Chapter 10 Mechanism Design and Postcontractual Hidden Knowledge

Transcription

1 Chapte 10 Mechanism Design and Postcontactual Hidden Knowledge 10.1 Mechanisms, Unavelling, Coss Checking, and the Revelation Pinciple A mechanism is a set of ules that one playe constucts and anothe feely accepts in ode to convey infomation fom the second playe to the fist. ð The mechanism contains an infomation epot by the second playe and a mapping fom each possible epot to some action by the fist.

2 Advese selection models can be viewed as poblems of mechanism design. ð The contact offes ae a mechanism fo getting the agents to tuthfully epot thei types. Mechanism design goes beyond simple advese selection. ð It can be useful even when playes begin a game with symmetic infomation o when both playes have hidden infomation that they would like to exchange.

3 Postcontactual Hidden Knowledge ð Moal hazad games complete infomation ð Moal hazad with hidden knowledge (also called postcontactual advese selection) symmetic infomation at the time of contacting asymmetic infomation afte a contact is signed The pincipal's concen is to give agents incentives to disclose thei types late.

4 The paticipation constaint is based on the agent's expected payoffs acoss the diffeent types of agent he might become. Thee is just one paticipation constaint even if thee ae eventually n possible types of agents in the model, athe than the n paticipation constaints that would be equied in a standad advese selection model.

5 ð What makes postcontactual hidden knowledge an ideal setting fo the paadigm of mechanism design is that the poblem is to set up a contact that induces the agent to make a tuthful epot to the pincipal, and is acceptable to both the pincipal and the agent.

6 Poduction Game VIII: Mechanism Design ð Playes the pincipal and the agent ð The ode of play 1 The pincipal offes the agent a wage contact of the fom wq (, m), whee q is output and m is a message to be sent by the agent. 2 The agent accepts o ejects the pincipal's contact.

7 3 Natue chooses the state of the wold s, accoding to pobability distibution F( s), whee the state s is good with pobability 0.5 and bad with pobability 0.5. The agent obseves s, but the pincipal does not. 4 If the agent accepted, he exets effot e unobseved by the pincipal, and sends message m { good, bad} to him. 5 Output is q( e, s), whee q( e, good) œ 3 e and q( e, bad) œ e, and the wage is paid.

8 ð Payoffs If the agent ejects the contact, _ then 1 œ U œ 0 and 1 œ 0. agent pincipal If the agent accepts the contact, then 1 agent œ Ue (, w, s) œ w e 2 and 1 pincipal œ Vq ( w) œ q w.

9 ð The agent does not know his type at the point in time at which he must accept o eject the contact. ð The message m is cheap talk it does not affect payoffs diectly and thee is no penalty fo lying. ð The pincipal cannot obseve effot, but can obseve output.

10 The pincipal implements a mechanism to extact the agent's infomation. ð In noncoopeative games, we odinaily assume that agents have no moal sense. ð Since the agent's wods ae wothless, the pincipal must ty to design a contact that eithe povides incentive fo tuth telling o takes lying into account.

11 The fist-best effot depends on the state of the wold. ð The pincipal can obseve the state of the wold and the agent's effot level. ð In the good state, the social suplus maximization poblem is Maximize 3 e e. g e g 2 g g * œ the optimal effot e 1.5 q g * œ 4.5

12 ð In the bad state, the social suplus maximization poblem is Maximize eb e e b 2. b b * œ the optimal effot e 0.5 q b * œ 0.5

13 The optimal contact ð The optimal contact must satisfy just one paticipation constaint, with the two incentive compatibility constaints. ð The pincipal must solve the poblem: Maximize [0.5 ( qg wg) 0.5 ( qb wb)] (10.1) q g, qb, w g, w b such that

14 the agent is paid unde a focing contact, ( q, w ), g g if he epots m œ good, and unde a focing contact, ( qb, wb), if he epots m œ bad, poducing a wong output fo a given contact esults in boiling in oil, and the contacts must induce paticipation and self selection.

