SUPPLEMENT TO IMPLEMENTATION WITH CONTINGENT CONTRACTS (Econometrica, Vol. 82, No. 6, November 2014, )

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1 Econometrca Supplementary Materal SUPPLEMENT TO IMPLEMENTATION WITH CONTINGENT CONTRACTS Econometrca, Vol. 82, No. 6, Noember 204, BY RAHUL DEB AND DEBASIS MISHRA IN THIS SUPPLEMENT, WE FIRST PRESENT AN EXAMPLE that shows that Theorem breas down when the type space s countably nnte. We then present a ew extensons to the results n the paper. Throughout ths supplement, we conduct the analyss or an arbtrary agent,x V, and, or notatonal conenence, we suppress the dependence on. Recall that we can do so because the ncente compatblty requrement s or each agent and all possble reports o the other agents. NONEQUIVALENCE WITH COUNTABLY INFINITE TYPES We construct an example o a type space wth countably nnte types and a socal choce uncton SCF where the latter can be mplemented by a contngent contract but not by a lnear contract. We begn by showng that acyclcty s sucent or mplementablty een when types are countably nnte. LEMMA A: Suppose the type space s countable. I an SCF s acyclc, t can be mplemented usng a contngent contract. PROOF: ConsderanSCF. We need to show that there exsts a contngent contract s such that or eery V,wehae s S s We wll dene an ncomplete bnary relaton s, s oer tuples { } or all V. These tuples correspond to a type mang a report o. We rst dene the relaton s0 and s : { { { { } { s0 } { s0 } { s } { s { } s0 { } } } } > < = = } or all We dene s as the transte closure o s0.formally,wesay{ } s {ˆ ˆ ˆ } there exsts a nte sequence {{ } 204 The Econometrc Socety DOI: /ECTA56

2 2 R. DEB AND D. MISHRA { K K K }} such that { } { R R K { K K K } R2 } RK+ {ˆ ˆ ˆ } where R { s0 s } and at least one R s0.itseasytoarguethatacyclcty o mples that the relaton s s rrelexe. Snce s s rrelexe and transte and V s countable, we can then use a standard representaton theorem Fshburn 970 that guarantees the exstence o a uncton s that respects s. Q.E.D. Note that scaled cycle monotoncty s.c.m. s sucent or mplementaton by a lnear contract een or uncountably nnte type spaces Proposton 2. In the example below, acyclcty s satsed but s.c.m. s not. EXAMPLE A: Consder a sngle agent wth the countably nnte type space V = { 2 3 } { } Suppose the set o alternates A has equal cardnalty and consder an SCF that satses or all Dene the type space such that 2 = 2 0 < = > = Fnally, we dene payo or type as 2 = < otherwse. It s easy to see that s acyclc. Ths s because or all < as 2 = > =

3 IMPLEMENTATION WITH CONTINGENT CONTRACTS 3 Moreoer, when < <, then Fnally,, as = < = 2 = 0 < = Lemma A shows that acyclcty remans a sucent condton or mplementablty the type space s countable and, hence, can be mplemented usng a contngent contract. We now show that cannot be mplemented by a lnear contract. Let us assume to the contrary that t s mplementable by r t. Then addng the two ncente compatblty condtons or types and msreportng as each other, we get r [ + r [ ] r r 2 0 = ] 0 Smlarly, ncente compatblty or types [ ] r + + r + [ + r + r and + mples ] = 2 + Multplyng nequaltes or succeedng = 2K, we get r K r 2 2K 3 K Combnng nequaltes, we get r r K K 2K 3 r 2

4 4 R. DEB AND D. MISHRA Tang the lmt K, we obsere that the rght sde derges, whch mples that r must be, whch s a contradcton. Hence, cannot be mplemented usng a lnear contract. PARTIALLY CONTRACTIBLE PAYOFFS We note that on many occasons the entre payo may not be contractble. Howeer, our results wll contnue to hold n some such stuatons. Suppose the payo o an agent has a contractble and a noncontractble component. We assume that the noncontractble component o the payo s a monotone uncton o the contractble component and the alternate chosen. Formally, type now relects the contractble component o agent s payo oer arous alternates. There s a map g : R A R that ges the noncontractble payo o agent. We assume that g s nondecreasng n the rst argument. Consder an SCF. Gen a contngent contract s, the net payo o agent by reportng wth true type s gen by s + g Here, we consder lnear contracts where the royalty term s not bounded rom aboe by or r : V 0. Gen a lnear contract r t, the net payo o agent by reportng wth true type s gen by r + g t We wll show that Theorem contnues to hold een under ths settng. Snce Theorem contnues to hold, wth an approprate redenton o aggregate payo maxmzers, we can also show that Theorem 2 holds. As beore, or any SCF, we dene the bnary relaton as ollows. For any V,wesay >. We also dene the bnary relaton as ollows. For any V,wesay. DEFINITION A: An SCF s acyclc or any sequence o types wth 2,wehae. As beore, we can show the necessty o acyclcty. LEMMA B: I an SCF s mplementable by a contngent contract, then t s acyclc.

