Balázs Szentes Contractible contracts in common agency problems

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1 Balázs Szentes Contractble contracts n common agency problems Artcle (Accepted verson) (Refereed) Orgnal ctaton: Szentes, Balázs (2015) Contractble contracts n common agency problems. The Revew of Economc Studes, 82 (1). pp ISSN DOI: /restud/rdu The Author Ths verson avalable at: Avalable n LSE Research Onlne: Aprl 2016 LSE has developed LSE Research Onlne so that users may access research output of the School. Copyrght and Moral Rghts for the papers on ths ste are retaned by the ndvdual authors and/or other copyrght owners. Users may download and/or prnt one copy of any artcle(s) n LSE Research Onlne to facltate ther prvate study or for non-commercal research. You may not engage n further dstrbuton of the materal or use t for any proft-makng actvtes or any commercal gan. You may freely dstrbute the URL ( of the LSE Research Onlne webste. Ths document s the author s fnal accepted verson of the journal artcle. There may be dfferences between ths verson and the publshed verson. You are advsed to consult the publsher s verson f you wsh to cte from t.

2 Contractble Contracts n Common Agency Problems Balázs Szentes December 16, 2013 Abstract Ths paper analyzes contractual stuatons between many prncpals and many agents. The agents have prvate nformaton, and the prncpals take actons. Prncpals have the ablty to contract not only on the reports of the agents but also on the contracts offered by other prncpals. Contracts are requred to be representable n a formal language. The man result of the paper s a characterzaton of the allocatons that can be mplemented as equlbra n our contractng game. When we restrct attenton to exclusve-contractng envronment, our characterzaton result mples that prncpals can collude to mplement the monopolst outcome. Fnally, n general, equlbrum contracts turn out to be ncomplete. That s, a contract wll restrct the acton space of a prncpal but wll not necessarly determne a sngle acton. 1 Introducton In many settngs, frms do not charge a fxed prce, but nstead make ther prces explctly condtonal on the prces offered by compettors n an apparent effort to attract customers. They often commt to prce relatonshp agreements, that s, they adopt polces whch are drectly lnked to the prce polces of other frms. Examples of these polces nclude meet-the-competton clauses, prce-beatng promses and lowest fare guarantees. It s not mmedately clear whether these polces are ndeed benefcal to consumers, or whether they smply enable colluson between frms. We are therefore motvated to explore n more general terms the possblty of contracts whch depend on the contracts offered by others. 1 Indeed, our goal n ths paper s to put forward a general common-agency model and then explore the consequences of allowng contractblty of contracts. Department of Economcs, London School of Economcs. 1 Recprocal trade agreements, such as GATT, also take the form of contractble contracts. A recprocal contract commts to settng a low tarff aganst a partcular country f that country s contract does the same. Fnally, tax treates sometmes have ths flavor for example, out of state resdents who work n Pennsylvana are exempt from payng Pennsylvana tax as long as they lve n a state that has a recprocal agreement exemptng Pennsylvana resdents from state taxes. See 1

3 The contrbuton of ths paper s threefold. Frst, we show that by allowng for contractblecontracts we are able to provde a full characterzaton of mplementable allocatons. Indeed, we prove a folk theorem. The ablty to contract on contracts gves prncpals the opportunty to collude, and thereby mplement a varety of outcomes, as n repeated games. Colluson s accomplshed through contracts whch punsh a prncpal f hs contract s not the one expected from hm n equlbrum. Snce contracts are contractble, a prncpal s able to commt to punshng a devator despte the fact that nteractons are not repeated. Our characterzaton theorem s presented n greater detal below. Second, we nvestgate the mplcatons of our general results when appled to exclusve-contractng envronments (where an agent can sgn a contract wth only one prncpal). Ths type of exclusvty exsts, for example, n the context of employer-worker relatonshps n whch the worker may accept only one job offer, and seller-buyer relatonshps n whch the buyer s nterested n purchasng a good or a servce from only one seller. If contracts were not contractble, the competton among prncpals would result n a Pareto eff cent outcome. In contrast, we show that the contractblty of contracts enables prncpals to collude and effectvely act as a monopolst, offsettng any eff cency gan generated by competton. Therefore, n such envronments, the prohbton of contractng on contracts emerges as a polcy mplcaton. Thrd, our theory provdes a ratonale for the ubquty of ncomplete contracts. A contract s referred to as ncomplete f some terms of the agreement between the prncpal and the agent are at the prncpal s dscreton even after the contract s sgned. In our model, whether or not we allow for the contractblty of contracts, equlbrum contracts wll, n general, be ncomplete. Incomplete contracts are common n many contexts. For example, labor contracts often specfy a fxed wage but allow the bonus to be at the employer s dscreton, adjustable rate mortgages permt the lender to change the nterest rate wthn pre-specfed bounds, and utlty and other long-term servce provders may unlaterally rase prces. In the specfc model analyzed n ths paper, there are several prncpals and several agents. Agents have types, and prncpals take actons. Each prncpal wshes to enter nto a contractual relatonshp wth each agent. Followng the usual approach n the lterature, we analyze equlbra n communcaton games. In a communcaton game, agents are endowed wth message spaces, and the game has three stages. At the frst stage, the prncpals offer contracts to the agents smultaneously. In our settng, a contract s a mappng from the messages of an agent and contract profles of the prncpals to the subsets of the acton space of the prncpal. 2 Durng the second stage, agents smultaneously send prvate messages to each prncpal. In the fnal stage, prncpals select actons from the subsets of actons determned by both the frst-stage contracts and the second-stage message profles. Our goal s to characterze the set of equlbrum outcomes of these 2 In exstng lterature on common agency models t s usually assumed that contracts are complete, that s, they determne a sngle acton for the prncpal as opposed to a subset. We show that ths assumpton results n a loss of generalty. 2

