INCENTIVE COMPATIBILITY AND MECHANISM DESIGN

Essenial Microeconomics -- INCENTIVE COMPATIBILITY AND MECHANISM DESIGN Signaling games 2 Incenive Compaibiliy 7 Spence s example: Educaional signaling 9 Single Crossing Propery Mechanism Design 7 Local and Global Incenive Compaibiliy 3 Coninuously disribued ypes 37

Essenial Microeconomics -2- Signaling Games Naure chooses player s ype Type agen chooses Uninformed players updae beliefs & he resuling payoff o player if he chooses is Fig -: Sequenial move signaling game Naure s decision node: Naure chooses player s ype Noaion: convenien o define = {,, T} and wrie he se of ypes as Θ { θ } Player s decision node: Player learns his ype θ Θ= { θ,, θt} and chooses his acion q Coninuaion of he game by responder(s) add specifics laer Q The oher players observe he acion ha player akes bu no his ype The res of he game is hen played ou The payoff o player, is r R

Essenial Microeconomics -3- Signaling game Naure chooses player s ype Type agen chooses Uninformed players updae beliefs & he resuling payoff o player if he chooses is Fig -: Sequenial move signaling game We will refer o ( q, r) Q R as a feasible oucome of he game (for player ) Called a signaling game since he choice of he informed player poenially reveals informaion abou his ype o he oher players and hus influences heir responses

Essenial Microeconomics -4- Signaling game Naure chooses player s ype Type agen chooses Uninformed players updae beliefs & he resuling payoff o player if he chooses is Fig -: Sequenial move signaling game We will refer o ( q, r) Q R as a feasible oucome of he game (for player ) Called a signaling game since he choice of he informed player poenially reveals informaion abou his ype o he oher players and hus influences heir responses Preferences of player represened by he uiliy funcion u( θ, qr, ) over Q R To simplify he exposiion, we will ofen rewrie he uiliy funcion of ype θ as U( qr, ) u( θ, qr, ) f ( θ ) probabiliy player is ype θ This probabiliy is common knowledge

Essenial Microeconomics -5- Naure chooses player s ype Type agen chooses Uninformed players updae beliefs & he resuling payoff o player if he chooses is Fig -: Sequenial move signaling game A ype θ player chooses an acion q Q We wrie his sraegy as { q } Responding players all updae heir beliefs Suppose ha all he responding players believe ha he sraegy of player is { q } From he acion hey observe, hey apply Bayes Rule o updae heir beliefs abou player s ype Thus each of he acions in player s sraegy { q } induces responses by he oher players which yields player a payoff r R *

Essenial Microeconomics -6- Naure chooses player s ype Type agen chooses Uninformed players updae beliefs & he resuling payoff o player if he chooses is Fig -: Sequenial move signaling game Player can also compue he bes responses given his sraegy and hus correcly infer he oucome of he game { q, r} The criical quesion hough becomes wheher player would choose he sraegy { q } knowing he oher player s responses (ie is { q, r} a bes response?) Suppose ha his ype is θ and insead of choosing acion q he chooses q s Since he responding players believe hey know his sraegy is { q }, he response is r s and so player s payoff is U( qs, rs) raher han U( q, r ) Then player has an incenive o deviae unless U( q, r) U( qs, rs)

Essenial Microeconomics -7- Incenive Compaibiliy The oucome for ype θ mus saisfy his inequaliy for each of he oher ypes Moreover hese incenive consrains mus hold for every possible ype U ( q, r) U ( q, r ), and s where s Incenive Consrains s s If all he incenive consrains are saisfied, he mapping M { q, r} is said o be incenive compaible In addiion, player mus be willing o play he game Le { U0 } be he uiliy associaed wih each ype s ouside opion Then, for an equilibrium in which player always paricipaes, he following consrain mus also be saisfied for each ype U ( q, r) U Paricipaion Consrains 0

Essenial Microeconomics -8- Bayesian Nash Equilibrium A mapping from ypes o oucomes, {( q, r)} is a BNE if (i) {( q, r)} saisfies he incenive and paricipaion consrains, and (ii) given he beliefs induced by player s sraegy, r is a bes response by he uninformed responders

Essenial Microeconomics -9- Example: Educaional Signaling A consulan is equally likely one of wo ypes θ Θ= { θ, θ2} The marginal values of he consulan o each of wo firms are m( θ ) = and m( θ 2) = 2 The cos of acion q is C ( q) = q/ θ If he is paid a fee of r his payoff is U( qr, ) = r q/ θ The consulan s ouside opporuniies are ( U, U ) = (, ) 0 02 4 4 The blue lines in he figure are indifference lines for a ype θ consulan The green line is an indifference line for a ype θ 2 consulan

