Incentive Compatible Transfers in Linear Environments

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1 Incentve Compatble Transfers n Lnear Envronments Nenad Kos Dept. of Economcs, IGIER Boccon Unversty Matthas Messner Dept. of Economcs, IGIER Boccon Unversty August 5, 2010 Abstract We study the mechansm desgn envronment wth lnear utlty functons and sngle dmensonal types, as n Myerson (1981), but allow for non-connected valuaton spaces. In ths envronment we characterze the whole set of transfers mplementng a gven mplementable allocaton rule. In partcular we show that every transfer rule mplementng a gven allocaton rule can be wrtten as a Remann-Steltjes ntegral wth respect to an extenson of the allocaton rule to the smallest nterval contanng the valuaton space. Ths extends the characterzaton of Myerson (1981) to general valuaton spaces and unfes the separate approaches used to deal wth mechansm desgn problems wth dscrete valuaton spaces and mechansm desgn wth nterval valuaton spaces. In the second part of the paper we explore condtons on type-dependent outsde optons under whch boundares of the set of transfers that mplement a gven allocaton rule satsfy ndvdual ratonalty. Ths enables us to characterze the set of allocaton rules that can be mplemented n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm, wthout requrng condtons that guarantee revenue equvalence. Envronments of Myerson and Satterthwate (1983) and Cramton, Gbbons, and Klemperer (1987) are specal cases n whch the valuaton space are ntervals and outsde optons lnear. JEL Code: C72,D44,D82. Keywords: Incentve Compatblty, Revenue Equvalence, Mechansm Desgn Department of Economcs and IGIER, Boccon Unversty, Va Roentgen 1, I Mlan, e-mal: nenad.kos@unboccon.t Department of Economcs and IGIER, Boccon Unversty, Va Roentgen 1, I Mlan, e-mal: matthas.messner@unboccon.t 1

2 1 Introducton We study the mechansm desgn envronment wth lnear utlty functons and sngle dmensonal types, as n Myerson (1981), but allow for non-connected valuaton spaces. In ths envronment we characterze the whole set of transfers mplementng a gven mplementable allocaton rule. In partcular we show that every transfer rule mplementng a gven allocaton rule can be wrtten as a Remann-Steltjes ntegral wth respect to an extenson of the allocaton rule to the smallest nterval contanng the valuaton space. Ths extends the characterzaton of Myerson (1981) to general valuaton spaces and unfes the separate approaches used to deal wth mechansm desgn problems wth dscrete valuaton spaces and mechansm desgn wth nterval valuaton spaces. For each mplementable allocaton rule we characterze, under several assumptons on the type dependent outsde optons, transfers that extract the most money from the agents n any ncentve compatble and ndvdually ratonal mechansm. Ths enables us to characterze the set of allocaton rules that can be mplemented n an ndvdually ratonal mechansm wth ex post budget balance. Some of the envronments that can be seen as specal cases of ours are the ones n Myerson and Satterthwate (1983) and Cramton, Gbbons, and Klemperer (1987). As opposed to the before mentoned papers, we do not requre that the type spaces are connected and dstrbutons have everywhere postve densty. The assumpton of nterval type spaces, rather than for example dscrete types, can crucally alter results on budget balanced mplementaton of effcent allocaton rules. Myerson and Satterthwate (1983) presented an example wth two types where ex post effcent allocaton rule can be mplemented n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm. Furthermore, as observed n Skreta (2006), mplementaton on general type spaces can play a crucal role when the nformaton structure s endogenous as n Bergemann and Pesendorfer (2007). Lterature: We buld on results n Kos and Messner (2010). They characterze the boundares of the set of transfers mplementng a gven allocaton rule n a quas lnear envronment wthout mposng any topologcal or other assumptons on the type space or the utlty functons (beyond the quas lnearty). The lnear structure n the present paper allows us to sharpen these characterzatons. The man thread of both papers s n showng that revenue equvalence, whle convenent, s n no sense necessary to deal wth most mechansm desgn problems. 2

3 Also close to our paper s Skreta (2006). She solves for the optmal auctons whle allowng for dsconnected type spaces. Our papers dffer both n contrbuton and n nterpretaton. Skreta (2006) characterzes the optmal mechansms. To do that she frst characterzes transfer rules that extract the most money gven an mplementable allocaton rules when outsde optons are constant. We, on the other hand, characterze the whole set of transfers mplementng an allocaton rule. Furthermore, we extend the analyss to more general outsde optons and mplementaton wth ex post budget balance, ssues that her paper does not deal wth. Moreover, whle Skreta (2006) solves the problem by mposng ncentve compatblty even for the types whch are not n the type space and thus recovers a form of payoff equvalence, our pont s that both revenue equvalence as well as mposng ncentve compatblty constrants for non exstent types s redundant snce the whole set of transfers mplementng an allocaton rule can be characterzed n any case. The body of lterature on auctons and mechansm desgn wth lnear utltes and undmensonal types s too vast for us to revew here. We wll attempt only a bref overvew. Myerson (1981) s, of course, the bass of our analyss. He showed that all the transfers mplementng a gven allocaton rule dffer at most by a constant by provdng an ntegral equaton for transfers; the revenue equvalence prncple. Revenue equvalence has snce been the workhorse of the lterature on auctons when valuaton spaces are ntervals. We show that a smlar type of analyss can be performed when the revenue equvalence does not apply. Even when all the transfers mplementng an allocaton rule do not dffer only by a constant one can wrte them n a smple ntegral form. A second strand of the lterature we buld upon s on mplementaton n budget balanced mechansms. Myerson and Satterthwate (1983) showed that even when there s scope for trade between a buyer and a seller wth valuatons n ntervals full effcency can not be acheved n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm (except n the trval case when t s common knowledge that the buyer s valuaton s above the seller s). They also presented an example n whch both the seller and the buyer have two possble valuatons and full effcency does obtan. Matsuo (1989) characterzed necessary and suffcent condtons on the dstrbutons wth two valuatons for the full effcency to obtan, whle Kos and Manea (2009) provded the characterzaton for dstrbutons wth a fnte number of valuatons. The present paper extends the characterzaton to general dstrbutons and, furthermore, allows for rcher envronments: arbtrary number of buyers and sellers (as for example n Makowsk 3