4 games. Such models gve rse to an nfnte regress problem, whch can make them dff cult to solve. Consder two prncpals whose payoffs both depend on the acton of the other. Each prncpal s acton s determned by hs contract wth the agents, so each prncpal would lke to offer a contract whch s contngent on the contract offered by the other prncpal. A prncpal s contract wll typcally be contngent on the contract of the other prncpal, whch, n turn, s contngent on the contract of the frst prncpal, and so on. One possble way of dealng wth ths herarchcal dependency s to nclude self-referental contracts n the contract space. In a self-referental contract, the acton a prncpal takes wll depend on whether the other prncpal offers the same self-referental contract. That s, the contract refers to tself. The use of ths type of contract allows us to collapse many statements about hgher order dependences nto a sngle self-referental statement. However, the constructon of a contract space whch ncludes such contracts s not mmedately obvous. Therefore, perhaps the most mportant feature of a common agency model s the set of contracts avalable to the prncpals. Followng the approach of Peters and Szentes (2012), ths paper models the space of contracts wth the set of defnable correspondences. Ths set s a generalzaton of recursve functons, and wll be dscussed n detal n the next secton. As wll be shown, the key feature of ths space s that t ncludes all sorts of self-referental mappngs. Our man result conssts of a folk theorem, whch asserts that an allocaton can be mplemented as an equlbrum n our contractng game f and only f the allocaton s subgame-mplementable and the nduced payoff of each prncpal s larger than hs mnmax value. We shall provde an explanaton for both subgame mplementablty and the mnmax value below. In order to do so, we frst defne an ordnary contract to be one that does not condton on the contracts of the other prncpals. 3 Consder a modfcaton of our contractng game n whch each prncpal must offer an ordnary contract at the frst stage. An allocaton s called subgame-mplementable f t s an equlbrum outcome n a subgame generated by some ordnary contract profle. Let us now provde a clear defnton of Prncpal q s mnmax value. Suppose that for each j q, Prncpal j s goal at the frst stage of the ordnary contractng game s to mnmze the payoff of Prncpal q. Defne Prncpal q s mnmax value to be hs lowest equlbrum payoff n ths game. Ths mnmax value s smlar to the standard defnton, except for the fact that prncpals can only punsh Prncpal q n the contractng stage; each player behaves strategcally n the subgame generated by the contract profle. In that subgame, others can only punsh Prncpal q by playng an equlbrum whch makes hm worst off. We characterze the set of subgame-mplementable allocatons n terms of the preferences of the agents and the prncpals. In partcular, we show that an allocaton s subgame-mplementable only f t s strongly ncentve compatble. To explan the noton of strong ncentve compatblty, recall that an allocaton n our model s a mappng from the agents type spaces to acton profles 3 That s, an ordnary contract s a mappng from the message profle of the agents to the subsets of the acton space of the prncpal. 3

5 of the prncpals. Each coordnate of the allocaton maps the vector of type profles of the agents to the acton space of a certan prncpal. Suppose that prncpals act smultaneously, each offerng a drect mechansm to mplement hs coordnate of the allocaton. An allocaton s sad to be strongly ncentve compatble f truth-tellng by all agents consttutes an equlbrum n the product of these drect mechansms. That s, an allocaton s strongly ncentve compatble f no agent s able to ncrease hs payoff by msreportng hs type to the prncpals. Ths defnton s stronger than the standard defnton of ncentve compatblty because, n our settng, agents may report dfferent types to dfferent prncpals. The most conceptually challengng aspect of our folk theorem concerns these mnmax values. Snce contracts are contractble, one mght magne that the punshment nflcted on a devatng prncpal mght depend on the actual devaton. If punshments could be made contngent on the devator s contract, then one mght suspect that a devator could be pushed below hs mnmax value, perhaps even to hs maxmn value. Ths argument turns out to be false. Despte the contractblty contracts, punshments can only depend on the devator s dentty, and not on hs contract. In other words, when the prncpals punsh a devator they make use of ordnary contracts, and do not condton the punshment tself on the other prncpal s contract. Ths fact s due to an argument based on mathematcal logc stated n Proposton 1. Fnally, n both ordnary and contractble contract settngs, we dentfy equlbrum allocatons whch can only be mplemented by contract profles whch do not pn down sngle actons for the prncpals n the last stage of the game. In ths sense, equlbrum contracts are often ncomplete. Ths s due to the exstence of a trade-off between commttng to a small set of actons and havng flexblty at the last stage of the game. On the one hand, more commtment can ncrease ex ante payoffs. On the other hand, more flexblty can deter certan devatons. Indeed, a devaton mght be more attractve f the devator knows exactly what actons hs opponents wll take at the last stage of the game. There s another sense n whch restrctng attenton to complete contracts results n a loss of generalty. We come across allocatons whch can be supported as an equlbrum f contracts are requred to be complete, but cannot be supported f contracts are allowed to be ncomplete. Ths s because a prncpal mght proftably devate by offerng an ncomplete contract, but there mght be no such a devaton n the form of a complete contract. These observatons mght provde new nsghts as to why contracts are often ncomplete n the real world. Lterature Revew There s a szeable appled lterature on the theory of prce relaton agreements. As n our paper, these papers tend to conclude that prce relaton agreements facltate tact colluson. Whle we analyze general prncpal-agent models, ths exstng lterature focuses only on prce competton among frms. It s typcally assumed that the contract of a frm conssts of a posted prce and a polcy whch maps the compettors posted prces nto prces. The actual prce pad by a consumer for a frm s product s a functon of the frm s posted prce and the frm s polcy evaluated at the others posted prces, see, for example, Salop (1986), Png and Hrshlefer (1987), Belton (1987), 4

