Implementation by decent mechanisms

Transcription

1 Implementaton by decent mechansms Jerne Copc and Clara Ponsat Aprl 003 Abstract We address the desgn of optmal mechansms for barganng problems subect to ncomplete nformaton on reservaton values. We characterze decent rules, those that are Pareto Optmal n the constraned set of rules satsfyng strategy profness, ndvdual ratonalty and weak ecency - a mld requrement of ex-post eccency. We prove that decent rules are mplemented by the Fltered Demands game, a natural procedure for dynamc barganng under ncomplete nformaton where players submt ther clams over tme to a passve agent who mnmzes transmsson of nformaton between the players. Copc sathss Dvson, Calforna Insttute of Technology (erne@caltech.edu) and Ponsat satinsttut d Anals conomca -CSIC and COD-Unverstat Autonoma de Barcelona (clara.ponsat@uab.es). Ths research was conducted n part whle Ponsat was at the Unversty of Toronto and Copc was at the Unverstat Autonoma de Barcelona. Fnancal support to Ponsat from DURSI grant SGR and DGCYT grant PB98-870; and to Copc from the US-Span cooperaton Commttee s gratefully acknowledged. We beneted from comments and dscussons wth J.Bruna, M. Jackson, J. Ledyard, V. Skreta and semnar audences at Cambrdge, Caltech, dnburgh, UAB, UCSD, UPF and WSC-Berln.

2 Introducton It s well known that n envronments where agents have prvate nformaton and act strategcally, achevng non-dctatoral rst best allocatons s often mpossble. Ths s true n domnant strategy settngs, as well as n Bayesan ones. How ecent can a gven ncentve compatble socal choce rule be? When addressng ths queston, t s mportant to dstngush between the rules that do well n the ex-ante sense, those that do well n the nterm, and the ones that do well ex-post. A rule that does well ex-ante, mght do very poorly ex-post, and vce-versa. In ths paper we nvestgate rules that satsfy an ex-post ecency property whch we call weak ecency (W). We take a smple envronment: a barganng problem where two agents must share a unt of surplus and have prvate reservaton shares (agents types). Ths problem s easly transformed 3 nto blateral trade or nto the problem of sharng the cost of a publc good. In the barganng context, possble outcomes are dened as all feasble dvsons of the surplus unon the dsagreement pont. The agents preferences over outcomes are determned by concave utltes over net surplus - the excess between an agents share and her (prvately known) reservaton share. 4 A rule s a socal choce functon, assgnng shares of the surplus to the agents, For domnant strategy envronments see Hurwcz [97], Gbbard ]973], Satterthwate [975], Green and Laont [977]. For Bayesan envronments see Myerson and Satterthwate [983]. Corchon [996] presents a uned treatment. In economc envronments wth quas-lnear preferences t has even been shown that requrng only Bayesan ncentve compatblty doesn t provde any mprovement n ecency, as compared to the domnant strategy ncentve compatblty. See Wllams [999] and Mookheree and Rechelsten [99]. A non-cooperatve game of ncomplete nformaton can be dvded nto three temporal stages. At the ex-ante stage each agent knows only the dstrbutons of types of all agents, ncludng hmself. At the nterm stage each agent knows her own type but stll knows only the dstrbuton of types of her opponents. At the ex-post stage the types of all agents are common knowledge. 3More precsely, n the blateral trade context, the problem can be translated nto one where the buyer has a valuaton of the good and the seller has a cost of producng the good. Smlarly, n the cost-sharng of a publc good, the problem s translated nto one where agents have prvate valuatons of the publc good and have to share ts cost. 4In the language of socal choce, the agents preferences le n the restrcted doman determned by each par of concave utltes over net surplus. Thus agents prvate nformaton on preferences s equvalent to prvate nformaton on reservaton values.

3 possbly n a probablstc way, for each par of reports on reservaton shares. Weak ecency mposes that whenever agents reservaton shares are compatble, the probablty that all the surplus s allocated s strctly postve. We focus on rules satsfyng weak ecency and the strongest ncentve requrements, strategy proofness and ex-post ndvdual ratonalty. We call the rules that are Pareto optmal n ths constraned set the decent barganng 5 rules. We fully characterze decent rules for all envronments where agents have concave utltes; thus we do not requre rsk-neutralty. Decent rules are always probablstc, the probablty of mplementng the dsagreement pont 6 beng the tool to elct truthful revelaton. An mportant characterstc of the decent rules s that the outcomes depend non-trvally on agents reports. Our characterzaton mples that, whenever t exsts, the decent rule s always unque. We prove that a decent net surplus rule, a rule n whch shares and probabltes depend only on (the sum of the revealed types), s necessarly one where agents obtan ther reservaton plus a x porton of the revealed net surplus. Furthermore, we completely dentfy the class of utltes for whch such decent rule exsts: t ncludes utltes wth constant relatve rsk averson (CRRA), but does not nclude for example, exponental utltes. For the doman of rsk-neutral preferences, Hagerty and Rogerson [987] prove n the context of blateral trade, that any strategy proof mechansm yelds the same ex-ante total surplus than a mechansm whose operaton s ndependent of agents reports. Such mechansms, n the barganng context 7 we call them posted splt mechansms, operate n the followng way. Frst, a dstrbuton functon s announced, and a splt of pe s selected accordng to a random draw from that dstrbuton. The splt s then announced publcly, and f both agents agree to t, t s mplemented. Otherwse the agents get ther dsagreement payos. It s easy to see that to maxmze ex-ante payos the posted splt should be selected wth a degenerate dstrbuton; for any gven dstrbuton of agents types, the mechansm must pck the splt that 5 Ths s unrelated to decency as a constrant on the behavor of agents as n Herrero and Corchon [003]. 6Ths follows from nexstence of ecent strategy-proof mechansms whch s mpled by Myerson and Satterthwate [98]. 7In the context of blateral trade Hagerty and Rogerson [987] call these mechansms posted-prce mechansms. The revelaton prncple for domnant strateges (see Corchon [996]) allows us to dentfy rules wth drect mechansms that mplement them. 3