Essenial Microeconomics -0- Single crossing Propery (SCP) The slope of an indifference line is smaller for a ype θ 2 consulan so he green indifference line crosses each indifference line for ype θonce Any pair of indifference maps wih his propery are said o have he single crossing propery I is his propery ha makes he cosly aciviy q a poenial signal There are wo oher players in he game They are firms bidding for he consulan s services They observe q and hen choose heir responses r, j > simulaneously j The consulan acceps he higher offer so r = Max{ r j > } j

Essenial Microeconomics -- NE Wih full informaion player 2 and 3 play a sub game This is a Berrand pricing game wih a unique equilibrium, r = m( θ ) j Thus he consulan s payoff is U( qm, ( θ )) = m( θ ) q/ θ This is maximized a q = 0 Now suppose ha he firms know only ha he consulan is equally likely o be ype θ or θ 2

Essenial Microeconomics -2- BNE Pure sraegy for player : { q( θ), q( θ 2)} q( θ ) is he acion of ype θ Pure sraegy for player j, j = 2,3: Bid rj ( q) for each possible q Q = + Sraegy profile: { q( θ}, ( ) θ Θ rj q j > Suppose ha { q( θ ), q( θ )} = {, } 2 4 2, q = 4 rj ( q) = 2, q = 2 0, oherwise Given hese responses he payoff o player is r = r ( q) j U ( q( θ ), r ( q( θ ))) =, U ( q( θ ), r ( q( θ ))) = 2 j 4 2 j 2 2 Thus his bes response is o choose q = q( θ), u ( q,0) = q

Essenial Microeconomics -3- The graph of he funcion rj ( q) is depiced in red The blue indifference line hrough ( q( θ), r ( q( θ))) lies above every oher poin on rj ( q ) Similarly for ype θ 2 Now consider he responding players j If q= q( θ) he consulan s value is m( θ ) = so heir equilibrium q bes responses are r = m( θ) = j If q = q( θ2), he consulans value is m( θ 2) = 2 so r = m( θ2) = No oher q is chosen in he sraegy profile so player j, j > can do no beer han bid zero Thus he proposed sraegy profile is a BNE sraegy profile j

Essenial Microeconomics -4- PBE Is he BNE a PBE? No Why? Separaing PBE q( θ) q( θ2) Responders equilibrium bes responses are r = m( θ ) = and r = m( θ ) = 2 We can choose any beliefs of he equilibrium pah 2 2 so choose he belief ha i is a ype θplayer wih probabiliy Then r 2 q r 3 q q q θ q θ 2 ( ) = ( ) =, { ( ), ( )} An example of a PBE is depiced

Essenial Microeconomics -5- Pooling PBE { q( θ ), q( θ )} = { q} 2 ˆ ˆq = 4 rq ( ˆ) = 5 r q r q q q 2( ) = ˆ 3( ) =, Fails he Inuiive Crierion Why? Assumpion: U( qr, ) = r q/ θ

Essenial Microeconomics -6- Mechanism Design A single uninformed player moves firs and proposes an assignmen of ypes o oucomes { q, r} (noe noaion change) Any such assignmen is called a mechanism and he firs mover is called he mechanism designer The oher player (or players), who knows his own ype, sees his lis of oucomes and chooses he oucome ha is bes, given his ype For he choices of each ype o coincide wih he proposed assignmen, { q, r} mus be incenive compaible Le d d U = fu ( q, r) be he expeced payoff of he mechanism designer As he firs mover, he designer chooses he mechanism ha maximizes his expeced payoff subjec o he incenive (and paricipaion ) consrains

Essenial Microeconomics -7- Necessary and sufficien condiions for incenive compaibiliy Noe ha if here are T differen ypes of player here are T incenive consrains ha mus be saisfied for each ype θ Θ For incenive compaibiliy his mus be rue for each of he T ypes of player so ha here are T ( T ) incenive consrains In addiion here are T paricipaion consrains In general, here is lile ha one can do wih a model ha has so many consrains To proceed, we need o impose furher srucure on he naure of he preferences of he differen ypes of player

Essenial Microeconomics -8- Le U ( qr, ) be he payoff funcion for ype θ We make he following very weak assumpion abou he payoff funcion Assumpion : For all he payoff funcion U ( qr, ) is coninuously differeniable on he closed U se Q Rand eiher U > 0 or < 0 r r The key o successful modeling is he assumpion ha he difference beween any pair of ypes reflecs differences in heir srengh of preference for q Definiion: Sronger preference for q Type θ has a sronger preference for q han ype θ s if for any ( qr ˆˆ, ) and ( qr, ) Q R, where q > qˆ, ( qr, ) ( qr ˆˆ, ) ( qr, ) ( qr ˆˆ, ) s U This is depiced below, firs wih > 0 r U and hen wih < 0 r

