Two-echelon supply chain coordination under information asymmetry with multiple types

Transcription

1 Two-echelon uly chain coordination under information aymmetry with multile tye.b.o. Kerkkam & W. van den Heuvel & A.P.M. Wagelman Econometric Intitute eort EI206-8 Abtract We analye a rincial-agent contracting model with aymmetric information between a ulier and a retailer. Both the ulier and the retailer have the claical non-linear economic ordering cot function coniting of ordering and holding cot. We aume that the retailer ha the market ower to enforce any order quantity. Furthermore, the retailer ha rivate holding cot. The ulier want to minimie hi exected cot by offering a menu of contract with ide ayment a an incentive mechanim. We conider a general number of dicrete ingledimenional retailer tye with tye-deendent default otion. A natural and common model formulation i non-convex, but we reent an equivalent convex formulation. Hence, the contracting model can be olved efficiently for a general number of retailer tye. We alo derive tructural roertie of the otimal menu of contract. In articular, we comletely characterie the otimum for two retailer tye and rovide a minimal lit of candidate contract for three tye. Finally, we rove a ufficient condition to guarantee unique contract in the otimal olution for a general number of retailer tye. Keyword: economic order quantity, mechanim deign, aymmetric information, hidden convexity Introduction We conider the claical 2-echelon Economic Order Quantity EOQ) etting with a ulier and a retailer. Both the ulier and the retailer act a fully rational individualitic entitie or agent) that want to minimie their own cot. It i well-known that uch individualitic viewoint are ubotimal for the entire uly chain. Thi lo of efficiency i often called the rice of anarchy, ee for examle Peraki and oel We aume that the uly chain ue a ull ordering trategy, i.e., the retailer lace order at the ulier. Therefore, the retailer ordering olicy i otimal for herelf. The ulier can decreae hi cot by omehow eruading the retailer to change to a different ordering olicy. One way the ulier can do o i by offering a contract to the retailer that tyically include a ide ayment or dicount. If the contract i acceted by the retailer, the cot for the entire uly chain decreae and the reulting rofit i divided between the two artie a agreed uon in the contract. Being elfih, the ulier want the larget oible hare of thi rofit. Deending on the tye of contract, it i non-trivial to determine a contract that maximie the ulier rofit and that i acceted by the retailer. The comlexity of the matter i increaed ignificantly if the retailer ha rivate information that i not hared with the ulier. For examle, the retailer cot tructure can be undicloed.

2 Econometric Intitute eort EI206-8 Furthermore, rivate information tyically lead to inefficiencie for the uly chain, ee for examle Inderfurth et al Thi artial cooeration between the ulier and the retailer lead to a rincial-agent otimiation roblem with aymmetric information. In the cae that the retailer hold rivate information, the ulier can ue mechanim deign or incentive theory to imrove hi ituation, ee Laffont and Martimort That i, he reent a menu of contract for the retailer to chooe from. We focu on contructing the otimal menu of contract that minimie the ulier exected cot, rovided that the retailer i not wore off by chooing one of thee contract.. Contracting model To further ecify the conidered otimiation roblem, we need to introduce the economical etting. The retailer face external demand with contant rate d >0, which mut be atified immediately, i.e., there i no backlogging. Placing an order at the ulier ha a fixed ordering cot of f >0 for the retailer. Delivery of the roduct i aumed to be intantaneou no lead time). Furthermore, the retailer ha inventory holding cot of h >0 er roduct and time unit. Since we aume that the retailer minimie her own cot, he lace an order if and only if her inventory i deleted the zero-inventory roerty). An order quantity of x >0 roduct lead to an average holding cot er time unit of 2 hx and an average ordering cot of df x. In total, the average cot er time unit for the retailer i given by φ x) = df x + 2 hx, which i minimied by ordering the well-known economic order quantity x = 2df/h ee for examle Banerjee 986). The minimal cot are φ = φ x ) = 2dfh. The cot tructure of the ulier i imilar: the ulier ha a fixed etu cot F >0 to handle an order and inventory holding cot H >0. Production take lace with contant rate d. To minimie hi own cot, the ulier roduce according to a jut-in-time lot-for-lot olicy. Per time unit the ulier ha average holding cot of 2 H d x and average etu cot of df x. Thi lead to a total cot for the ulier of φ S x) = df x + 2 H d x, which i minimied if the order quantity i x S = 2F /H. The ulier and retailer both have their own otimal order quantity and either olicy i ubotimal for the entire uly chain unle x = x S ). From the erective of the uly chain, the ulier and retailer can cooerate to lower the total joint cot. The joint cot are given by ) φ J x) = df + F ) x + 2 h + H d x, with otimal joint order quantity x J 2df = + F )/h + H d ). It i not difficult to verify that x J alway lie between x and x S ee Lemma A.). Therefore, lower joint cot can be achieved by deviating from the individually otimal order quantitie. Whether uch coordination take lace deend on further aumtion on ower relation and market otion. A mentioned before, we aume that both the ulier and the retailer behave rationally and want to minimie their own cot. Furthermore, aume that the retailer ha the market ower to enforce any order quantity on the ulier. Conequently, the retailer chooe her own otimal 2