4 and Mezzett (1994)) and more general outsde optons whch among other allow for multple agent ownershp of an object lke n Cramton, Gbbons, and Klemperer (1987). 2 Framework, notaton and basc concepts There are I agents. An allocaton s a par (x, τ) [0, 1] I R I +. The component x s to be nterpreted as the non-monetary component of the socal decson; τ nstead s a vector of monetary transfers to be pad by the agents. Wth a slght abuse of termnology we wll often use the term allocaton to ndcate the non-monetary component x of a par (x, τ). Agent only cares about the -th component of (x, τ). Hs preferences are descrbed by a lnear (Bernoull) utlty functon. That s, at (x, τ) agent s utlty s v x τ. The parameter v R s agent s prvately observed preference/valuaton type. All players know that ths valuaton s drawn from the pdf F defned on the real lne. Agent s preference type s stochastcally ndependent of all other agent s valuatons. Moreover, we assume that for all the support of F, whch s denoted by V, s compact but not necessarly connected. V thus has a smallest and a largest element whch we denote by v and v, respectvely. V = I =1 V s the set of all preference profles. We wrte v for a typcal element of V. By the Revelaton Prncple t s wthout loss of generalty to restrct our attenton to drect mechansms. 1 A drect mechansm s a par of functons (Q, T ), where Q : V [0, 1] I, I =1 Q (v) 1, and T : V R I. The nterpretaton s as follows: The planner asks the agents to report ther types (reveal ther prvate nformaton) and promses to mplement the socal decson Q(v) and the transfers T (v) f the profle of reports s v. We wll often refer to the functons Q and T as allocaton rule and transfer scheme, respectvely. q : V [0, 1] denotes agent s nterm expected allocaton rule. It s defned by q (v ) = E v [Q (v)], Smlarly, agent s nterm expected transfer scheme, t : V [0, 1] s defned by t (v ) = E v [T (v)]. We say that a mechansm (Q, T ) s ncentve compatble f q (v )v t (v ) q (v )v t (v ), (1) 1 See Dasgupta, Hammond, and Maskn (1979) and Myerson (1979). 4

5 for all and v, v V. That s, (Q, T ) s ncentve compatble f truthful reportng s a Bayesan Nash equlbrum of the game nduced by (Q, T ). The allocaton rule Q s mplementable f there exsts a transfer scheme T such that (Q, T ) s ncentve compatble. In ths case we also say that T mplements Q. Smlarly, an expected allocaton rule of agent, q, s sad to be mplemented by the expected transfer scheme t f the par (q, t ) satsfes condton (1). If for a gven q there exsts an expected transfer scheme by whch t s mplemented, then q s mplementable. Of course, Q s mplementable f and only f each q s so. An mplementable expected allocaton rule for agent, q, s sad to satsfy Revenue Equvalence f any two expected transfers, that mplement q dffer only by a constant. That s, f q s mplemented both by t and t and satsfes Revenue Equvalence then there exsts a c R such that t (v ) t (v ) = c, for all v V. An mplementable allocaton rule Q s sad to satsfy Revenue Equvalence f for each agent s assocated expected allocaton rule Revenue Equvalence holds Agents may decde not to partcpate n a mechansm. The payoff agent realzes by dong so s descrbed by the functon Φ : V R. That s, he obtans the payoff Φ (v) f the profle of valuatons of all agents s v. We say that Φ descrbes the value of agent s outsde opton. The (nterm) expected value of agent s outsde opton f hs own type s v s denoted by φ (v ). That s, φ (v ) = E v [Φ (v)]. The mechansm (Q, T ) s sad to be (nterm) ndvdually ratonal f under the assumpton of truthful reportng each type of each agent realzes a larger (nterm) expected payoff by partcpatng n the mechansm than by optng out of the mechansm. That s, (Q, T ) s ndvdually ratonal f q (v )v t (v ) φ (v ), (2) for all and v V. 3 Incentve Compatble Transfers It s well known that n lnear envronments as the one we are consderng n ths paper an allocaton rule s mplementable f and only f each agent s expected allocaton rule s non- 5

6 decreasng. Our objectve n ths secton s to characterze the set of transfer schemes whch mplement a gven allocaton rule. In the frst step we wll show that extremal transfers of agent wth type v for the allocaton rule q, gven that type ˆv s transfer s fxed to 0, can be expressed as t (v ; ˆv, q ) = ˆv sdq (s; ˆv ) and t (v ; ˆv, q ) = ˆv sd q (s; ˆv ), where q ( ; ˆv ) and q ( ; ˆv ) are functons whch extend q from V to the smallest nterval contanng V. Ths result s remnscent of the characterzaton result of Myerson (1981). He shows that when the type sets are ntervals (.e. when V = [v, v ]) every transfer rule T whch mplements the allocaton rule Q satsfes the condton t (v ) = v q (v ) v q (v ) v q (s)ds + c, for some c R. An mportant mplcaton of ths s that every mplementable allocaton rule satsfes Revenue Equvalence. We generalze Myerson s characterzaton result to envronments where type sets are not connected. In dong so we rely on the technques developed n Kos and Messner (2010). Kos and Messner (2010) ntroduce the concept of extremal transfers. The extremal transfers for an mplementable (expected) allocaton rule for agent, q, defne the lower (t ( ; ˆv, q )) and upper ( t ( ; ˆv, q )) boundary of the set of all transfers whch ) mplement q and ) are equal to zero for a pre-specfed type ˆv. The type ˆv for whom the transfer payment s normalzed to zero, s referred to as the anchor type. For later reference t s convenent to recall the defntons of the extremal transfers and the general characterzaton result of Kos and Messner (2010). In order to do so, let S (v ; ˆv ) be the set of all fnte sequences n V whch start at ˆv and end at v ;.e. sequences {v j }n j=0 wth v0 = ˆv and v n = v. Gven an expected allocaton rule q the extremal transfers, t ( ; ˆv, q ), t( ; ˆv, q ) : V R, are defned by t (v ; ˆv, q ) = nf {v j } S (v ;ˆv ) j=0 v j [q (v j ) q (v j 1 )], (3) and for all v V. t (v ; ˆv, q ) = sup {v j } S (v ;ˆv ) j=0 v j 1 [q (v j ) q (v j 1 )], (4) 6

