An Introduction to Mechanism Design

Transcription

1 An Introducton to Mechansm Desgn Felx Munoz-Garca School of Economc Scences Washngton State Unversty 1 1 Introducton In ths chapter, we consder stuatons n whch some central authorty wshes to mplement a decson that depends on the prvate nformaton of a set of players. Here are two standard examples: A government agency may wsh to choose the desgn of a publc-works project (e.g., a brdge) based on preferences of ts ctzens. These preferences are, however, unobserved by the government as they are each ctzen s own prvate nformaton. A monopolstc rm may wsh to dentfy the consumers wllngness to pay for the product t produces wth the goal of maxmzng ts pro ts. A seller (auctoneer) sellng an object (e.g., a pantng) to a group of ndvduals, wthout beng able to observe ther wllngness to pay for the object. Mechansm desgn s the study of what knds of mechansms the central authorty (or the monopolst, or the seller, n the above examples) can devse n order to nduce players (e.g., ctzens or consumers n the above examples) to reveal ther prvate nformaton (e.g., preferences for a brdge, or wllngness to pay for a product). For compactness, the central authorty s often referred to as the "mechansm desgner". Model: Mechansms as Bayesan Games Players: Each player = f1; ; :::; ng prvately observes hs type whch determnes hs preferences over the publc project (or hs wllngness to pay over the object for sale n an aucton). The pro le of types for all n players, = ( 1 ; ; : : : ; n ) ; s often referred to as the "state of the world". State s drawn randomly from the state space 1 n. The draw of s accordng to some pror dstrbuton () over. Whle the spec c draw s player s prvate nformaton, the dstrbuton () s common knowledge among all players. Many applcatons assume that every player has quaslnear preferences, whch elmnates wealth e ects. In partcular, a common utlty functon consders that player s utlty s v (x; t; ) = u (x; ) + t 1 I apprecate the suggestons and comments of several students, specally Pak Cho. 1

2 where u (x; ) ndcates player s utlty from consumng x unts of the good (e.g., publc project or good beng sold at an aucton) gven hs ndvdual preference for such good, as captured by parameter. Functon u () could be ncreasng (decreasng) n x X when x represents a good (bad, respectvely), and made concave or convex n x dependng on the applcaton we seek to study. Transfer t s the amount of money gven to (or taken away from) ndvdual : Such a transfer can thus be postve, but can also be negatve f money s taken away from ndvdual (e.g., he pays t to the central authorty n order to fund the publc project). An outcome would be represented as y = (x; t 1 ; ; t N ), whch descrbes, for nstance, the amount of publc project to be provded, x, and the pro le of transfers to each ndvdual (whch allows for some of them to be postve,.e., subsdes, whle other can be negatve,.e., taxes). Mechansm Desgner: that depends on the types of players. The mechansm desgner has the objectve of achevng an outcome For nstance, the seller n an aucton seeks to maxmze hs revenue wthout beng able to observe the valuatons that each bdder has for the good; or a government o cal consderng the constructon of a brdge would lke to maxmze a socal welfare functon wthout observng the preferences of hs consttuents for that brdge. Hence, most of our subsequent dscusson deals wth the ncentves that mechansm desgners can provde to prvately nformed agents (e.g., bdders or ctzens n the above two examples) n order for them to voluntarly reveal ther prvate nformaton. We assume that the mechansm desgner does not have a source of funds to pay the players. That s, the monetary payments have to be self- nanced, whch mples that P n =1 t 0. Hence, when P n =1 t < 0; the mechansm desgner keeps some of the money that he rases from players; np whle f, nstead, t = 0; all negatve transfers collected from some players end up dstrbuted to =1 other players, that s, the budget s balanced. Snce, as de ned above, an outcome s represented as a vector y = (x; t 1 ; ; t N ), the set of outcomes s Y = ( (x; t 1 ; ; t N ) : x X; t R for all N; ) nx t 0 In words, an outcome s an alternatve x X and a transfer pro le (t 1 ; t :::; t n ) such that P n =1 t 0 holds. Fnally, the mechansm desgner s objectve s gven by a choce rule f () = (x () ; t 1 () ; ; t N ()) ; That s, for every pro le of players preferences ; the choce rule f() selects an alternatve x() X and a transfer pro le (t 1 (); t (); :::; t n ()) satsfyng P n =1 t 0: In aucton settngs, x X represents the assgnment of the object for sale, thus becomng a vector x = (0; :::; 0; 1; 0; :::; 0) where 0 ndcates that ndvdual 1; ::; 1 dd not receve the object for sale, as so dd ndvduals + 1; :::; N; whle a 1 ndcates that ndvdual receved the object. For ths reason, n auctons x s referred to as an assgnment or allocaton of the object. =1

3 .1 The Mechansm Game Indrect revelaton mechansm. An ndrect revelaton mechansm (IRM) = fs 1 ; S ; : : : ; S n ; g ()g s a collecton of n acton sets S 1 ; S ; : : : ; S n and an outcome functon g : S 1 S S n! Y that maps the actons chosen by the players nto an outcome of the game: In ths context, a pure strategy for player n the mechansm s a functon that maps hs type nto an acton s S, that s, s :! S : The payo s of the players are then gven by v (g (s) ; ) ; whch depends on the outcome that emerges from the game g(s) when the acton pro le s s; and on player 0 s type (e.g., hs preferences for a publc project). Snce the mechansm rst maps players types nto ther actons, and then ther actons nto a spec c outcome, ths type of mechansm s often referred to as "ndrect revelaton mechansm"; as depcted n gure 11.1a. Fgure 11.1(a). Indrect revelaton mechansm. In a specal class of mechansms, each player 0 s strategy space S s restrcted to concde wth hs set of types,.e., S =. Drect revelaton mechansm. A drect revelaton mechansm (DRM) conssts of = ( 1 ; ; ; N ) and a socal choce functon f() mappng every pro le of types, where = ( 1 ; ; ; N ), nto an outcome x X, f :! X As mentoned above, DRMs can be understood as a specal class of mechansms, n whch each player 0 s strategy space S s restrcted to concde wth hs set of types,.e., S = : In contrast,

