On the use of menus in sequential common agency

Transcription

1 Games and Economc Behavor 6 (2008) Note On the use of menus n sequental common agency Gacomo Calzolar a, Alessandro Pavan b, a Department of Economcs, Unversty of Bologna, Pazza Scaravll 2, 026, Bologna, Italy b Department of Economcs, Northwestern Unversty, 200 Sherdan Road, Evanston, IL 60208, USA Receved 6 Aprl 2007 Avalable onlne 2 February 2008 Abstract We llustrate, by means of two examples, why assumng the prncpals offer smple menus (.e. collectons of payoff-relevant alternatves) as opposed to more general mechansms may preclude a complete characterzaton of the set of equlbrum outcomes n certan sequental contractng envronments. We then dscuss how refnements of the soluton concept, or enrchments of the menus that allow for recommendatons, may restore the possblty of usng menus to obtan a complete equlbrum characterzaton Elsever Inc. All rghts reserved. JEL classfcaton: D89; C72 Keywords: Sequental contractng; Mechansm desgn; Menus theorems. Introducton When multple prncpals contract smultaneously wth the same agent, Peters (200) and Martmort and Stole (2002) have proved the followng result: For any equlbrum relatve to any game wth arbtrary sets of mechansms for the prncpals, there exsts an equlbrum n the game n whch the prncpals are restrcted to offer the agent the menus of payoff-relevant alternatves they could have offered n the orgnal game that sustans the same outcomes. Ths result s referred to n the lterature as the Delegaton Prncple (or the Menu Theorem) and has proved useful n applcatons. A prevous verson was enttled On the Valdty of Menu Theorems n Sequental Common Agency Games. * Correspondng author. E-mal addresses: gacomo.calzolar@unbo.t (G. Calzolar), alepavan@northwestern.edu (A. Pavan) /$ see front matter 2008 Elsever Inc. All rghts reserved. do:0.06/j.geb

2 330 Note / Games and Economc Behavor 6 (2008) Recent years have wtnessed nterest n envronments n whch contractng s sequental, n the sense that the agent contracts wth hs multple prncpals at dfferent ponts n tme. In lght of ths fast growng lterature, t s mportant to understand whether restrctng the prncpals to offer menus of payoff-relevant alternatves, as opposed to more general mechansms, s wthout loss of generalty also when contractng s sequental. In ths note we construct two examples that llustrate why smple menus may fal to sustan all equlbrum outcomes n certan sequental contractng envronments. The frst example features a stuaton n whch the prncpals observe the payoff-relevant decsons taken upstream, but not necessarly the mechansms used to select them. In ths settng, restrctng the prncpals to offer menus may mean restrctng the extent to whch dfferent prncpals can have dvergng belefs about the specfc mechansm used upstream to select an off-equlbrum decson. When the agent s strategy s not Markov (.e. t may depend on the entre upstream hstory), ths means mposng restrctons on the prncpals expectatons about the agent s behavor downstream. Such restrctons may preclude the possblty of sustanng certan outcomes. The second example features a stuaton n whch all prncpals observe both the mechansms and the payoff-relevant decsons selected upstream. The reason why smple menus fal n ths example s that they do not permt the prncpals to use payoff-rrelevant nformaton as a devce to correlate ther decsons. In the absence of alternatve nstruments such as sunspots or cheap talk messages, ths means restrctng the possble outcomes. These examples warn aganst the use of smple menus n certan sequental contractng envronments. However, there are stuatons of nterest for applcatons n whch the problems ndcated by these examples never arse. Furthermore, there are ways of enrchng the menus (for example allowng the prncpals to send each other, and/or the agent, recommendatons about the decsons to take downstream) that may restore the possblty of usng menus to sustan all equlbrum outcomes. We dscuss some of these ssues at the end of the note. 2. Smple menus Ths secton contans two examples that llustrate why smple menus may not sustan all equlbrum outcomes when contractng s sequental. 2.. Out-of-equlbrum belefs Consder a game n whch four prncpals contract sequentally wth the same agent, A. The game has four stages. At each stage, a dfferent prncpal, P, =,...,, contracts wth A. Each prncpal must select a payoff-relevant alternatve a (also referred to as a decson) from aseta of feasble alternatves. Dependng on the applcaton of nterest, a can be a polcy, a level of trade, or the decson to undertake a project. The selecton of a s obtaned through a mechansm; the latter conssts of a set of possble messages M along wth a mappng φ : M A such that, when A sends the message m M, P responds by selectng the alternatve a A. In ths example, the sets of feasble alternatves are A ={b,c },for =,...,3, and A ={d,e,f}. The prncpals and the agent s payoffs are descrbed by the quntuples We refer the reader to Pavan and Calzolar (2007) for a dscusson of the relevance of sequental contractng n applcatons.