7 Theorem 1. Kos and Messner (2010) Let (Q, T ) be ncentve compatble. Then for any ˆv V, t (v ; ˆv, q ) t (v ) t (ˆv ) t (v ; ˆv, q ), for all and v V. Moreover, (Q, t( ; ˆv, q)) and (Q, t( ; ˆv, q)) are ncentve compatble. Reader should consult Kos and Messner (2010) for the proof of the above Theorem. It s mportant to pont out that the extremal transfers may depend n a non-trval way on the anchor type ˆv and that t (ˆv ; ˆv, q ) = t(ˆv ; ˆv, q ) = 0, for every ˆv and every mplementable allocaton rule q. Extremal transfers wth dfferent anchor valuatons wll typcally not just dffer by a constant. We llustrate ths n the followng example. Example 1. Let V = {v L, vh }, wth vl < v H, and suppose that agent faces an nterm allocaton rule q = (q L, qh ), wth ql q H. If the transfer of type vl s set equal to zero then ncentve compatblty requres that the transfer for type v H does not exceed v H(qH q L ). On the other hand, f the payment of type v H cannot exceed v L(qL q H ). Therefore s normalzed to zero, then the transfer of type v L t(v H ; v H, q ) t(v L ; v H, q ) = v L (q H q L ) v H (q H q L ) = t(v H ; v L, q ) t(v L ; v L, q ). Thus the two extremal transfers dffer by a constant f and only f the allocaton rule s constant,.e. q H = q L. Theorem 1 does not requre the lnearty assumpton whch we mpose here on preferences. However, under lnearty the expressons that defne the extremal transfers t and t can be seen as Remann-Steltjes ntegrals of the dentty functon on [v, v ] wth respect to two functons whch extend q from V to [v, v ]. In order to see ths we start by defnng two famles of extensons of the allocaton rule q, q and q, both ndexed by an anchor type ˆv V : nf{q (ṽ ) : ṽ V, ṽ v } f v ˆv q (v ; ˆv ) =, sup{q (ṽ ) : ṽ V, ṽ v } f v < ˆv and q (v ; ˆv ) = sup{q (ṽ ) : ṽ V, ṽ v } f v ˆv. nf{q (ṽ ) : ṽ V, ṽ v } f v < ˆv Observe that q ( ; ˆv ) and q ( ; ˆv ) preserve the monotoncty property of q. We are now ready to state the central result of ths secton. In the followng theorem q and q are merely auxlary tools whch allow us to express the extremal transfers n a smple and convenent way. partcular, ther use does not requre mposng ncentve compatblty outsde V. In 7

8 Theorem 2. Let q be an mplementable allocaton rule. Then for any ˆv V, t (v ; ˆv, q ) = for all v V. sd q (s; ˆv ) and t (v ; ˆv, q ) = sdq (s; ˆv ), ˆv ˆv The followng example llustrates the result. Example 2. Let V = {v L, vm, v H}, where vj R and vl < v M < v H and assume that q s strctly ncreasng,.e. q L < q M < q H (where of course q s stands for q (v s ), s = L, M, H). If we fx v M as the anchor type, then the extensons q ( ; ˆv M ) and q ( ; ˆv M ) are gven by q L f v = v L q L f v L v < v M q (v ; ˆv M ) = q M f v L < v < v H and q (v ; ˆv M ) = q M f v = v M q H f v = v H q H f v M < v v H. Integratng the agent s valuaton wth respect to these two functons yelds t (v ; v M, q ) = v L[qM q L] f v = v L 0 f v (v L, vh ) v H[qH q M ] f v = v H and t (v ; v M, q ) = v M [q M q L] f v < v M 0 f v = v M v M [q H q M ] f v M < v. Notce that n the example the extremal transfer t ( ; ˆv M, q ) satsfes the condtons v H q H t (v H ; v M, q ) = v H q M t (v M ; v M, q ) v L q L t (v L ; v M, q ) = v L q M t (v M ; v M, q ). Ths means that between the anchor type and the type above the downward ncentve constrant s bndng, and between the anchor type and the type below the upward ncentve constrant s bndng. Ths observaton does not only apply to the above example but generalzes to more general settngs. For each v V above the anchor type ˆv there s an ncreasng sequence of types, {v j } such that vj < v for all j, sup j {v j } = sup{v V : v < v } and v q (v ) t (v ; ˆv, q ) = lm j q (v j )v t (v j ; ˆv, q ). 8