4 IRMs requre that, rst, every player chooses a strategy s S ; such as a bd or a producton level, and then all players strateges are mapped nto an outcome. Fgure 11.1b below depcts a DRM, whch could be understood as drectly connectng the two unconnected balloons n the upper part of gure 11.1(a) rather than dong the "de-tour" of rst mappng strateges nto actons, and then actons nto outcomes. Fgure 11.1(b). Drect revelaton mechansm. Examples of DRMs The followng examples explore a settng where a seller (agent 0) seeks to sell an ndvsble object to one of the two buyers (agents 1 and ) so that the set of players s N = f0; 1; g. The set of feasble outcomes s X = f(y 0 ; y 1 ; y ; t 0 ; t 1 ; t ) : y f0; 1g where X y = 1 and t R 8 Ng; =0 In words, the object s assgned to ether the seller, y 0 = 1, buyer 1, y 1 = 1, or buyer, y = 1; and a transfer t s proposed to player, f t > 0, or a tax s mposed on hm, f t < 0. (At ths pont, we do not requre the mechansm to be budget balanced, whch would mply that postve and negatve transfers o set each other at the aggregate level, P =1 t = 0. We return to the budget balance property n further sectons.) For an outcome x n the above set of feasble outcomes,.e., x X, every buyer s utlty s u (x ; ) = y + t for all = f1; g where represents the buyer s valuaton for the object (whch buyer only enjoys f the object s assgned to hm,.e., y = 1); and t s the postve (or negatve) transfer he receves (or pays). Example Drect revelaton mechansm: Consder a settng n whch the seller asks buyers 1 and to smultaneously and ndependently reveal ther types (ther valuaton for the object), ^ 1 and ^, and the seller assgns the object to the agent wth the hghest revealed valuaton ^. Wthout loss of generalty, we assum that f there s a te, the object s assgned to buyer 1. More formally, for every pro le of announced types, 4

5 ^ = (^1 ; ^ ), the assgnment rule of ths drect revelaton mechansm s y 0 (^) = 0 and y (^) = ( 1 f ^ ^ j 0 otherwse where = f1; g and the transfer (or payment) rule s t (^) = ^ y (^) where = f1; g and t 0 (^) = [t 1 (^) + t (^)] = ^ 1 y 1 (^) + ^ y (^) In words, f player player reports a larger valuaton than hs rval, ^ ^ j, he s assgned the object, y (^) = 1, payng a transfer equal to hs reported valuaton ^,.e., t (^) = ^ 1 = ^. In contrast, hs rval j does not receve the object, y j (^) = 0, thus entalng a zero transfer t j (^) = 0. Fnally, the seller receves the sum of the transfers, whch n ths settng s equvalent to the transfer pad by the ndvdual who receves the object, that s, t 0 (^) = t (^). Example 1. - Drect revelaton mechansm (varaton of Example 1.1) Buyer 1 and report ^ 1 and ^ to the seller, the seller assgns the object to the buyer wth the hghest announced report ^ (that s, we use the same allocaton rule y (^) for = f0; 1; g as n the prevous example), but the payment rule d ers: t (^) = ^ j y (^) and t 0 (^) = [t 1 (^) + t (^)] Intutvely, f player reports a larger valuaton than hs rval, ^ ^ j, he s assgned the object, y () = 1, but pays the second hghest reported valuaton, ^ j. A smlar argument extends to settngs wth N players, where t (^) = max j6= f^ j g y (),.e., player, f he s assgned the object, pays a prce equal to the hghest competng reported valuaton. Example 1. - Procurement contract Consder a seller (0) and buyers 1 and, wth the set of outcomes X beng the same as that n all prevous examples, and the same utlty functon. However, the assgnment rule s now reversed, as the seller seeks to assgn the servce (e.g., publc water management) to the rm reportng the lowest cost. That s, the assgnment rule spec es y 0 (^) = 0 5

6 mplyng that the seller never keeps the object, and y (^) = ( 1 f ^ ^ j 0 otherwse for every = f1; g That s, the procurement contract s assgned to the rm announcng the lowest cost, ^ ^ j : Fnally, the transfer rule concdes wth that n Example 1.1 (f the wnnng agent s pad hs costs) or wth that n Example 1. (f the wnnng agent s pad the cost of the losng rm). Example.1 - Fundng a publc project A set of ndvduals N = f1; ; ; ng seek to buld a brdge. Let k = 1 ndcate that the brdge s bult, and k = 0 that t s not. The cost of the project s C > 0. Let t be a transfer to agent, so t s a tax pad by agent. The project s then bult, k = 1, f total tax collecton exceeds the brdge s total cost C P n =1 t, but t s not buld otherwse. (Alternatvely, kc P n =1 t captures both the case n whch the brdge s bult and the case t s not.) The set of outcomes, X, n ths settng s then X = ( (k; t 1 ; t ; ; t n ) : k f0; 1g; t R; and kc ) nx t where N where, as usual n other sets of outcomes, spec es the assgnment rule k followed by transfer rule to each agent N (whch are allowed to be taxes snce t R s not restrcted to be postve). Utlty functon for every agent s =1 u (k; t ; ) = k + t where can be nterpreted as agent s valuaton of the project. Note that agent only enjoys such a valuaton f the brdge s bult, k = 1, and that we allow for agent to pay taxes f t < 0. Example Drect revelaton mechansm n the publc project In ths case, the mechansm asks agents to drectly report ther types (.e., ther prvate valuaton for the brdge). In other words, the game restrcts every player s strategy set to concde wth hs set of types, S =. In ths settng, the socal choce functon maps the reported (announced) pro le of types ^ (^ 1 ; ^ ; ; ^ n ) nto an assgnment rule and a transfer rule. In partcular, the assgnment rule spec es k(^) = ( 1 f P n =1 ^ C 0 otherwse.e., the project s bult f and only f the aggregate reported valuaton of all agents exceeds the 6

7 project s cost. In addton, the transfer rule of ths mechansm s t (^) = C n k(^).e., f the project s bult, k(^) = 1, then every agent bears an equal share of ts cost, C n ; but f the project s not bult k(^) = 0; no agent has to pay anythng,.e., t (^) = 0 for all agents : Implementaton.1 Testng the mplementablty of SCF n drect revelaton mechansm Let us test the mplementablty of the scf descrbed n Example 1.1 above. Suppose 1, U[0; 1] and..d. In order to test f truthfully reportng hs type 1 = ^ 1 ; s a weakly domnant strategy for player 1, let s assume that player truthfully reports hs type, so hs equlbrum strategy s ^ s ( ) = and check for pro table devatons for player 1. (Recall that ths s the standard approach to test whether a strategy pro le s an equlbrum, where we x the strateges of all N 1 players and check f the remanng player has ncentves to devate from the proposed equlbrum strategy.) In partcular, player 1 solves max ^1 ( 1 p) probfwng = ( 1 ^1 ) probf ^ 1 g where 1 ^1 represents the margn that player 1 keeps by under-reportng hs valuaton of the object (whch helps hm obtan the good at a lower prce), whle probf ^ 1 g denotes the probablty that player 1 wns the object because he reveals a larger valuaton than player to the seller. Snce U[0; 1], then probf ^ 1 g = F (^) = ^ 1, whch reduces player 1 s problem to max ( 1 ^1 ) ^ 1 = 1^1 ^ 1 ^1 Takng FOCs wth respect to ^ 1 yelds 1 ^ 1 = 0. Solvng for ^, we obtan an optmal announcement of ^1 = 1 (An analogous argument apples to player : f player 1 truthfully reports hs type, ^ 1 = 1, then player s optmal report s ^ =.) Hence, the SCF n Example 1.1 s not mplementable as a DRM snce t doesn t nduce every player to truthfully report hs type to the seller. 7