4 8 maxmzes expected gans from trade. Because the decent mechansm s necessarly probablstc, under rsk neutralty t must yeld the same exante payos as a posted splt rule wth a non-degenerate dstrbuton. Ths payo equvalence no longer holds under rsk averson. In fact, f agents are sucently rsk averse, the ex-ante payos of the decent rule domnate those of any posted splt rule. In general, the decent rule does not maxmze the sum of expected payos. However, desgnng ex-ante optmal rules requres knowledge of the dstrbuton of agents types. Wthout ths nformaton, the ecency of rules that are not weakly ecent - posted splt rules, for example - wll depend on luck alone, and ther necency can be severe. A decent mechansm, although possbly not optmal ex-ante, wll always be decent n terms of ecency, 9 regardless of the dstrbuton of agents types. Another reason we may be nterested n the decent rule s that mplementng a socal choce rule requres strong commtment by the agents. They have to obey the mechansm, and abstan from renegotatng the outcomes, even f 0 t s revealed that mutually agreeable mprovements were possble. Posted splt rules do not account for such renegotaton consderatons because they operate at the ex-ante stage. In other words, n a posted splt mechansm t may well happen that the two agents have mutually compatble shares whch are not compatble wth the announced dvson. Thus, the agents mght be nclned to renegotate. But f the outcomes can be renegotated ex-post, and the agents know t ex-ante, ths wll change the agents ncentves. Weak ecency can be nterpreted as a weak requrement on renegotaton-proofness. The motvaton for weak ecency s most apparent when a barganng rules are nterpreted as the drect revelaton mplementaton of equlbra n dynamc barganng games among mpatent agents. In these settngs, the probablty of dsagreement s equvalent to delay. Weak ecency s 8 The characterzatons of strategy proof rules by Barbera and Jackson [995] and Sprumont [99] are also related. Barbera and Jackson [995] show that n two-person economes strategy proofness and ndvdual ratonalty mply that trade s at xed proportons. For the dvson problems wth sngle peaked preferences, when the dsagreement s not an alternatve, Sprumont [99] proves that the unform rule s the unque rule that s ecent, symmetrc and strategy proof. 9 0 See Secton 7 for concrete examples. See Hurwcz [983], Jackson and Palfrey [00] and Rubnsten and Wolnsky [99] for dscussons on the problem of enforceablty of mechansms. On renegotaton see for nstance Maskn and Moore[999]. 4

5 then the property that an agreement occurs, sooner or later, when types are compatble and never otherwse. In a decent mechansm, agents agree as soon as possble, whle the correct ncentves are preserved. We formalze ths nterpretaton n the second part of the paper, where we provde a natural dynamc game mplementng decent rules. We call ths game the Fltered Demands (FD) game. The FD game s the smplest possble barganng game n contnuous tme, ts man feature beng that the nformaton ow between the agents s mnmzed. It can be envsoned as a market wth a completely closed order book where both agents keep postng ther demands. In contnuous tme, the agents keep sendng ther changng demands (they have a common dscount functon) to a central agent, the Flter. The Flter s role s to record these messages secretly, makng them publc only when the agreement s possble. Then the game ends wth the agents obtanng ther agreed shares. Thus, the agents recognze how much net surplus s avalable only when they reach an agreement, and at that moment they share all the remanng surplus. Hence, at no moment of agreement can they renegotate to a better outcome ex-post. In the FD game we dene an equlbrum n regular strateges as a Bayesan equlbrum whch s undomnated and n whch the agents strateges satsfy some smoothness requrements. We prove that the regular Bayesan equlbra of the FD game mplement precsely all the decent rules. The proof s mpled by our result that no common pror s needed to play a regular equlbrum n the FD game. The mplementaton result that we obtan s qute strong. Frst, there s a one-to-one correspondence between the regular equlbra of the FD game and the decent rules. Second, the FD game s n a way more general than the decent mechansms. To construct a decent mechansm, the desgner (as well as both agents) has to know the restrcted doman of preferences (.e. the forms of the utltes of both agents, but not Another way to address ths s the followng. If an agent wshed a strategy proof mechansm that gave her as much as possble for all her types and n any possble draw of the opponent s types, then a decent rule s what she should subscrbe to. It should thus come as no surprse that the more rsk averse the agents are, the more ecent are the decent rules. See also the remark n Secton 6. By the rst order condton, all the regular equlbra of the FD game are belefndependent, hence they are ex-post equlbra (see Ledyard[978] or Bergemann and Morrs [00] ). The nformatonal assumpton that we need s the condtonal ndependence of belefs, also called the spannng property. 5

6 ther prvate types). On the other hand, n the FD game, only the agents need to know the forms of each other s utlty functons. The mplementaton va the FD game proves that the best desgn for a dynamc game mplementng a strategy-proof, weakly ecent rule s to completely restrct the communcaton between the agents. The FD game also provdes a strong lnk to the lterature on non-cooperatve barganng games wth ncomplete nformaton. Whle a great deal s known about the barganng games where two agents alternate oers, and there s one-sded ncomplete nformaton, characterzng the set of equlbra wth 3 two-sded ncomplete nformaton has remaned an elusve task. When mpatent agents bargan non-cooperatvely over tme, agreements are delayed because ecent equlbra are mpossble under two-sded ncomplete nformaton. In such barganng process, agents must learn what aspratons are reasonable before they are ready for an agreement; learnng requres communcaton and tme. But learnng s a double-edged sword. On one hand, as agreements are more easly attaned when the partes know well the lmts of what s agreeable, t s mportant that the agents credbly communcate what they cannot accept. On the other hand, when an agent learns of her opponent s readness for agreement, such knowledge may ncrease the agent s aspratons. In these crcumstances, ratonal learnng actually shrnks the room for agreement nstead of wdenng t. Thus, when the agents bargan face to face, drectly exchangng proposals and reples, the scope for useful credble communcaton s severely lmted or nexstent. For nstance, separatng equlbra n statonary strateges exhbt the undesrable property that, n the lmt as the tme nterval between the oers vanshes, the probablty of agreement vanshes too. Thus, when barganng s face to face, 4 a smooth learnng process conducve to agreement s dcult, f not utterly mpossble. Our results demonstrate what can be attaned when face to face barganng s ether ruled out, or the agents are cogntvely constraned and are cannot update ther belefs. Thanks to the Flter, barganers learn only what ther opponent cannot yeld. Over tme, learnng smoothly decreases players aspratons. The dea of drastcally lterng communcaton has prevously been explored n Jarque, Ponsat and Sakovcs [003]. There, nstead of the present 3 4 See Ausubel, Cramton and Denekere [00] for an excellent survey and references. See Theorem n Ausubel and Denekere [99]. 6