5 IMPLEMENTATION WITH CONTINGENT CONTRACTS 5 PROOF: Suppose SCF s mplementable by a contngent contract s.consder any sequence o types wth 2. Choose { }.Snce s mplementable by s, we get that s s s g g + + g where the second nequalty used the act that +, s s ncreasng n the rst argument, and g s nondecreasng n the rst argument. Hence, we get that or any { },wehae S2 s s g + + g Addng nequalty S2orall { } and telescopng, we get s + g S3 s + g Snce s mplementable, we hae s + g s +g. Ths along wth nequalty S3gesus s + g S4 s + g Now assume, or contradcton,.then >. Snces s strctly ncreasng n the rst argument and g s nondecreasng n the rst argument, we get that s + g S5 >s + g Ths s a contradcton to nequalty S4. Q.E.D. We now proceed to show that the remander o the proo o Theorem can be adapted straghtorwardly. Frst, we dene some termnology. For any V,let d :=

6 6 R. DEB AND D. MISHRA and d := g g DEFINITION B: An SCF s generalzed scaled cycle monotone there exsts λ : V 0 such that or eery sequence o types +, we hae [ λ d = + ] + d + 0 PROPOSITION A: An SCF s mplementable by a lnear contract and only t s generalzed scaled cycle monotone. PROOF: The necessty o generalzed scaled cycle monotoncty ollows by addng any cycle o ncente constrants. For sucency, suppose satses generalzed scaled cycle monotoncty. Let λ : V 0 be the correspondng multpler. Then, by the Rochet Rocaellar theorem, there exsts a map W : V R such that or eery V,wehae S6 W W [ λ d + d ] Now, or any V,let t := λ + g W Now, substtutng n nequalty S6, we get or eery V, W W = λ + g t λ g + t λ λ + g g Cancelng terms, we get λ λ + g t + g t Ths ges us the desred ncente constrants. Q.E.D. We wll now show that s acyclc, then t s generalzed scaled cycle monotone. To do so, we rst obsere that s acyclc, then we can apply Lemma 2 to clam that the type space can be -order-parttoned. Now, we can use ths to

7 IMPLEMENTATION WITH CONTINGENT CONTRACTS 7 construct a λ : V 0 map recursely. Let V V K be an -ordered partton o V.Frst,wesetorall V K λ := Hang dened λ or all V + Let C be any cycle o types q, V K,wedeneλ or all V V K wth V K.LetC be the nolng types n V at least one type n V andatleastonetypenv + set o all such cycles. Now dene or each cycle C q q+ LC := C V + V K λ d + + q = d +. C, and lc := V C d + Now, consder two possble cases. I LC 0orallC C, then set λ = orall V. I LC < 0orsomeC C, we proceed as ollows. Snce V s -orderparttoned, or eery V and + V V K, wehaed >0 Property P2. Smlarly, or eery V,wehaed 0Property P. Then, or eery C C, wemusthaelc > 0, snce t noles at least one type rom V andatleastonetyperomv + V K.Now,oreery V,dene λ := max C C LC lc We thus recursely dene the λ map. PROPOSITION B: I s acyclc, then λ maes generalzed scaled cycle monotone. PROOF: Consder any cycle C q q+. We wll show that S7 q = λ d + + d + 0 I C V K, then d + 0, d + 0, and λ = λ + or all + C. Hence, nequalty S7 holds. Now, suppose nequalty S7 s true or all cycles C V + V K. Consder a cycle C q