Essenial Microeconomics -9- case (i), case (ii) Fig -2: Type has a sronger preference for q Noe ha in each case as q increases, he indifference curve for ype θ crosses from he upper conour se for ype θ s o he lower conour se Since here can be no crossing from he lower conour se o

Essenial Microeconomics -20- he upper conour se, he assumpion ha θ has a sronger preference for q is called he single crossing propery Assumpion 2: Single crossing propery (SCP) Higher indexed ypes have a sronger preference for q

Essenial Microeconomics -2- We hen have he following lemma Lemma -: Preference rankings under he SCP Suppose ha ˆq < q If he SCP holds hen (i) ( qr ˆˆ, ) ( qr, ) ( qr ˆˆ, ) ( qr, ), > s s (ii) ( qr ˆˆ, ) ( qr, ) ( qr ˆˆ, ) ( qr, ), s< s Proof: Condiion (i) is simply a resaemen of he definiion of sronger preferences given he SCP To confirm condiion (ii), suppose ha for some θs < θ, (ii) is false Tha is (a)( qr ˆˆ, ) ( qr, ) and ( b) ( qr ˆˆ, ) ( qr, ), forsomeθ s < θ Bu by he SCP, if (b) holds hen ( qr ˆˆ, ) ( qr, ) hus conradicing (a) s QED

Essenial Microeconomics -22- Consider he indifference curve for ype θ hrough ( qr ˆˆ, ) We define he marginal rae of subsiuion (MRS) in he usual way: U q MRS ( ˆˆ q, r) = U r I is emping o conclude ha if he SCP holds, hen he MRS of he wo ypes mus be everywhere differen However, his is no quie rue Insead, we have he following resul Lemma -2: Single Crossing Suppose ha ype θ has a sronger preference for q han θ s on Q R If U / r > 0, τ, hen for all ( qr, ) Q R, MRS ( q, r) MRS ( q, r) τ If U / r < 0, τ, hen for all ( qr, ) Q R, MRS ( q, r) MRS ( q, r) s s This is a echnical poin As a pracical maer, economiss usually simply assume sric monooniciy of he MRS wih respec o ype

Essenial Microeconomics -23- MRS ( qˆˆ, r) < MRS ( qˆˆ, r) MRS ( qˆˆ, r) > MRS ( qˆˆ, r) s s case (i), case (ii) Fig -2: Type has a sronger preference for q

Essenial Microeconomics -24- Example : Educaional Signaling (Spence) A consulan of ype θ has a value m( θ ) o poenial cliens The consulan knows his ype bu his is no observed by he cliens A ype θ consulan can obain an educaion of observable qualiy q a a cos of C ( q ) Higher valued ypes have a lower marginal cos of educaion, >s ha is, if θ > θs C ( q) < C s ( q) Cliens offer differen fees, r, for differen educaion levels An offer can hen be wrien as ( qr, ) Le {( q, r)} be he Fig 3: Educaional signaling offers seleced by differen ypes A ype θ consulan s payoff is U ( q, r) = r C ( q ) and he profi of he firm (he second mover) ha makes he successful offer is m( θ ) r

Essenial Microeconomics -25- Example 2: Insurance (Rohschild-Sigliz) An individual insuree seeks o purchase coverage in he even of a loss, L, from a risk neural insurer The insuree s probabiliy of no incurring a loss akes on possible values θ Θ= { θ,, θt} The no loss probabiliy θ is known only o he insuree An insurance policy is a commimen o cover he loss less some deducible q The maximum deducible is he oal loss L so Q= [0, L] In reurn for he insurance coverage, he insuree mus pay a premium r, regardless of wheher here is a loss The vecor ( qr, ) hen compleely describes an insurance policy

Essenial Microeconomics -26- Insuree s expeced uiliy In boh he loss and no loss saes he insuree pays he premium r In he loss sae he insuree incurs a loss L and he insurer pays ou L q Given a von Neumann-Morgensern uiliy funcion u, he expeced uiliy of a ype θ insuree is U( qr, ) = θuw ( r) + ( θ ) uw ( q r) Risk neural insurer If an insuree who purchases he policy ( qr, ) is of ype θ, he join probabiliy ha he insuree is of ype θ and incurs a loss is f( θ) Given he deducible q, he insurer pays ou L q Then if a ype θ insuree chooses a policy ( q, r ),he expeced profi of he insurer is U0 = f( r ( θ)( L q) T