7 assumpton that every dvson s possble and that concessons must be smooth, t s assumed that concessons must take place n dscrete steps. Thus, only a few ntermedate agreements can be reached. Ths characterstc s often a natural feature n realstc stuatons. However, the dscretzaton of the set of parttons of the surplus comes at the expense of great techncal problems. The set of equlbra s very large; they all depend on the dstrbuton of types, and ther exstence and ecency performance cannot be establshed wthout the detaled nformaton about the dstrbuton of 5 types. The rest of the paper s organzed as follows. In Secton we formally descrbe the mechansm desgn problem. We dene and characterze decent rules. In Secton 3 we present the Fltered Demands game. In Secton 4 we characterze ts equlbra. In Secton 5 we show that the equlbra of the FD game mplement decent rules and that all decent rules are attaned va the FD game. For envronments of CRRA utltes we explctly compute the equlbrum. In Secton 6 we llustrate our results wth an applcaton to blateral trade. In Secton 7 we provde some welfare comparsons. In Secton 8 we conclude and dscuss the extensons. Most of the proofs are n the Appendx. Decent Barganng Rules Two agents, =,, bargan over a unt of surplus. Denote by the share of the good that gets allocated to. Index wll always ndcate the agent other than,.e. =, for =,. To denote a vector ( x,x ), we wll wrte x. The set of barganng alternatves, A, s the set feasble dvsons unon the dsagreement pont. Formally, A = { + } d. An agents type s [0, ] represents her reservaton share, the share that leaves her nderent to dsagreement, and s her prvate nformaton. The preferences of agent over A are represented by a utlty functon u( s), wherewhere u ()s. a twce derentable, strctly ncreasng, and concave functon, wth 5 If we drop the regularty requrement n the equlbra of the FD game, then the set of Bayesan equlbra also contans the equlbra whch are smlar to those n Jarque, Ponsat, and Sakovcs[003]; f an agent beleves that the opponent wll only concede n dscrete steps, then t only makes sense to concede at the complementary splts. 7

8 u(0) = u( d s ) = 0. The payo from dsagreement s normalzed to 0. Agents preferences over ( A), the set of lotteres over A, are represented by ther expected utltes. From now on we x u = ( u,u). We dene a probablstc barganng rule ( Y,P) to be a drect revelaton 6 mechansm, mappng pars of reports z = ( z,z) nto two-pont lotteres P( z) Y( z) +( P( z)) d. Thus the rule ( Y,P) prescrbes dsagreement wth probablty ( P( z )), and agreement at Y( z) = ( Y( z ),Y( z)) wth probablty P( z ), where Y( z) s the share of the surplus allocated to agent, 0 Y( z) and Y( z) + Y( z ) =. Gven ( Y,P) the expected utlty of agent of type s upon reports z, U ( s,z), s U ( s,z) = u ( Y ( z) s ) P ( z ). We consder the followng propertes of barganng rules.. Strategy Proofness (SP): ( Y,P) s strategy-proof f truthful reports consttute a domnant-strategy equlbrum. That s: U( s,s) U ( s,z,s ) for every z = s, all s,s [0, ], =,.. x post Indvdual Ratonalty (IR): ( Y,P) s ex-post ndvdually ratonal f P( s ) > 0 mples that u( Y( s) s) 0 for all s [0, ] [0, ], =,. 3. Weak ffcency(w) : ( Y,P) satses weak ecency f s + s < P ( s,s ) > 0. Denton A quas-decent barganng rule s one that satses to 3. A decent barganng rule s a Pareto optmal rule among the quas-decent barganng rules. 6 Ths s a slght abuse of termnology. In a general framework, we should dene a bargang rule to be a socal choce functon, mappng pars of utlty functons nto ( A). We wll be dealng wth the rules that are not manpulable n domnant strateges. Hence, we can appeal to the revelaton prncple for the domnant strategy envronments and dentfy the set of non-manpulable socal choce functons wth the set of drect revelaton mechansms that mplement them. Also note n our settng, the prvate nformaton s restrcted to be over a one-dmensonal parameter, whch s thus the only thng that an agent reports to the mechansm. 8

9 The rst example shows that W s not a trval requrement. xample Consder a rule whch prescrbes a splt at xed shares, a postedsplt rule, and the agents dvde the surplus f they both agree to such dvson. Thus, Y( z) = y, Y( z) = y = y, where y [0, ]. Clearly, such a rule satses SP and IR, but does not satsfy the W. W mposes an open ended constrant. Perhaps one could desgn a W mechansm as a combnaton of an ex-ante ecent mechansm that swtches wth a slght probablty to a W mechansm. In such case, decency would not be well dened. The next example hnts that such swtchng mechansms fal n ether W or IR, and sometmes also n SP. A formal proof of ths s a smple consequence of Lemma 5. The fact that decency s a well dened concept s a trval consequence of Theorem 7. xample 3 Take a rule prescrbng a posted-splt wth probablty p< 0, and gvng the whole surplus to agent wth probablty p, where p+ p+ p =. Such a rule satses SP, but doesn t satsfy the IR. To see that W also fals observe that snce the draw of how the surplus gets allocated s made ex-ante, t s not true that the probablty of a Pareto-ecent outcome s postve whenever types are compatble. Consder now the rule where rst a posted-splt s announced, and f the agents don t both agree to t, then wth probablty p the whole surplus goes to agent, p + p =, p [0, ]. Ths rule always satses W, never satses IR, and sometmes satses SP, dependng on utltes of the agents, on p, and on the posted splt. Fnally, we gve a smple example of a decent mechansm, whch provdes most of the ntuton for what we do n the rest of ths secton. Proposton generalzes the next example. xample 4 Assume that the utltes of the agents are lnear: s. Then for each [0, ] a mechansm dened by { ( s s) ;fs + s < P ()= s, 0; otherwse u ( s ) = Y ()= s (+ s s ), =,, s quas-decent. To see that compute U ( z,s ; s ) = ( z s ) z s s ( + ) 9

