
the insurance company has profit Y1 - y1 in high endowment state of nature and
Y2 – y2 in low endowment state of nature. The insurance could guess that the
individual is high risk with the probability of ρ and low risk with probability of 1 - ρ.
What source of contract will the insurance company offer to this individual to make
the largest possible profit.
We can draw the indifference curve for the consumer in the ( y1 , y2 ) space with the
form of ( 1- πi ) U(y1 ) + πi U(y2 ) = k . Πi is the probability of having low level of
endowment when individual’s risk is i= L , H . We could find y2 as a function of y1
from the indifference curve function as y2 (y1 ) . Let v be the inverse function of U ;
y (y ) V(
2
1
k  (1   i )U ( y )
i
1
)
taking derivative with respect to y1 we will get the slope of the indifference
curve as ;
2
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
k  (1   i )U ( y ) 1  
dy2
i
1
 V '(
)(
)U '( y1 )
or ;
dy1
i
i
k  (1  i )U ( y )
1 i
dy2
1
for
 V '(U ( y2 ))(
)U '( y1 )
y2  V (
)
dy1
i
i

1- if y2 (y1 ) = y1 , the slope of indifference curve through ( y1 , y2 ) equals to
–(1-πi )/πi , since V’(U(y2)) = 1/U’(y2) , and U’(y2) = U’(y1)
y1 = y2
y2
When πi = 1/2 The
indifference curve is symmetric
with respect to the 450 line .
Initial endowment point
Y2
450
3
Y1
Slope = - ( 1- πi ) / Πi
y1
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
2- The indifference curve of low risk consumer is steeper than the indifference curve
of high risk consumer . dy2 /dy1 decrease when πi increase [( 1- πi )/ πi decrease].
y1 = y2
y2
Single crossing indifference
curve is assumed
Low risk
Y2
High risk
Y1
4
y1
If we start from (Y1 , Y2 ) point and
decrease Y1 a little bit , it take a smaller
increase in Y2 to compensate the high
risk consumer than to compensate the
low risk consumer. The high risk
consumer has a greater chance of
having the low endowment state prevail
, hence additional income in that state is
worth more to him so he needs a
smaller increase in y2 compared to low
risk .
THE REVELATION PRICIPLE AND
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
Now consider the iso-expected profit linePARTY
for the risk neutral firm is ;
( 1- π ) ( Y1 - y1 ) + π ( Y2 – y2 ) where ;
Yi = endowment of the consumer at state of i
yi = The income which will be left ( guaranteed) for individual in state of i .
(Yi - yi ) = p if i=1
=the profit left for the firm in state of 1
(Yi - yi ) = P-B if i =2
= the profit left for the firm in state of 2
π is the firm’s overall assessment as to how likely it is that a consumer will have a low level of
endowment. π could be either high (πH )or low (πL )
Slope of the iso-profit curve = -(1- π )/π .
Suppose that the firm knows that it is dealing with a high risk consumer, that is π=πH . What
contract does the firm will offer to the consumer to maximize its profit. The consumer is not
going to accept any contract which leaves him a utility less than before ( at initial endowment
point). So the firm is constrained to offer the individual a contract should be in the shaded area
in the figure below . The firm will also want to maximize its profit . So the optimal point is point
A where slope of iso-profit line (-(1- π )/π ) and the lower bond indifference curve are the
same . We know that when slope of indifference curve is equal -(1-πH )/πH we will have y1 = y2
If we repeat the same process for low risk consumer π=πL , we will get a point like point B
which is higher on the 450 line . This means that the low risk individual needs to be guaranteed
a higher level of income , or in other words he likes to buy a cheaper contract .
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Optimal contract for low risk individual
Y1 = Y2
y2
C
B
A
Y2
450
Y1
consumer
y1
Iso-profit line for insurance company
when π=πL .
firm
Initial endowment point
Iso-profit line for insurance company
when π=πH . ( y0 = πH y2 + (1- πH )y1 )
18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
High risk indifference curve
Optimal contract for high risk individual
6
Low risk indifference curve
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Lemma . Suppose the firm offers a contract ( y1 , y2 ) that is accepted by some type
of consumer . If we change that contract in a way that keeps the utility of this type
of consumer constant and we move in the direction full insurance , y1 = y2 , then the
expected profits of the insurance firm ( from this type of consumer accepting the
contract ) increase . we can prove this by comparing points A and C on the
indifference curve in the previous page
. Now suppose that the insurance company can not tell whether the consumer is high
risk or low risk. If the firm offer the either of optimal contracts ( contracts A or B in
page 6 ) for both low and high risk people and let them to choose the proper contract
, either both will choose the contact designed for low risk if A is offered , or only
high risk choose B if B is offered. What firm can do ? The firm should take into
account the incentives of the high and low risk people as showed in the next page.
As a solution compare two contracts A and B at the figure in the next page ; .
The insurance firm could offer these two contracts to both low and high risk
individuals taking in to account the utility levels indicated by the two indifference
curves. the low risk will chose contract A ,because he has higher utility in A
compared to B . High risk will choose contract B, since he has higher utility in B
compared to A . In this example The high risk consumer will get full insurance , and
low risk a partial insurance. This is optimal for the insurance company .
18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
Low risk
consumer
Y2
Y1 = Y2
High risk
consumer
Contract chosen by high risk
consumer . UA > UB > UC
A
y2
Contract chosen by low risk
consumer UB > UA > UC
H
y2L
B
Y2
C
y1H
y1L Y1
Y1
Initial endowment point .
No insurance point .
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
We can solve the firm’s analttical problem as follows ;
Firm offer two contracts ; ( y1H , y2H ) for high risk consumer and ( y1L , y2L ) for low
risk consumer . If we assume the consumers chooses the contracts designed for them
correctly , then the expected profit of the firm is equal to
E(П) = ρ[( 1 – πH )(Y1 - y1H ) + πH (Y2 - y2H )] + (1- ρ) [( 1 – πL )(Y1 - y1L )+ π L (Y2 - y2L )]
The maximization problem of the firm ;
Max E(П)
S.T.(1- πH)U(y1H) + πH U(y2H ) ≥ (1- πH)U(Y1) + πH U(Y2 )
( P H)
. (1- πH)U(y1H) + πH U(y2H ) ≥ (1- πH)U( y1L) + πH U( y2L )
(IH)
(1- πL)U( y1L) + πL U( y2L ) ≥ (1- πL)U(Y1) + πL U(Y2 )
(P L)
H
(1- πL)U( y1L) + πL U( y2L ) ≥ (1- πL)U(y1H) + πL U(y2H)
(IL)
The firm offers different contracts for each type of consumers , why can’t the firm
pool the two types of consumers and offer a single contract . Or is it possible that the
firm will wish to offer a contract that only one type and not the other accept. .