Essenial Microeconomics -27- SCP The indifference curves hrough 0 0 ( q, r ), for premium ypes θ and θs > θ, are depiced Noe ha he slope of a ype θ insuree s indifference curve hrough 0 0 ( q, r ) is dr U U = / dq q r U θ u w q r = θu ( w r ) ( ) u ( w q r ) 0 0 ( ) ( ) 0 0 0 + θ = 0 θ u ( w r ) ( ) + 0 0 θ u ( w q r ) deducible Fig 4: Signaling by purchasing less coverage As θ increases he denominaor increases and so he slope of he indifference curve becomes less negaive Thus he SCP holds

Essenial Microeconomics -28- Example 3: Bidding A bidder s value of a single iem for sale is θ Θ The bidder is risk neural Bidders compee for he iem by making bids Le q be he probabiliy of winning and le r be he expeced cos of bidding If he bidder has a value θ, his expeced payoff is U( qr, ) = qθ r The marginal rae of subsiuion is hen dr U U = / = θ dq q r U Since his is higher for a buyer wih a higher value he SCP holds

Essenial Microeconomics -29- We now show ha if Assumpions and 2 hold, hen he problem of idenifying incenive compaible acions and oucomes {( q, r)} is considerably simplified The following wo proposiions are he key insighs Proposiion -3: Monooniciy If he SCP holds, hen he local incenive consrains imply ha q qt Proof: Suppose ha for some ypes θ and θ +, q > q + The local upward incenive consrain is ( q +, r+ ) ( q, r) Since q > q + i follows from Lemma - ha ( q, r ) ( q, r) Bu his is + + + impossible since i violaes he local downward incenive consrain for ype θ + QED

Essenial Microeconomics -30- Proposiion -4: Local incenive compaibiliy implies global incenive compaibiliy If Assumpions and 2 hold and all he local incenive consrains are saisfied, hen he mapping M {( q, r)} saisfies all he incenive consrains Uτ Proof: I will be assumed ha > 0, τ r By hypohesis he local upward consrain holds for ype θ and he local downward consrain holds for θ +, ha is (i) ( q, r) ( q+, r+ ) and (ii) ( q, r) ( q+, r+ ) + Consider any s and such ha s< and τ { ss, +,, } From Proposiion -3 he monooniciy propery mus be saisfied so qτ qτ+ Case (a) qτ = qτ+ I follows from (i) and (ii) ha rτ = rτ+ Then ( qτ, rτ) ( qτ+, rτ+ ) τ

Essenial Microeconomics -3- Case (b) q < q τ τ+ The local downward consrain holds for ype θ τ +, ha is ( qτ, rτ) ( qτ+, rτ+ ) Since τ + i follows from saemen (i) of Lemma - ha ( qτ, rτ) ( qτ+, rτ+ ) Then we may conclude ha in eiher case ( q, r ) ( q, r ) τ τ τ+ τ+ Since his preference relaion holds for each τ { ss, +,, }, i follows ha τ + ( q, r ) ( q, r ) ( q, r) s s s+ s+ By an almos idenical argumen, i follows from saemen (ii) of Lemma - ha ( q, r ) ( q, r ) ( q, r) s s s+ s+ s s s We have herefore shown ha for all s and > s, ( q, r ) ( q, r) and ( q, r ) ( q, r) Thus all he incenive consrains hold s s s s s QED

Essenial Microeconomics -32- As we shall see, here are imporan special cases when eiher he local upward or downward consrain is binding for each ype, ha is, for { q, r} T eiher (i) ( q, r) ( q, r ) or (ii) ( q, r) ( q, r ) + + Since i is a necessary condiion for incenive compaibiliy, we assume monooniciy, ha is, q q + + + + Suppose ha (i) holds so ha he local upward consrain holds Appealing o he SCP, i follows ha ( q, r) ( q, r ) so he local downward consrain mus also hold + + + Suppose nex ha (ii) holds Then he local downward consrain holds Suppose ha he local upward consrain is violaed, ha is, ( q, r) ( q+, r+ ) Bu his canno be rue because by he SCP his implies ha ( q, r) ( q, r ), conradicing (ii) + + + Then ( q, r) ( q+, r+ ) so he local upward consrain mus also hold

Essenial Microeconomics -33- The following resul hen follows from Proposiion -4 Corollary -5: Incenive consrains wih binding local consrains Suppose ha { q, r} is increasing in q, he SCP holds and for each ype eiher he local upward or downward consrain is binding Then all he oher local consrains mus also hold and so all he incenive consrains are saisfied