10 It s mmedate to see that ths quadratc functon has a maxmum precsely s. Theorem 7 below wll show that these are all the quas-decent at z mechansms for ths case. Hence the unque decent one s obtaned by settng. Lemma 5 () If a barganng rule Y, P sats es () and () then, for,, Y s, s s monotoncally ncreasng n s, P s, s s monotons Y P s, s cally decre u Y s, s, s > Y P Proof. s s Y, P, s P s Y s P s s s s s Y s s Y, P Lemma 6 s s s > P s. s s < P Y s s P s P s s s s u s s s u s s, < P s P,,,.,

11 Proof. See Appendx. In the man theorem of ths secton we summarze the full strength of quas-decency. Theorem 7 Whenever a decent rule exsts, t s unque. Moreover f ( Y,P) s the decent rule for a gven vector of utltes, then all the quas-decent rules are gven by ( Y,P), (0, ]. Proof. See Appendx. Remark Observe that quas-decent rules are nvarant wth respect to multplyng utltes by constants. That s, f ( P, Y ) s a quas-decent rule for utltes ( u,u ) then t s also a quas decent rule for ( Cu,Cu ) where C > 0, =,. We already saw n xample 4 that the decent rule exsts when utltes are lnear. Does t exst for any other class of utltes? We focus our search for the decent rules among those that satsfy the followng smplfyng dentons. ns ns Denton 8 A net surplus rule s a rule ( Y,P ) where the probablty of agreement and the share of the net surplus assgned to each agent depend only on the net surplus. as Thus, wth some abuse of notaton, a net surplus rule can be expressed ns Y () s = s + (), =,, { ns P ( ), > 0, P () s = 0, otherwse. Denton 9 A xed net surplus rule s a rule for whch some constant, 0. ( s,s ) = for We now show that among the quas-decent rules, the net surplus rules are necessarly constant net surplus rules. Lemma 0 A quas-decent rule that s net surplus must be a constant net surplus rule.

12 Proof. See Appendx A. In the next proposton we fully characterze the class of utltes for whch decent xed share mechansms exst: Proposton The decent rule exsts and t s a constant net surplus rule f and only f ether of the followng holds:. u( x) = Cu( x), C > 0, for every x [0, ]. In ths case the unque decent rule s the constant net surplus mechansm, gven by = = = u P ( ) = u =,, 7. For agent has a utlty of the form : ( ) D s, u ( s ) = C ( s ) e C > 0, (0, ], D [, ] P = ( ) = In ths case a unque decent rule s the constant net surplus rule, gven by + { DsDs e ; s + s 0; otherwse. Proof. See Appendx A. 7 It s easy to see that u s ncreasng and concave on [0, ] f and only f (0, ] and D,. Notce also that when D = 0 ths utlty functon s u( s) =( s) whch s the constant relatve rsk averson (CRRA) utlty. Also note that n ths case, the mechansm s a net surplus mechansm.

13 3 The Fltered Demands Game In ths secton we propose a dynamc barganng game mplementng decent barganng rules. The game. The Fltered Demands Game (FD game) s a contnuoustme game. The agents send prvate messages clamng some share of the good to the Flter. The Flter s a dummy player whose only role s to receve clams, keepng them secret whle they are ncompatble, and to announce the agreement as soon as t s reached. As tme goes by, the agents can contnuously decrease ther demands at any moment. Thus the agents revse ther clams untl they become mutually compatble. Then the Flter announces that agreement has been reached, the agents receve the agreed shares, and the game ends. Strateges. A strategy (.,.) of player safuncton mappng her type s and tme t nto a share, (.,.) : [0, ] [0, ) [0, ], =, Thus ( s,t) s the share agent of type s clams for herself at t 0. Strctly speakng, a strategy s a functon mappng each type and each hstory nto a proposal at every moment. However, gven her type s, the hstory at tme t only depends on t, as the agent s not able to see the proposals of her opponent. The rules of the game are such that the followng condtons have to be satsed by the players strateges:. ( s,t) s non-ncreasng n t for all t (0, ) and all s [0, ]. 8. ( s,.) s a rght contnuous functon of tme. The rst condton assures that outcomes are well dened. The second condton s nnocuous: an agent can only observe agreement (or dsagreement) so t s plausble that she wll ether keep her ntal demand or 8 monotoncally concede to her opponent. Outcomes. If two strateges are such that ( s, 0) + ( s, 0) <,.e. the demands are more than compatble at t = 0, then agreement between Suppose an agent decded to play a strategy that would at tme t requre a share for herself. Then t would make lttle sense to demand 0 < at an earler tme t 0 < t. The agent mght ust as well stck to at t0. Condton allows the agents to forget about such consderatons. 3

14 ( s, ( s))+ ( s, ( s )). t ( s, 0) ( s, 0) 9 types ( s,s )occurs at t= 0 at shares ( s, 0) +. Gven a par of types s a strategy prole determnes a unque outcome of the game denoted by ( x(,s ),x(,s ), (,s)), where x and x are the shares of agents and, and s the tme of agreement,.e. Inter-temporal Utltes. The statc utlty of agent s gven by the functon u( s), satsfyng the same requrements as n Secton. Agents dscount the future exponentally. Thus upon agreement at t 0, the payo of agent s gven by U (,s,t) = e u ( s ). Clearly, n the event of perpetual dsagreement the payos are zero. Ths s for convenence and can be relaxed to a general class of dscountng crtera (),where t (.) s a strctly p t t Informaton and Belefs. s F f,, 0 t f s s,t s t t f s t equlbrum. 9 Our results are ndependent of the excess sharng rule, as long as t gves postve shares to both agents. 0Our results hold for any F wth postve densty f on a square [ s,s] [ s,s ],s< / <s. Ths s equvalent to the requrement that F has support on [ s,s] [ s,s], and the condtonal belefs of agents are ndependent. In the lterature, ths condton also appears as the spannng condton (see for nstance Mookheree and Rechelsten[99], p395).