15 ð The self-selection constaints in the good state 1 agent q g w g good w g q g (, l ) œ ( Î3) 2 (10.2) 2 b b 1agent b b w ( q Î3) œ ( q, w l good) in the bad state 1 agent q b w b bad w b qb (, l ) œ 2 (10.3) 2 g g 1agent g g w q œ ( q, w l bad)

16 ð The single paticipation constaint At the time of contacting, the agent does not know what the state will be ( q, w l good) ( q, w lbad) (10.4) agent g g agent b b 2 2 g g b b œ 0.5 { w ( q Î3) } 0.5 ( w q ) 0.

17 ð The single paticipation constaint (10.4) is binding. The pincipal wants to pay the agent as little as possible. 2 2 g g b b 0.5 { w ( q Î3) } 0.5 ( w q ) œ 0

18 ð The good state's self-selection constaint (10.2) will be binding. In the good state, the agent will be tempted to take the easie contact appopiate fo the bad state, and so the pincipal has to incease the agent's payoff fom the good-state contact to yield him at least as much as in the bad state. 2 2 g g b b w ( qî3) œ w ( qî3)

19 ð Fom the two binding constaints, we obtain the following expessions fo w and w. 2 w œ (5Î9) q b b w œ (1Î9) q (4Î9) q g b g 2 g 2 b ð The bad state's self-selection constaint (10.3) will not be binding. Let the agent not be tempted to poduce a lage amount fo a lage wage. w q w q b 2 2 b g g Solve the elaxed poblem without this constaint, and then check that this constaint is indeed satisfied.

20 The second-best contact ð The pincipal's maximization poblem (10.1) ewitten Maximize [0.5 { qg (1Î9) q q qb q q q g (4Î9) b} 0.5 { (5Î9) b}] g, b Eliminate w and w fom the maximand b using the two binding constaints, and pefom the unconstained maximization. g ** ** g b ð q œ 4.5 q œ 0.5 w ** ** g b 2.36 w 0.14

21 ð The bad state's self-selection constaint (10.3) is satisfied. ** ** 2 b b ** g ** 2 g w ( q ) w ( q ) ð Note that, if the ealization of the state of the wold is the bad state, then the agent's payoff is negative. Does a beach of the contact o enegotiation occu? ð In both states, effot is at the fist-best level. ð The agent does not ean infomational ents. At the time of contacting, he has no pivate infomation.

22 ð The pincipal in Poduction Game VIII is less constained, compaed to Poduction Game VII, and thus able to come close to the fist-best when the state is bad, and educe the ents to the agent.

23 Obsevable but Nonveifiable Infomation and the Maskin Matching Scheme ð Thee playes involved in the contacting situation the pincipal who offes the contact the agent who accepts it the cout that enfoces it ð We say that the vaiable s is nonveifiable if contacts based on it cannot be enfoced.

24 ð What if the state is obsevable by both the pincipal and the agent, but is not public infomation? nonveifiable Mutual obsevability can help. Maskin (1977) suggests coss checking.

25 ð Coss checking fo Poduction Game VIII 1 Pincipal and agent simultaneously send messages m and m to the cout saying whethe the state is good o bad. If m Á m, p a then no contact is chosen and both playes ean zeo payoffs. If m œ m, the cout enfoces pat 2 of the scheme. p a p a 2 The agent is paid the wage ( w l q) with eithe the good-state focing contact (2.25 l 4.5) o the bad-state focing contact (0.25 l 0.5), depending on his epot m a, o is boiled in oil if the output is inappopiate to his epot.

26 Thee exists an equilibium in which both playes ae willing to send tuthful messages, because a deviation would esult in zeo payoffs. The agent eans a payoff of zeo, because the pincipal has all of the bagaining powe. The pincipal's payoff is positive, and effots ae at the fist-best level.

27 ð Usually this kind of scheme has multiple equilibia. pevese ones in which both playes send false messages which match and inefficient actions esult ð A bigge poblem than the multiplicity of equilibia is enegotiation due to playes' inability to commit to the mechanism.

28 Unavelling: Infomation Disclosue When Lying Is Pohibited ð Anothe special case in which hidden infomation can be foced into the open when the agent is pohibited fom lying and only has a choice between telling the thuth o emaining silent ð Poduction Game VIII m œ bad in the bad state If m œ silent, the pincipal knows the state must be good. The option to emain silent is wothless to the agent.