6 Logan and Lutter (1989), Baye and Kovenock (1994), Chen (1995) and Zhang (1995). In other words, certan parts of the contracts are contractble,.e. posted prces, but other parts of the contracts are not contractble,.e. polces. It s unclear whether the conclusons of ths lterature are vald only takng nto account these restrctons or whether they can be generalzed to arbtrary contractble contracts. In addton, there are two drawbacks of these contract spaces. Frst, such contracts make sense only n the smplest prce-settng contexts, and t s not obvous how they mght be adapted to envronments where the contractble objects are more complex than smple prces. Second, these contracts do not accommodate any communcaton between prncpals and agents, and hence cannot be used for screenng. Indeed, the agents (buyers) n the models mentoned above are essentally non-strategc and possess no prvate nformaton. In contrast, the contract space n our model s not context-specfc, and can handle arbtrarly complex contractble decsons as well as adverse selecton among agents. In terms of results, our paper generalzes nsghts from ths exstng lterature to arbtrary common-agency envronments. Indeed, we show that the contractblty of contracts mght lead to a softenng of competton, and reduce welfare n an exclusve-contractng envronment. In the context of prce-settng, collusve contracts specfy hgh prces f other prncpals also offer collusve contracts and trgger low prces f a prncpal devates. Ths has a smlar flavor to the meet-thecompetton clause, whch enables a frm to lower ts prce n response to undercuttng by ts compettor. The emprcal lterature on the meet-the-competton clause s ample and s consstent wth our results. The semnal paper s by Hess and Gerstner (1991) whch analyzes the competton between two large supermarkets, Wnn Dxe and Food Lon, n North Carolna. The authors document that Wnn Dxe s adopton of the meet-the-competton clause led to coordnated prces whch ultmately reduced consumer welfare. Arbatskaya et al. (2004) analyze the relatonshp between prces and advertsed prce-matchng promses of retalers across varous ndustres. The authors confrm that prce-matchng promses typcally soften competton. Arbatskaya et al. (2004) draws smlar conclusons n the context of tre prces and prce-match guarantees advertsed n local newspapers. Colluson among prncpals does not emerge as a unque equlbrum n our model. The contractblty of contracts does facltate colluson, but also allows prncpals to behave compettvely. In partcular, a contract may specfy a relatvely low prce rrespectve of the prcng polces of compettors. An example of such a contract s a prce-beatng guarantee whch strctly undercuts the prces of compettors. Not surprsngly, most emprcal studes fnd that such guarantees do not lead to softenng of the competton, see, for example, Arbatskaya et al. (2004, 2006) and Manez (2006). Whether or not prce-relatonshp contracts across sellers should be prohbted s subject to ongong debate. The emprcal fndngs suggestng that a meet-the-competton clause softens competton but prce-beatng promses do not are nterpreted as mxed evdence n favor of makng these agreements llegal, see for example Aguzzon et al. (2012). We adopt a dfferent vew; we propose that contractng on contracts should be dsallowed because t fosters colluson 5

7 among prncpals, even though t does not necessarly lead to t. Our paper s also related to the theoretcal lterature on common agency. Our most mportant departure from ths lterature s the contractblty of contracts. Although the meet-the-competton example s frequently used as motvaton, ths lterature usually assume that the contracts cannot be contracted upon drectly, but only through the reports of the agents. In order for the agents to communcate ther contracts to the prncpals, ther message spaces must be at least as large as the space of contracts. Snce the contracts are mappngs from the message spaces, t s not straghtforward to construct such a message space. Epsten and Peters (1999) show that there exsts a unversal message space that s rch enough for agents to communcate ther prvate nformaton as well as the contracts offered by the prncpals. They show that any equlbrum n a communcaton game wth a large enough message space can be mplemented as an equlbrum n the game wth the unversal message space. 4 Peters (2001) and Martmort and Stole (2002) show that a verson of the Taxaton Prncple holds for common agency games. That s, any equlbrum n any communcaton game can be mplemented as an equlbrum n a game where the prncpals offer menus of ordnary contracts. An ordnary contract s one whch maps reports of types to outcomes. The agent then selects tems from the menu of each prncpal. One of the shortcomngs of the lterature s a lack of characterzaton of these allocatons. Perhaps the man contrbuton of our paper to ths lterature s the full characterzaton of the equlbrum allocatons. Fnally, our paper s also related to the lterature on mutually dependent commtment devces, see for example Tennenholtz (2004), Kala et al. (2010) and Peters and Szentes (2012). Ths lterature consders two-stage games n whch players submt commtment devces at the frst stage, and play a normal form game at the second stage. A player s commtment devce s a restrcton of hs acton space as a functon of the commtment devces of the other players. Varous folk theorems have been proven n these stuatons. The equlbrum constructon s usually based on a self-referental commtment devce, smlar to the concept n our paper. Tennenholtz (2004) analyzes complete nformaton games and models the commtment devce space as the set of Turng machnes. Tennenholtz (2004) proves one drecton of a pure-strategy folk theorem. That s, he shows the mplementablty of any outcome n whch each player receves at least hs mnmax payoff. Kala et al. (2010) characterzes mxed-strategy equlbra n complete nformaton envronments. Ther man theorem states that any correlated outcome of the second-stage normal form game can be mplemented by commtment devces n whch all players payoffs exceed ther mnmax payoffs. Peters and Szentes (2012) departs from complete nformaton envronments and nvestgates Bayesan games wth commtment devces. As n ths paper, Peters and Szentes (2012) models commtment devces as defnable functons, and also show that ths space ncludes self-referental devces and that a player s payoff cannot be pushed below hs mnmax value. In contrast, the 4 Calzolar and Pavan (2006) and Yamashta (2010) smplfy the unversal message space, whch makes t possble to characterze equlbra n specal cases. 6

8 problem of compettve screenng does not arse n the model of Peters and Szentes (2012) because they do not have agents. The key trade-off n Peters and Szentes (2012) s related to the nformaton content of the devces. On the one hand, a player benefts from offerng type-contngent devces because dfferent types prefer to commt to dfferent actons. On the other hand, a player mght be hurt at the second stage f he reveals too much nformaton through hs devces. Therefore, dfferent types of players mght prefer to offer the same devce n order to dsclose less nformaton. A player s equlbrum devces balance these countervalng ncentves and generate a partton of the player s type space. Two dfferent types of players submt the same devce f and only f they belong to the same partton element. Let us emphasze that no such trade-off s present n ths paper, as the prncpals have no nformaton to start wth. The man result of Peters and Szentes (2012) states that the set of allocatons mplementable wth ther contractng game s the same as the set of allocatons mplementable wth publc message mechansms. A publc message mechansm s smlar to a standard drect mechansm except that messages are publcly observable and non-partcpants can arbtrarly restrct ther acton spaces as a functon of others reports. 2 An Example The goal of ths secton s to explan the space of contracts and provde an llustraton of our approach n the context of an olgopoly example. Suppose there are two frms (1 and 2) and a sngle consumer. Each frm can produce a partcular good at no cost. The goods are close substtutes but not dentcal. The consumer has one of two equally lkely types, A and B. The consumer s valuatons for the goods are summarzed n the followng table: A B Frm Frm That s, f the consumer s type s A, he values Frm 1 s good at 9 and Frm 2 s good at 8.5. If hs type s B, he values Frm 2 s good at 9 and Frm 1 s good at 8.5. Hs margnal value for a second good s zero. The acton space of each frm conssts of settng a prce from the set {0,..., 10}. The frms maxmze ther profts, and the agent wants to maxmze hs value for the good he purchases mnus the prce. Consder the followng game. Frst, frms submt contracts smultaneously. A contract specfes a prce as a functon of the contract of the other frm and the message of the consumer. Contracts are publcly observable. Second, the consumer sends messages to each frm. Fnally, the consumer decdes whch product to buy f any. If the frms were to set prces smultaneously wthout beng able to contract on contracts, the market prces and the jont proft would be at most 2. If the frms could collude, they would maxmze ther jont proft by settng a prce of 9 and the consumer would buy the good from Frm 1 f hs type s A and from Frm 2 f hs type s B. Next, we explan the contract space and show that t s possble 7