3 Econometric Intitute eort EI206-8 order quantity x by default, called the default ordering olicy or default otion. By uing incentive mechanim, the ulier can eruade the retailer to deviate from the default olicy. We analye uing a ide ayment z a an incentive mechanim for cooeration. Note that in the literature ide ayment are ometime called quantity dicount. The air x, z) of an order quantity x and a ide ayment z i called a contract. The reented contract x, z) mut be contructed uch that the retailer i not wore off than with her default otion: φ x) z φ. Thi condition i called the Individual ationality I) contraint or articiation contraint. If the offered contract lead to the ame cot for the retailer a her default otion, we aume that the retailer i indifferent and that the ulier can convince the retailer to chooe the contract referred by the ulier. By aumtion, the ulier can do o without any additional cot. Hence, the retailer alway accet the reented contract if it atifie the I contraint. If the ulier ha comlete information of the uly chain, it i traightforward to determine that the otimal contract offer the joint order quantity x = x J and minimal ide ayment z = φ x J ) φ. The reulting ordering olicy lead to erfect uly chain coordination: it i otimal for the entire uly chain, a if there i a central deciion maker. However, we tudy the cae that the retailer ha rivate information on her cot tructure: either the ordering cot or the holding cot i rivate but not both). We conider the cae that the ulier i uncertain about the retailer holding cot, which i without lo of generality a will be hown in Section 2.. The ulier ha narrowed the retailer real holding cot down to one of K N oible cenario. Each cenario correond to a o-called retailer tye. Tye k K = {,..., K} ha cot function φ k x) = df x + 2 h kx, where 0 < h < h 2 < < h K < h K are the oible holding cot. We aume that the default otion deend on the tye. A uch, we add the index k K to our notation to dicern between retailer tye. For examle, for tye k K the default order quantity i x k = 2df/h k with correonding cot φ k = φk xk ). The ulier deign a menu of K contract for the retailer to chooe from, one for each retailer tye. For each tye k K the ulier contruct a contract x k, z k ) that i individually rational for that ecific tye, imilar to before. However, the retailer can lie about her tye and chooe any of the reented contract. Thi ituation i alo called a contracting or creening game in the literature, ee Laffont and Martimort Furthermore, the ulier aign an objective weight ω k >0 to each tye k K, indicating it likelihood, and minimie hi exected cot. Without lo of generality, ω i a robability ditribution k K ω k = ), but thi i not required for the model and our reult. Thi lead to the following non-linear otimiation roblem: min ω k φ S x k ) + z k ),.).t. k K φ k x k ) z k φ k, k K,.2) x k, z k ) {x, z ),..., x K, z K )}, k K,.3) φ k x k ) z k φ k x l ) z l, k, l K,.4) x k 0, k K. The deigned contract x k, z k ) mut atify the I contraint.2). The air x k, z k ) denote the choen contract by retailer tye k K, which mut be one of the reented contract, ee 3