9 For each type v V above the anchor the local ncentve constrants wth respect to smaller types are bndng or come arbtrarly close to beng so. We therefore say that above ˆv the local ncentve constrants are downward bndng. Below the anchor type ˆv the local ncentve constrants are upward bndng. That s, for each type v V below ˆv the ncentve constrants wth respect to larger neghbors are bndng or arbtrarly close to bndng. Expressed n formal terms, for each v V such that v < ˆv there s a decreasng sequence {v j } such that v < v j, nf j{v j } = nf{v V : v > v } and v q (v ) t (v ; ˆv, q ) = lm j q (v j )v t (v j ; ˆv, q ). When the type set s fnte and the anchor type s the lowest type, v, then we obtan the famlar result that the largest transfer s defned as the one for whch the downward adjacent ncentve constrants are bndng. In the next secton we wll encounter stuatons where restrctng attenton to v as anchor type s not wthout loss of generalty. Observe that f V s tself an nterval, then q ( ; ˆv ) and q ( ; ˆv ) concde wth q. Consequently, for each ˆv V the lower bound on transfer, t ( ; ˆv, q ), must be equal to the upper bound, t ( ; ˆv, q ). In fact, both of them must be equal to the Remann-Steltjes ntegral of the dentty functon on V wth respect to the allocaton rule q. The fact that n envronments wth connected type sets for every mplementable q the extremal transfers concde amounts to sayng that for every such allocaton rule Revenue Equvalence s satsfed. Ths s a well known result that has frst been shown by Myerson (1981). We fnd t nstructve to restate t here for our approach yelds a smple alternatve proof. Corollary 1 (Myerson (1981)). Let V = [v, v ], and let (q, t ) be ncentve compatble. Then there exsts a c R such that t (v ) = c + sdq (s), v for all v V. Proof. The result follows drectly from Theorem 2 after notcng that when V s an nterval q = q = q regardless of the choce of ˆv. For expostonal purposes we present a complete proof here. Let v V. Incentve compatblty mples t (v ) t (v ) v [q (v ) q (v )]. Now let v 1 (v, v ). Incentve compatblty yelds t (v ) t (v 1 ) v [q (v ) q (v 1 )], t (v 1 ) t (v ) v 1 [q (v 1 ) q (v )]. 9

10 Summng the last two nequaltes yelds t (v ) t (v ) v [q (v ) q (v 1 )] + v 1 [q (v 1 ) q (v )]. Extendng the same knd of reasonng one obtans t (v ) t (v ) j=0 v j [q (v j ) q (v j 1 )], for any ncreasng sequence {v j }n j=0 such that v0 = v and v n = v. Consequently t (v ) t (v ) nf j=0 v j [q (v j ) q (v j 1 )], where the nfmum s over all ncreasng sequences as descrbed above. The rght hand sde of the nequalty s a Remann-Steltjes ntegral, therefore Incentve compatblty also mples t (v ) t (ˆv ) v [q (v ) q (v )] t (v ) t (v ). By the same type of reasonng as above one obtans whch together wth equaton (5) yelds v sdq (s) t (v ) t (v ), t (v ) t (v ) = v sdq (s). (5) v sdq (s). Extremal transfers are pnned down only on the nterm level. There are many ways to support these nterm transfers wth ex post transfers. So far we have been usng so called all-pay transfers. In partcular we defned ex post transfers T wth the nterm transfers: T (v) = t (v ; ˆv, q ), for all v V. There are other ex post transfers that yeld the same nterm transfers. Kos and Manea (2009) study ths ssue extensvely n the dscrete type settng. The nterestng nsght s that the expected externalty transfers (agent pays the externalty he mposes on the remanng agents) does n general not extract the most money from the agent when the ex post effcent 10

11 allocaton rule s beng mplemented. Perhaps the smplest non trval example s wth two bdders bddng for an object where one of them has valuatons 0 and 3 wth equal probablty and the other one s commonly known to have valuaton 1. Whle the expected externalty transfers mplementng the effcent allocaton rule would requre the frst bdder to pay 1 when he wns, a transfer that yelds the extremal transfer wth the anchor 0 for the frst bdder s the one that requres hm to pay the smallest valuaton he could have, report truthfully, and wn gven the report of the second bdder. Thus payng 3 when wnng. For an n-depth treatment see Kos and Manea (2009). Theorem 2 shows how the two extensons of q, q ( ; ˆv ) and q ( ; ˆv ), can be used n order to obtan an ntegral representaton of the extremal transfers t ( ; ˆv, q ) and t ( ; ˆv, q ). Next we show that ntegratng wth respect to other approprately defned extensons of q to [v, v ] yelds ncentve compatble transfers as well. Indeed, any ncentve compatble transfer scheme for an allocaton rule q can be obtaned n such a way. In order to show ths we frst defne the class of approprate extensons of q. Defnton 1. A functon q : [v, v ] [0, 1] s called a monotonc range preservng extenson (henceforth, MRP extenson) of an mplementable allocaton rule q : V [0, 1] f t concdes wth q on V, s non-decreasng, and ts range concdes wth the range of q. The followng example llustrates the concept of MRP extensons. Example 3. Let V 1 = {0, 1} and q 1 (0) = 0, q 1 (1) = 1/2. q 1 s ncreasng and thus mplementable. A MRP extenson of q 1 s a weakly ncreasng functon whch maps the nterval [0, 1] nto the set {0, 1/2}. Any such functon can be characterzed by a par (α, x) [0, 1] {0, 1/2}, where α ndcates the pont at whch the functon jumps from 0 to 1/2 and x s the value the functon takes at α. If x = 0 the extenson s left contnuous, whle x = 1/2 means that the extenson s rght contnuous. The followng fgure shows the extensons characterzed by the pars (1/4, 0) (green) and (3/4, 1/2) (blue). Of course, also the prevously consdered functons q ( ; ˆv ) and q ( ; ˆv ) are MRP extensons of q. They are represented by the pars (0, 0) (q ( ; 0) and q ( ; 1)) and (1, 1/2) (q ( ; 1) and q ( ; 0)). We are now ready to show that every transfer scheme whch mplements a gven allocaton rule q can be descrbed as a Remann-Steltjes ntegral wth respect to an approprately chosen MRP extenson of q. It s nstructve to start wth the partcularly smple case n whch V conssts of only two valuatons. 11