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
Whatever scheme the insurance company plans to adopt, it will end with the
consumer taking some contract or another. A contract will be taken by the low risk
consumer and a contract will be taken by the high risk people . These contracts could
be the same , they could be different , and either or both could be the endowment
point
If (y1H , y2H ) = ( Y1 , Y2 ) , the firm will offer the high risk the original
endowment and leaving the high risk be like the case of not being insured.
If (y1L , y2 L ) = ( Y1 , Y2 ) , the firm will offer the low risk the original
endowment and leaving the low risk be like the case of not being insured.
if y1H = y1L , y2H = y2 L , then the two types are pooled .
So far we have assumed that the insurance firm offers two types of contracts ;
If the consumer anticipates all the things the firm will do in marketing insurance , and
respond to it rationally the menu of two contracts is without loss of generality.
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MECHANISM DESIGN
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
We assume that the consumer can work
through the choices she will be offered , and
respond optimally. Given this, the two choice menu scheme will give the optimal
scheme for the insurance firm .
Preposition 1. At the final solution of this problem , the constraints ( PL) and (IH)
will be binding, and the high-risk contract will be a full-insurance contract . Or
Low risk contract curve should lie on the intersection of low and high risk consumer curve. High
risk contract curve should lie on the intersection of 45 degree line and high risk consumer
curve
y2
High risk
contract
This can be proved taking into account
the following steps ;
High risk
consumer
Low risk
consumer
Step 1 .The high risk contract (y1H ,
y2H) must lie in the shaded area
Low risk
contract
y1
11
compared to low risk contract to fulfill
(IH) and (IL) constraint. High risk
consumer will not accept the low risk
contract and low risk consumer will not
accept the high risk contract since they
loose utility .
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MECHANISM DESIGN
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y2
18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
High risk
contract
High risk consumer
A
Low risk consumer
B
C
v
y1
Low risk
contract
12
Step 2 ; the constraint (PL) binds . In
particular , (Y1,Y2 ) must lie along the
low risk indifference curve through (y1L
,y2L) and to the right of ( y1L ,y2L )
To see this consider the figure . We
can see that the (Y1,Y2 ) point should
lie in the shaded area to satisfy the
(PH) and (PL) constraint . We should
also note that (Y1,Y2 ) should lie along
the boundary between shaded and unshaded area ( ABC border). Otherwise
we could increase the firm’s profit by
lowering the payments ( lowering the
utility level of individuals and
increasing the iso-profit line of
company ) and still satisfying the (IH)
and (IL) constraint .
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MECHANISM DESIGN
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
Now suppose that (Y1,Y2 ) lies to the left
of (y1H , y2H) as shown in the figure below ,
iso-profit curve passing through high risk-contract is lower than the one passing
through no-contract situation. All the constraint are satisfied if we move (y1H , y2H)
point to(Y1,Y2 ).Expected profit will increase by the consequence of the lemma(since
Y1 > Y2 ) and utility remain constant.
y2
Y1 = Y2
Low risk consumer
High risk consumer
High risk contract
450
y1
Initial endowment
points . No contract
situation
13
Low risk contract
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MECHANISM DESIGN
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
Now suppose that (Y1,Y2 ) lies at or to the
right of (y1H , y2H) and the left of ( y1L , y2L ) as
shown in the figure below. all the constraints are satisfied if we move ( y1L , y2L ) point
towards (Y1,Y2 ) and firm’s profit will increase since Y1 > Y2 , and the shift involves
movement along the low risk indifference curve in the direction of full insurance which will
increase the expected profit by the lemma mentioned. Hence (Y1,Y2 ) should lie on the low
indifference contract curve and through the ( y1L , y2L ) and to the right of it .
y2
Y1 = Y2
Low risk consumer
High risk consumer
High risk contract
450
y1
Initial endowment
points . No contract
situation
14
Low risk contract
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
Step 3- the constraint (IH) should bind . By steps 1 and 2 we can see that the figure
should be like the one below. But in this figure the (IH) constraint is not binding .
Then the movement of high-risk contract down and to the left and staying within the
wedge satisfy all the constraints and increase the expected profit. The key point is
the positioning of no contract situation which implies as long as high risk contract
stays within the edge the (PH) constraint is satisfied.
y2
High risk consumer
Low risk consumer
Low risk contract
High risk contract
Initial endowment
points . No contract
situation
y1
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
Step 4 - the contract ( y1L , y2L ) does not involve over insurance that is y1L ≤ y2L .
To see this consider figure below ; if the low risk contract is moved along the low
risk indifference curve towards the full insurance , expected profit will increase by
the lemma .
y2
High risk consumer
High risk contract
Low risk contract
Low risk consumer
y1
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
H
H
Location of High risk contract
Step 5 – the high risk contract (y1 , y2 ) involves full insurance. By steps 1
through 4 the figure should like the figure below , where high risk contract lies along
the high risk indifference curve and to the left of low risk contract and the full
insurance line lies on or to the left of low risk contract point . All points along the
high risk indifference curve and to the left of low risk contract maintain feasibility
and by the lemma full insurance point will maximizes the profit along the heavy
marked line on the high indifference curve .
17
y2
High risk consumer
Low risk consumer
Maximum expected
profit for insurance
company for for high
risk contract
Low risk contract
y1
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
Consider figure below ; the high risk contract curve must involve full insurance
contract and must lie between the high and low indifference curve and low
indifference curve should pass through the endowment point.
Once the location of high risk contract is selected (point A ) , we know the (IH) and
(IL) constraints should be binding , so the location of low risk contract ( point C ) is
forced to be at the intersection of high risk indifference curve through high risk
contract and low risk indifference curve which is passing through the no contract
position(point D )
y2
High risk consumer
A
B
C
Low risk contract
D
Y2
Low risk consumer
18
High risk contract
E
We can see that this figure
satisfies the preposition 1 on
page 11 [ PL , IH is binding
and High risk contract is a full
contract ].
Y1
Initial endowment points
. No contract situation
y1
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
1- when high risk contract is at the first best location or highest expected profit for
the firm and full insurance for the high risk consumer, (move from A to B ), then by
the reasoning of steps 1 to 5 the low risk contract should be located on the no
insurance or initial endowment point ( point C moves to point D ). So, when there is
first best insurance for the high-risk consumer there will be no insurance for the
low risk consumer.
2- if we put the low risk contract at the first best location (from firm’s point of view)
( on the Y1 = Y2 line ) , we will see that low and high risk contract is the same ( points
C and A move to point E) . This correspond to the full insurance for both types at the
term that are best for the low risk contract ( from firm’s perspective).
3- other wise we will get full insurance for the high risk and partial insurance for the
low risk.