15 or are unobservable to the opponent; own devaton from a B cannot be optmal at any t. Hence, a formal denton of B wll suce. s Let U (,F) denote the expected payo of player of type s at the strategy prole when types are dstrbuted accordng to F: s (,s,z) U (,F) = u( x(,s,u) s) e df ( s,u ). for all s [0, ], =,, =. s s U (,F) U,,F,, Observe that for each B prole, a prole constructed by addng a [0,T),.e. ( s,t + T) = ( s,t ), s a B as well, for ( s,t) t ( s,t) s 0 Denote by the set of strateges for player. A strategy prole consttutes abayesan equlbrum f and only f stand stll nterval any T<. As the opponent does not concede any postve amount untl T, no concesson pror to T s useful. Regardless of T, such strategy proles are weakly domnated. In the next secton we wll show that the type of player who makes the earlest relevant oer s s = 0. Type s = 0 has nothng to lose f she starts movng at 0, snce she has no reason to expect some other type to start movng any earler. Ths, n turn, would provoke other types to start movng as well. We say that a B s undomnated f t does not have a stand stll nterval. Snce types and dates take values n a contnuum, and the range of strateges s also a contnuum, natural patterns of behavor should rule out dramatc changes when types change only margnally. We say that a strategy s regular provded that:. exsts and s contnuous for all t [0, ), s [0, ];. exsts and s contnuous for all t [0, ), s [0, ]; 3. lm ( s,t) s a left-contnuous functon of s. t When the support of types s [ sl, sh ], s L < 0, the delay that a negatve type s ready to endure, rather than agree to 0 and obtan at least s L > 0, s bounded above. In these case stand- stll PB can be sustaned only for T [0,T L ), T L <. 5

16 The rst condton means that players do not change her demand by a postve amount n 0 tme (recall that she s only allowed to ncrease what she s wllng to oer to her opponent). Ths condton elmnates strateges are as those descrbed by Jarque et. al [00], where demands are step functons takng only ntely many values. When condton does not hold, the agents mght n a sequentally ratonal way beleve that the opponent wll almost surely only bd ntely many ntermedate agreement ponts between the two extreme agreements. Best response to such a strategy s to bd only the complementary ntermedate agreement ponts (snce any other bd s essentally rrelevant). The second condton requres smoothness wth respect to types. In equlbrum, t wll mply that player s strategy s fully separatng. The last condton s roughly an nderence breakng rule: f an agent of some type s at the horzon nderent between two concessons to the opponent, she wll concede more (see also the footnote n the proof of Lemma 4 n the Apendx). Ths condton s enough to assure that the contnuty of the demands wth respect to types s preserved at the tme horzon. Denton From now on, an equlbrum of the FD game s a B n undomnated and regular strateges. 4 qulbra n the FD Game We wll show that there s a one to one correspondence between the set of equlbra of the FD game and the set of decent rules. Frst we ntroduce some prelmnary results that wll be useful n characterzng equlbra of the FD game. The rst obvous observaton s that agents prefer dsagreement to negatve payos at every moment. Lemma 3 x-post Indvdual Ratonalty: s for all tand all s. In equlbrum ( s,t) In the next lemma we state that clamed shares asymptotcally approach the reservaton values. The ntuton s clear: f the agent of a gven type doesn t reach agreement n a very long tme, the opponent was probably of 6

17 a relatvely hgh type. Thus the agent should lower her demand, and she would only keep lowerng t untl her type. Lemma 4 Asymptotc demands: for all s [0, ]. In equlbrum lm ( s,t) = s t Proof. See Appendx. We next assert that agents wth hgh reservaton values, tougher agents, never demand less than softer ones. Moreover, ndvduals wth derent types never make at the same tme the same relevant concesson (one that nstantaneously leads to an agreement wth some of the opponent s types). ( s,t) Type Monotoncty: 0 s,. Moreover, for each type and f ( s,t) for some then s Lemma 5 In equlbrum,, t (0, ) s [0, ] s t > 0, ( s,t) = ( s,t) s [0, ] > 0. Proof. See Appendx. We now dscuss the optmzaton problem of the agents when ther opponent uses a strategy that s regular and strctly ncreasng n types. After two lemmas dervng the ntal condtons for the optmal strateges of the agents, we state the dynamc optmzaton program that the agents are facng. The man proposton of ths secton follows. There we derve the rst order condton, whch turns out to be belef ndependent. We rst focus on the ntal condtons for the agents strateges. From Lemma 5, t follows that any s starts partcpatng n the negotatons once her demand becomes feasble wth the demand of s = 0. Before that moment the agent must know that she s demandng too much to agree even wth the lowest type of the opponent. Therefore the queston s: should an agent enter n the game already at t = 0 (and wth what demand), or should she wat untl the eld softens up a bt. The answer s provded by the followng lemma. We denote by g( s) the startng pont of the demand of type s: g ( s ) = lm ( s,t). t 0 Lemma 6 g (0) =. Intal Condton: In any equlbrum t must hold that g (0)+ Ths argument s only vald f the reservaton demand of the toughest type s very hgh - that s f. s H 7

18 Proof. In an equlbrum the type s = 0 at tme 0 demands a share that wll gve her a postve probablty of agreement n at least a very short tme - otherwse each type of every agent would know that there was some dead delay at the start where the only thng that would happen would be that agents would lower ther demands up to the pont where the lowest types could agree. On the other hand, t could not be that she would demand a 0 share whch would meet the demand of some type s > 0 of player - meanng 0 that g(0)+ g s =. Ths follows from the excess prot sharng rule snce then an agent s = 0 could protably devate by startng wth a demand that met type s = 0. Then she would rp o all the excess agreement prots 0 by lowerng her demand very rapdly to g s. By makng her move fast enough t s clear that such devaton could be protable. Thus for all types except the lowest type t s n equlbrum optmal to wat wth a hgh demand for a whle. It means that there wll necessarly be delays wth probablty. Now we dene the entry tme of of type s, t ( s), as the rst moment that agent makes a realstc proposal. That s t ( s ) = nf { t > 0 ( s,t) + ( s,t) for some s [0, ] }. The followng corollary to the above lemma establshes that t ( s) s the moment when the demand of type s s compatble exactly wth the lowest type of the opponent. The proof s exactly the same as the proof of Lemma 6. The remark that follows s equally smple. Corollary 7 Tmng of ntry: In equlbrum, 0,t ( s), for =,, =, and all s [0, ]. s,t ( s) = Remark Notce that n any equlbrum t ( s ) < f and only f s <. Otherwse the strategy of s would be strctly domnated. We are now ready to turn attenton to the dynamc optmzaton. In equlbrum, agents select a strategy amng at the hghest possble payo, gven the type-contngent strateges of the other player. Thus agents are pckng optmal functons ( s, ), =,. Ths means that agent of type s decdes how her concessons of the good to the other sde should optmally change wth tme. 8