Essenial Microeconomics -34- Paricipaion As we shall see, here are some applicaions for which he following assumpion is naural Assumpion 3: Single ouside alernaive For all ypes he paricipaion consrain can be expressed as follows: U( qr, ) U(0, r0 ) Suppose ha Assumpion 3 and he SCP holds and Q + Suppose also ha he paricipaion consrain holds for he lowes ype, ha is U ( q, r) U ( q, r ) = U (0, r ) 0 0 0 Suppose firs ha q = 0 and ha U ( qr, ) is sricly increasing in r Then r r 0, and so for all, U( q, r) U(0, r0) An idenical argumen holds if uiliy is decreasing in r Suppose nex ha q > 0 Then by he SCP, U( q, r) > U(0, r0) for all Thus o check he paricipaion consrain, we need only check he consrain for ype

Essenial Microeconomics -35- Noe finally ha he paricipaion consrain for ype θ is also a local downward consrain We herefore have he following generalizaion of Proposiion -4 Proposiion -6: If Assumpions -3 hold and he local upward and downward incenive consrains are saisfied (including he downward consrain for he lowes ype), hen { q, r} T saisfies all he incenive and paricipaion consrains

Essenial Microeconomics -36- Coninuously disribued ypes In he coninuous version of he model, a ype θ player has uiliy funcion u= u( θ, qr, ) where θ Θ= [ αβ, ] Assumpions and 2 can hen be rewrien as follows Assumpion : The payoff funcion u( θ, qr, ) is coninuously differeniable and eiher u/ r > 0 or u/ r < 0 for all ( θ, qr, ) Θ Q R Assumpion 2 : SCP Suppose ha q > q If 0 0 0 0 0 u( θ, q, r ) u( θ, q, r ), hen for all φ > θ, u( φ, q, r ) > u( φ, q, r ) For he coninuous model we consider oucome profiles {( q( θ), r( θ))} where q( θ ) and r( θ ) θ Θ are piecewise coninuously differeniable Such a profile is incenive compaible if u( θ, q( x), r( x)) u( θ, q( θ), r( θ)) θ and x Θ (IC) We define U( θ, x) u( θ, qx ( ), rx ( )) and V( θ) = U( θθ, )

Essenial Microeconomics -37- Then we have he following proposiion Proposiion -7: Incenive compaibiliy If Assumpions and 2 hold and q () and r () are coninuous and piecewise differeniable, hen for {( q( θ), r( θ))} θ Θ o be incenive compaible: (i) q( θ ) mus be non-decreasing and dv u = dθ θ (ii) ( θ, q( θ), r( θ)) where V( θ) = u( θ, q( θ), rq ( )) Togeher hese condiions are sufficien for incenive compaibiliy

Essenial Microeconomics -38- Proof: If he SCP holds, hen arguing exacly as in he finie ypes case, q() is increasing so (i) is rue For any θ and θ2 > θ, U( θ, θ2) U( θ, θ) hence U( θ2, θ2) U( θ, θ2) U( θ2, θ2) U( θ, θ) Also, U( θ2, θ) U( θ2, θ2) hence U( θ2, θ2) U( θ, θ) U( θ2, θ) U( θ, θ) Combining hese inequaliies, U( θ2, θ2) U( θ, θ2) U( θ2, θ2) U( θ, θ2) U( θ2, θ) U( θ, θ) θ θ θ θ θ θ 2 2 2 If { q( θ), r( θ)} θ Θ is coninuous, he upper and lower bounds converge in he limi Therefore d V ( θ) = U( θθ, ) = uθ ( θ, q( θ), r( θ) dθ This proves saemen (ii) of he proposiion

Essenial Microeconomics -39- Also θ = arg Max{ u( θ, q( x), r( x))} x Θ The firs order condiion for a maximum is uq ( θ) + ur ( θ) = 0 q r Thus r ( θ) = MRS( θ, q( θ), r( θ)) q ( θ) (*) Finally noe ha U x uq ( θ, x) = uqq ( x) + urr ( x) = ur[ r ( x) + q ( x)] = ur[ r ( x) + MRS( θ, q( x), r( x)) q ( x)] u r Subsiuing from (*), U (, x) ur[ MRS( x, q( x), r( x)) MRS(, q( x), r( x))] x θ = θ q ( x) From Lemma -2, if he SCP holds hen u [ MRS( θ, q( x), r( x) MRS( x, q( x), r( x)] 0 r Also q( θ ) is non-decreasing Thus he righ hand side is non-negaive for x < θ and non-posiive for x > θ Thus U( θ, x) akes on is maximum a x = θ and so he necessary condiions are indeed sufficien QED