4- taking into account ;
E(П) = ρ[( 1 – πH )(Y1 - y1H ) + πH (Y2 - y2H )] + (1- ρ) [( 1 – πL )(Y1 - y1L )+ πL (Y2 - y2L )]
a - When ρ=1 , the consumer is sure that the consumer is high-risk, it is obvious
that the optimum is to provide the first best insurance for the high risk consumer.
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
b – when ρ=0 , it is clear that the optimum is to provide the first-best insurance for
the low-risk consumer .
C- it can be shown that as ρ decreases from 1 to zero , then contract moves up the
full insurance line from point B to point E .
d- if we know that for any ρ>0 , the first-best contract for the low risk indifference
curve is not the profit maximizing contract for the firm, we know that pooling does
not occur.
Variation, Extension and Complication ;
1- the party is presented with a menu of options, one option for every possible
piece of private information that the party might possess. In the literature, you will
find reference to various types of party instead of two types of low and high risk.
Each piece of private information that the party possess correspond to a different
type.( married & single , employed & unemployed , good or bad track record)
2- the criterion used in the text was maximization of profit while other criterion like
maximization of the utilities of shareholders or maximizing some sort of weighted
some of consumer’s expected utilities might be taken into account.
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
3- another variation concerns problems
that combine moral hazard with adverse
selection . For example , in the life insurance , the insuree might take better or worse
care of herself, affecting her lifetime period and the insurance company can not
enforceable include in an insurance policy rule about the insuree’s diet or exercise
regime. The insurance company would then wish to design a menu of contract that
not only caused the insuree to identify herself on the base of her health, but also
provided her with an incentive to take care of herself.
4- we might imagine that the consumer purchases partial insurance from a number
of different firms, getting terms better than those she can get by buying complete
insurance from one firm.
5- the insurance company deal not with a single individual but with many individuals
of varying types. So we can suppose that the insurance company is extending the
insurance to many consumers of varying types. The question is what sort of
contracts the firm should offer to maximize his expected profit. To answer the
question we need to ask two more question :
a- can the firm tell the level of the risk of each consumer( how risky is he)
b– and if it can is it able to act on that information in the sense that it can offer a
different contract to each individual according to the individual’s type.
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18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL
PARTY
6- in case of insurance contracts it seems reasonable to assume that the choice of
contract from a menu by one insuree is unaffected by choices made by other
insurees. But if we relax this assumption , other methods should be taken into
account.
18.2 – Optimal contract for interacting parties
in this section we try to design optimal contracts for parties whose action and choices
interact.
Toy Problem ;
Imagine that some party( government) must procure 100 units of a particular item
(airplane). Two firms could supply these items i= 1,2 . MCi is fixed for each firm ,
i= 1,2 . C1 and C2 are independent from each other . While firm I knows its cost ,
neither the other firm nor the third party ( government ) does not know about it but
they know that C1 is either 1 or 2 with probability ½ .
It is obvious that If the government knew the cost of two firms , its procurement
problem would be easily solved. The government buys from the cheaper firm ..
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
Since the government does not know the unit cost of the firm it might try the
following scheme :
The firm voluntarily are asked to reveal their cost . If the revealed costs are the same ;
the government will split the order between them . If one names the lower the
government buys from him. If the firms are restricted to name their costs either 1 or
2 , both want to announce their marginal cost equal to 2 regardless of their costs1
Since each of them could recieve a higher price if name its cost equal to MC=2. So the
cost of government will be 200 ( 100  2 = 200 ) .
Clearly the government needs to provide firms with an incentive for revealing that
their cost is 1 when it is 1 . Consider the following scheme ;
a - If both firms name their MC equal to 2 , the order is split and the government
pays 2 per unit . The cost of government is 200
b - If both firms name their cost equal to 1 , the order is split and government pays an
amount equal to 2 > X > 1 per unit to induce them to announce MC=1 . The cost of
government is 100 x .
c- If one names 1 and the other names 2 all the order goes to the one with MC=1
and the government pays an amount equal to 2 > y > 1 .
What will be the amount of X and Y to force the firms truthfully reveal their
costs.
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23
Assume that firm 1 believes that firm 2 will truthfully reveal its cost . If the firm 1
cost is equal to 1 ( MC1 =1 ) , then ;
A- firm 1 can name its cost equal to 2 and get profit equal to 50 [ =( 2-1)(100/2) ]
with probability 1/2 , if other firm’s cost is equal to 2 and zero if it is equal to 1 .
B –Or firm 1 can name its cost equal to 1 and have profits of 50( x -1 ) with
probability 1/2 if the other firm’s cost is 1 ,
C- or firm 1 can name its cost equal to 1 and have profits of 100( y -1 ) with
probability 1/2 if the other firms cost is 2 .
Firm’s 1 truthfully reveal its cost when MC1 =1 if
Expected Profit when he claims MC=1 ≥ Expected profit when he claims MC=2
(1/2) (50)(x -1 ) + (1/2) (100)( Y -1 ) ≥ (1/2) 50 (2-1) this constraint is true for
the second consumer too.
by nature of the problem, It is a Nash equilibrium between two firms to reveal their
true cost . Let us proceed under the assumption that the two firms reveal their costs ;
If MC1 =1 with P =1/2 , then → MC2 = 1 with p= 1/2 or MC2 = 2 with P= ½ .
If MC1 =2 with P =1/2 , then → MC2 = 1 with p= 1/2 orMC2 = 2 with P= ½ .
Then , the government pays x per unit with probability 1/4 and y per unit with
probability 1/2 and pays 2 with probability 1/4 .
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The government objective function is
Min 100 ( (1/4)x + (1/2)y + (1/4)2 ) =25 x + 50 y + 50 government expected cost
S.T. 100( y -1 ) + 50( x -1 ) ≥ 50 or 25 x + 50 y ≥ 100
truth telling constraint .
It is evident that any selection of x and y ( for example x=2, y=1, are the solutions)that satisfy the constraint
with equality gives the minimum expected cost to government and that expected cost is 150
Suppose the government chooses
x=2 ,y=1 . Is truth telling the only Nash equilibrium
Suppose that firm 2 reports cost of 2 regardless of what its cost are .
A-If firm 1 reports a cost of 2 when its cost are 1 , then it is sure that to make 50 . Since it
gets half of the order for sure ( 50 ) and it is paid 2 per unit when its cost is 1.
B-If firm 1 reports costs of 1 when its cost is 1 , it gets the entire order ( 100 )but gets
nothing since it will be paid 1 dollars ( y=1) and its cost is also 1 dollar.
C-When firm’s 1 cost is 2 , it will surely reports cost of 2 .and receive 2 per unit and gets
half of the order and make no profit.