19 From now on, let ( s,t; ) be the functon gvng the type of agent wth whom agent of type s enters n agreement at moment t f prole s played. We wll omt n the arguments of (.). Formally, for any ( s,t ) [0, ] [ t ( s ), ), ( s,t) s the soluton of the equaton Proposton 8 Optmzaton Program: In equlbrum, agent of type s solves the followng optmzaton program 3 = ( ( s,t),t)+ ( s,t). () t ( s,t) Max e u( ( s,t) s) f( ( s,t)) dt, ( s, ) t [ t ( s ), ) s.t. () and s,t ( s) = 0 denes t ( s). Proof. Fx the type of agent to be s. When enterng nto negotatons at t ( s), she decdes her optmal concesson plan ( s,t), t > t ( s), n order to maxmze her expected dscounted future payo. Denote by P ( t) the probablty of type s reachng agreement up to tme t (for smplcty we omt the parameter s n P ( t)). Agent s solvng the followng program t Max e u( ( s,t) s) dp ( t) ( s, ) [ t( s ), ) A consequence of Lemma 5 and the mplct functon theorem s that such s well dened. Ths can also be seen from the proof of the followng proposton. But the possblty of reachng an agreement at some t>t ( s) s exactly the possblty that agent wll at t meet the demand of some type of agent.for any t t ( s), recall that ( s,t) s the type of agent wth whom reaches agreement at moment t. Thus ( s,t) s mplctly dened from the relaton ( ( s,t ),t)+ ( s,t ) =. 3 By denton and Lemma 5, s,t ( s) = 0, and by Lemma 4 lm t ( s,t) = s. Takng the dervatve wth respect to t, we can The corollary mples that at every nstant there wll be only one type reachng an agreement wth any partcular type of the other agent. 9

20 express Lemma 9 Frst order condton: Fx an equlbrum and a type s. For t>t ( s) the functon ( s,., ) =,, satses the followng rst order condton ) (,t) u(,s) = u ( s) +. (3) s t t 4 ( ( s,t ),t) ( s,t) t ( ( s,t),t) s ( s,t) t + = t By assumpton, and are both nte and non-postve. Hence we see t t from Lemma 5, and the mplct functon theorem, that for any t t ( s), ( s,t) ( s,t ) s a well dened derentable functon of tme, wth 0 < t.inother words, at any t t ( s) there exsts exactly one type ( s,t) of player, wth whom s would reach agreement at that moment. These facts have two consequences. Frst, the probablty of reachng an agreement by t, P ( t), has no mass ponts because the dstrbuton of types of player has no mass ponts. Second, the margnal ncrease n P ( t),.e. dp ( t), s equal to the margnal ncrease of the mass of types of player, that player would agree wth by moment t. Also, agent knows that before t ( s) her proposals were unrealstc, so she cannot update her belefs untl that moment. Snce s derentable wth respect to tme, the belefs are updated contnuously and derentably from t ( s) on. In other words, we have establshed that at t ( s) the belef of agent s exactly F( s), and at every moment dp ( t) = ( s,t) df( ( s, t)) = f( ( s, t)) dt. Ths completes the proof. t The optmzaton problem stated n Proposton 8 can be best approached as a problem where s choosng two unknown functons ( s, ) and ( s,. ) whch are bound by the constrant (), where (, ) s a gven and xed 4 functon (the strateges of all possble types of agent ). The optmalty condton at the lower boundary of optmzaton s gven by denton of t ( s) - mplctly wrtten t s ( s,t ( s)) = 0. In the followng lemma we provde the rst order condton of the optmzaton program of agent, for t>t ( s). Weare omttng most of the arguments n the functons. The arguments are: = ( s,t), except when derentatng wth respect to, and = ( s,t). Our manual for the calculus of varatons s lsgolts [970].. 0

21 Proof. See Appendx. Lemma 9 yelds a condton that s ndependent of the belefs of player about the types of player. Ths remarkable property s of mportance type s s s,t for any gven to every type of the other player must be playng a best response 5 Proposton 0 General Dscountng functons Suppose that dscountng s gven by a general functon t where. s a strctly postve, monotoncally decreasng functon wth, and t t. Denote the equlbrum strateges for exponental dscountng by s,. Then{ the equlbrum strateges } for dscountng t are gven by s,t, s, t. Proof. () s Y s s, s, P s e. 5 5 xstence of qulbra and Implementaton Implemented Barganng rule. s s, s s, s () s e mplements Y,P f and only f the outcome assocated to s such that and Ths ndependence wth respect to dscountng could prove useful when desgnng experments.