8 8 R. DEB AND D. MISHRA nolng types n V V K. IeachtypenC s n V, then agan d + 0, d + 0, and λ = λ + or all + C. Hence, nequalty S7 holds. By our hypothess, all types n C belong to q+ V + V K cycle that noles at least one type rom V V + V K C [ λ, then agan nequalty S7 holds. So assume that C s a and at least one type rom.letλ = μ or all V. By denton, ] d + d + = LC + μlc 0 + where the last nequalty ollows rom the denton o μ. Hence, nequalty S7 agan holds. Proceedng le ths nductely, we complete the proo. Q.E.D. To summarze, we hae shown the ollowng result. THEOREM A: Consder the partally contractble enronment and suppose the type space s nte. Then, or any SCF, the ollowng statements are equalent: s mplementable by a contngent contract. s acyclc. s generalzed scaled cycle monotone. s mplementable by a lnear contract. AN INFINITE TYPE SPACE WHERE THE EQUIVALENCE HOLDS We now show that the equalence establshed n Theorem extends to a model wth uncountably nnte type spaces under addtonal condtons. We mae the ollowng assumptons. B. The set o alternates A s a metrc space. B2. The set o types V s a compact metrc space and each V s contnuous n a. B3. The SCF s contnuous n. THEOREM B: SupposeassumptonsB B3 hold and, addtonally, supposean SCF can be mplemented by a contngent contract s that s twce contnuously partally derentable n the rst argument. Then can also be mplemented by a lnear contract. PROOF: We are gen that can be mplemented by a contngent contract s : R V R that s twce contnuously derentable n the rst argument. We rst show that ths mples that can also be mplemented by a contngent contract s that s conex n the rst argument. Consder the ollowng transormaton o s: s = e γs where γ>0

9 IMPLEMENTATION WITH CONTINGENT CONTRACTS 9 Clearly, snce s s a monotone transormaton o s, t s both strctly ncreasng n the rst argument and ncente compatble. Thereore, t also mplements. We denote partal derates o s wth respect to the rst argument by s u. Snce s s twce derentable n the rst argument, so s s and ts second partal derate s gen by 2 s 2 = γe γs s 2 s / u 2 u s / u + γ 2 u 2 Now, snce V s compact, s contnuous, and s s twce contnuously derentable n the rst argument, ths mples that 2 s / u2 s / u 2 s bounded rom below or all V Ths s because the aboe uncton s contnuous on the compact set V V and, hence, must attan a mnmum. Ths n turn mples that there exsts a large and nte γ>0 such that s s conex n the rst argument. Incente compatblty and conexty o s appled n turn then yeld the nequalty, or all V, s s s s + [ u ] Now set multplers λ = s u > 0orall V and notce that wll satsy scaled cycle monotoncty wth these multplers. O course, ths mples that can be mplemented by a lnear contract Proposton 2, whch completes the proo. Q.E.D. IMPLEMENTATION IN A LINEAR ONE DIMENSIONAL UNCOUNTABLE MODEL In ths secton, we descrbe a smple model o a one dmensonal type space wth uncountable types, where 2-acyclcty s sucent. We assume that the set o alternates A s nte. Addtonally, we assume that types are one dmensonal and lnear. Formally, or eery alternate a A, there exsts κ a 0andγ a such that or all, a = κ a + γ a where V R The ollowng theorem s the characterzaton result.

10 0 R. DEB AND D. MISHRA THEOREM C: Suppose A s nte and the types are one dmensonal and lnear. Then the ollowng condtons on an SCF are equalent: satses 2-acyclcty. s scaled 2-cycle monotone. s mplementable by a lnear contract. s mplementable by a contngent contract. PROOF:. Dene the map ν : A R + as ollows. For eery a A, κ νa = a 0, κ a 0 κ a = 0. Further, dene ν := max a A νa and V 0 := { V : κ = 0}. Now dene r : V 0 ] as ollows. Fx an ε 0 ]. For eery V, { ε V 0 r = ν ν V \ V 0 Now note that V 0, then r κ = 0, and V \ V 0, then r κ =. Hence, or eery V 0 ν and V \ V 0,wehae S8 r κ >r κ Now consder any V.Snce s 2-acyclc, t means mples. Equalently, κ 0 mples κ 0. Equalently, > and κ = 0, then κ = 0. Ths urther means that V 0 and <, then V 0. Hence, usng nequalty S8, we get that >, then S9 r κ r κ Now, or any V wth >, scaled 2-cycle monotoncty requres that S0 r κ r κ 0 Ths s true because o nequalty S9.. Usng Proposton 2, t s enough to show that s scaled 2-cycle monotone, then t s scaled cycle monotone. Because satses scaled 2-cycle monotoncty, or any >,nequaltys0 s satsed. But ths mples that nequalty S9 s satsed. Assume or contradcton that als scaled cycle monotoncty. Let be the smallest nteger such that als scaled -cycle monotoncty. Snce satses