Hence if firm 2 always reports cost of 2 , then it is best reponse for firm one always report cost
of 2 . also if firm 1 always reports cost of 2 , then it is best repose for firm 1 always report cost
of 2 . So it is a Nash equilibrium for both firms to report costs of 2 , and each will have an
expected profit of 25 . Because with p=1/2 firm 1 will have MC=1, and will be paid 2 and
with p=1/2 firm 1 will have MC=2 and will be paid 2 , so firm’s 1 expected profit is equal to
1/2 (100/2)(2-1) + 1/2 (100/2) (2-2)= 25. the same is true for firm 2.
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25
By comparison at the Nash equilibrium where both firms always tell the truth , each
has expected profit of 12.5 . .
If we believe that firms will find their way to Nash equilibrium that are best for them,
the government pays 2 per unit ( x=2) . Then x=2 , y=1 , does not seem so good for
the government. Since the government should pay 2 per unit and the cost is 200 .
To avoid this , we clearly want to make y as large as possible because in this case
the one with lower cost has more incentive to announce his cost truly and 100
units cost for the government less than 2.
Consider the other end that is x=0 , and y=2 ; this means that if both are
reporting low cost each should provide half of the order free. Now suppose that firm 2
always names cost of 2 regardless of its cost . Then firm’s 1 will announce 1 when
its costs are 1 . Since it gets 100 by announcing MC=1 ( y=2 , 100 ( 2-1) ), and 50 by
announcing MC=2 ( 50 (2-1) ) . But now suppose that one firm always adopt the
strategy of reporting MC=1 regardless of the other firm announcement. The other
firm can never do better than announcing MC=2 . Since if it announce MC=1 , it
should provide half the order free. So this is another Nash equilibrium. Since when
any of the firms announce MC=1, the other one will announce MC=2 and the one
with MC=1 will get the order and the expected cost of the government is still 100 .
26
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So y is too high . Try y=1.9 and x=0.2 .
A-If a firm’s costs are 2 it will never report it 1 no matter what the other firm will do
B-If a firm’s cost are 1 , it must assume that the rival will announce 2 when its cost
are 2 . and if he thinks that the rival will announce 1 truthfully when its costs
are 1, then he will be indifferent between announcing 1 or 2 . { because ,when it
announce 1 he will not obtain benefit for the 50 units he produces, he looses(1- 0.2)
for every unit , so it is better not to produce anything . and when he announce 2 he
will not receive any order so he will not make any benefit }. C- if he thinks its rival
will report 2 when its costs are 1 , then it strictly preferred to report cost of 1 .
Because reporting 2 give him 50 = (50 ( 2-1)) while reporting 1 will provide him
with
100 ( 1.9 -1)=90 . Now if we increase the numbers a little bit by y= 1.91
and x= 0.21 , then the mechanism designed will force the participant to reveal their
cost truthfully. The government expected cost is 150.5. { 100[ ¼ x + ½ y + ¼ 2] }
Consider the following scheme ;
1- the government announce that if both firms claim that their costs are 2 , the order
will be split between the two and each will be paid 2 per unit
2- if both claim that their cost are 1 , the order will be split between them and each
will be paid 1.01 .
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
If one claims that its cost are 1 and the other claims that its costs are 2 , the one
which with low cost will get the entire order and will be paid 1.51 per unit ,
4- each will be paid 0.01 for participating at all .
I- Suppose firm 1 has cost of 2 , no matter what it think the firm 2 will do it is
better for him to announce cost of 2, and receive a bonus of 0.01 for
participating.
II-Suppose that firm one has MC=1 ; a- if firm two names MC=1 , it better for
firm one to announce MC=1 , and gets 0.01 bonus rather than announcing MC=2
and gets nothing . b- if firm two names MC=2 , then it better for firm one to
announce MC=1 rather than MC=2 . Because with MC=2 , it will get 50 ( 2-1)=50 ,
and with MC=1 , it will get 100 ( 1.51 -1 ) = 51 . That is truth telling is a strictly
dominant strategy for each of the firms. Now the expected cost of the government is
equal to c= 100[ (1/4)( x= 1.01) +(1/2)(y=1.51) + (1/4)2 ) + 0.02 = 150.77 . It is
possible to make the bonus small enough to reduce the expected cost of government
close enough to 150.
Summarizing the process , We found a scheme that truth telling is a Nash equilibrium
costing government an expected cost of 150, but this scheme has multiple Nash
equilibrium , one of which is better
for two firms than the truth-telling equilibrium.
THE REVELATION PRICIPLE AND
28
MECHANISM DESIGN
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
So we can modified the scheme in which truth telling is the unique Nash equilibrium.
but truth telling is not the dominant strategy in in this case . So we modified the
scheme again to make truth telling the dominant strategy for each firm.
OPTIMAL DIRECT REVELATION MECHANISM;
In the analysis , we made two special assumptions about the mechanism ;
First: we made a qualitative assumption that the government would ask firms to
name their costs, with the contract amounts and corresponding payments
depending on the pair of named cost.
Second ; we made a quantitative assumption and assume that the contract
amounts and corresponding payments took a particular form based upon the named
amount of the named costs of the two firms.
Over the next two sections we will find that the scheme which we found ( truth
telling dominant strategy ) is the best one in terms of lowering the cost. We will see
that we can not find any other scheme which can lower that government’s cost
below 150 units .
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A direct revelation mechanism ;
For each of the four pairs mn such that m=1,2 and n = 1,2 there is a given four
numbers ( x1mn , x2mn , t1mn , t2mn ) such that x1mn + x2mn ≥ 100 . In our example
two firms will participate and simultaneously and independently announce whether
their marginal costs are 1 or 2 .
What we mean by Direct mechanism is as follows ; let m= announcement of firm
1 , n = announcement of firm 2 , then Firm i must produce ximn units of item
for which it receives a payment of timn . ci = true MC of firm i . So firm’s net
profit = timn - ci ximn .