8 . Incentve Compatblty Therefore, player 1 shades hs valuaton n half, not truthfully reportng hs type to the seller, so ^1 s 1 ( 1) 6= 1. As suggested by Example 1.1, players may not have ncentves to truthfully report ther types n DRMs. Ths s, however, a desrable property that the mechansm desgner wll try to guarantee n order to extract nformaton from the agents. When a SCF nduces prvately nformed players to truthfully report ther types n equlbrum, we refer to such SCF as Incentve Compatble. We can, nonetheless, consder two types of ncentve compatbltes dependng on whether truthtellng s an equlbrum n domnant strateges, or a Bayesan Nash Equlbrum (BNE) of the ncomplete nformaton game. Bayesan Incentve Compatblty, BIC: A SCF f() s BIC f the DRM D = (( ) N ; f()) has a BNE (s 1 ( 1); ; s n( n )) n whch s ( ) = for all and all N. That s, every player nds truthtellng optmal, gven hs belefs about hs opponents types, and gven that all hs opponents strateges are xed at truthtellng, s ( ) =. More formally, BIC entals that for every player N and every type, and, E [u (f( ; ); )j ] E u (f( 0 ; ); )j for every msreport 0 6=. Ths nequalty just says that player ; prefers to truthfully report hs type ; yeldng an outcome f( ; ) than msreportng hs type to be 0 6= ; whch would yeld an outcome f( 0 ; ): Importantly, player prefers to truthfully reveal hs type n expectaton, as he doesn t observe the pro le of types of hs rvals : As a consequence, the above de nton could allow player to nd truthtellng optmal for some values of hs rvals types ; but not for others as long as n expectaton he prefers to truthfully report hs type : The followng verson of ncentve compatblty s more demandng, as t requres player to nd truthtellng optmal regardless of the spec c realzaton of ths rvals types ; and regardless of hs rvals announcements. That s, we next focus on SCFs for whch truthtellng becomes a domnant strategy for every player N: Domnant Strategy Incentve Compatblty, DSIC: A SCF f() s DSIC f the DRM D = (( ) N ; f()) has a domnant strategy equlbrum (s 1 ( 1); ; s n( n )) n whch s ( ) = for all and all N. Therefore, every player nds truthtellng optmal regardless of hs belefs about hs opponents types, and ndependently on hs opponents strateges n equlbrum,.e., both when they truthfully report ther types, s ( ) =, and when they do not, s ( ) 6=. More formally, DSIC entals that for every player N and every type he may have, u (f( ; s ); ) u (f( 0 ; s ); ) 8

9 where s S, for all 0 6=. Then, DSIC s a more demandng property than BIC, n partcular, DSIC requres that players nd truthtellng optmal regardless of the spec c types of ther opponents and ndependently on ther spec c actons n equlbrum. In contrast, BIC asks for truthtellng to be utlty maxmzng only n expectaton and gven that all other players are truthfully reportng ther types. In addton, note that DSIC requres that player nds t optmal to truthfully reveal hs type both when hs rvals choose equlbrum strateges,.e., when they truthfully report ther types and thus s =, but also when they don t,.e., when they msreport ther types, s 6=. Fnally, DSIC s often referred to as "strategy-proof" or "truthful", snce players cannot nd an alternatve strategy (msreportng ther types) that would yeld a larger payo. 4 Indrect Revelaton Mechansm An ndrect revelaton mechansm (IRM) allows strategy spaces to d er from a drect announcement of types,.e., S 6=, or to concde, S =, for every player N. In that regard, a DRM can then be nterpreted as a specal case of IRM whereby players strateges are restrcted to concde wth ther type space,.e., when S = we only allow players to report a type (ether truthfully or msreportng) but they cannot do anythng else. In contrast, n an IRM players can potentally choose from a rcher strategy space. Once every player chooses hs strategy S, and a pro le of strateges emerges s = (s 1 ; s ; :::; s n ), the IRM maps such strategy pro le s (s 1 ; s ; ; s n ) nto an outcome g(s). The equlbrum that arses n the IRM has every player ; choosng a strategy as a functon of prvately observed type, s ( ); yeldng an equlbrum strategy pro le s () = (s 1 ( 1); ; s n( n )) : Such strategy pro le entals an equlbrum outcome g(s ()): A natural queston s whether the equlbrum oucome g(s ()) emergng from the IRM, whereby everyplayer freely choose an acton whch ultmately gves rse to an outcome of the game. For completeness, we explore ths concdence n outcomes (whch s referred to as that the IRM mplements the planner s SCF) rst usng domnant strateges and then usng BNE (as for ncentve compatblty). 4.1 Implementaton n Domnant Strateges A mechansm M = ((S ) N ; g()) mplements the SCF f() n domnant strategy equlbrum f there s a weakly domnant strategy pro le s () = (s ( 1); ; s n( n )) of the Bayesan game nduced by the mechansm M such that g (s ()) = f() for all Example: Second-prce auctons mplement the SCF n Example 1. n weakly domnant strat- 9

10 egy equlbrum. In partcular, the strategy set for every bdder s hs set of feasble bds, whch n the case of postve bds wthout the exstence of a reservaton prze smpl es to S = R +. In ths context, we showed that every bdder nds that a bd of s ( ) = (bds concdng wth hs valuaton) consttutes a weakly domnated strategy n the second-prce aucton,.e., he would choose t regardless of hs opponents valuatons for the object and ndependently of ther bddng pro le s. Hence, the object s assgned to the bdder submttng the hghest bd, who pays a prce equal to the second hghest bd. Ths outcome that concdes wth the SCF n Example 1. whereby the socal planner could observe all bdders valuatons,. 4. Implementaton n BNE A mechansm M = ((S ) N ; g()) mplements the SCF f() n BNEs f there s a BNE strategy pro le s () = (s ( 1); ; s n( n )) of the Bayesan game nduced by the mechansm M such that g (s ()) = f() for all Example: Recall that the SCF of Example 1. s BIC, snce for every, the equlbrum strategy sats es truthtellng,.e., s ( ) = for all N. In addton, we can use FPA as an IRM that mplements SCF of Example 1. n ts BNE. The above dscusson suggests a connecton between the outcomes of a DRM that nduces truthtellng and an IRM. In partcular, we mght wonder f, for a gven SCF mappng pro les of types nto socally desrable outcomes, we can desgn a clever game (a IRM) n whch equlbrum play would yeld the exact same outcome as that dent ed by the SCF. The answer s postve (although we dscuss some dsadvantages later), and t s known n the lterature as the Revelaton Prncple. The next sectons separately present t for the cases of BNE and domnant strateges. Fgure 11. depcts the revelaton prncple by combnng left and rght panels of gure In the upper part of the gure llustrates a drect revelaton mechansm mappng types nto outcomes through a socal choce functon. The lower part, n contrast, takes an "ndrect route" by rst allowng every player to map hs own type nto a strategy,.e., s ( ) for every N; and then takng the acton pro le and mappng nto an outcome of the game. The queston that the revelaton prncple asks s then whether we can nd game rules that provde players wth the ncentves to choose strateges that ultmately lead to outcomes concdng wth those selected by a socal choce 10