3 Note / Games and Economc Behavor 6 (2008) Fg.. Payoffs. (u,u 2,u 3,u,u A ) n Fg. ; note that, when all players payoffs are ndependent of the decsons taken after perod t =,...,3, the tree has been cut to hghlght drectly the fnal payoffs. Before choosng her mechansm, each downstream prncpal observes all the payoff-relevant decsons taken upstream. Furthermore each downstream prncpal observes all upstream mechansms, wth the excepton of the mechansm selected by P. None of the prncpals observes the messages sent by the agent to the other prncpals. Now consder the game Γ n whch the sets of feasble mechansms are Φ ={φ b,φc, φ c, φ b,c } for P, Φ ={φ b,φ c,φ b,c } for P 2 and P 3, and Φ ={φ d,φe,φf,φd,e,φ d,f,φ e,f, φ d,e,f } for P. Superscrpts denote the mage (.e. the range) of the mechansm; for example, Im(φ d,e ) ={d,e}. Note that, for any, and any menu of payoff-relevant alternatves ψ 2 A \{ } there exsts one and only one mechansm φ ψ Φ such that Im(φ ψ ) = ψ. In contrast, for =, there are two mechansms φ c and φ c whose range s {c }. In ths game, the outcome (b,b 2,b 3,d)can be sustaned as a pure-strategy perfect Bayesan equlbrum. It suffces to take any strategy profle wth the followng propertes: P offers the mechansm φ b ; regardless of a, P 2 offers φ b 2 2 ; regardless of (a,a 2,φ 2 ), P 3 offers φ b 3 3 ;regardless of (φ 2,φ 3 ), P offers φ d,e,f f (a,a 2,a 3 ) = (c,b 2,b 3 ) and φ d otherwse; at t =, A chooses c f P offers φ b,c ; at t = 2, A chooses c 2 f P 2 offers φ b 2,c 2 2 ;att = 3, A chooses c 3 f P 3 offers φ b 3,c 3 3 ;att =, when offered the mechansm φ d,e,f,achooses e f φ = φ c, f f φ = φ c and d f ether φ = φ b or φ = φ b,c. Clearly, ths s only a partal descrpton of the strategy profle; however, such a descrpton contans all nformaton that s relevant for the result we want to establsh. The aforementoned strategy profle s sustaned by the followng out-ofequlbrum belefs: after observng c, P 2 beleves that φ = φ c, P 3 beleves that φ = φ b,c, and P beleves that φ = φ c. We clam that the outcome (b,b 2,b 3,d) cannot be sustaned n the menu game Γ M n whch the prncpals are restrcted to offer the menus of payoff-relevant alternatves they could have offered n Γ and delegate to the agent the choce of the decsons. We prove the result by showng that ths outcome requres that, after observng a devaton to c, the supports of P 2 s, P 3 s, and P s belefs about the mechansm used by P not overlap, whch clearly cannot be the case n Γ M. To see ths, note that P (weakly) prefers b to c f and only f c s followed by b 2 and b 3. Hence, for (b,b 2,b 3,d)to be an equlbrum outcome n Γ M, t must be that, after observng a