Suppose the government restrict attention to direct mechanisms with the property that
agreeing to participate and then truthfully revealing costs constitute a Nash
equilibrium for the two firms in the corresponding game of incomplete information
between them. What is the lowest expected payment that the government must make to
the firms using such a mechanism.
this problem can be solved by the help of linear programming technique.
to begin with we considered what is required if participation and truth telling are
to be a Nash-equilibrium for a given direct revelation mechanism. For each of the
firms we have four constraints , THE
forREVELATION
firm 1 they
are;
PRICIPLE AND
30
MECHANISM DESIGN
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
Firm’s 1 constraints ;
1 -(P1-1) if firm 2 participate and tell the truth , and if firm 1’s cost are 1 , then firm 1
is wiling to participate and tell the truth instead of refusing to participate if
(1/2) [ t111 - (1) x111 ] + (1/2) [ t112 - (1) x112 ] ≥ 0
2 -(P1-2) if firm 2 participate and tell the truth , and if firm 1’s cost are 2 , then firm 1
is wiling to participate and tell the truth instead of refusing to participate if
(1/2) [ t121 - (2) x121 ] + (1/2) [ t122 - (2) x122 ] ≥ 0
3 -( I1-1) if firm 2 participate and tell the truth , and if firm 1’s cost are 1 , then firm 1
is wiling to participate and tell the truth instead of participating and falsely claiming
that its costs are 2 if ,
•(1/2)[t111 - (1) x111 ]+ (1/2) [t112 - (1) x112 ] ≥ (1/2) [t121 - (1) x121]+(1/2)[t122 - (1)x122]
4 -(I1-2) if firm 2 participate and tell the truth , and if firm 1’s cost are 2 , then firm 1
is wiling to participate and tell the truth instead of participating and falsely claiming
that its costs are 1 if ,
(1/2)[t121 - (2) x121 ] + (1/2) [t122 - (2) x122 ] ≥ (1/2) [t111 - (2) x111]+(1/2)[t112 – (2)x112
]
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the first two constraints shows the conditions for participation in the game and telling
truth and the next two constraints shows the conditions for participating in the game
and choosing the right announcement which is consistent with the true one .
There are four similar constraint for firm two ,
(1/2) [ t211 - (1) x211 ] + (1/2) [ t221 - (1) x221 ] ≥ 0
(1/2) [ t212 - (2) x212 ] + (1/2) [ t222 - (2) x222 ] ≥ 0
(1/2)[t211 - (1) x211 ] + (1/2) [t221 - (1) x221 ] ≥ (1/2) [t212 - (1) x212]+(1/2)[t222 - (1)x222 ]
(1/2)[t212 - (2) x212 ] + (1/2) [t222 - (2) x122 ] ≥ (1/2) [t211 - (2) x211]+(1/2)[t221 – (2)x221 ]
the variables are nonnegative and four constraints that says that government obtains its
100 units ;
x111 + x211 ≥ 100 .
x112 + x212 ≥ 100
x121 + x221 ≥ 100
x122 + x222 ≥ 100
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So the optimization problem of the government is to minimize the expected cost ;
Min (1/4) [ t111 + t211 ] + (1/4) [t121 + t221 ] +( 1/4) [ t112 + t212 ]+ (1/4) [t122 + t222 ]
S.T. 12 constraints mentioned .
Unknowns are 16 variables for i , m, n = 1,2 in ximn and timn .
The solution is the direct revelation mechanism for which participation and truth
telling is a Nash equilibrium .
Suppose that the solution is as follows :
x111 = x112 = x22 1 = x222 = 100 , t112 = t221 = t222 =200 , all other variables
equal to zero . Taking in account the following constraints it means that firm 1 is
assigned all the production if it announce costs its costs of 1 ; other wise firm 2
assigned to produce all 100 units . Firm 1 is paid 200 if it announces its costs
equal to 1 and firm 2 announces cost of 2 . And firm 2 is paid 200 if firm 1
announces that its cost are 2 . This cost the government 150.
This solution is not the only possible answer to induce truth telling as a Nash
equilibrium in a direct revelation game . Here is another solution ;
x111 = x211 = x122 = x222 = 50 , x112 = x221 = 100 , t111 = t211 =50 , t122 = t222 =100 ,
t112 = t221 =150 ,all other variables equal to zero.This solution corresponds to paying
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
1 Dollar per unit when both firms name cost of 1 , 2 dollar per unit when
both firms name cost of 2 , and 1.5 dollar per unit goes to the firm that
names cost of 1 . We will call this solution the nice direct revelation
principle.
There are many direct revelation schemes for which participation and truth
telling give a Nash equilibrium in this situation and which cost the government
150. But there is no scheme for which participation and truth telling is a Nash
equilibrium that produces a lower expected cost .
In this problem we have assumed that for each pair of reports mn by the two
firms a single value ( x1mn , x2mn , t1mn , t2mn ) is implemented , later on we can
generalize the problem by relating a probability distribution μmn to each of
,( x1mn x2mn , t1mn , t2mn )
In the context of this problem , allowing for the random direct revelation
mechanism does not change anything . Since the firm and government are risk
neutral and the firms have linear production technologies , replacing any lottery
by the ( x1mn , x2mn , t1mn , t2mn ) that its mathematical expectation does not
affect any party's expected profit . Hence the best the government can do with
random direct revelation mechanism is the best it can do without it , that is
expected cost of 150 .
THE REVELATION PRICIPLE AND
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34
MECHANISM DESIGN
18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
We observed that the government can not do any better than an expected cost of 150
by using a direct revelation mechanism that induces participation and truth telling as a
Nash equilibrium. The revelation principle confirms that we can not do better with
any other complex sort of mechanism.
General mechanism;
Government design a game in which it ask to firm to participate. We can think of this
general form as a finite extensive form tree for the two firms, possibly incorporating
moves by the nature. The government designs and presents this game tree to the two
firms and ask them if they would like to play.
Because the government is allowed to present any finite game tree at all for its
mechanism, this formalism encompasses many mechanism .
For every general mechanism of this sort there is a corresponding game of
incomplete information in which each firm begins with private information of its
cost.
Information sets are used to mimic the idea that the firms know their own costs but not
the cost of other firm.
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
A mechanism designed by the government
Node ..
Firm 1
Accept
offer
Firm 2
Firm 1
Firm 1 reject the
Firm 2 reject
offer
the offer
Accept
offer
100 , 0 , 100 , 0
0 , 100 , 0, 150
Firm 1 reject the
offer
50 , 50, 10, 10,
100 , 0 , 200 , 0,
First number Production designed for firm 1
Second number Production designed for firm 2
Third number Payment made to firm 1
Fourth number Payment made to firm 2
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
-40, -90
100,0
A
R
-40, -40
1
A
0,-50
R
A
4 mechanisms
and 4 nodes for
2
each mechanism.
0,0
R
In each
A
mechanism the
1
first number at the
-100,0
1 W12 W11
end points shows
profit of the first
W22 1
W
21
A
firm and second
0,-50
1
number shows
A
R
profit for the
-100,0
A
second firm.
0,0
R
-90,-90
37
2
100,0
R
2
Firm 2
A
R
0,50
A
Firm 1
0,0
R
2
A
0,50
A
R
R
1
R
1
1
A
R
Wmn shows the mechanism in which the
cost of fist firm is m and second firm is n
.
0,0
-90,-40
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
The Revelation Principle
Consider any mechanism of the sort described above , the associated game of incomplete
information between the two firms, and any Nash equilibrium of this game in which both
agree to participate. The expected cost to the government and the assignment of the
production levels and payment to the firms in this equilibrium , viewed as a function
of their costs, can be precisely obtained by participation and truth-telling in a direct
revelation game for which participation and truth –telling give a Nash equilibrium.