11 functon. Fgure 11.. The Revelaton Prncple 4. Revelaton Prncple - I: BNE Approach A mechansm M mplements f() n BNE f and only f f() s BIC. Proof: Snce the "f and only f" clause means that: (1) Mechansm M mplements f(:) n BNE ) f(:) s BIC; and () f(:) s BIC ) mechansm M mplements f(:) n BNE, we next show both lnes of mplcaton. (() If f() s BIC, then t can also be mplementable n BNE by the drect revelaton mechansm n whch we restrct every player s strategy set to concde wth hs set of types, S =. ()) If mechansm M mplements f() n BNE, then there exsts a BNE of the IRM (s 1 ( 1); ; s n( n )) such that g (s 1( 1 ); ; s n( n )) = f( 1 ; ; n ) for all : Snce strategy pro le (s 1 ( 1); ; s n( n )) s a BNE, then E u g s ( ); s ( ) ; j E u g s ; s ( ) ; j for all s S, all, and all N. Note that a devatng strategy s on the rght hand sde of the nequalty could be s (0 ) so player uses the same strategy functon as n the left-hand sde but evaluatng t at a msreported type 0 6=. Combnng the above two nequaltes yelds E [u (f( ; ); ) j ] E u f( 0 ; ); j 11

12 for all 0 6=, all, and all N, whch s exactly the condton that we need for SCF f() to be BIC. (Q.E.D.) 4.4 Revelaton Prncple - II: DSIC Approach A mechansm M mplements f() n domnant strategy equlbrum f and only f f() s DSIC. Proof: In ths case we also need to show both drectons of the "f and only f" clause. (() Identcal as the rst step of the above proof. ()) Smlar to the prevous proof, but we do not need that every player takes expectatons of hs opponents types, and we don t need hm to x hs opponents strateges n equlbrum, s ( ), but nstead he consders any strategy of hs opponents, s ( ). As a practce, let us develop the proof. If M mplements f() n domnant strategy equlbrum (DSE), there exsts a weakly domnant BNE, (s 1 ( 1); ; s n( n )) such that g (s 1( 1 ); ; s n( n )) = f( 1 ; ; n ) for all : Snce strategy pro le (s 1 ( 1); ; s n( n )) s a domnant strategy equlbrum of the mechansm M, u (g (s 1( 1 ); s ( ))) u (g (s ; s ( )) ; ) for all s S, all, all, all s S, and all N. In words, player does not have ncentves to devate,.e., of choosng a strategy s = s ( ); for any type he may have, any pro le of types hs opponents may have, ; and for any strategy pro le they may choose s ( ): Smlarly as n the above proof, the devatng strategy s on the rght-hand sde of the nequalty could be s (0 ) whereby player uses the same functon as n the left-hand sde but evaluated at a msreported type 0 6=. Combnng the above condtons yelds u (f( ; ); ) u f( 0 ; ); whch exactly concdes wth the condton that we need for the SCF to be DSIC. (Q.E.D.) In summary, the revelaton prncple n ts two versons tells use that A mechansm M mplements f() n BNE () f() s BIC A mechansm M mplements f() n DSE () f() s DSIC Hence, f a mechansm s not BIC or DSIC, (.e., tellng the truth s not an equlbrum n the 1

13 DRM), then we cannot nd a clever game or nsttutonal settng (an IRM) that mplements such a SCF f(). Alternatvely, f a mechansm M s BIC, we can nd an IRM that mplements f() n BNE. Smlarly, f a mechansm s DSIC, we can nd an IRM that mplements f() n DSE. 5 VCG mechansm In most of the sectons hereafter we consder the followng quaslnear preferences v (k; ) = u (k; ) + w + t where k K descrbes, as usual, the assgnment rule (e.g. k = f0; 1g representng f a publc project s mplemented, k = 1; or not k = 0). In standard settngs,, wealth s strctly postve, w > 0, and t > 0 denotes that player receves a net transfer whle t < 0 ndcates that he pays to the system. In addton, P N t 0 ndcates budget balance. In partcular, f such condton holds wth equalty, we refer to t as strong budget balance, whle otherwse we refer to t as weak budget balance snce t allows for the system to run a de ct (or a surplus) at the aggregate level. 5.1 Allocatve e cency We say that a SCF f() = (k(); t 1 (); ; t n ()) sats es allocatve e cency f, for every pro le of types, the allocaton functon k() sats es k() X arg max u (k; ) kk N That s, k() allocates objects (or publc projects) n order to maxmze aggregate payo s for each pro le of types,. The followng examples test whether the allocaton functon k() n two d erent SCF sats es allocatve e cency (AE). Example.1 - Publc project wth an allocatve e cent SCF Consder a settng wth two agents N = f1; g each wth two types = f1; g. Ther utlty functon s = f0; 60g for all u (k; ) = k( 5) whch ndcates that f the project s not mplemented, k = 0, agents utltes are zero; but f t s mplemented, k = 1, both agents bear an equal cost of 5. Consder the followng allocaton Hence, allocatve e cency s analog to Pareto e cency. However, snce most mechansm desgn problems deal wth the allocaton of property rghts (e.g., auctons and procurement contracts) and the mplementaton of publc projects, we normally use the concept of allocatve e cency. 1