12 1/ /4 3/4 1 v Fgure 1: A left-contnuous (green) and a rght-contnuous (blue) MRP extenson of q Lemma 1. Suppose that V = {v, v }, and let q : V [0, 1] be nondecreasng. Transfer t mplements q f and only f there exsts a constant c and a MRP extenson q of q such that for every v V. t (v ) = c + v sd q (s), (6) Proof of Lemma 1. ( ) Let q be some MRP extenson of q wth a jump at the pont α [v, v ] (see the dscusson n the precedng example). Moreover, take any c R and let t be defned by t (v ) = c + v v sd q (s). Then v [q ( v ) q (v )] α[q ( v ) q (v )] = t ( v ) t (v ) = α[q ( v ) q (v )] v [q ( v ) q (v )], thus renderng (q, t ) ncentve compatble. ( ) As for the other drecton assume that (q, t ) s ncentve compatble so that v [q ( v ) q (v )] t ( v ) t(v ) v [q ( v ) q (v )]. But then there exsts an α [v, v ] such that t ( v ) t (v ) = α[q ( v ) q (v )], and a MRP extenson q of q wth a jump at α. c s set to t (v ). The deas on whch the proof of the precedng result s bult, carry over to the more general case where V may contan more than two elements. Proposton 1. Let q be an mplementable allocaton rule. Transfer t mplements q f and only f there exsts an MRP extenson q of q and a constant c such that t (v ) = c + v sd q (s), (7) for every v V. Moreover, the MRP extenson q s unque up to a set of (Lebesgue) measure zero. 12

13 Proof. See Appendx. 4 Extremal Transfers and Indvdual Ratonalty We explore mechansm desgn envronments where the planner cannot force the agents to partcpate n the mechansm but needs to make sure that for each type of each agent partcpaton s voluntary (nterm ndvdual ratonalty). In the precedng secton we have characterzed the boundares of the set of transfers whch mplement a gven allocaton rule. In ths secton we nvestgate the queston under whch condtons the concept of extremal transfers can also be used to descrbe the boundares of the set of all transfers whch are both ncentve compatble and ndvdually ratonal. We wll present condtons on the envronment guaranteeng that an extremal transfer extracts the most money from an agent gven the requrement of ncentve compatblty and ndvdual ratonalty. 2 The frst two condtons have been ntroduced n Kos and Messner (2010) for more general envronments. Our goal here s to dentfy smpler condtons on the prmtves guaranteeng that at least one of the frst two assumptons holds. Assumpton 1. Q s mplementable and for each there exsts a type ˆv V such that for all v V. ˆv q (ˆv ) t (ˆv ; ˆv, q ) φ (ˆv ) v q (v ) t (v ; ˆv, q ) φ (v ), A type ˆv that satsfes the condton n Assumpton 1 s called a crtcal type of agent for the (expected) allocaton rule q. Assumpton 2. For every there exsts a type ˆv such that for all v V and x X. ˆv x φ(ˆv ) v x φ(v ) It s easy to verfy that Assumpton 2 mples Assumpton 1 for all mplementable allocaton rules. A proof of ths and the followng result can be found n Kos and Messner (2010). We chose to present both assumptons although they are nested. Whle Assumpton 1 s weaker t 2 Extremal transfers, gven an anchor type, are the transfers that extract the most and the transfers that extract the least money gven ncentve compatblty. It s easy to construct an example wth outsde optons such that none of the extremal transfers s ndvdually ratonal. In that case one needs to resort to analyss as n Julen (2000). 13

14 s somewhat more tedous to verfy, and moreover, s allocaton rule dependent. Assumpton 2, whle more restrctve, s smpler and ndependent of a partcular mechansm. Proposton 2. Let (Q, T ) be an ncentve compatble and ndvdually ratonal mechansm. Moreover, assume that Q satsfes Assumpton 1. If ˆv s a profle of crtcal types, then (Q, T ), where T : V R, T (v ) = t (v ; ˆv, q ) + q (ˆv )ˆv φ(ˆv ) s ncentve compatble and ndvdually ratonal. Furthermore, for all v V. t (v ) T (v ), The transfer rule T = ( T ) extracts more money from each agent than does any other transfer that mplements the allocaton rule Q n an ncentve compatble and ndvdually ratonal mechansm. Snce, as mentoned before, Assumpton 2 mples Assumpton 1 the above proposton also holds f the latter assumpton s replaced by the former. In what follows we present two assumptons specfc to the lnear envronment. Assumpton 3. For every there exsts a functon h on [v, v ] whch concdes wth φ on V and ˆv V such that every x [0, 1] s a supergradent of h at ˆv. 3 It s easy to verfy that Assumpton 3 mples Assumpton 2. Moreover they concde when V = [v, v ]; n that case h = φ. Furthermore, ths mples that f φ s lnear Assumpton 2 can be satsfed f only f the coeffcent n φ s ether smaller or equal to 0, and then ˆv = 0, or larger or equal to 1 n whch case ˆv = 1. Proposton 3. Suppose for some, φ (v ) = kv + n, where k, n R. Assumpton 2 s satsfed f and only f ether k 0 n whch case ˆv = v or k 1 and then ˆv = v. Condtons of Proposton 3 are satsfed, for example, n Myerson (1981) and Myerson and Satterthwate (1983). In former φ s equal to zero for all agents and all valuatons and n the latter t s equal to zero for the buyer and φ(v) = v for the seller. On the other hand, Assumpton 1 requres that for each agent the correspondence f (ˆv ) = arg mn v V v q (v ) t (v ; ˆv, q ) φ (v ), (8) admts a fxed pont n V. The defnton of f nvolves the extremal transfer t. Thus, Assumpton 1 s not an assumpton drectly on the fundamentals of the envronment but nvolves 3 As a remnder, x s a supergradent of h at ˆv f h(v) h(ˆv) (v ˆv)x for all v [v, v]. 14