Ϭij = strategy of firm i when its costs are j . i = 1,2 . j= m, n
Take a direct revelation game as follows,
For the pair corresponding to the costs m for firm 1 and n for firm 2 , construct
the probability distribution over outcome in the mechanism tree that results from
firm 1 playing ϭ1m firm 2 playing ϭ2n . This gives a probability distribution over
four-tuples , which in direct revelation game is μmn .
The claim is that in this direct revelation game, truth telling and participation is a
Nash-equilibrium that induces the same outcomes as the Nash equilibrium in the
original game.
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In the original game , a firm can always act as if its costs were “ the other value” .
The statement that we have a Nash equilibrium means that it has no positive
incentive to lie . Direct revelation game is constructed essentially as follows; the two
firms simultaneously and independently report their cost to the third party. This
third party then implements in the original general mechanism what the firm would
have done in the original equilibrium as a function of cost they report . That is if firm
1 reports cost m and firm 2 reports cost n , the third party implements play of ϭ1m
against play of ϭ2n . Since a firm would never choose in the original game to act as
if its costs were other than what they are , so the firm would not wish to have the
third party “ misplay” on its behalf.
Sometimes heard rephrases of the revelation principle as follows; anything that
the government can achieve in a Nash equilibrium between the two firm in a
general mechanism can be achieved by a direct revelation mechanism where truth
telling is a Nash equilibrium.
The first objection could be seen as the government might be able to devise a
mechanism so complex that the firms do not see their way to a Nash equilibrium in it,
and their behavior may then give the government lower expected costs than it gets in
a truth telling equilibrium of a direct revelation game.
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
The second objection comes trough this point that the two firms may interact
repeatedly , and say, collude implicitly. If this so , it is not entirely reasonable to
suppose that the firms in one particular encounter will behave according to the dictate
of Nash equilibrium. In such a situation the party designing the mechanism might do
well to think about how to design a mechanism that reduces the firm’s abilities to
collude .
In other words , the direct revelation mechanism could be seen as a tool of analysis for
finding the limits of what outcomes can be implemented , without reference to how
best to implement a particular outcome.
In some context of direct revelation, there will be situations ex post where the party in
the role of the government knows that it can obtain further gains from trade from one
or more of the parties who participated.
We could see the example in the context of insurance company with adverse
selection, who might be able to restrain itself from renegotiating with a consumer
who identifies herself as being low risk. Similarly in many applications of revelation
principle, the party in the role of mechanism designer must be able to commit
credibly to no subsequent renegotiation once it learns the types of the parties with
which it is dealing.
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What do we do with the government ordering 100 plans to the two firms. The best it
can do( in terms of minimizing the cost) with a truth telling in a direct revelation
mechanism is to get cost down to 150. if it can do better with some other mechanism ,
it has to such that to confuse the two firms that they do not play to a Nash equilibrium
Multiple Equilibrium and dominant Implementation
We can note once more that in many direct revelation mechanisms truth telling is a
Nash Equilibrium that cost the government 150 in expectation, and some are better
than the others .
Best of all mechanism we constructed , is a mechanism ( nice mechanism) in which
truth-telling was a dominant strategy for each of the firms.
Every thing else held equal , it seems natural to prefer a mechanism in which the
equilibrium the designer wishes is the only Nash equilibrium or, at least , in which
all the equilibrium are no worse for the designer than the one being aimed for.
Moreover the mechanism designer might worry that the players won’t settle on an
equilibrium at all , for example because the government mechanism is to complex for
the firms to find their ways towards equilibrium.
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
If the desired behavior is a strictly dominant strategy for each player , then it is of
course a unique Nash equilibrium.
These considerations motivates the notion of dominant strategy mechanism , a
mechanism in which each participant , for each possible value of her private
information , has one course of action that dominates all the others , no matter what
the other participant in the mechanism do.
what should be done to reach to the dominant strategy mechanism .
We should seek for which direct revelation mechanism ( ximn , ti mn ) is
participation and truth telling a dominant strategy.
There are more constraints than before. For firm 1 we have four participation
constraints and four incentive constraints. for example a participation constraint for
firm 1 is t121 - 2 x121 ≥ 0 , which shows that firm 1 must not sustain a loss when it
is truthfully reports its cost of 2 and firm 2 reports cost of 1 (truthfully or not ). We
can write three more participation constraint for firm 1 which makes the constraints
equal to 4 as follows ;
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
t111 - 1 x111 ≥ 0
t112 - 1 x112 ≥ 0
t122 - 2 x122 ≥ 0
t121 - 2 x121 ≥ 0
we can write down the four participation constraints for firm 2 as follows
t2 11 - 1 x2 11 ≥ 0
t2 21 - 1 x2 21 ≥ 0
t2 22 - 2 x2 22 ≥ 0
t2 12 - 2 x2 12 ≥ 0
one incentive constraint for firm 1 can be shown as follows ;
t121 - 2 x121 ≥ t111 - 2 x111 , in which the firm 1 when its costs are 2 and when its
rival report cost of 1 , prefers to report truthfully that its costs are 2 rather than
misrepresenting its costs as 1 .
the four incentive constraints for firm 1 are as follows ;
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1
18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
t121 - 2 x121 ≥ t111 - 2 x111
t122 - 2 x122 ≥ t112 - 2 x112
t111 - 1 x111 ≥ t121 - 1 x121
t112 - 1 x112 ≥ t122 - 1 x122
the four participation incentive constraints for firm 2 is ;
t221 - 1 x221 ≥ t222 - 1 x222
t211 - 1 x111 ≥ t212 - 1 x212
t212 - 2 x212 ≥ t211 - 2 x211
t222 - 2 x222 ≥ t221 - 2 x221
if we satisfy all eight constraints for firm 1 and for firm 2 , then we will have a direct
revelation mechanism in which participation and truth-telling are dominant ( weakly
dominant). the government then wants to minimize its expected cost subject to the
above 16 constraints ;
exp(cost) = (1/4) ( t111+t211)+(1/4)(t121+t221)+(1/4)(t112+t212)+(1/4)(t1 22 + t2 22 )
We should also satisfy the constraint x1mn + x2mn ≥ 100 for m,n = 1,2 .
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The solution to this linear programming is minimum cost of 150 for the government .
Also ; we will have as a solution
x211 = x212 = x122 = x112 = 100 , t112 = t122 = 200 , t211 = t221 = 100 , every thing else
equal to zero .
Firm 2 makes all the 100 units if it reports cost of 1 and it will be paid 100 .
If firm 2 reports cost 2 , firm 1 is given the assignment to make 100 units and is
paid 200 .
We know that the government can not do better with some more complex dominant
strategy mechanism in terms of reducing the expected cost
The outcome of any dominant strategy mechanism can be achieved in a direct
revelation mechanism for which truth-telling and participation is a dominant
strategy.