14 functon k( 1 ; ) = ( 0 f 1 = = 0 1 otherwse thus ndcatng that f both ndvduals valuatons are low (0), then the project s not mplemented, but f at least one ndvdual s valuaton s hgh (60), then the project s mplemented. The next table consders all possble type pro les, and the utltes the agents obtan gven the above allocatve functon k(). ( 1 ; ) k() u 1 (0; 1 ) u (0; ) u 1 (1; 1 ) u (1; ) u 1 (1; 1 ) + u (1; ) (0; 0) (0; 60) (60; 0) (60; 60) Hence, the SCF wth the above allocaton functon k( 1 ; ) and transfer functon ( 0 f 1 = = 0 t ( 1 ; ) = 5 otherwse s allocatve e cent. To see ths, note that for pro le of types ( 1 ; ) = (0; 0) (n the rst row), the total utlty of mplementng the publc project s negatve and thus lower than that of not mplementng t (whch s zero). The allocaton functon k() correctly selects k() = 0 snce n ths case not mplementng the publc project s welfare maxmzng. In contrast, for all remanng type pro les (rows to 4), the total welfare from mplementng the project s postve, and thus larger than from not mplementng t. In all of these type pro les, the allocaton functon selects k() = 1, thus mplementng the project. Example. - Publc project wth an allocatve ne cent SCF Consder now a utlty functon u (k; ) = k for all agents = f1; g: That s, we stll consder the same quaslnear preference as n the above example, but the project s now costless,.e., t () = 0 for all and all N. Gven such a change n the transfer functon (and thus n the SCF), the SCF s no longer allocatve e cent. For the SCF to be allocatve e cent, t should mplement the project, k() = 1, regardless of the type pro le. For nstance, when ( 1 ; ) = (0; 0), the allocaton functon determnes that k() = 0, whch yelds P N u (0; ) = However, mplementng k() = 1 would yeld n ths case a total welfare of P N u (1; ) = = 40 snce the project s now costless. Intutvely, even f agent s don t assgn a hgh value to the project, the aggregate value they obtan s stll postve (.e., 40 or hgher), whch would always exceed ts (zero) cost from developng the project. Snce the allocaton 14

15 functon k() descrbed above does not mplement the project when both ndvduals valuatons are low,.e., when ( 1 ; ) = (0; 0); we can conclude that allocaton functon k(), and thus the SCF, are not allocatve e cent. 5. Ex-post e cency and Quaslnear preferences We say that a SCF f() s ex-post e cent f, for every type pro le, the outcome chosen by the SCF, f() = x; maxmzes the sum of all agents utltes. That s, X u (f(); ) X u (x; ); for all feasble outcomes x X N N We check that from an ex-post perspectve: after observng all players types n vector. An nterestng property of ex-post e cency s that, under quaslnear preferences, t s equvalent to sayng that the SCF s allocatve e cent and budget balanced, as we show n Appendx 1. 6 Examples of common mechansms We next present some famous mechansms extensvely used n theoretcal and appled lterature. In partcular, we are nterested n showng that the SCF they mplement sats es AE,.e., we cannot nd alternatve outcomes that could ncrease socal surplus, and DSIC,.e., agents nd t optmal to truthfully reveal ther prvate nformaton to the mechansm desgner ndependently on what ther rvals do. 6.1 Groves Theorem Let the SCF f() = (k(); t 1 (); ; t n ()) satsfy AE. Then f() sats es DSIC f transfer functons can be represented by t ( ; ) = X u j (k(); j ) + h ( ) j6= where h :! R s an arbtrary functon. Intutvely, the transfer that player receves depends on the utlty that all other agents experence from the pro le of announced types,.e., the externalty that player s announcement causes on ther well-beng (as the allocaton rule consders the entre pro le of preferences ), plus a functon h ( ) whch s ndependent on player s announcement. If player changes hs report from 15

16 to 0, hs transfer changes n the externalty that he mposes on all other agents. In partcular, t ( ; ) t ( 0 ; ) = X j6= uj (k( ; ); j ) u j k( 0 ; ); j Let us now show that such a transfer functon entals DSIC. Proof: By contradcton. Suppose that a SCF f() sats es AE and ts transfer functon can be represented à la Groves as stated above, but t s not DSIC. That s, there s at least one agent for whch msreportng hs type s convenent, that s, u f( 0 ; ); > u (f( ; ); ) n at least one of hs types, and one pro le of hs rvals types, where 0 6=. Gven quaslnearty, we can expand ths nequalty yeldng u k( 0 ; ); + t ( 0 ; ) + w > u (k( ; ); ) + t ( ; ) + w We can now plug the transfer from the Groves theorem, t ( 0 ; ) = X j6= u j k( 0 ; ); j + h ( ) and smlarly for t ( ; ). Hence, the above nequalty becomes u whch smpl es to k( 0 ; ); + X j6= u j k( 0 ; ); j > u (k( ; ); ) + X u j (k( ; ); j ) j6= t ( 0 ; ) X N u k( 0 X ; ); > u (k( ; ); ) N t ( ; ) entalng that the SCF f() s not AE snce t doesn t maxmze total surplus,.e., allocaton k( 0 ; ) yelds a larger socal welfare. Hence, f SCF f() s AE and transfers can be expressed a la Groves, the SCF s DSIC. (Q.E.D.) For nstance, f player changes hs report from to 0, hs transfer changes n the externalty 16

17 that he mposes on all other agents. In partcular, 4 X u j (k( ; ); j ) + h ( ) 5 j6= 4 X u j (k( 0 ; ); j ) + h ( ) 5 j6= = X j6= u j (k( ; ); j ) = X j6= uj (k( ; ); j ) X u j (k( 0 ; ); j ) j6= u j (k( 0 ; ); j ) 6. Clarke (Pvotal) mechansms Ths type of mechansms consttute a specal class of Groves mechansms descrbed above, n whch the functon h ( ) takes the form h ( ) = X u j (k ( ); j ) for all ; and for all N j6= where k ( ) denotes the allocaton that the SCF selects when consderng all agents j 6=,.e., as f player was absent. Hence, the transfer becomes t () = X u j (k(); j ) + h ( ) j6= = X j6= u j (k(); j ) X u j (k ( ); j ) j6= Clarke h ( ) functon for all N Intutvely, the rst term represents the total value that all j 6= agents obtan when the seller (mechansm desgner) consders player s preferences when allocaton k() s beng determned. The second term, n contrast, descrbes the total value that they obtan when the seller gnores player s preferences, so the allocaton becomes k( ). Therefore, the d erence between both terms captures the margnal contrbuton that player s preferences have on the mechansm s allocaton. In ths sense, the Clarke mechansm s pvotal, as every ndvdual plays a pvotal role n determnng the transfer that other players receve (or pay) by havng player partcpatng n the mechansm. Example of VCG mechansm - I Consder 5 bdders partcpatng n a second prce aucton (SPA), whose valuatons v 1 = 0; v = 15; v = 1; v 4 = 10; v 5 = 6 Hence, submttng a bd equal to hs valuaton, b (v ) = v for all v and all N; s a BNE of the 17