15 endogenous objects. Notce however that n envronments where V s connected and thus Revenue Equvalence apples for any par ˆv, ˆv V, the extremal transfers t ( ; ˆv, q ) and t ( ; ˆv, q ) dffer only by a constant. Moreover, the transfer can be expressed as ntegral of the agent s type wth respect to the allocaton rule. Combnng these two observatons allows us to conclude that n ths case the mappng f s a constant whch s defned n terms of exogenous varables only. As for Assumpton 1 ths means that t s satsfed f and only f the mnmzaton problem mn q v v V v sdq (s) φ (v ) (9) has a soluton. Of course, exstence of a soluton to ths problem s guaranteed f the agent s outsde opton s a contnuous functon of hs type. A problem of the form (9) s solved n Cramton, Gbbons, and Klemperer (1987). In partcular, they consder a set up n whch the agents outsde optons are lnear n ther respectve types. When Revenue Equvalence s not satsfed Assumpton 1 s n general harder to verfy. Yet, our lnear setup allows for smplfcatons. Assumpton 4. Q s an mplementable allocaton rule and for each there exsts a ˆv V such that for all v V. φ (v ) φ (ˆv ) ˆv q (s; ˆv )ds, (10) Proposton 4. Assumpton 1 s equvalent to Assumpton 4. Moreover, the set of valuatons ˆv satsfyng Assumpton 1 concdes wth the set of valuatons ˆv satsfyng Assumpton 4. Smpler suffcent condtons on φ and q for Assumpton 4 to hold can be gven. Proposton 5. Suppose for each there exsts an extenson φ of φ to [v, v ] whch s absolutely contnuous and dfferentable on (v, v ). Furthermore, assume that for each there exsts a w V such that q (v ; w ) φ (v ) for v > w, and q (v ; w ) φ (v ) for v < w. Then for all v V. φ (v ) φ (w ) w q (s; w )ds, (11) An envronment where the assumptons of Proposton 5 are satsfed s the one from Cramton, Gbbons, and Klemperer (1987), where the outsde optons are lnear, φ (v ) = r v. Moreover, the assumptons are satsfed for every mplementable allocaton rule. Our result s more general n that t allows for more general outsde optons and does not requre the valuaton space to be connected. 15

16 5 Ex Post Budget Balance In ths secton we explore what allocaton rules are mplementable n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm. Theorem 3. Let Q be mplementable and suppose Assumpton 4 holds for all. Then Q can be mplemented n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm f and only f [ˆv q (ˆv ; ˆv ) φ (ˆv ) E sd q (s, ˆv )] 0. ˆv Put smply, an mplementable allocaton rule Q can be mplemented n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm f and only f the ncentve compatble and ndvdually ratonal mechansm wth allocaton rule Q that extracts the most money from each agent does not run an ex ante defct. Or yet dfferently, f and only f Q can be mplemented n an ncentve compatble and ndvdually ratonal mechansm whch runs an ex ante budget surplus. Theorem 3 s a restatement of a more general theorem n Kos and Messner (2010). The equaton for the ex ante surplus of the partcular mechansm s somewhat smpler here, however, due to the addtonal structure that lnearty permts. Smlar nequaltes, though under revenue equvalence, have been obtaned and used to argue that the ex post effcent allocaton rule can or cannot be mplemented n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm. For the case of blateral trade and the mpossblty result the classc reference s Myerson and Satterthwate (1983), whle some possblty results have been obtaned n Cramton, Gbbons, and Klemperer (1987) and Makowsk and Mezzett (1994). A specal case of the above equaton was obtaned n Kos and Manea (2009) who characterzed t for the case of dscrete types. Fnally, we present an example of how Theorem 3 can be put to use. Example 4. We consder a blateral trade envronment of Myerson and Satterthwate (1983), where the buyer has wth probablty 1 p valuatons dstrbuted unformly on the nterval [0, 2] and wth probablty p hs valuaton s c, where c > 3. The seller has valuatons dstrbuted unformly on the nterval [1, 3]. Agent s {B, S} utlty s v X T, where X s the probablty that agent obtans the object and T s the transfer. The outsde opton for the buyer s φ B (v B ) = 0 and for the seller φ S (v S ) = v S. The effcent allocaton rule Q allocates the object to the agent wth the hgher valuaton. Smple computaton gves q B (c) = 1, q B (v B) = 0.5v B 0.5 for v B [1, 2] and q B (v B) = 0 for v B < 1. On the other hand, q S (v S) = 1 p for v S 2 and q S (v S) = v S 1 p 2 for v S [1, 2]. 16

17 It s easy to verfy that Assumpton 2 s satsfed wth ˆv B = 0 and ˆv S = 3. Snce V S s an nterval q S ( ; 3) = q S ( ). On the other hand, q B ( ; 0) = q B ( ) on [0, 2] and q B (v B; 0) = 0.5 for v B (2, c). Usng Proposton 2, and ntegraton by parts one can compute the relevant extremal transfers by the formula: t (v ) = v q(v ) φ (ˆv ) ˆv q (s; ˆv )ds, for all v V and {B, S}. Somewhat tedous computatons yeld 1 2 c f v B = c t B (v B ) = 1 4 v2 B 1 4 f v B [1, 2], 0 f v B [0, 1] and t S (v S ) = { 3p f vs [2, 3] 1 2p (1 p)v2 S f v S [1, 2]. After takng expectatons one obtans Et B = p[ c ] + 1 6, and Et S = p. The ex post effcent allocaton rule can be mplemented n an ncentve compatble, ndvdually ratonal, and ex post budget balanced mechansm f and only f Et B + Et S = 1 53 p(c 2 12 ) Ths n turn means that c needs to be at least 9 2. Kos and Manea (2009) show, n an envronment wth dscrete types, that a mechansm whch prescrbes the buyer to pay the smallest valuaton he could have, report truthfully and obtan the object gven the seller s report and gves to the seller the hghest value he could have, report truthfully and sell the object mplements the ex post effcent allocaton rule wth extremal transfers. The mechansm extracts the most money from the buyer and gves the least money to the seller among all ex post effcent, ncentve compatble and ndvdually ratonal mechansms. It s easy to verfy that the same type of transfers wll work n ths case. As an example, f the buyer reports valuaton c and the seller reports 2.5 the buyer pays c and the seller obtans 3. If the buyer reported any other valuaton he could have had there would be no trade n the effcent mechansm, thus c s the smallest valuaton he could have, report truthfully and wn, therefore he has to pay t. Seller, on the other hand, could report 3 and stll nduce trade, thus that s what he gets. If, however, the seller s valuaton s 1.5 and the buyer s c, then the buyer pays 1.5 and the seller obtans 3. 17