General discussion
we can generalize the mechanism in Toy problem in many different aspects . And in
this way asked that which mechanism among all possible mechanisms is optimal for
achieving some given end ?
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The problem of finding an optimal direct-revelation mechanism in which truth telling
is a Nash equilibrium or is a dominant strategy equilibrium is fairly a simple
mathematical programming problem, because the conditions that “ truth-telling is
Nash” and “truth-telling is Dominant” can be expressed in terms of inequalities.
This continues to hold in problems more general than the toy problem;
1- in the toy problem each firm had only a finite number types . This can be
generalized with a continuum of types .
2- the linearity of the cost structure in the toy problem meant that all the constraints
were linear. If instead the firms have non-linear cost function we have to deal with
nonlinear programming technique.
3- in the toy problem we assumed that it costs the government no more to have a
mechanism in which truth telling is dominant than to have one in which truthtelling constitute a Nash-equilibrium. If the firms have three possible costs , and if
those costs are not independently distributed, then it can be costly for the government
to insist on dominant strategy implementation.
Then we should see if we can restrict attention to truth-telling in a direct revelation
game, to see which outcome can be implemented in general mechanism.
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We should note that an outcome in this sense is some thing more than just the
expected cost to the government. An outcome specifies how much each firm
produces and what transfer is made to the firm as a function of the firm’s cost.
We should also look at the implement ability of the outcomes as follows’
1 -There is some mechanism with a Nash equilibrium which gives desired outcome.
2 - there is some mechanism for which all Nash equilibrium give desired outcome
3 - there is a mechanism for which there is a unique Nash equilibrium and this
Nash-equilibrium gives this outcome.
4 - there is a mechanism in which the outcome arises from the play of dominant
strategies.
If a mechanism admits several Nash equilibriums, some of which are worse for the
designer than is the equilibrium desired, then one worries that the participants find
their ways towards the wrong equilibrium.
If the desired outcome is the product of dominant strategy choices by the
participants , then the mechanism designer can have the greatest faith in the
mechanism she has designed.
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We have seen two revelation principles, which claim that anything that can be
implemented in the sense of (1) or (4 ) can be implemented in the corresponding
sense in the form of truth-telling in direct revelation games. These results are true in
substantial generality.
The correspondence between truth-telling in direct revelation games and outcomes
of general mechanism breaks down when the solution concept is adapted from (2 )
or (3).
But it is possible to augment a direct revelation game , adding signals that the
participant send beyond the declaration of type and then to obtain to obtain augmented
revelation principle that speaks to the implementation in sense of 2 and 3 .
Mixture of adverse selection and moral hazard is also possible in a general form.
We might imagine , for example, that our two firms’ costs are not determined entirely
endogenously but result as well from R&D and investment decisions made by the
firms. Then the government besides buying its hundred units , might seek to put in
place a mechanism that induces firms to invest optimally in plant and R&D where
optimality means that make the government’s cost as low as possible. In principle one
can use the revelation principle to tackle such complex problems.
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
THE PIVOT MECHANISM
Several farmers live on a bank of a stream. i= 1,2,3……n .
i=I
K cost of the building a bridge.
Ui the monetary value each farmer attaches to the bridge or the utility derived from
building the bridge by individual i . Or his willingness for building the bridge .
K/I the amount of money each individual actually has to pay if bridge has to be built.
ti the amount of money ( transfer of money ) taken from i if ti >0 ,(or given to i , ti
<0 ) in addition to K/I or some sort of surplus. Transfer is allowed even if the bridge
has not been built for those who really want the bridge to be built.
farmer’s utility = ui - K/I - ti if the bridge is built
farmer’s utility = - ti
if the bridge is not built , ui can be negative or positive
vi = ui - K/I =farmer’s valuation of the bridge net of his contribution for building it.
vi – ti = farmer’s utility if the bridge is built
Take into account the following mechanism;
Every one will write his name on a paper and an amount of money (pledge) which he
should have to pay after the bridge is built. Every one knows only his and not the
others. Negative pledges are also allowed.
49
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If the sum of pledges ( social surplus) exceeds 0, the bridge will be built and every
farmer must contribute the amount of his pledges plus his share , K/I . If the pledges
amount to less than 0 , then the bridge is not built and no transfer ( ti ) will be made.
Taking into account this mechanism and the free rider problem associated with this
mechanism , it is highly probable that total pledge will be less than Σi vi and the
bridge may not be built when it should be.
Consider the tax system for building the bridge which has the following properties ;
A – the bridge will be built if it is socially efficient to do so ; if Σi vi >0 .
B- the optimal action of each farmer in the mechanism ( as a function of the
farmer’s private valuation vi ) should dominate any other actions the farmer
might take, no matter what his fellow farmers will do.
C-in the optimal play the farmer should not end up with utility less than Min {vi , 0}.
If vi < 0 , then individual i will be heart by building the bridge which is not allowed
by the tax system.
D – the taxes collected must be nonnegative. ( less any subsidies , and not including
the building tax K/I per farmer , if the bridge is built )
Condition B says that we wish to design a general mechanism in which each farmer ,
as a function of his personal valuation vi , has a dominant strategy to play. With the
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revelation principle , we can restrict ourself to the direct revelation principle. Because
as the result of revelation principle we know that anything we can do with a general
dominant strategy mechanism can be done with a direct revelation mechanism in
which truth-telling is a dominant strategy.
Each farmer is asked to reveal his personal valuation ( vi )
ˆi
What the farmer reveal is v
As a function of vˆ  (vˆ1 ,..., vˆ I ) a decision is made whether to build the bridge or
not , and taxes on each farmer is determined.
 (vˆ)  0 the bridge is not built .
 (vˆ)  1 the bridge is built .
tax imposed on farmer i as a function of revealed values.
t i (vˆ)
vˆ  i  (vˆ 1 ,..., vˆi1 , vˆi 1 ,..., vˆI ) vector of reported valuations for allfarmers except i
t i (vˆ)  t i (vˆi , vˆi ) tax on farmer i is a function of all revealed values on farmer i
and others.