9 to mplement the collusve outcome wth contracts. We endow each market partcpant wth a formal language. We requre each contract offered by a frm and each message sent by the consumer to be a text wrtten n ths language, where a text s a fnte strng of symbols. A frm s contract gves precse nstructons on how to determne the prce as a functon of the texts submtted by the frm and the consumer. Below, we construct a contract for each frm, c 1 and c 2, whch mplements the allocaton that maxmzes the jont proft. These contracts wll be cross-referental and take the followng form 9 f c 2 = c 2, and m = A 9 f c 1 = c 1, and m = B c 1 (c 2, m) = 10 f c 2 = c 2, and m A and c 2 (c 1, m) = 10 f c 1 = c 1, and m B 0 otherwse, 0 otherwse, where c 1 and c 2 denote the contracts of Frms 1 and 2, respectvely, and m s the consumer s message. The contract c 1, for example, specfes a prce of 9 f Frm 2 s contract s c 2 and the consumer s report s A and a prce of 10 f Frm 2 s contract s c 2 and the consumer s report s not A. If the contract of Frm 2 s dfferent from c 2 then the contract c 1 forces Frm 1 to set a prce of zero. If the frms offer c 1 and c 2, the best response of the consumer s to report her type truthfully. Also note that f Frm offers c, the best response of Frm j (j ) s to offer c j because any other contract would trgger a prce of zero by Frm. So the contract profle (c 1, c 2) ndeed mplements an allocaton whch maxmzes the jont proft. The problem s that c 1 explctly depends on c 2 and c 2 explctly depends on c 1. So, the text correspondng to c 1 has to nclude a descrpton of the text descrbng c 2, whch, n turn, has to nclude the descrpton of c 1 tself. Below, we explan how to construct these texts. We take advantage of the fact that languages can be coded. That s, there exsts a bjecton from the set of fnte texts nto the set of ntegers, so each text can be coded by a unque nteger. One such mappng s called the Gödel Codng. So, to any text whch descrbes a mappng from texts to prces there s a correspondng text whch descrbes a mappng from codes of texts nto prces. Snce all the codes are ntegers, ths latter text s a descrpton of an arthmetc mappng. An arthmetc functon whch can be descrbed n a formal language s called defnable. Ths set s formally defned n the next secton. Snce the Gödel Codng can also be descrbed usng a text, we can dentfy the space of contracts wth the set of defnable functons from N 2 {0,..., 10}, where the frst argument of these functons s the code for the other frm s contract and the second argument s the code for the message of the consumer. The range for these functons s the set of prces. 5 In what follows, we use Gödel Codng to construct texts correspondng to c 1 and c 2 n (1). Let [ϕ] denote the Gödel code of the text ϕ. Consder the followng contract for Frm 1: 9 f [c 2 ] = n 2, and [m] = [A] c n2 1 ([c 2], [m]) = 10 f [c 2 ] = n 2, and [m] [A] 0 otherwse. 5 In general, the actons of a prncpal do not correspond to ntegers. Then the range of these functons are the codes of the actons. (1) 8

10 Ths contract says that f the Gödel code of Frm 2 s contract s n 2 and the consumer reports type A, then Frm 1 wll set the prce at 9. If the Gödel code of Frm 2 s contract s n 2 but the consumer does not report type A, then the prce wll be 10. Otherwse, the prce wll be zero. Smlarly, defne Frm 2 s contract as follows: c n1 2 ([c 1], [m]) = 9 f [c 1 ] = n 1, and [m] = [B] 10 f [c 1 ] = n 1, and [m] [B] 0 otherwse. Notce that f [c n2 1 ] = n 1 and [c n1 2 ] = n 2 then these contracts correspond to c 1 and c 2. Therefore, we have reduced the problem of constructng cross-referental contracts to fndng a soluton to the fxed-pont problem (n 1, n 2 ) = ([c n2 1 ], [cn1 2 ]). Before we proceed, we ntroduce two peces of notaton. Frst, the functon <. > s the Gödel codng nverse operaton. That s, < n > s the text whose Gödel code s n. Second, we shall make use of free varables to express statements such as x > y n texts. Integers can be substtuted for the free varables n order to make statements about these ntegers. If φ s a text, then φ (n1,n2) denotes the same text as φ, except that f φ contaned the free varables x or y, then the value of the free varable x s set to be n 1 and the value of the free varable y s set to be n 2. For example, f φ x > y then φ (3,2) 3 > 2. Now, consder the followng two texts: 9 f [c 2 ] = [ < y > (x,y)], and [m] = [A] c x,y 1 ([c 2 ], [m]) = 10 f [c 2 ] = [ < y > (x,y)], and [m] [A] 0 otherwse, c x,y 2 ([c 1 ], [m]) = 9 f [c 1 ] = [ < x > (x,y)], and [m] = [B] 10 f [c 1 ] = [ < x > (x,y)], and [m] [B] 0 otherwse. These texts are not contracts, because they contan free varables. However, they become contracts when we evaluate these free varables at any par of ntegers. Let γ 1 and γ 2 denote the Gödel codes of these two texts respectvely. Then 9 f [c 2 ] = [ < γ 2 > (γ 1,γ )] 2, and [m] = [A] c γ 1,γ 2 1 ([c 2 ], [m]) = 10 f [c 2 ] = [ < γ 2 > (γ 1,γ )] 2, and [m] [A] 0 otherwse, (2) and c γ 1,γ 2 2 ([c 1 ], [m]) = 9 f [c 1 ] = [ < γ 1 > (γ 1,γ 2 )], and [m] = [B] 10 f [c 1 ] = [ < γ 1 > (γ 1,γ 2 )], and [m] [B] 0 otherwse. (3) Recall that γ 1 s the Gödel of c x,y 1, so < γ 1 > (γ 1,γ 2 ) s c γ 1,γ 2 1. Smlarly, < γ 2 > (γ 1,γ 2 ) s just c γ 1,γ 2 2. Therefore, one can replace < γ 1 > (γ 1,γ 2 ) and < γ 2 > (γ 1,γ 2 ) wth c γ 1,γ 2 1 and c γ 1,γ 2 2 n (2) and (3) 9