18 game. If, nstead, a VCG mechansm was used, player 1 s transfer would be t 1 () = X j6=1 u j (k(); j ) = 0 15 = 15 X u j (k 1 ( 1 ); j ) j6=1 In the rst term, the allocaton rule consders the valuaton of all the bdders. Then, the object would be assgned to bdder 1 entalng a value of =0 to the other j 6= 1 bdders. The second term, n contrast, gnores bdder 1 s preferences (valuaton), thus assgnng t to bdder (as he s now the player wth the hghest valuaton). Bdder s utlty from recevng the good s 15, mplyng that the sum of valuatons s now =15. The d erence between the two terms yelds a transfer of t 1 () = 0 15 = 15, thus ndcatng that player 1 pays 15,.e., the second largest valuaton. A smlar argument apples to all other players. However, snce ther valuatons are lower than that of player 1, ther transfers become t () = 0 0 = 0 for all 6= 1 (show t as a practce). Importantly, the VCG mechansm leads to the same outcome (the object s allocated to the bdder wth hghest valuaton) and transfer pro le (the ndvdual recevng the object pays a transfer equal to the valuaton of the ndvdual wth the second hghest valuaton, whle everyone else pays zero) as the SPA (whch s an IRM). Example of VCG mechansm - II Consder the same bdders as n the prevous example, wth the same valuatons. However, allow for dentcal tems to be avalable n the aucton. Each bdder wants only one tem. In ths context, the transfer to player 1 becomes t 1 () = X X u j (k(); j ) u j (k 1 ( 1 ); j ) j6=1 j6=1 = (15 + 1) ( ) = 10 When the valuaton pro les of all players s taken nto account n the allocaton rule, k(); the three avalable tems are assgned to the players wth the hghest valuaton: player 1, and. The rst term, however, measures the utlty that players j 6= 1 obtan from such an allocaton,.e., the valuatons of player and, (15 + 1). In the second term, we stll measure the utlty of players j 6= 1 but gnorng player 1 s preferences. In ths case, the three tems go to the player wth the hghest valuaton (player, and 4) yeldng a total utlty of ( ). As a result, the transfer that player 1 has to pay s -$10, ndcatng that, f hs preferences were consdered he would mpose a negatve externalty of -$10 on the remanng players. Ths externalty captures the utlty loss that player 4 su ers as he would get one object when player 1 s preferences are gnored (enjoys a utlty of 10) but he does not receve any object when the preferences of player 1 are consdered. Example of VCG mechansm - III. See Tadels, pp

19 7 Groves mechansm and budget balance (techncal) Is the Groves mechansm budget balanced? Not necessarly. As the next result from Green and La ont 4 (1979) shows, f the set of possble types s su cently rch, no socal choce functon sats es DSIC and ex-post e cent (whch would requre a k() functon maxmzng total surplus and transfers beng budget balanced, P N t () = 0.) Green-La ont mpossblty theorem. Suppose that for each agent N, that F = fv (; ) such that g: that s, every possble valuaton functon from k to R arses for some : Then, there s no SCF that s DSIC and ex-post e cent (EPE). In other words, ether agents have to overpay, P N t () < 0 for some, or have an ne cent project selecton,.e., a project for whch we could nd an alternatve allocaton k 0 6= k () that yelds a larger total surplus. Some good news: f the preferences of at least one agent are common knowledge (such as the seller n an aucton), then we can nd SCFs that satsfy DSIC and EPE (and hence BB), as we next show. Budget balance of Groves mechansms: If there s at least one agent whose preferences are known (that s, hs type set s a sngleton) then t s possble to dentfy a functon h () n the Groves mechansm that yelds BB,.e., P N t () = 0. Proof: Let agent 0 s preferences be known, 0 = f 0 g. In ths settng, EPE holds when we choose transfer functons (t 1 () ; :::; t N ()) for the N agents whose preferences are unknown, as long as they satsfy t 0 () = X t () for all 6=0 That s, f P N t () < 0 then agent 0 receves the total transfers of all other N ndvduals, and f P N t () > 0 agent 0 pays the de ct n contrbutons by the N ndvduals. Intutvely, agent 0 can be understood as the a government agency that absorbs surpluses or compensates for de cts. (Q.E.D.) 8 Partcpaton constrants Thus far we assumed that all agents partcpated n the mechansm, as f partcpaton was compulsory by some government agency. But what f ther partcpaton s voluntary? We then need to add partcpaton constrants (PC) to each agent wth type. 4 Green, J.R. and La ong, J. J. (1979). Incentves n Publc Decson Makng (Amsterdam: North-Holland). 19

20 We wll next present d erent approaches to wrte the PC, dependng on the nformaton that the agent knows when the PC constrant s de ned: Before he knows hs type (ex-ante stage); After knowng hs type, but wthout observng hs opponents type (nterm stage); and After knowng hs type, and the announcements of all other ndvduals (ex-post stage) Usng u ( ) to denote agent s reservaton utlty, the PC n the above three stages becomes Ex-ante PC: E [u (g( ; ); )] E [u ( )] Interm PC: E [u (g( ; )j )] u ( ) for all Ex-post PC: u (g( ; ); ) u ( ) for all ( ; ) At the ex-ante stage, ndvdual takes expectatons of both hs own type, ; and hs rvals, ; snce he could not observe hs own type yet. At the nterm stage, he only takes the expectatons of hs rvals types, ; whle at the ex-post stage he does not need to take expectatons snce all the type pro les = ( ; ) have been revealed. As you can antcpate, for any SCF g () Ex-post PC ) Interm PC ) Ex-ante PC whch occurs because the ex-post de nton s more demandng (for all ( ; ) pars) than the nterm de nton (for all ), and both are more demandng than the ex-ante de nton. In the followng subsectons we apply the above PC de ntons to d erent settngs, such as under a groves mechansm, and under a Clarke mechansm, among others. 8.1 Partcpaton constrants n the VCG mechansm Example 1 - Publc good project Consder a socety wth two ndvduals N = f1; g. A publc project s ether mplemented or not, k = f0; 1g; and both ndvduals prvate valuatons for the project are drawn from 1 = = f0; 60g. Fnally, the total cost of buldng the project s 50. In ths settng, the set of feasble outcomes s X = f(k; t 1 ; t ) : k = f0; 1g; t 1 ; t R; (t 1 + t ) 50g That s, allocaton rules k = f0; 1g and transfer rules that guarantee total payments of $50. Consder the allocaton functon we consdered n prevous sectons for ths example (where the 0

21 project s mplemented f at least the valuaton of one ndvdual s 60), whch we reproduce below: ( k 0 f 1 = = 0 ( 1 ; ) = 1 otherwse and de ne the same valuaton functon as n prevous secton v (k ( 1 ; ) ; ) = k ( 1 ; ).& 1 0 ( 5) margn for all 1 ; Recall from prevous sectons that such allocaton rule s AE. From the Groves theorem, we know that f the transfer functon s à la Groves then the resultng SCF sats es DSIC. Let us now check f, despte beng DSIC, such SCF volates ex-post PC. In partcular, assume that reservaton utlty s u ( ) = 0 for all and for all N. Hence, for ex-post PC, we need u (g ( ; ) ; ) 0 for all 1 1 ; and all In the case that ( 1 ; ) = (0; 60), such condton requres whch reduces to v 1 (k (0; 60) ; 0) + t 1 (0; 60) t 1 (0; 60) 0, or t 1 (0; 60) 5. Now consder a d erent pro le of types ( 1 ; ) = (60; 60). Snce SCF s DSIC, we need truthtellng, v 1 (k (60; 60) ; 60) =5 + t 1 (60; 60) 5 v 1 (k (0; 60) ; 0) =5 =) + t 1 (0; 60) # 5 Intutvely, msreportng doesn t a ect the probablty of the publc project beng mplemented, nor player 1 s valuaton for the publc project. Hence, the project s nfeasble snce total transfers fall short of the total cost, t 1 (0; 60) + t (60; 60) 10 < 50 = total cost. 8. Partcpaton constrants n Clarke mechansm Clarke mechansms satsfy ex-post PC f they satsfy the followng propertes: 1. Reservaton utlty s zero, u ( ) = 0 for all. The mechansm sats es "choce set monotoncty": The set of feasble outcomes X weakly grows n N. The ntuton behnd ths assumpton s that the choce set X becomes wder as more agents enter the populaton. 1