4 332 Note / Games and Economc Behavor 6 (2008) devaton to c, P 2 and P 3 offer menus that contan, respectvely, b 2 and b 3, and that A selects b 2 and b 3 n each of these menus. Furthermore, because A strctly prefers c 2 to b 2 after c and strctly prefers c 3 to b 3 after (c,b 2 ), t must be that P 2 offers the (degenerate) menu {b 2 } after observng c and that P 3 offers the (degenerate) menu {b 3 } after observng the menu {b 2 } and the decsons (c,b 2 ). Because P 2 can always guarantee herself a payoff of 2 by choosng c 2 after c, for her to offer the menu {b 2 } t must be that she expects A to choose b 3 wth P 3 and f wth P. Smlarly, because P 3 can always guarantee herself a payoff of 2 by choosng c 3 after (c,b 2 ), for her to offer the menu {b 3 } t must be that she expects A to choose d wth P. Lastly, because P can always guarantee herself a payoff of 2 by choosng e after (c,b 2,b 3 ), for her to offer any other menu, t must be that ths menu contans e and that she expects A to choose e from the menu. We conclude that any strategy profle that sustans (b,b 2,b 3,d) as an equlbrum n Γ M must satsfy the followng propertes: P 2 offers the menu {b 2 } after observng c ; P 3 offers the menu {b 3 } after observng the menu {b 2 } and the decsons (c,b 2 ); P offers the menu {e,f,g} after observng the menus {b 2 } and {b 3 } and the decsons (c,b 2,b 3 ). Now, gven the decsons (c,b 2,b 3 ) and the upstream menus {b 2 } and {b 3 }, the agent s behavor at t = may vary only on the bass of the partcular menu contanng c offered at t =. Because n Γ M there are only two such menus, there are only two possble behavoral strateges that A can follow at t = gven the decsons (c,b 2,b 3 ) and the menus {b 2 } and {b 3 }. It s thus mpossble that P 2, P 3 and P expect A to choose respectvely f, d, and e wth probablty one when offered the menu {e,f,g}. The outcome (b,b 2,b 3,d) can thus be sustaned n Γ but not n the menu game Γ M Correlaton devces We now llustrate a second reason why menus may not sustan all possble outcomes. Consder an envronment n whch three prncpals contract sequentally wth the same agent. The sets of feasble alternatves are A ={t} and A 2 = A 3 =[0, ]. For smplcty, assume P s and P 3 s payoffs are constant over A A A 2 A 3, whereas P 2 s and A s payoffs are respectvely u 2 = 2a 2 a 3 + ( a 2 )( a 3 ) and u A = a 3 ( + a 2 ). In ths envronment, all prncpals observe all mechansms selected upstream. Whether they also observe the payoff-relevant decsons taken n these mechansms s not mportant n ths example but, to fx deas, assume they do. Now consder a game Γ n whch the sets of feasble mechansms are Φ ={ φ,φ } and Φ ={φ a } a A, = 2, 3, where φ a s a mechansm such that Im(φ a ) = a. The followng s a perfect Bayesan equlbrum n Γ : P randomzes over φ and φ wth probablty q (0, ) and q, respectvely; gven φ, P 2 chooses a 2 =, whereas gven φ she chooses a 2 = 0; at t = 3, regardless of (φ 2,a 2 ), P 3 chooses a 3 = fφ = φ and a 3 = 0 f φ = φ. The equlbrum outcome s (t,, ) wth probablty q and (t, 0, 0) wth probablty q. Ths outcome cannot be sustaned n the menu game Γ M n whch the prncpals offer the menus of payoff-relevant alternatves they could have offered n Γ and delegate to the agent the choce of the decsons. The reason s that Γ M does not permt P to correlate the other prncpals decsons. The role of P as a correlaton devce s key to sustan the outcome descrbed above and cannot be replcated by P 2. In fact, P 2 fnds t optmal to match the decson taken by P 3 and hence to respond to the mechansm selected by P antcpatng how P 3 responds to t but s never wllng to mx over A 2, for she strctly prefers (a 2,a 3 ) = (, ) to (a 2,a 3 ) = (0, 0).