11 and conclude that c γ 1,γ 2 1 and c γ 1,γ 2 2 are ndeed the cross-referental contracts correspondng to (1). What s more, each prncpal s contract s now well-defned: each contract gves precse nstructons as to how the consumer s message and the exact content of the other prncpal s contract together dctate the prce that a prncpal wll offer. 3 The Model 3.1 The Physcal Envronment There are n prncpals and k agents. Each prncpal has a fnte acton space. The set of actons avalable to Prncpal j s A j = k =1 A j, where A j denotes the set of actons of Prncpal j whch affects the payoff of Agent. Let A and A denote n j=1 A j and n j=1 A j, respectvely. The fnte type space of Agent s T, and T denotes k =1 T. The jont dstrbuton of types s common knowledge. The payoff to Prncpal j s gven by u j : T A R. The payoff to Agent s v : T A R. Prncpals and agents all maxmze expected utlty. 3.2 The Language and the Gödel Codng We consder a formal language that s suff cently rch to allow ts user to state any arthmetc proposton. Ths mples that one can express, for example, that there exst Pythagorean trples: x, y, z [ (n 3) (x 0) (y 0) (z 0) (x 2 + y 2 = z 2 ) ]. In addton, statements that nvolve any fnte number of free varables can be expressed. For example, x < 4 s a sentence n our language and the symbol x s a free varable n the statement. One can substtute any nteger nto x and then the predcate s ether true or false. Ths partcular sentence s true f x = 1, 2, 3 and false otherwse. Defnton 1 The functon f : N k 2 N s sad to be defnable f there exsts a frst-order arthmetc statement, φ, n k + 1 free varables such that b f (a 1,..., a k ) f and only f φ (a 1,..., a k, b) s true. We provde the formal defnton for a frst-order arthmetc statement n the Appendx. The reader should keep n mnd that a correspondence s defnable f t can be explaned n a language. To better understand the defnton, consder the followng correspondence: f (n) = {n, n + 1} for all n N. In order to show that ths correspondence s defnable, we must construct the statement requred by the prevous defnton. Let φ (x, y) (y = x) (y = x + 1). Notce that for any par of ntegers, a and b, φ (a, b) s true f and only f b s ether a or a + 1. Therefore, the predcate φ ndeed defnes f. 10

12 Let L be the set of all sentences n our formal language. Each of ts elements s a fnte strng of symbols. It s well known that one can construct a one-to-one functon mappng L N. Let [ϕ] be the value ths functon takes at ϕ L, and call t the Gödel Code of the text ϕ. 3.3 The contractng game Each prncpal offers a contract to each agent. The set of feasble contracts s the set of defnable mappngs from N nk N 2 N. The frst nk arguments are the Gödel codes for the prncpals contracts. The last argument s the code for the message sent by the agent to whom the contract s offered. We denote the set of contracts Prncpal j can offer to Agent by C j, and set C j = k =1 C j, C = n j=1 C j. The tmng of the game s as follows. Prncpals smultaneously submt contracts (c 1,..., c n ) C. These contracts are publcly observable. Then, agents send messages to the prncpals prvately. Let m j denote the message sent by Agent to Prncpal j. Fnally, prncpals smultaneously take actons chosen from the subsets of ther acton spaces determned by the contracts and messages. That s, Prncpal j can take acton a j = ( a 1 j,..., ak j ) Aj only f for all = 1,..., k [ ] ( a j c j [c1 ],..., [c n ], [ m ]) j, where [c q ] denotes ([ c 1 q],..., [ c k q ]) for all q = 1,..., n. In the name of transparency, we wll abuse notaton and replace codes wth actons, wrtng c j : Nnk N 2 A j \ { } whle stll thnkng of c j as a defnable functon. These contracts can be mplemented by cross-referental menus. The tems on a menu of Prncpal j offered to Agent are subsets of A j. The menu correspondng to the contract c j { ( s c j [c1 ],..., [c n ], [ ]) m j : m j N } gven c. Agent s report can be nterpreted as choce from ths menu. We restrct attenton to pure-strategy perfect Bayesan equlbra (PBE). That s, the prncpals and agents are requred to play a Bayesan equlbrum n every subgame generated by a contract profle. 6 The man result of ths paper does not actually depend on the equlbrum concept, so long as players play some equlbrum n the subgames generated by the frst-stage contracts. In partcular, the set of sequental equlbra would be characterzed by essentally the same constrants. We also pont out that the exstence of an equlbrum s only guaranteed f mxed strateges are allowed at the second and thrd stages. The restrcton to pure strateges s purely for notatonal convenence. Allowng for mxed strateges has no substantve consequence on our analyss. 6 In order to guarantee that these subgames exst, one should descrbe the game such that the types of the agents are determned only after the contracts are offered by the prncpals. Ths way of modelng the game has no strategc mplcatons but makes our termnology precse. 11

