18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY

1. 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 risk-averse 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 y2where 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.

2. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY the insurance company has profit Y1 - y1in 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 ρand 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 ; taking derivative with respect to y1 we will get the slope of the indifference curve as ; THE REVELATION PRICIPLE AND MECHANISM DESIGN

3. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY y1 = y2 y2 When πi = 1/2 The indifference curve is symmetric with respect to the 450 line . or ; for 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) Initial endowment point Y2 ΠiSlope = - ( 1- πi ) / 450 y1 Y1 THE REVELATION PRICIPLE AND MECHANISM DESIGN

4. y1 = y2 y2 Single crossing indifference curve is assumed 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY Low risk 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 . Y2 High risk 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 Y1 THE REVELATION PRICIPLE AND MECHANISM DESIGN

5. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY Now consider the iso-expected profit line 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 . THE REVELATION PRICIPLE AND MECHANISM DESIGN

6. Optimal contract for low risk individual firm Y1 = Y2 y2 C 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY Iso-profit line for insurance company when π=πH . ( y0 = πH y2 + (1- πH )y1 ) B Iso-profit line for insurance company when π=πL . Initial endowment point A Y2 450 y1 consumer Y1 High risk indifference curve Optimal contract for high risk individual Low risk indifference curve THE REVELATION PRICIPLE AND MECHANISM DESIGN

7. 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 .

8. Low risk consumer Y2 Y1 = Y2 High risk consumer Contract chosen by high risk consumer . UA > UB > UC 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY A Contract chosen by low risk consumer UB > UA > UC y2H y2L B Y2 C Y1 y1H y1L Y1 Initial endowment point . No insurance point . THE REVELATION PRICIPLE AND MECHANISM DESIGN

9. 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 )+ πL (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 ) ( I H ) (1- πL)U( y1L) + πL U( y2L ) ≥ (1- πL)U(Y1) + πL U(Y2) (P L) (1- πL)U( y1L) + πL U( y2L ) ≥ (1- πL)U(y1H) + πL U(y2H) H ( I L ) 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. . THE REVELATION PRICIPLE AND MECHANISM DESIGN

10. 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. ذHE REVELATION PRICIPLE AND MECHANISM DESIGN

11. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY 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 This can be proved taking into account the following steps ; y2 High risk consumer Step 1 .The high risk contract (y1H, y2H) must lie in the shaded area 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 . 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 consumer Low risk contract High risk contract y1 THE REVELATION PRICIPLE AND MECHANISM DESIGN

12. y2 High risk contract 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 un-shaded 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 . High risk consumer 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY A Low risk consumer B C v y1 Low risk contract THE REVELATION PRICIPLE AND MECHANISM DESIGN

13. y2 Y1 = Y2 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY Low risk consumer High risk consumer 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. High risk contract 450 y1 Initial endowment points . No contract situation Low risk contract THE REVELATION PRICIPLE AND MECHANISM DESIGN

14. 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 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY Y1 = Y2 Low risk consumer High risk consumer High risk contract 450 y1 Initial endowment points . No contract situation Low risk contract 14 THE REVELATION PRICIPLE AND MECHANISM DESIGN THE REVELATION PRICIPLE AND MECHANISM DESIGN kreps

15. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY y2 High risk consumer Low risk consumer 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. Low risk contract High risk contract Initial endowment points . No contract situation y1 THE REVELATION PRICIPLE AND MECHANISM DESIGN

16. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY High risk consumer y2 High risk contract 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 . Low risk contract Low risk consumer y1 THE REVELATION PRICIPLE AND MECHANISM DESIGN

17. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY High risk consumer y2 Low risk consumer Step 5 – the high risk contract (y1H, y2H) 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 . Maximum expected profit for insurance company for for high risk contract Location of High risk contract Low risk contract y1 THE REVELATION PRICIPLE AND MECHANISM DESIGN

18. 18.1 OPTIMAL CONTRACT DESIGNED FOR A SINGAL PARTY y2 High risk consumer 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 ]. 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 ) High risk contract A E B C Low risk contract D Initial endowment points . No contract situation Y2 y1 Y1 Low risk consumer THE REVELATION PRICIPLE AND MECHANISM DESIGN

19. 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

20. 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

21. 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

22. 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 .. THE REVELATION PRICIPLE AND MECHANISM DESIGN

23. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES Since thegovernment 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 costs1Since 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.

24. 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 2and 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 .

25. 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.

26. 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 .

27. 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 . THE REVELATION PRICIPLE AND MECHANISM DESIGN

28. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES If one claims that itscost 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 MECHANISM DESIGN

29. 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 . THE REVELATION PRICIPLE AND MECHANISM DESIGN

30. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES A direct revelation mechanism ; For each of the four pairs mnsuch 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 , for firm 1 they are; THE REVELATION PRICIPLE AND MECHANISM DESIGN

31. 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 ] THE REVELATION PRICIPLE AND MECHANISM DESIGN

32. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 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 THE REVELATION PRICIPLE AND MECHANISM DESIGN

33. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 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 THE REVELATION PRICIPLE AND MECHANISM DESIGN

34. 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 MECHANISM DESIGN

35. 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

36. A mechanism designed by the government Node .. Firm 1 Firm 2 Firm 1 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 50 , 50, 10, 10, Firm 1 reject the offer Firm 1 reject the offer Firm 2 reject the offer Accept offer Accept offer 100 , 0 , 100 , 0 0 , 100 , 0, 150 100 , 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 THE REVELATION PRICIPLE AND MECHANISM DESIGN

37. -40, -90 100,0 -40, -40 A R R 1 A 1 0,-50 R A 100,0 4 mechanisms and 4 nodes for each mechanism. In each mechanism the first number at the end points shows profit of the first firm and second number shows profit for the second firm. R 2 2 Firm 2 A 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 0,0 R A R 0,50 1 -100,0 1 W12 W11 A Firm 1 0,0 W22 1 W21 A 0,-50 1 R A R -100,0 2 A 2 0,0 R A R 0,50 A 1 1 A R R 0,0 Wmn shows the mechanism in which the cost of fist firm is m and second firm is n . -90,-90 -90,-40

38. 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

39. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

40. 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

41. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

42. 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 ; THE REVELATION PRICIPLE AND MECHANISM DESIGN

43. 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 t211 - 1 x211 ≥ 0 t221 - 1 x221 ≥ 0 t222 - 2 x222 ≥ 0 t212 - 2 x212 ≥ 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 ; THE REVELATION PRICIPLE AND MECHANISM DESIGN

44. 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 . THE REVELATION PRICIPLE AND MECHANISM DESIGN

45. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 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 ? THE REVELATION PRICIPLE AND MECHANISM DESIGN

46. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 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 truth-tellingconstitute 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

47. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES We should note that an outcome in this sense is some thing more than just the expected cost to the government. Anoutcomespecifieshow much each firm produces andwhat 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 issomemechanism 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 isa 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

48. 18.2 OPTIMAL CONTRACT FOR INTERACTING PARTIES 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. THE REVELATION PRICIPLE AND MECHANISM DESIGN

49. 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.

50. 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 THE REVELATION PRICIPLE AND MECHANISM DESIGN