22 . The mechansm sats es "no negatve externalty": Formally, the utlty that player obtans a postve utlty when hs preferences are gnored, v k ( ) ; 0 where allocaton k ( ) s AE for all ; all ; and all N. In words, player obtans a postve value from the allocaton that emerges when hs preferences are gnored. Otherwse, the preferences of all other agents would lead to an allocaton k ( ) that mposes a negatve externalty on player. Let us next show why the above three propertes help guarantee that the Clarke Mechansm sats es ex-post PC. Proof: Recall that, gven the transfer functon n the Clarke mechansm, the utlty functon u (g(); ) becomes u (g (); ) = v (k (); ) + 4 X X v j (k (); j ) v j k ( ); j 5 j6= j6= = X v j (k (); j ) j the rst two terms n the above expresson X j6= v j t ( ; ) k ( ); j From choce set monotoncty, the choce wth agent, k (), must generate the same or more total value than the choce wthout hm, k ( ). Hence, the above expresson becomes (where we only changed the rst term n the rght-hand sde) u (g (); ) X j v j k ( ); j X j6= v j k ( ); j In addton, the rght-hand sde smpl es to v (k ( ); ) snce the rst term n the rght-hand sde of the above expresson ncludes utlty of agent whle the second term does not. Therefore, the above expresson reduces to u (g (); ) v (k ( ); ) 0 = u ( ) where the 0" nequalty orgnates from the "no negatve externalty" property. Hence, u (g (); ) u ( ) holds for all, as requred for the SCF to satsfy ex-post PC. (Q.E.D.) Examples of ex-post mechansms are the rst prce aucton, and the second prce aucton (whch, as shown above, s a specal case of Clarke mechansm). Check the ex-post e cency of these two aucton formats as a practce.

23 8. dagva (expected externalty) mechansms From our prevous dscusson, mechansms satsfyng all three propertes, DSIC+AE+BB, were really d cult to nd. We could relax AE or BB, as n some prevous examples, but why not relax DSIC, replacng t wth the mlder requrement BIC? Recall that, ntutvely, DSIC requres every player to nd truthtellng optmal for all of hs opponents types and strateges,.e., even f they choose o the equlbrum strateges. However, under BIC, every player nds truthtellng optmal when hs opponents strateges are n equlbrum, and when he takes the expectaton of hs utlty over all possble types of hs opponents. As a consequence, BIC can hold even f DSIC does not for some values of or some strateges s S : Ths s the approach of d Aspremont, Gerard-Varet and Arrow mechansm (dagva, for compactness). Consderng, for smplcty, a quaslnear envronment where agents types are..d., the dagva mechansm guarantees AE, BB and BIC. dagva Theorem. Let a SCF be AE and types be..d. Ths SCF s BIC f the transfer functon can be expressed as t ( ; ) = " ( ) + h ( ) for all and all N where " ( ) = E 6 4 X j6= v j k ( ); j same as the rst term n the transfer functon of the Groves mechansm expectaton of such a transfer over all possble pro les of s opponents types, 7 5 and where h ( ) s the same arbtrary functon as n the Groves mechansm. Proof: We seek to prove that, f a SCF s AE, types are..d and t () has the above dagva representaton, then the SCF s BIC, that s E [u (g ( ; ) ; ) j j] E u g 0 ; ; j j for all, all 0 6= and every player N. Frst, note that the LHS of the above nequalty can be rewrtten n our quaslnear envronment as E [u (g ( ; ) ; ) j j] = E [v (k ( ; ) ; ) + t ( ; ) j j] where we do not need to condton player s expectaton on hs type snce types are..d. Substtutng the dagva transfer functon nto t ( ; ) (.e., the last term at the rght-hand

24 sde) yelds E 6 v (k ( ; ) ; ) + h ( ) + E 4 X v j (k ( ; ); j ) j6= 5 t ( ; ) whch smpl es to E 4 X v j (k ( ; ); j ) 5 + E [h ( )] jn We can now use the property that allocaton k () s AE, thus mplyng a larger total surplus X v j (k ( ; ); j ) X jn jn v j k ( 0 ; ); j for all 0 6=. (In words, total surplus when all agents truthfully report ther types s larger than when agent, or more agents, msreports ther types.) Combnng the nequalty of the AE property wth the above expected payo, we obtan E 4 X jn v j (k ( ; ); j ) 5 + E [h ( )] E 4 X jn v j k ( 0 ; ); j 5 + E [h ( )] whch mples that, under dagva transfer functons, the expected utlty that player obtans from truthfully reportng hs type s hgher than that from msreportng hs type (announcng 0 6= ). More formally, E [u (g ( ; ) ; )] E u g 0 ; ; for all, all 0 6=, and all N. Ths s exactly the BIC property that we sought to prove. (Q.E.D.) For compactness, we use "dagva mechansm" (or "expected externalty mechansm") to refer to drect revelaton mechansm D = () N =1 ; g () where the SCF g () = (k () ; t 1 () ; :::; t N ()) has dagva transfer functons. 4

25 8..1 dagva and Budget Balance We can easly show that a proper choce of the h ( ) functon yelds a dagva mechansm that s strct BB,.e., P N t () = 0. In partcular, consder a transfer t ( ; ) = E 4 X v j (k ( ; ); j ) 5 j6= " ( ) + 1 X " j ( j ) N 1 j6= h ( ) whch can be rewrtten as t ( ; ) = " ( ) 1 N 1 X " j ( j ) j6= Summng over all N on both sdes yelds X t ( ; ) N = X " ( ) N = X " ( ) N 1 X X " j ( j ) N 1 N j6= P N 1 N 1 N (N 1)" ( ) X " ( ) = 0 N Therefore, we obtan as requred for strct BB. (Q.E.D.) X t ( ; ) = 0 N Example of dagva and strct BB. Consder a settng wth three agents N = f1; ; g. Accordng to the above transfer functon that guarantees strct BB, we have t ( ; ) = " ( ) 1 [" j ( j ) + " l ( l )] for every agent k 6= l 6= You can easly check that X t ( ; ) = " 1 ( 1 ) =1 1 [" ( ) + " ( )] + " ( ) = " 1 ( 1 ) + " ( ) + " ( ) 1 [" 1 ( 1 ) + " ( )] + " ( ) 1 [" 1 ( 1 ) + " ( ) + " ( )] = 0 1 [" 1 ( 1 ) + " ( )] Example of dagva - Blateral trade. Consder a seller wth equaly lkely valuatons 1 = f10; 0g and a buyer wth equaly lkely valuatons = f10; 0g. Every agent smultaneously and ndependently announces hs type, and trade occurs f and only f 1 (the buyer s 5