13 4 Equlbrum Characterzaton We seek to characterze the set of allocatons whch can be mplemented as PBE of the contractng game. A determnstc allocaton s a mappng from the type profle of the agents to the acton profles of the prncpals. Our strategy s to frst analyze equlbra n games where contracts are observable but not contractble. We call these games ordnary contractng games. The analyss of these games leads to a full characterzaton of the contractble contractng games. However, these games are nterestng n ther own rght. In fact, these are the communcaton games analyzed n the common agency lterature. In addton, we am to dentfy envronments where the ablty to contract on contracts can lead to neff cency. In order to do so, we have to characterze the set of equlbra n the ordnary contractng games. 4.1 Ordnary Contractng Games The set of ordnary contracts s the set of defnable mappngs from N 2 N. The doman of these functons are the Gödel codes of the messages sent by the agent to whom the contract s offered. Let Dj denote the set of contracts Prncpal j can offer to Agent, and let D j = k =1 D j and D = n j=1 D j. The tmng of the ordnary contractng game s as follows. Prncpals smultaneously select contracts (d 1,..., d n ) D. These contracts are publcly observable. Then, agents send messages to the prncpals prvately, { m 1,..., m k} N k. Fnally, prncpals take actons smultaneously, such that Prncpal j can take acton a j = ( a 1 j,..., ) ak j Aj f [ ] ([ ]) a j d j m j for all = 1,..., k. Agan, for smplcty we use actons of the prncpals nstead of ther codes and wrte d j : N 2Aj pure-strategy PBE of ths game. whle stll thnkng of d j as a defnable functon. We restrct attenton to We characterze the equlbra n these games by descrbng the best-response constrants of the prncpals and the agents. Notce that when an agent decdes what messages to send to the prncpals, he knows hs type and already observed the contract profle of the prncpals. Hence, the messages of the agents are functons of these two objects. Let β : T D L n denote the strategy of Agent, and let β j denote the jth coordnate of β, that s, the message ( sent to ) Prncpal j by Agents. Let β j denote the messages receved by Prncpal j, that s, β 1 j,..., β k j. Prncpal j s acton at the last stage of the game can depend on both the frst-stage contract profle and the messages sent to hm by the agents. Let α j = ( α 1 j,..., ) αk j, α j : L k D A j for all, denote the strategy of the prncpals at the last stage. Snce Prncpal j s acton must be consstent wth hs contract, α j (m ([ ]) j, d) d j m j must hold for all, mj L k, and for all d = (d j, d j ) D. As usual, α j denotes the acton profle of prncpals other than Prncpal j, and β denotes the message profle of agents other than Agent. In what follows, we defne PBE n terms of three sets of constrants. The frst constrant 12

14 guarantees that each prncpal takes an acton at the last stage whch maxmzes hs payoff. For all j, d D : α j (m j, d) arg max E t [u j (t, a j, α j ) d, m j, β, α j ], (4) a j d j([m j]) for all m j L k and d D. The expectatons are formed accordng to Bayes Rule f the message profle sent by the agents, m j, s consstent wth ther equlbrum behavor. However, PBE mposes no restrcton on the belef of Prncpal j f m j s off the equlbrum path. 7 The second constrant ensures that each agent maxmzes hs payoff by hs message n every subgame generated by a contract profle. For all, t T, and d D, β ( t, d ) arg max m L n E t [ v ( t, α (( m, β ), d )) d, t ]. (5) The last constrant guarantees that no prncpal wants to devate from hs equlbrum contract n the frst stage of the game. Let (d 1,..., d n) = d denote the equlbrum contract profle. Then, for all j: We are ready to defne PBE as follows: d ( ( ( ( j arg max E t uj t, α β, dj, d )))) j. (6) d j D j Defnton 2 The strategy profle (d, β, α) consttutes a PBE n the Ordnary Contractng Game f and only f (4), (5), and (6) are satsfed. It turns out to be useful to defne the set of those allocatons that can be mplemented n a subgame generated by some ordnary contract profle. To ths end, let σ d denote the set of those (α, β) pars for whch both (4) and (5) are satsfed. Then the set of allocatons that can be mplemented n some subgame s defned as follows: A = { g : T A : d D, (α, β) σ d s.t. g (t) = α (β (t, d), d) }. We refer to A as the set of subgame-mlementable allocatons. Next, we characterze ths set n terms of the preferences of the agents and the prncpals. Frst, we fx an allocaton g and explore the mplcatons of g A for the preferences of the agents. Let (α, β) σ d for some d D such that g (t) α (β (t, d), d). Consder Agent wth type t and fx an arbtrary vector ( ( t 1,..., tn) ) T n. Then, by (5), Agent s better off sendng the message profle β ( t, d ) as opposed to β ( j t j, d ) to each Prncpal j. Ths mples E t [ v ( t, ( g ( t, t ))) t ] E t [ v ( t, ( g 1 ( t 1, t ),..., g n ( t n, t ))) t ]. (7) Indeed, the left-hand sde of ths nequalty s the expected payoff of Agent condtonal on t n the subgame generated by d and gven (α, β). The rght-hand sde s the expected payoff of Agent condtonal on t f he devates and sends message β ( j t j, d ) to Prncpal j nstead of β ( j t, d ). The nequalty (7) motvates the followng 7 A stronger equlbrum refnement concept mposes restrctons on the belefs accordng to whch the expectatons are formed n (4), but has no other mpact on our characterzaton result. 13

15 Defnton 3 Let gj : T A j for all j = 1,..., n, = 1,..., k and let g = ( g1,..., gn). Then the allocaton g = ( g 1,..., g k) s called strongly ncentve compatble f for all {1,..., k}, t T, and ( ( t 1,..., tn) ) T n the nequalty (7) s satsfed. Ths defnton s smply the standard noton of ncentve compatblty extended to a multprncpal settng. Indeed, ths defnton would concde wth the standard defnton f the nequalty (7) were requred to hold only for those type vectors, ( ( t 1,...tn) ) T n, where t 1 = t 2 =... = t n. Such a constrant would requre that no agent be able to beneft from mmckng another of hs type. In our mult-prncpal model, however, we must take more complex devatons nto account. In partcular, the messages of the agents are prvate, and therefore, an agent may report dfferent types to dfferent prncpals. Of course, any strongly ncentve compatble allocaton wll also be ncentve compatble. The followng example shows that the converse s not true. Example 1. Suppose that n = 2, k = 1, and A 1 = A 2 = {a 1, a 2 }. The agent has two equally lkely types, T = {1, 2}. The payoffs to the agent are descrbed by the followng matrx: a 1 a 2 a a The allocaton g, defned by g (t) = (a t, a t ) for t = 1, 2, s obvously ncentve compatble but not strongly ncentve compatble. Next, we turn our attenton to the prncpals. In the subgame generated by the contract profle d, Agents wth dfferent types mght send the same message to Prncpal j. The reportng strategy of Agent generates a partton on T denoted by τ j : T 2 T \ { }. 8 Let τ j denote n =1 τ j. After recevng the messages, Prncpal j learns only τ j (t) but not t. Snce the acton of Prncpal j can depend only on nformaton he knows, the strategy profle (α, β) mplements the allocaton g only f the functon g j (t) s measurable wth respect to τ j, that s, g j (t) = g j (t ) whenever τ j (t) = τ j (t ). In addton, the acton of Prncpal j must be consstent wth hs contract, that s, ( α j m j, d ) ([ ]) ([ d j m j. As a consequence, the set d j β j (t, d) ]) ( must contan gj t, t ) for all t T. Therefore, an mplcaton of (4) s that g j (t) arg max a j {g j (t,t ) : t T } E t [u j (t, a j, g j (t)) τ j (t)]. (8) To summarze, we have argued that f g A then g s strongly ncentve compatble, and there s a partton of the type space for whch (8) holds. Next, we show that the converse s also true. The followng lemma fully characterzes the set of subgame-mplementable allocatons. Lemma 1 The allocaton g : T A s an element of A f and only f () g s strongly ncentve compatble and () there exsts a partton, τ j : T 2 T \ { } for all (, j) such that g j s τ j -measurable and (8) s satsfed for all j = 1,..., n. 8 That s, τ ( ) ( ) ( j t 1 = τ j t 2 f and only f β j t 1, d ) = β ( j t 2, d ). 14