29 ð s µ U [0, 10] The agent's payoff is inceasing in the pincipal's estimate of s. The agent cannot lie but he can conceal infomation. The pincipal would continue this pocess of logical unavelling to conclude that s œ 2. The pincipal would make the same deduction fom m 2 as fom m œ 2.

30 ð The unique equilibium must be fully sepaating. Somebody would deviate fom any patially pooling equilibium. ð Pefect unavelling is paadoxical.

31 The Revelation Pinciple ð A pinciple can design and offe a contact that induces his agent to lie in equilibium. He can take lying into account. This complicates the analysis. ð The evelation pinciple helps us simplify contact design.

32 ð The evelation pinciple Fo evey contact wq (, m) that leads to lying (i.e., to má s ), * thee is a contact w ( q, m) with the same payoff fo evey s but no incentive fo the agent to lie. ð Thee ae two levels of simplification in mechanism design. If thee ae n possible types of agent, we can estict the agent's message to take only n values. We can equie the mechanism to be constucted to elicit tuthful messages fom the agent.

33 ð Diect and indiect mechanisms If a mechanism esticts the agent's messages to the set of types, it is called a diect mechanism. If a mechanism allows moe possible messages than types, it is called a indiect mechanism. ð We can add a thid constaint to the incentive compatibility and paticipation constaints to help calculate the equilibium. tuth-telling The equilibium contact makes the agent willing to choose m œ s.

34 ð The evelation pinciple depends heavily on the following assumption. The pincipal cannot beach his contact. ð Thoughout this chapte, we will be assuming that the pincipal can commit to his mechanism. He can commit to not using all the infomation he eceives fom the agent.

35 The Sende-Receive Game of Cawfod and Sobel: Coase Infomation Tansmission ð Even if the infomed and uninfomed playes have diffeent incentives, can lie, and can't commit to a mechanism, if thei incentives ae close enough, tuthful (if impefect) messages can be sent in equilibium.

36 The Cawfod-Sobel Sende-Receive Game ð Playes the sende (the infomed playe) the eceive (the uninfomed playe) ð The ode of play 0 Natue chooses the sende's type to be t µ U [0, 10]. 1 The sende chooses message m [0, 10]. 2 The eceive chooses action a [0, 10].

37 ð Payoffs The payoffs ae quadatic loss functions in which each playe has an ideal point and wants a to be close to that ideal point. 2 1sende œ α { a ( t 1)} 1eceive œ α ( a t) 2

38 Equilibia ð Thee is no fully sepaating equilibium in which each type of sende epots a diffeent message. Pefect tuthtelling cannot happen in equilibium.

39 ð Pooling Equilibium 1 Sende: Send m œ 10 egadless of t. Receive: Choose a œ 5 egadless of m. Out-of-equilibium belief: If the sende sends m 10, the eceive uses passive conjectues and still believes that t µ U [0, 10].

40 ð Pooling Equilibium 2 Sende: Send m using a mixed-stategy distibution independent of t that has the suppot [0, 10] with positive density eveywhee. Receive: Choose a œ 5 egadless of m. Out-of-equilibium belief: Unnecessay, since any message might be obseved in equilibium.

41 ð In each of these two equilibia, the sende's message conveys no infomation, and is ignoed by the eceive. ð Aveaging ove all possible t, both thei payoffs ae lowe than if the sende could commit to tuthtelling.

42 ð Patial Pooling Equilibium 3 Sende: Send m œ 0 if t [0, 3] o m œ 10 if t [3, 10]. Receive: Choose a œ 1.5 if m 3 and a œ 6.5 if m 3. Out-of-equilibium belief: If m is something othe than 0 o 10, then t µ U [0, 3] if m [0, 3) and t µ U [3, 10] if m [3, 10].

43 In the Sende-Receive Game, the eceive cannot commit to the way he eacts to the message, so this is not a mechanism design poblem. ð Instead, this is a cheap-talk game, so called because of these absences: m does not affect the payoff diectly, the playes cannot commit to futue actions, and lying bings no diect penalty.

44 The sende and the eceive's inteests ae simila but not identical, and they could both benefit fom some tansfe of infomation. ð If expectations ae appopiate, they do so, in the patially pooling equilibium. ð If they do not expect the cheap talk to be infomative, howeve, it will not be, and coodination will fail.