18 Tatur (2005) provdes a smlar example observng that ex post effcency, ncentve compatblty, ndvdual ratonalty and ex post budget balance can be obtaned f c s large enough but does not provde an explct characterzaton of cut off values of c and p. 6 Concluson We use the mechansm desgn approach developed by Kos and Messner (2010) to study mplcatons of ncentve compatblty, ndvdual ratonalty and ex post budget balance n a lnear envronment lke n Myerson (1981) but wth possbly dsconnected valuaton spaces and type dependent outsde optons. Our approach dffers from most of the lterature on mechansm desgn n that ncentve compatblty s mposed globally and yelds an mmedate characterzaton of ncentve compatble transfers as opposed to frst dervng the local mplcatons of ncentve compatblty, for example condtons on the dervate of the ndrect utlty functon, and then expandng them globally. Ths enables us to provde a characterzaton of the whole set of transfers mplementng a gven allocaton rule regardless of whether the condtons guaranteeng the revenue equvalence (connectedness of the valuaton space) are satsfed or not. Our approach s then extended to characterze the set of allocaton rules that can be mplemented n an ncentve compatble, ndvdually ratonal and ex post budget balanced mechansm. Thus generalzng the characterzatons of, for example, Myerson and Satterthwate (1983) and Cramton, Gbbons, and Klemperer (1987) to non connected valuaton space and more general outsde optons. 18

19 7 Appendx Proof of Theorem 2. We prove that for any gven ˆv V, t (v ; ˆv, q ) = v ˆv sd q (s; ˆv ) holds for all v V. The followng arguments can easly be adapted to show that on V, t ( ; ˆv, q ) concdes wth v ˆv sdq (s; ˆv ). The reader should keep n mnd that v ˆv when ˆv > v. We start by showng that for all v, ˆv V = ˆv v and thus negatve ˆv sd q (s; ˆv ) t(v ; ˆv, q ). (12) For each par v, ˆv V, defne S (v ; ˆv ) as the set of all fnte monotonc sequences n V whch start at ˆv and end at v. Observe that t (v ; ˆv, q ) = nf v j [q (v j ) q (v j 1 )] (13) {v j } S (v ;ˆv ) j=0 nf {v j } S (v ;ˆv ) j=0 = nf {v j } S (v ;ˆv ) j=0 v j [q (v j ) q (v j 1 )] v j [ q (v j ; ˆv ) q (v j 1 ; ˆv )]. The frst equaton s defntonal and the nequalty s mpled by the fact that S (v ; ˆv ) S (v ; ˆv ). The second equalty follows from the fact that q and q ( ; ˆv ) concde on V. We next show that nf v j {v j } S (v [ q (v j ; ˆv ) q (v j 1 ; ˆv )] sd q (s; ˆv ). ;ˆv ) ˆv j=0 By ts defnton the Remann-Steltjes ntegral on the rght hand sde of ths nequalty s the greatest lower bound of the set of numbers whch can be generated as sums of the form j=1 v j [ q (v j ; ˆv ) q (v j 1 ; ˆv )] when all sequences n the nterval contaned between the ponts ˆv and v are consdered whch are monotonc and whose endponts concde wth the endponts of the nterval. Thus the two sdes of the nequalty dffer only n the set of sequences over whch the respectve nfmum operatons are performed. Suppose that contrary to our hypothess nf {v j } S (v ;ˆv ) j=0 v j [ q (v j ; ˆv ) q (v j 1 ; ˆv )] ˆv sd q (s; ˆv ) > 0. (14) By the defnton of the Remann-Steltjes ntegral there exsts a monotonc sequence {v j }n j=0 n the nterval defned by the ponts ˆv and v, wth v 0, vn {ˆv, v } such that [ ] 0 v j q (v j ; ˆv ) q (v j 1 ; ˆv ) sd q (s; ˆv ) < /2. (15) ˆv j=1 19