16 Proof. The only f part of the proof s already establshed n the text. To prove the f part, suppose that () and () are satsfed. Consder the followng contract offered to Agent by Prncpal j: d j { { ( ([ ]) m g j = j t, t ) : t T } f m j = τ ( ) j t 9 and otherwse, a j where a j s an arbtrary element of A j. By (), the functon g j s measurable wth respect to τ j, so ths contract s well-defned. Snce the allocaton g s strongly ncentve compatble, truth-tellng by the agents consttutes an equlbrum n the subgame. (That s, m j ( t, d ) ( = τ ) j t for all, t and j s an equlbrum.) Fnally, by (8), Prncpal j optmally chooses acton g j (t) f the type profle of the agents s t. Ths equlbrum obvously mplements g. In the case wth a sngle agent, the contracts constructed n the proof of the prevous lemma determne sngle actons for the prncpals as a functon of the agent s message profle. In the subgames generated by these contracts, the prncpals do not make any decsons, and hence, part () s always satsfed. Therefore, we clam the followng Remark 1 Suppose that k = 1. Then the allocaton g : T A s an element of A f and only f g s strongly ncentve compatble. We further nvestgate the propertes of equlbra of the ordnary contractng games n Secton 6. Next, however, we use Lemma 1 to characterze the set of equlbra n contractble contractng games. 4.2 Contractble Contractng Games Ths secton s devoted to the characterzaton of the equlbra n the contractble contractng game. We prove a folk theorem and show that an allocaton s mplementable f and only f t subgame-mplementable and the payoff of each prncpal s larger than hs mnmax value, to be defned later. To see that the allocaton must be subgame-mplementable, we frst argue that any contract profle generates an ordnary contract profle. To ths end, suppose that (c 1,..., c n) s an equlbrum contract profle. For each (j, ), defne d j Dj, such that d j (l) = c j ([c 1],..., [c n], l) for all l N and let d j denote ( ) d 1 j,..., dk j. Notce that d = (d 1,..., d n) s an ordnary contract profle, and the subgame generated by c n the contractble contractng game s the same as the subgame generated by d n the ordnary contractng game. Snce players are requred to play an equlbrum n the subgame generated by the frst-stage contract profle, we can conclude that any allocaton that can be mplemented as a PBE n the contractble contractng game must belong subgame-mplementable. The dff cult part of the theorem s to pn down the mnmax values of the prncpals. mnmax value of Prncpal j s the lowest possble value that he can get n the ordnary contractng 9 To be more precse, m j s a text descrbng τ ( j t ). The 15

17 game f the goal of the other prncpals at the frst stage of the game s to mnmze hs payoff. Formally, we shall prove that the mnmax value of Prncpal j, u j, s: u j = mn d j D j d j D j max mn E t (u j (t, α (β)) (d j, d j )). (9) (α,β) σ (d j,d j) The meanng of ths expresson can be explaned as follows. All the prncpals other than Prncpal j offer ordnary contracts at the frst stage of the game n order to mnmze the payoff of Prncpal j. Prncpal j also offers an ordnary contract whch s a best response to the contracts of the others. These contracts generate a subgame n whch there can be multple equlbra. In ths subgame, the prncpals and agents play an equlbrum whch s the worst one for Prncpal j. The fact that Prncpal j can only be punshed by playng the worst equlbrum n the subgame s obvous because PBE requres the players to play an equlbrum n any subgame generated by a contract profle. The nontrval part of our man result s the rest of the defnton of u j. As we explaned at the begnnng of ths secton, the equlbrum contracts and a frst-stage devaton of Prncpal j determnes an ordnary contract profle. The formula n (9) essentally says that the ordnary contract profle of the prncpals other than Prncpal j does not depend on the devaton of Prncpal j, and hence, Prncpal j can best-respond to t. Snce contracts are contractble, the ordnary contract profle of the prncpals other than Prncpal j can depend on the devaton of Prncpal j. Therefore, one mght conjecture that the prncpals mght be able to push Prncpal j s value below u j. For example, f Prncpal j would be restrcted to offer ordnary contracts then the others could always offer contracts whch are contngent on the ordnary contract of Prncpal j. Beng able to offer these contngent contracts, s smlar to beng able to move after observng Prncpal j s contract, and hence, hs lowest value would be max mn d j D d j D j mn E t (u j (t, α (β)) : (d j, d j )). (α,β) σ (d j,d j) Of course, Prncpal j s not restrcted to offer ordnary contracts, and hs contract can be contngent on the contracts offered by the other prncpals, whch are contngent on hs contract etc. In fact, because of ths nfnte regress problem, t s not even clear that the lowest value of Prncpal j s well-defned. Nevertheless, we show that ths value s well-defned and, nterestngly, the most severe punshment nflcted on Prncpal j can be assumed to be nvarant to hs devaton. To be more specfc, Proposton 1 shows that no matter what the contract profle of the prncpals s, there always exsts an ordnary contract profle d j D j, such that for all d j D j, there s a way for Prncpal j to wrte a contract so that the generated ordnary contract profle s (d j, d j ). But then t s wthout the loss of generalty to assume that the prncpals use the ordnary contract profle d j to punsh Prncpal j. We are ready to state our man result formally. 16