ˆ   Ij 1vˆi
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ˆ i   j i vˆ j
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
We are looking for a direct revelation mechanism in which truth telling is a
dominant strategy and in which the bridge is built if and only if Σi vi ≥ 0 . This
restriction combines with the notion of truth telling will be dominant tells us what the
function of α must be ;
 (vˆ)  1, if ˆ  0
 (vˆ)  0, if ˆ  0
Lemma 1 ; the taxes paid by farmer i must take the form of
t i (vˆi , vˆ i )  t i (vˆ i ), if ˆ  0
t i (vˆi , vˆ i )  t i (vˆ i ), if ˆ  0
The idea is that what a farmer pays in taxes cannot depend on what he himself
reveals as his valuation , except in so far as his revelation changes the decision
whether or not to build the bridge. To see this suppose that vi and wi two valuation
for individual i . Suppose that ;
v  ˆ i (  j i vˆ j )  0
i
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wi  ˆ i (  j i vˆ j )  0
THE REVELATION PRICIPLE AND
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t i (vi , vˆ i )  t i ( wi , vˆ i )
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Whether i announces vi or wi the bridge will be built but the tax is higher with vi . So
this mechanism does not lead to truth telling since individual i prefers to reveal his
preferences as wi , when his true preferences is vi .
t i (vˆ) should depends only on vˆ  i .
Lemma 2 :
t i (vˆ i )  t i (vˆ i )  ˆ i
The difference between the tax on individual i when the bridge is built and when it is
not is equal to the announced valuation of the other individuals except i .
ˆ i for fixed value of vˆ i . That is the tax
Proof ; Consider the case when vi = 
difference on individual i is equal to sum of revealed valuation of other individuals
except i .
Revealing vi by individual i means that the bridge will be built, because
ˆ  vi  
ˆ i  0

Comparing vi to revealing vi - ε (when individual i reveals less than his true value vi ),
we can see that when the revelation is vi - ε the bridge will not be built, since
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(ˆ   )  vi    ˆ i  0
18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
i
i
i
Revealing vi the bridge is built and the farmer gain v  t (vˆ ) and revealing vi - ε
means the bridge will not be built and farmers gain t i (vˆ i ) so for truth telling we
should have
vi  ˆ i  t i (vˆ i )  t i (vˆ i )
now consider the case where the farmer reveals vi which is ;
vi    ˆ i
Then in this case truthful revelation of vi cause the bridge not to be built, since
so the bridge will not be built. Not revealing truth
ˆ  vi  ˆ i  0
where vi    ˆ i when farmer falsely reveals vi + ε and ˆ  0 the bridge will be
built and the farmer has net utility vi  t i (vˆ i ) . In the truth telling we should have
i
i
his utility equal to  t (vˆ )
In order to have truth telling we should have
v
i
 t. i (vˆi ) ≤
 t i (vˆi )
If ε→0, we could
see that
54
.
_
or
.
vi =   ˆ i  t i (vˆ i )  t i (vˆ i )
ˆ i  t i (vˆ i )  t i (vˆ i )
THE REVELATION PRICIPLE AND
MECHANISM DESIGN
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
_
_
ˆ  i  t i (vˆ  i )  t i (vˆ i )

_
ˆ i  t i (vˆ i )  t i (vˆ i )
Lemma 3 and 4;
ˆ i  t i (vˆ i )  t i (vˆ i )
i
i
i
ˆ
ˆ
ˆ
t (v )   if   0
i
i
ˆ
t (vˆ )  0if   0
i
i
The amount of the tax when bridge is not built ( t i (vˆi ) ) is
equal to the sum of revealed value of other individuals except
ˆ  i ) ,if it is greater or equal to zero , and it is equal to zero
i (
when the sum of revealed value of other individuals except
I( 
ˆ  i ) is less than zero . (proof not needed) 1
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THE REVELATION PRICIPLE AND
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
Proposition 2 ;
There is only one direct revelation mechanism for which (a) to (d) [in page 50] hold
, namely the direct revelation mechanism defined by (1) , (2) , (3), (4 ). This
mechanism is alternatively defined as
 (vˆ)  1, if ˆ  0
 (vˆ)  0, if ˆ  0
i
ˆ
ˆ
ˆ
1- t (v )  0, if ;   0, and ...  0
i
ˆ  0, and ...ˆ i  0
2- t (vˆ)  0, if ; 
ˆ  i , if ; ˆ  0, and ...ˆ i  0
3- t i (vˆ)  
ˆ i , if ; ˆ  0, and ...ˆ i  0
4- t i (vˆ)  
i
This particular mechanism is called pivot
mechanism because a tax is paid by farmer i only
if his valuation changes the decision from what it
would be if he reported zero.
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Moreover, when i’s valuation is pivotal
, i is taxed an amount that is just equal
to the “social distress” his pivotal
valuation causes. If he causes the
bridge to be built when it otherwise
wouldn’t be , he pays (he will be
subsidizes) 
ˆ i which is the
monetary cost to the other members of
the community. And if he causes the
bridge not to built when it otherwise
ˆ  i , which is the
would be , he pays 
monetary benefit to others .
THE REVELATION PRICIPLE AND
MECHANISM DESIGN
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
Condition A ( page 50) – the bridge will be built if it is socially efficient to do so ; if
Σi vi >0 .
Says that we wish to achieve a social optimum which in this example where utility is
linear equals to maximizing the sum of individual utilities. But the mechanism does
not achieve a social optimum if there were any pivotal individuals , because it
produces a positive net collection of taxes ( case 3 page 56).
This raises the question , if there are pivotal individuals, so a net surplus of funds is
collected, what happens with this surplus. If the surplus is used in any fashion that
gives utility to the farmers, and if farmers anticipate this , then the direct revelation
mechanism is not what we described. If we wish to achieve (a) through (d) we must
find a use for the surplus that is of no benefit to the farmers.
So condition ( A) , does not guarantee that a social optimum is reached. The
decision to built the bridge will be done optimally , but at a waste of other social
resources if there were pivotal individuals.
Condition ( A) would imply that a social optimum is reached if it were joined to the
following modification of condition D ;
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THE REVELATION PRICIPLE AND
MECHANISM DESIGN
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18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES
The sum of taxes and transfers , (not including the taxes collected to build the bridge if
the bridge is built), totals zero precisely. This condition will be known as the balance
budget condition.
And we can investigate that it is impossible to satisfy this condition with (a) , (b), (c).
In other words it is impossible to achieve Pareto optimality in this example of closed
economy with a dominant strategy mechanism.
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THE REVELATION PRICIPLE AND
MECHANISM DESIGN
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Plan 1 ; if both announce the same cost 50 units allocates for each . If one announce
MC=1 and the other MC=2 , all 100 units allocates to the cheaper one
If MCiT = 2 never MCiT = 1 , otherwise loose , i =
1,2
MC2A =1 , π1 = 100/2 ( 1-1) =
0
MC1A =1
If MC1T = 1
MC2A =2 , π1 = 100( 1-1) = 0
MC2A =1 , π1 =0
MC1A = 2
MC2A =2 , π1 = 100/2( 2-1) = 50
Firm 1 always announce MC=2 whether his true cost is 2 or 1 , the same for
firm 2 . Government cost is 100(2) = 200 and ordering 50 to each .
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THE REVELATION PRICIPLE AND
MECHANISM DESIGN
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