45 10.2 Myeson Mechanism Design ð Depending on who offes the contact and when it is offeed, vaious games esult. ð We will look at one in which the selle makes the offe, and does so befoe he knows whethe his quality is high o low.

46 The Myeson Tading Game: Postcontactual Hidden Knowledge ð Playes a buye and a selle ð The ode of play 1 The selle offes the buye a contact { qh, ph, ql, pl} unde which the selle will declae his quality m to be high o low, and the buye will then buy q o q units of the 100 the selle has available, at pice p o p. The contact is { qmpm ( ) ( ), qm ( )}. Zeo is paid if the wong output is deliveed. l l h h

47 2 The buye accepts o ejects the contact. 3 Natue chooses whethe the type of the selle's good, s, is High quality (pobability 0.2) o Low (pobability 0.8), unobseved by the buye. 4 If the contact was accepted by both sides, the selle declaes his type to be L o H and sells at the appopiate quantity and pice as stated in the contact.

48 ð Payoffs If the buye ejects the contact, 1 œ 0, 1 œ , and 1 selle L œ buye selle H If the buye accepts the contact and the selle declaes a type that has pice p and quantity q, then 1 buye L l œ (30 p) q, 1 buye H l œ (50 p) q, 1 selle H œ 40 (100 q) pq, and 1 selle L œ 20 (100 q) pq.

49 The selle has an oppotunity cost (a pesonal value o poduction cost) of 40 pe high-quality unit and 20 pe low-quality unit. ð Fo efficiency, all of the good should be tansfeed fom the selle to the buye. ð The only way to get the selle to tuthfully eveal the quality of the good, howeve, is fo the buye to say that if the selle admits the quality is bad, he will buy moe units than if the selle claims it is good.

50 The fist-best quantities * * h l ð q œ 100 and q œ 100 The optimal contact ð The selle wants to design a contact subject to two sets of constaints. ð The paticipation constaint fo the buye buye ll buye lh 0.8 (30 p ) q 0.2 (50 p ) q 0 l l h h

51 This constaint will be binding. p œ 30 and p œ 50 l h ð We do not need to wite out the selle's paticipation constaint sepaately. the acceptable (if vacuous) null contact { qh, ph, ql, pl} œ {0, 0, 0, 0}

52 ð Two incentive compatibility constaints fo the selle himself The selle must design a contact that will induce himself to tell the tuth late once he discoves his type. The selle is tying to sell not just a good, but a contact, and so he must make the contact to be attactive to the buye. when he has low quality 1 ( q, p ) 1 ( q, p ) selle L l l selle L h h 20 (100 q ) 30 q 20 (100 q ) 50 q l l h h ql 3q h Ê q l qh

53 when he has high quality 1 ( q, p ) 1 ( q, p ) selle H h h selle H l l 40 (100 q ) 50 q 40 (100 q ) 30 q h h l l q h q l Ê satisfied fo all possible q and l q h

54 ð The selle's maximization poblem q œ 3 q at the optimum l h (fom the low-quality incentive compatibility constaint) The selle's payoff function 1 œ ( q, p ) ( q, p ) s selle L l l selle H h h œ 0.8 {20 (100 q ) 30 q } 0.2 {40 (100 q ) 50 q } l l h h

55 The selle must solve the poblem: Maximize {0.8 (2, ql) 0.2 (4, qh)} q q l, h subject to q œ 3 q, q Ÿ 100, and q Ÿ 100. l h l h ** ð q h œ 100Î3 q ** l œ 100

56 The equilibium follows the geneal patten fo these games. ð The paticipation constaint is binding (fo the buye). ð The incentive compatibility constaint is binding fo the type with the most temptation to lie, and not fo the othe type. ð Using the two binding constaints, we can solve out fo the values of some of the choice vaiables in tems of othe choice vaiables. ð We can maximize the payoff of the playe making the offe (the selle) to solve fo values of those emaining vaiables.

57 The mechanism will not wok if futhe offes can be made afte the end of the game. The mechanism is not fist-best efficient. The impotance of commitment is a geneal featue of mechanisms. We could have set it up instead as ( wq, ), a total pice amount w fo the quantity q. ð That would be moe in the style of mechanism design.