18 Theorem 1 An allocaton g : T A s mplementable as an equlbrum n the contractble contractng game f and only f () g s subgame-mplementable, and () for all j {1,..., n} E t u j (t, g (t)) u j. We break the proof of the theorem nto two parts. The f part s based on the same arguments as the ones used n the example of Secton 2. We shall construct cross-referental contracts whch support the desred allocaton. Essentally, the contract of Prncpal j (for all j) specfes target codes, k for each of the other prncpals. If the Gödel codes of the contracts of Prncpal q are the same as hs target codes for all q, then Prncpal j cooperates. If Prncpal q devates, and the codes of hs contracts are dfferent from hs target codes, the contract of Prncpal j prescrbes an ordnary contract whch s used to mnmax Prncpal q. The set of equlbrum contracts are cross-referental because the Gödel codes of Prncpal j s contracts, whch we have just descrbed, are exactly the same as hs target codes specfed n the contracts of all the other prncpals. Recall two peces of notaton from the ntroducton. Frst, f l N then < l > denotes the text whose Gödel code s l. That s, [< l >] = l. Second, for any text ϕ and (l 1,..., l n ), let ϕ (l1,...,ln) denote the text where f the letter x q stands for a free varable n ϕ then x q s substtuted for l q n ϕ for q = 1,..., n. For example, f ϕ s x 1 < x 2, l 1 = 1, and l 2 = 2 then ϕ (l1,l2) s 1 < Consder now the followng text n n free varable: < x q > (x1,...,xn), where q n. Snce the Gödel codng s a bjecton, < l q > s a text for each l q N. Snce ϕ (l1,...,ln) s defned for all ϕ and (l 1,..., l n ) N n, < l q > (l1,...,ln) s a text for all (l 1,..., l n ) N n. It s a well-known result n Mathematcal Logc that f f (l 1,..., l n ) = [ < l q > (l1,...,ln)], then f s a defnable functon. Proof of the f part of Therorem 1. Snce the allocaton g s n ( A there exsts an ordnary contract profle d = (d 1,..., d n), a strategy profle of the agents, β = β 1,..., β k), a thrd-stage strategy profle of the prncpals, α = (α 1,..., α n), such that g (t) = α (β (t, d ), d ) and both (4) and (5) are satsfed, that s, (α, β ) σ d. In addton, let d j,q denote the contracts of Prncpal j whch he uses to mnmax Prncpal q. That s, the contract profle d q,q solves mn d q D q d q D q max mn E t (u q (t, α (β)) : (d q, d q )). (10) (α,β) σ (dq,d q) Let x m = ( x 1 m,..., x k m) a vector of free varables for all m = 1,..., n. Consder the followng text of Prncpal j, c,x1,...,xn j, n nk free varables: { ([ ]) d j m ([ j ]) m j d j,q c,x1,...,xn j ( ([cl ]) n l=1, [ m ]) j = f { [ l : s.t. < x l > (x1,...,xn)] [ ] } c l 1, f { [ l : s.t. < x l > (x1,...,xn)] [ } (11) cl] = {q}, 10 Of course, t s possble that the text ϕ does not contan some of the symbols {x 1,..., x n}. In that case, there s no substtuton for the mssng letters n ϕ (l 1,...,l n). For example, f ϕ s x 2 > 2, then ϕ (3,4) s 4 > 2, because x 1 does not appear n ϕ. 17

19 for all [ m j] N. Ths expresson (11) s not a contract, but rather a contract wth free varables. However, c x1,...,xn j would become a contract f the free varables (x 1,..., x n ) are replaced by ntegers. Each of these contracts wth free varables has a Gödel code, so let γ j ] [c =,x1,...,xn and γ j = ( γ 1 j,..., ) { γk j. The functons contracts. Notce that = c,γ 1,...,γ n j { d j The contract c,γ 1,...,γ n j d j,q c,γ 1,...,γ n j } ( ([cl ]) n l=1, [ m j]) ([ m j ]) ([ m j ]),j f { l : s.t. f { l : s.t. have no free varables, so they consttute a set of [ < γ l > (γ 1,...,γ )] m [ ] } c l 1, [ < γ l > (γ 1,...,γ )] m [ ] } c l = {q}. j (12) s defnable because d j, d j,q and f (l 1,..., l n ) = [ < l q > (l1,...,ln)] are all defnable. Observe what happens when Prncpal q offers contract c,γ 1,...,γ n q for all q,. Prncpal j needs to check whether the Gödel code of < γ q > (γ 1,...,γ m ) s equal to the Gödel code of c,γ 1,...,γ n q. The nteger γ q s the Gödel code of the contract wth free varables cq,x1,...,xn. Prncpal j s contract says to take ths contract wth the free varables, fx the free varables at γ 1,..., γ n (whch gves the contract c,γ 1,...,γ n q ), then evaluate ts Gödel code. Ths s what s to be compared wth the Gödel code of the contract offered by Prncpal q to Agent. Of course, f Prncpal q offers c,γ 1,...,γ n q Agent these are the same. In fact, f Prncpal q offers c,γ 1,...,γ n q to for all (q, ) then Prncpal j ends up wth the ordnary contract d j accordng to the frst lne of (12). Therefore, f Prncpal j offers contract c,γ 1,...,γ n j for all (j, ) then the resultng subgame s generated by the ordnary contract profle d. Defne the strateges of the agents and the prncpals as β (, d ) and α (, d ). These strateges obvously support the allocaton g. It remans to specfy the strateges of the players off the equlbrum path and show that no player can proftably devate. Next we defne the second-stage strateges of the agents and the thrd-stage strateges of the prncpals off the equlbrum path. It s enough to defne these strateges n subgames whch result from a devaton of a sngle prncpal. Suppose ( that Prncpal q offers a contract c q nstead of c,γ 1,...,γ n q to Agent. Let d q ( ) denote c q [c q ], ([ c γ 1,...,γ n ])j q ),. As a result of ths devaton, accordng to the second lne of (12), Prncpal j wll end up wth the ordnary contracts d j,q for all j q. Therefore, the subgame resultng from the devaton of Prncpal q s generated by the ordnary contract profle d = (d q, d q,q ). Defne the strateges of the agents and the prncpals, α (d) and β (d), so that the expected payoff of Prncpal q s mnmzed. That s, (α (d), β (d)) solves j mn E t (u q (t, α (β)) : d). (13) (α,β) σ d Fnally, we argue that nether the prncpals nor the agents have ncentves to devate from the equlbrum strateges. Frst, f Prncpal j offers contract c,γ 1,...,γ n j for all (j, ), then no player can proftably devate n the subgame generated by the ordnary contract profle d because (α, β ) σ d. In fact, we have defned the strateges of the players, α (d) and β (d), n any relevant subgame generated by an ordnary contract profle, d, such that (α, β) σ d. Therefore, we only 18