22 Snce tme plays a crucal role n the present setup, a natural nterpretaton s that rather than selectng outcomes stochastcally, barganng rules allocate agreements over tme. We may thus vew the FD game as a dynamc barganng rule ( and ts assocated dynamc drect revelaton mechansm), ( X, ), where ndvduals report ther type at t = 0 and are nstructed to mplement an agreement wth shares X ( s) only at date ( s ) [0, ]. If ()= s, then the prescrbed outcome s dsagreement. The FD game n equlbrum mplements decent barganng rules. We wll show that ths s always true, ndependently of agents utltes. Clearly, the queston s whether any equlbra of FD game exst at all (the agents strategy sets are non-compact). We wll prove that there s a one-to-one correspondence between the set of equlbra of the FD game and the set of decent rules. Hence, for any utltes, an equlbrum of the FD game exst f and only f a decent mechansm exsts. By the unqueness theorem for decent mechansms, we know that the equlbrum of the FD game wll always be unque. All of ths s summarzed n the followng lemmas and propostons. qulbra of the FD game mple- Lemma Indvdual Ratonalty: ment rules satsfyng IR. Proof. Ths s a drect consequence of Lemma3. Lemma Weak ffcency: qulbra of the FD game mplement rules satsfyng weak ecency. Proof. Lemmas 4 and 5 mply that all pars that produce a postve net surplus reach agreement at a nte date, whch translates nto W. Lemma 3 Strategy proofness: rules satsfyng SP. qulbra of the FD game mplement Proof. By Lemma 9 equlbra of the FD game are belef ndependent. Ths mples that the mplemented rule must be strategy proof (see Ledyard[978]). Lemma 6 then mples that the FD game mplements precsely decent rules. As we show n the next proposton, there s a one-to-one correspondence between the set of the equlbra of the FD game and the set of decent mechansms.

23 Proposton 4 Implementaton: An equlbrum of the FD game mplements a decent barganng rule. Conversely, any decent barganng rule s mplementable as an equlbrum of the FD game. Proof. The prevous three lemmas show that the rule mplemented n the equlbrum of the FD game must be quas-decent. Decency s then mpled by the fact that equlbra of the FD game are by denton undomnated, hence the probablty of the Pareto-ecent outcome s always set to be maxmal. For the converse, see the Appendx. Corollary 5 FD-Unqueness: Whenever the equlbrum n the FD game exsts, t s unque. Proof. Ths s a drect consequence of Theorem 7 and the prevous proposton. These are the central results of ths secton. Proposton 6 shows how to calculate all decent rules, gven one-parametrc utlty functons. Proposton 4 goes much further: regardless of the utltes, an equlbrum of the FD game mplements a decent barganng rule, as long as the two agents know each others utltes. The desgner does not need to know ths nformaton. Unless the set of equlbra of the FD game s empty, ths game must mplement a decent barganng rule. Computng an equlbrum requres solvng self-referental equatons () and (3). For the envronments where the decent rule s known, t s by the Proposton 4 easy to compute the equlbra of the FD game. In the next proposton we provde the strateges of the agents f they both have CRRA utltes. A smlar exercse can be repeated for the other cases where the decent exsts and t s a constant net surplus rule. Proposton 6 xstence: If agents have CRRA utltes, then the followng type-contngent strateges are the unque equlbrum and they mple- 6 ment the -constant net surplus rule : t ( s,t ) = mn,s + e, =,, 6 t s u u e Notce that at the tme when = + ( ( ) ) ths volates our assumpton on derentablty of strateges, but the strateges are stll derentable a.e. Moreover, at that pont, the demand of agent of type s s rrelevant, hence we can modfy t slghtly to make t smooth. 3

24 where and =. = (4) + Proof. Foraproof that these strateges satsfy the FOC of the FD game see Appendx A. The rest follows from Propostons and 4, and the prevous corollary. 6 Blateral trade and ex-ante ecency In ths secton we evaluate the ex-ante performance of the decent rule n derent scenaros, and we compare t to alternatve mechansms. To carry out ths exercse we focus attenton to problems of blateral trade: A seller can produce the good at a cost s, the buyer values the good t at b. Upon agreement at date t, on a prce p, the seller obtans us( p s) e t 7 and the buyer obtans ub( b p) e. In ths context, the FD game s a closed-book dynamc double aucton, agents contnuosly submt ask and bd prces, A( s,t) and B( b,t ), that are not dsplayed unless trade proceeds. The Flter can be nterpreted as a computer wth two nput nodes (one for each agent) and two output screens where ether No trade yet or Trade appear n the correspondng states of the world. Rsk neutralty: When both agents are rsk neutral ndvdually ratonal rules maxmzng the ex-ante sum of payos subect to Bayesan ncentve compatblty and strategy proofness are known thanks to Myerson and Satterthwate [983] and Hagerty and Rogerson [987] respectvely. We can thus compare the expected surplus attaned under these two rules to the expected payos of the decent rule. xample 7 Potental Welfare under unform dstrbutons. Assume that both agents are rsk neutral, and that costs and valuatons are ndependent and unformly dstrbuted n [0, ]. 7 To relate to our prevous notaton smply dentfy player wth the seller and player wth the buyer and set s = s, b = s. 4

25 . The unconstraned potental welfare s U W = ( b s) dbds =. 6 0 s. Myerson and Satterthwate [983] prove that the optmal rule under Bayesan ncentve compatblty and ndvdual ratonalty s to trade for sure f b s+ and abstan from trade otherwse. Ths yelds 4 expected total surplus MS W = ( b s) dbds = s+ 4 Hence, approxmately 85% of the potental expected gans attaned. W U are 3. Hagerty and Rogerson [987] show that under strategy proofness, the ex-ante surplus that obtans wth any gven rule can be attaned wth aposted-prce rule. It s mmedate to check that expected surplus s maxmzed at the xed prce p =. Ths yelds U whch s 75% of. W HR W = ( b s) dsdb = 0. 5, 0 4. Fnally, the decent rule, trade at prce wth probablty ( b c) f and only f b>, yelds expected surplus b+ c b d W = 0 0 b+ c ( b c ) dcdb =, U whch s only of W. The next example compares decent mechansms wth posted prces when the desgner s poorly nformed on the dstrbuton of costs and values. 5