5 Note / Games and Economc Behavor 6 (2008) Dscusson The equlbrum constructed n the frst example s a (weak) perfect Bayesan equlbrum but nether a sequental nor a Markov-perfect equlbrum. Imposng such refnements may restore the possblty of usng menus to sustan all outcomes. In ths respect, the example hghlghts an mportant dfference between smultaneous and sequental common agency. In the former, the valdty of the Delegaton Prncple s ndependent of whether one s nterested n all perfect Bayesan equlbrum outcomes or only n outcomes that can be sustaned by mposng refnements such as sequental or Markov-perfect equlbrum. Ths s not the case n the latter. Next, consder the second example. If publc sunspots are avalable, then restrctng the prncpals to offer smple menus may not pose any problem. The role of the example s to warn aganst the use of smple menus n envronments n whch alternatve correlaton devces are not avalable another mportant dfference wth respect to the smultaneous case. Also note that the noton of menus consdered here s the one used n applcatons: a menu s a collecton of payoff-relevant alternatves. 2 The problems hghlghted by our examples vansh f one consders more general menus that allow the prncpals to send each other, and/or the agent, recommendatons about the decsons to take downstream. For nstance, n the frst example, the two mechansms ψ ={c } and ψ ={c } can be replaced by two menus that contan the same payoff-relevant decson but two dfferent recommendatons to the agent about the strategy to follow downstream. Provded that these recommendatons are prvate (n the sense that they are not observed by the downstream prncpals), then the outcome (b,b 2,b 3,d) can be sustaned also n the menu game wth recommendatons. 3 Smlarly, the outcome n the second example can be sustaned by lettng the frst prncpal send publc (perfectly correlated) recommendatons to the downstream prncpals. These enrched menus are more smlar to Myerson (982) generalzed drect revelaton mechansms than to smple menus such as prce-quantty schedules, as typcally used n applcatons. As shown n Peters (200), allowng for such enrched menus may be necessary when the agent exerts some effort after communcatng wth the prncpals. In Peters envronment, contractng s smultaneous and recommendatons are used to fashon the agent s belefs about the prncpal s response to the agent s effort. Because ths s the only role that recommendatons play n smultaneous games, such recommendatons can be dspensed wth f one allows the prncpals to offer menus of lotteres over contracts (as opposed to menus of determnstc decsons). The role of recommendatons n sequental contractng s dfferent: n the frst example, recommendatons are used to permt the downstream prncpals to have dvergng belefs about the agent s behavor n downstream relatonshps. In the second example, recommendatons are used to correlate the prncpals decsons. Furthermore, t may not suffce to ntroduce lotteres to dspense wth such recommendatons, as t can be seen by consderng the second example where there s a sngle such lottery. Whle our examples warn aganst the use of smple menus n certan envronments, there are stuatons n whch smple menus do sustan all equlbrum outcomes. As shown n Pavan and Calzolar (2007), ths s always the case when contractng s prvate,.e. when downstream prncpals observe nether the mechansms nor the payoff-relevant decsons selected upstream. 2 Ths s also the noton used by Martmort and Stole (2002) to establsh the Delegaton Prncple. 3 We thank a referee for suggestng ths possblty. In a moral hazard settng, a decson should be nterpreted as a contract that specfes the prncpal s acton as a functon of some verfable performance measure correlated wth the agent s effort.

6 33 Note / Games and Economc Behavor 6 (2008) Furthermore, even when contractng s not prvate, all equlbrum outcomes sustaned by Markov strateges can be sustaned wth smple menus. Because Markov strateges are often consdered a focal class, smple menus may retan a sgnfcant appeal n many applcatons, even f they do not sustan all equlbrum outcomes. Acknowledgments For comments and useful suggestons, we thank Perpaolo Battgall, Edde Dekel, Mke Whnston, Asher Wolnsky, an assocated edtor, and two anonymous referees. References Martmort, D., Stole, L., The revelaton and delegaton prncples n common agency games. Econometrca 70, Myerson, R., 982. Optmal coordnaton mechansms n generalzed prncpal agent problems. J. Math. Econ. 0, Pavan A., Calzolar, G., Sequental contractng wth multple prncpals. Mmeo, Northwestern Unversty and the Unversty of Bologna. Peters, M., 200. Common agency and the revelaton prncple. Econometrca 69,