26 announced valuaton s weakly larger than that of the seller), whch entals an allocaton functon k ( 1 ; ) that s AE. Let us next nd the valuaton functon for each pro le of types ( 1 ; ). In partcular, for the seller, v 1 (k (10; 10) ; 10) = 10; v 1 (k (0; 10) ; 0) = 0 v 1 (k (10; 0) ; 10) = 10; v 1 (k (0; 0) ; 0) = 0 and for the buyer, v (k (10; 10) ; 10) = 10; v (k (0; 10) ; 10) = 0 v (k (10; 0) ; 0) = 0; v (k (0; 0) ; 0) = 0 Intutvely, when the announcement of buyer and seller s 10, trade occurs, entalng a loss (gan) of 10 for the seller (buyer, respectvely) gross of transfers,.e., v 1 = 10 but v 1 = 10. If, nstead, the seller announces a valuaton of 0 whle the buyer announces a lower valuaton of 10,.e., (0; 10), trade does not take place, entalng that the seller keeps the object wth valuaton v 1 = 0 whle the buyer s s v = 0. 5 We can now compute the " ( ) values, re ectng the expected externalty of every agent. Frst, for the seller the values of " 1 ( 1 ) are " 1 (10) = 1 v (k (10; 10); 10) + 1 v (k (10; 0); 0) = 1 (10 + 0) = 15 " 1 (0) = 1 v (k (0; 10); 10) + 1 v (k (0; 0); 0) = 1 (0 + 0) = 10 Smlarly, for the buyer (agent ), the values of " ( ) are " (10) = 1 ( 10) + 1 (0) = 5 " (0) = 1 ( 10) + 1 ( 0) = 15 Therefore, the transfers for the seller become t 1 (10; 10) = " 1 (10) " (10) = 15 ( 5) = 0 t 1 (10; 0) = " 1 (10) " (0) = 15 ( 15) = 0 t 1 (0; 10) = " 1 (0) " (10) = 10 ( 5) = 15 t 1 (0; 0) = " 1 (0) " (0) = 10 ( 15) = 5 5 In the opposte case, where the seller announces a valuaton of 10 whle the buyer announces a hgher valuaton of 0,.e., (10; 0), trade takes place, yeldng a loss for the seller of v 1 = 10 and a gan for the buyer s of v = 0. 6

27 The transfer for the buyer wll be exactly the reverse,.e., t ( 1 ; ) = ( 1 ; )-par. (Q.E.D.) t 1 ( 1 ; ) for every You probably notced n the prevous example that we can nd pro les of types for whch PC does not hold. For nstance, f ( 1 ; ) = (0; 0), the buyer s utlty becomes u (0; 0) = v (k (0; 0) ; 0) + t (0; 0) = 0 + ( 5) = 5 Ths s actually a general property of blateral tradng settngs, as shown by Myerson and Satterthwate. Myerson-Satterthwate Theorem. Consder a blateral tradng settng n whch the buyer and seller are rsk neutral, wth valuatons 1 and beng..d., and drawn from ntervals [ 1 ; 1 ] R and [ ; ] R wth strctly postve denstes, and ( 1 ; 1 ) \ ( ; ) 6=,.e., the two ntervals of types overlap n at least some types. Then, there s no SCF satsfyng BIC that also sats es expost e cency. (For a parametrc example on ths result, see Fudenberg and Trole (1991, Chapter 8). Intutvely, the nformaton rent that s requred to guarantee thuthtellng n BIC, makes the mechansm desgner sacr ce ex-post e cency.) Proof: See MWG, pages Lnear utlty Ths s a specal case of the quas-lnear utlty envronment, where u (x; ) = v (k) + m + t (Indeed, the only d erence wth respect to the quaslnear envronment s that the v (k; ) functon (the rst term on the rght-hand sde) s now,v (k; ) = v (k).) For smplcty, we also assume that: 1) Types are n the nterval [ ; ] R, where < ; and ) Types are..d. wth postve denstes for all [ ; ] In ths context, consder a SCF f() (k(); t 1 (); ; t N ()), and de ne expected transfers nd valuatons as follows: 1. t (^ ) E [t (^ ; )], that s, agent s expected transfer when he reports ^ and all other agents truthfully report ther types. As a practce, note that agent s expected transfer from truthtellng, reportng hs type s then t ( ), snce we evaluated t (^ ) at ^ = : 7

28 . v (^ ) E [v (^ ; )], that s, agent s expected valuaton when he reports ^ and all other agents truthfully report ther types. Agan, we can then express hs expected value from truthtellng as v ( ).. u (^ j ) E [u (f(^ ; ); )j ] = v (^ ) + t (^ ), that s, agent s expected utlty (n a lnear envronment) when he reports ^ whle all other agents truthfully report ther types. Fnally, f agent truthfully reports hs type,.e., ^ = ; hs expected utlty becomes u () = u ( j ) = v( ) + t ( ) We next present under whch condtons a SCF n ths lnear envronment sats es BIC; a result orgnally presented by Myerson. 9.1 Myerson Characterzaton Theorem In a lnear envronment, a SCF s BIC f and only f for every agent N, 1. v ( ) s nondecreasng n, and. Functon v ( ) can be expressed as Z v ( ) = v ( ) + v (s) ds for all Proof: See MWG, pp Intutvely, we can dentfy all SCFs satsfyng BIC n two steps: Frst, dentfy allocaton functons k() that lead every agent s expected bene t functon v ( ) to be weakly ncreasng n hs type ; second, among these allocaton functons, choose the expected transfer functon t ( ) that entals an expected utlty whch can be expressed n terms of the second condton of the theorem. Substtutng for v ( ) n the above condton yelds an expected transfer t ( ) of for some constant t ( ). Z t ( ) = t ( ) + v ( ) v ( ) + v (s) ds Snce many studes n the aucton theory and ndustral organzaton consder lnear envronments for smplcty, Myerson s characterzaton result has been appled to many applcatons. We next present one of the most famous applcatons, n the aucton theory, to show that, under relatvely general condtons, the expected revenue from sellng an object usng d erent aucton formats would concde. 8