11 IMPLEMENTATION WITH CONTINGENT CONTRACTS scaled 2-cycle monotoncty, 3. Ths means or eery r : V 0 ] and or some nte sequence o types,wehae = l r + < 0 where + and l r + := r [ + ]. Consder a r : V 0 ]. Let > p or all p {}\{}. We wll show that l r + l r + l r + 0. To see ths, + l r l r l r [ = r [ r + κ r [ r + + κ + r κ + r κ = [ + r κ r 0 ] κ + ] κ ] κ where the last nequalty ollows rom the act that > + and applyng nequalty S9. Snce satses scaled -cycle monotoncty, we now that l r 2 + +lr 2 + l r + + l r l r 0. But because o the last nequalty, we must hae = l r + 0 whch ges us a contradcton. O course, and Lemma establshes that. Ths concludes the proo. Q.E.D. REMARK: A closer loo at the proo o Theorem C reeals that κ a > 0 or all a A, then or eery SCF, V 0 =, and, hence, eery SCF satses 2-acyclcty acuously. Thus, eery SCF can be mplemented usng a lnear contract. ] SUFFICIENCY OF 2-ACYCLICITY IN A LINEAR TWO DIMENSIONAL COUNTABLE MODEL In ths secton, we consder a lnear two dmensonal generalzaton o the model n the preous secton. Formally, or eery alternate a A, there ex-

12 2 R. DEB AND D. MISHRA sts κ a 0, κ a2 > 0, and γ a such that or all, a = κ a + 2 κ a2 + γ a where V R The proo wll use the normalzed ector or an alternate a A, whchwe denote by κa κ a = κ a2 Note that snce we hae restrcted κ a2 > 0, the aboe normalzed ector s well dened. The next result shows that 2-acyclcty s sucent or mplementablty n the lnear two dmensonal enronment. The proo shows that 2-acyclcty mples acyclcty. Thus, rom Lemma A, we can conclude that n countable type spaces, 2-acyclcty s sucent or mplementablty. THEOREM D: Suppose the type space s countable. Then an SCF s mplementable n the lnear two dmensonal enronment and only t s 2-acyclc. PROOF: We need to show that 2-acyclcty mples -acyclcty or all. We wll proceed by nducton on. The base case o = 2 s trally true. As the nducton hypothess, we assume the mplcaton holds or some >2. We wll now show the nducton step that 2-acyclcty mples + -acyclcty. Suppose s 2-acyclc. Consder a sequence + wth the ollowng propertes. For all { }, each element s wealy greater than the succeedng element and no element s strctly greater than any preous element n the sequence. Formally, + and + or all {} whch s equalent to κ and κ + 0 orall {} Addtonally, wthout loss o generalty, we can tae the nequalty to be strct or : 2 or that κ 2 2 > 0 Ths assumpton s requred or the result. It s possble to construct a smple counterexample we allow κ a2 = 0.

13 IMPLEMENTATION WITH CONTINGENT CONTRACTS 3 The nducton hypothess -acyclcty then mples that or that κ < 0 or all {2} Fnally, + s such that + or κ + The nducton hypothess mples that or all {2} or κ + 0 orall {2} It s sucent to show that or such sequences t must be that + or κ + < 0 We consder two cases, dependng on how the rst component o the normalzed ector κ compares to the rst components o the ectors κ or {2+ }. CASE I: The rst component o κ s the largest or smallest n the sequence. Ths mples that ether the rst component o κ + les between the rst components o κ and κ 2 or that the rst component κ 2 les between the rst components o κ and κ +. Consder subcase rst. Here there must be an α [0] such that κ + = ακ + ακ 2. Then t must be that +, whch can be seen rom the seres o nequaltes κ + + = κ + + κ + + ακ + ακ 2 = ακ + ακ ακ 2 2 > 0 Note that the strctness ollows rom the act that ether or both o the rst two terms n the aboe must be strctly poste dependng on the alue o α. But then applyng 2-acyclcty to the sequence { + } mples that +.

14 4 R. DEB AND D. MISHRA Now consder subcase. Here there must be an α [0] such that κ 2 = ακ + ακ +. Obsere that κ 2 + = κ κ > 0 whchnturnmplesthat ακ + + ακ + + > 0 Hence, t must be that ether κ + + 0andκ + >0, n whch case ths subcase s completed, or that κ + + >0. In the latter case, we can once agan apply 2-acyclcty to the sequence { + } and get the desred relaton +. CASE II: The rato o the components n κ les between some κ and κ +,where {2}. Then there must be an α [0] such that κ = ακ + ακ +.Then κ + = κ + κ + > ακ + ακ + + ακ + + = ακ ακ whch completes the proo. Q.E.D. REFERENCE FISHBURN, P. C. 970: Utlty Theory or Decson Mang. New Yor: Wley. [2] Unersty o Toronto, 50 St. George St., Toronto, ON M5S 3G7, Canada; rahul.deb@utoronto.ca and Indan Statstcal Insttute, Delh, New Delh 006, Inda; dmshra@ sd.ac.n. Manuscrpt receed Aprl, 203; nal reson receed July, 204.