20 Next, consder the sequence {ṽ j }n j=0 defned as follows: ṽj = vj for all j such that v j V ; otherwse set { max{v ṽ j = V : v < v j } f vj ˆv mn{v V : v > v j } f vj < ˆv. Notce that {ṽ j }n j=0 s well defned (snce V s closed) and entrely contaned n V. Ths means that q (ṽ j ; ˆv ) = q (ṽ j ) for all j. Furthermore, by the defnton of q, and the constructon of {ṽ j }n j=0 we have q (v j ; ˆv ) = q (ṽ j ; ˆv ) for all j. Fnally, notce that v j ṽj for all j f v < ˆv and v j ṽj for all j f v > ˆv. In addton, Q beng mplementable mples that q s nondecreasng along the ncreasng sequence from ˆv to v, when ˆv < v and nonncreasng along the decreasng sequence from ˆv to v, when ˆv > v. Combnng these observatons we get j=1 ṽ j [ ] q (ṽ j ; ˆv ) q (ṽ j 1 ; ˆv ) Usng (14), (15) and (16) we obtan = nf {v j } S (v ;ˆv ) j=0 j=1 j=1 ṽ j v j j=1 v j v j [ q (v j ; ˆv ) q (v j 1 ; ˆv )] [ q (ṽ j ; ˆv ) q (ṽ j 1 ; ˆv ) ] [ ] q (v j ; ˆv ) q (v j 1 ; ˆv ). (16) ˆv sd q (s; ˆv ) (17) sd q (s; ˆv ) (18) ˆv [ ] q (v j ) q (v j 1 ) sd q (s; ˆv ) < ˆv 2, (19) whch s a contradcton. Thus we can conclude that sd q (s; ˆv ) nf v j ˆv {v j } S (v [ q (v j ; ˆv ) q (v j 1 ; ˆv )]. (20) ;ˆv ) j=0 Inequaltes (13) and (20) yeld nequalty (12). Next we show that for any gven par v, ˆv V, ˆv sd q (s; ˆv ) t(v ; ˆv, q ). Recall that t ( ; ˆv, q ) s the largest transfer schedule whch ) mplements q and ) s equal to zero at ˆv. Thus we can verfy the above nequalty by showng that t (v ) = v ˆv sd q (s; ˆv ) consttutes a transfer whch mplements q and vanshes at ˆv. The condton ˆv ˆv sd q (s; ˆv ) = 0 s satsfed by constructon. So t remans to be shown that (q, t ) s ncentve compatble. Suppose that contrary to our hypothess t fals to mplement q. Then there must exst types v, v V such that v q (v ) v ˆv sd q (s; ˆv ) < v q (v ) 20 v ˆv sd q (s; ˆv ),

21 or equvalently v v [q (v ) q (v v )] < sd q (s; ˆv ) sd q (s; ˆv ) = ˆv ˆv v v sd q (s; ˆv ). The ntegral on the rght hand sde of ths nequalty s bounded above by v [q (v ) q (v )] whch s ncompatble wth the strct nequalty sgn. Ths allows us to conclude that (q, t ) s ncentve compatble and therefore ˆv sd q (s; ˆv ) t (v ; ˆv, q ) for all v V. Proof of Proposton 1. ( ) Let q be some ntegrable extenson of q and let t be defned by 7. We need to show that (q, t ) s ncentve compatble. Let v, v V, and v > v (the other case s handled smlarly), then v q (v ) t (v ) [v q (v ) t (v )] = v [q (v ) q (v )] v [q (v ) q (v )] v v v sd q (s) v d q (s) = v [q (v ) q (v )] v [ q (v ) q (v )] = 0, where the frst lne follows from the defnton of t, the second from the fact that q s nondecreasng, and the fourth by the fact that q and q concde on V. ( ) Let (q, t ) be ncentve compatble. Interval [v, v ] can be wrtten as a dsjont unon of V and a countable set of dsjont open ntervals, whch we call holes. Indeed, denote the complement of V n R by V C. The latter s open, thus can be wrtten as a countable unon of dsjont ntervals (we are operatng on the real lne). But then so can be [v, v ]\V. Lkewse there can only be countably many dsjont nontrval connected closed ntervals contaned n V. By Proposton 1 on each of these subntervals, say [w, w ], the transfer dfference t (w ) t(w ), can be wrtten as an ntegral. But now, for every v V the transfer dfferental, t (v ) t(v ) can be splt nto a countable sum of transfer dfferences of whch each ether corresponds to a closed nterval or to boundares of a hole. In the frst case the ntegral s taken wth respect to the actual allocaton rule. In the second case we get the extenson by Lemma 1. The fact that all MRP extensons whch yeld the desred transfer are equvalent almost everywhere follows from nspecton of Lemma 1. Ths lemma mples that the extensons whch delver the same transfer dfferental can only dffer at the pont of dscontnuty,.e. the pont α. Snce the number of holes s countable, as argued n Step 1, the two extensons can only dffer on a countable set of ponts. 21

22 Proof of Proposton 3. Suppose Assumpton 2 s satsfed. Furthermore, suppose ˆv = v. Snce, v x kv n v x kv n for all v V and x X, we have x k for all x, and therefore k 0. Now, suppose ˆv > v. Then, the nequalty v x kv n ˆv x kˆv n, for all x mples k x for all x, thus k 1. For k 1, however, we obtan (ˆv v ) x(ˆv v ) for all x and v, whch s only true f ˆv v for all v. Thus the only two possble cases are k 0 and ˆv = v and k 1 and ˆv = v. To prove the other drecton one can easly verfy that n both cases Assumpton 2 s satsfed. Proof of Proposton 4. Suppose Assumpton 1 holds. Then, for every there exsts a ˆv V such that ˆv q (ˆv ) t (ˆv ; ˆv, q ) φ (ˆv ) v q (v ) t (v ; ˆv, q ) φ (v ), (21) for all v V. Usng the characterzaton from Theorem 2 nequalty 21 s equvalent to φ (v ) φ (ˆv ) v q (v ) ˆv q (ˆv ) ˆv sd q (s; ˆv ), (22) for all v V. Integratng by parts the rght hand sde yelds the desred result. Snce the nequaltes are equvalent the two assumptons are equvalent, and clearly ˆv s from the two assumptons concde. Proof of Proposton 5. Suppose v > w. Then φ (v ) φ (w ) = w v w φ (s)ds q (s; w )ds, where the frst lne follows from φ beng an absoloutely contnuous extenson of φ on [v, v ] and the second from the assumptons of the proposton. Smlarly, for v < w φ (v ) φ (w ) = [φ (w ) φ (v )] = = w v w v v w φ (s)ds q (s; w )ds q (s; w )ds. 22

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