26 xample 8 Msspecfcaton of the Dstrbuton: Let agents be rsk neutral. Assume that the desgner had no knowledge about the dstrbuton of costs and valuatons and that by the prncple of nsucent reason he assumed that t was symmetrc for both agents. He would then use the p = posted-prce rule. However, assume that the dstrbutons of agents reservaton shares were n fact asymmetrc. In partcular, take an ε> 0, and assume that the reservaton shares of the agents were ndependent and ther denstes had the followng forms: ε for 0 s fs ()= s + ε for < s + ε ε ε for + ε s { ε for b < ε fb ()= b + ε for ε b ε Some tedous, straght-forward calculus shows that then the gans from trade under the posted-prce rule are equal to ε(3 4 ε+ ε ) and the gans from trade under the decent rule are (6 7 ε + 60ε 53 ε + ε + 4 ε ). Clearly, 4 for ε small enough, the decent rule extracts a szeable porton of the possble gans from trade, whereas the posted-splt rule extracts almost none. Rsk Averson. When both agents have CRRA utltes s s p s,b s b, b sb =, wth probablty P ( s,b) = ( b s) f and only f b > s. b+ bs Observe s that as the agents become more rsk averse,.e. and go to b b u ( ps) = ( ps ),u ( bp) = ( bp ), the decent rule for blateral trade mplements trade at prce ( ) = ( + ) 0, the probablty of trade when b>sncreases, approachng full ecency n the lmt. Therefore, under sucent rsk averson, the decent mechansm domnates the best posted prce rule ex-ante. xample 9 The decent rule domnates posted prces: that both agents are equaly rsk averse, Assume s u ( ps) = ( ps ),u ( bp) = ( bp ), b 6

27 and that costs and valuatons are ndependent and unformly dstrbuted n [0, ]. Inthe absence of ncentve constrants the total surplus s maxmzed when agents trade at a prce whenever b>s. Ths yelds ex-ante payos b+ s ) b s = pp W = b + s dbds = ( +)( +) + 0 W d ) b s = 0 s W U 0 s dbds. The ex-ante payos under the optmals posted prce rule and under the decent rule are easly computed: ( b s) dbds = ( +)( + ) It s easly checked that f <. 9 then the decent mechansm performs better ex-ante than the optmal posted prce mechansm. 7 Concluson and xtensons We have addressed the desgn of mechansms for the barganng problem where the dsagreement ponts are prvate nformaton. For the envronments wth concave utltes, we have fully characterzed barganng rules that we call decent - those that are Pareto Optmal n the constraned set of rules satsfyng ndvdual ratonalty, weak ecency, and strategy proofness. We have proved that when t exsts, the decent rule s unque; by constructon we have proved the exstence for a large set of utltes. We have proposed a smple dynamc game, the FD game, whch always mplements the decent rule, regardless of the agents utltes and dscountng crteron. Ths mplementaton result s due to the fact that the equlbra of the FD game do not depend on agents belefs. The game protocol tself s smple. Nether a dctatoral prncpal desgnng complex contracts, nor strong commtments to assure the agents obedence over tme are requred. The dynamc game thus provdes a lnk between the weak ecency and the renegotaton-proofness. It also provdes a sharp predcton to the stuatons of blateral barganng 7

28 under ncomplete nformaton when the agents behavor s regular and ther updatng constraned. Our characterzaton of the decent rules can be nterpreted n the sprt of the classcal axomatc approach to barganng. Takng as the startng pont a barganng problem n whch the dsagreement pont s not common knowledge, we characterze the rule that nduces agents to reveal ther prvate nformaton, and assgns wth a postve probablty a Pareto-optmal soluton to the revealed problem. Gven that we requre strategy proofness, ths probablty has to non-trvally depend on the agents prvate nformaton. The present work can be extended to address stuatons wth more than two agents. In Copc and Ponsat [00a] we address multlateral barganng wth prvate reservaton shares. Ths generalzaton s approprate to address the problems of when to supply, and how to share the cost of a publc good 8 when there are many agents. In ths case, the Flter can be envsoned as a central agent admnsterng a publc account. Indvduals pledge ther contrbutons towards the cost of the publc good, and can ncrease ther pledge at any tme. The Flter assures that contrbutons are not publcly dsclosed untl the necessary amount has been pledged. Payments are made only f and when the proect s carred out. In Copc and Ponsat [00b] we dscuss markets wth many partcpants. There we generalze the FD game to a dynamc double aucton wth many sellers and buyers wth prvate valuatons of the obect. In that case the FD game can be magned as a market wth contnuous tradng and a closed lmt order book. References [] Ausubel, L. M., P. Cramton, and R. J. Denekere [00]: Barganng wth Incomplete Informaton, n Handbook of Game Theory, edted by R. J. Aumann and S. Hart, Amsterdam: lsever Scence B. V., forthcomng. [] Ausubel, L. M., and R. J. Denekere [99]: Durable Goods Monopoly wth Incomplete Informaton, Revew of conomc Studes, 59, Malath and Postlewate [990] address the queston of Bayesan ncentve-compatble mechansms for the envronments wth rsk-neutral agents. 8

29 [3] Barbera, S. and M.O. Jackson [995]: Strategy-Proof xchange, conometrca, 63, [4] Copc, J. and C. Ponsat [00a]: Dynamc Revelaton of Preferences for Publc Goods, mmeo. [5] Copc, J. and C. Ponsat [00b]: Dynamc Double Auctons, mmeo. [6] Corchon, L. [996]: The Theory of Implementaton of Socally Optmal Decsons n conomcs, London, MacMllan Press. [7] Corchon, L. and C. Herrero [003]: ADecent Proposal, mmeo. [8] lsgolts, L.: Derental quatons and the Calculus of Varatons, Mr Publshers, Moscow, 973. [9] Gbbard, A. [973]: Manpulaton of votng schemes, conometrca, 4, [0] Green, J. and J.-J. Laffont [977]: Characterzaton of satsfactory mechansms for the revelaton of preferences for publc goods, conometrca, 4, [] Hagerty, K. M. and W. P. Rogerson [987]: Robust Tradng Mechansms, Journal of conomc Theory, 4, [] Hurwcz, L. [97]: On Informatonally Decentralzed Systems Decson and Organzaton: A Volume n honor of Jacob Marshak, edted by R. Radner and C. B. McGure, Amsterdam, North Holland. [3] Hurwcz, L. [994]: conomc Desgn, Adustment Processes, Mechansms and Insttutons conomc Desgn,, -4. [4] Jackson, M.O. and T. R. Palfrey [00]: Voluntary Implementaton Journal of conomc Theory, 98, -5. [5] Jarque, X., C. Ponsat and J. Sakovcs [003]: Medaton:Incomplete Informaton Barganng wth Fltered Communcaton, Journal of Mathematcal conomcs, forthcomng. 9