4 Econometric Intitute eort EI206-8 contraint.3). The retailer chooe the mot beneficial contract for herelf by oibly lying, which i enforced by contraint.4). The ulier objective i to minimie hi exected cot including ide ayment, ee.). Conider an otimal olution to the non-linear roblem and uoe that the retailer lie about her true tye. By relabelling the reented contract, we can contruct another otimal olution for which the retailer will never lie about her tye, i.e., x k, z k ) = x k, z k ) for all k K. Thi i alo known a the revelation rincile. For examle, uoe the retailer tye i K lie being tye j K. Thi imlie that x i, z i ) = x j, z j ) and in articular φ i x j ) z j = φ i x i ) z i.4) φ i x i ) z i.2) φ i. So, contract x j, z j ) i individually rational for tye i. elabelling or redefining x i, z i ) to be equal to x j, z j ) lead to an equivalent feaible olution where tye i doe not lie. A direct conequence i that we can ue the following equivalent imler non-linear model: min ω k φ S x k ) + z k ),.t. k K φ k x k ) z k φ k, k K,.5) φ k x k ) z k φ k x l ) z l, k, l K,.6) x k 0, k K. We call thi imler model the default contracting model. Here,.6) are the Incentive Comatibility IC) contraint to revent tye from lying. Thee enable u to imlicitly et x k, z k ) = x k, z k ) and dro the choice of contract comletely from the model. Note that the menu of contract with x k, z k ) = x k, 0) for all k K i a feaible olution..2 Connection to the literature Similar model have been tudied in the literature and there are many variation. One variation i to conider a continuou range of retailer tye uch a in Corbett and De Groote 2000; Corbett, Zhou et al Pinar 205 analye the model with tructurally different cot function. In Cakanyildirim et al. 202 the role of the ulier and retailer are waed: the ulier ha rivate information and the retailer deign a menu of contract. We focu on literature that cloely relate to our model, ee alo Table for a comarion. In thi aer, we aume that only one cot arameter of the retailer i rivate, which lead to o-called ingle-dimenional tye. Pihchulov and ichter 206 analye the ame etting, but with two-dimenional retailer tye. That i, both the ordering cot and the holding cot are uncertain. Their reearch rovide a comlete analyi of the model in Sucky 2006, who conider the ame roblem. Both ue otimality condition to determine a lit of candidate for the otimal olution. However, the analyi i retricted to only two retailer tye, wherea we conider a general number of tye, albeit ingle-dimenional tye. From our reult we ee different qualitative roertie of the otimal olution for two tye veru more than two tye. Li et al. 202 incororate a controllable lead time into the contracting model. The retailer ha additional afety tock roortional to the quare root of the lead-time demand. Only two retailer tye are conidered. The two tye are two-dimenional, but the tye with low cot ha lower ordering and holding cot than the tye with high cot. 4

5 Econometric Intitute eort EI206-8 In Voigt and Inderfurth 20 the ulier etu cot i an additional deciion variable in the contracting model. The ulier ha to decide whether to lower hi etu cot at the cot of lot invetment oortunitie. Furthermore, the ulier ha no holding cot and the retailer no ordering cot. Beide the difference in cot function, their model aume the ame default otion for all retailer tye. To our knowledge, Voigt and Inderfurth 20 i the only aer with a related model that conider a general number of retailer tye, although the author do aume a certain condition on the ditribution of the retailer tye. Another model imilar to our i dicued in Zii et al. 205, but there are only two retailer tye. Furthermore, the ulier ha no holding cot, which reduce the number of otimal menu of contract that can occur. Since we analye the cae for two tye in detail, our reult generalie their derived tructural roertie of the otimal menu of contract. In light of the reviou reference, we emhaie that the incluion of both ordering/etu cot and holding cot for the retailer and ulier reult in tructurally different otimal menu of contract. Thi i becaue both involved artie have a finite individually otimal order quantity. Deviating from that quantity lead to higher cot. Thi i not true if only one tye of cot ordering or holding) i included, ince then the individually otimal order quantity i either zero or infinity. Furthermore, in the literature it i common to aume that the ulier refer a larger order quantity than the retailer. We do not make thi aumtion and therefore alo rovide inight into contract when the ulier refer maller order quantitie. Paer Sulier cot: etailer cot: Number of tye: Tye-deendent Dimenion of tye: Setu Holding Ordering Holding Two Multile default otion One Two Sucky 2006 Voigt and Inderfurth 20 Li et al. 202 Zii et al. 205 Pihchulov and ichter 206 Our aer Table : Comarion of related literature..3 Contribution We conider a rincial-agent contracting model with aymmetric information under the EOQ etting. Our model ditinguihe itelf from the literature by having a general number of retailer tye with tye-deendent default otion. Furthermore, the ulier and the retailer tye have both ordering/etu cot and holding cot. Conequently, a tyical analyi uing otimality condition i comlex and doe not aear to lead to a generaliable olution method. Our main contribution are a follow. Firt, we how that thi non-convex model ha a hidden convexity, which i achieved by a change of deciion variable. Hence, in ractice we can numerically olve our model to otimality for a general number of retailer tye uing variou efficient technique. Second, we determine tructural roertie of the otimal olution for a general number of retailer tye. The analyi how ignificant difference in the tructure of otimal menu of contract for two tye comared to more than two tye. Third, we rove a ufficient condition to guarantee unique contract in the otimal olution. We rovide counterexamle when thi condition i omitted. In articular, we ue the tructural roertie to analye the difference between two and three retailer tye. To do o, we analytically olve the model for thee two cae. We rovide a comlete characteriation of the otimal olution for the cae with two retailer tye. The derived cloedform formula of the otimal olution are not only imler than thoe found in related literature, 5

6 Econometric Intitute eort EI206-8 they alo how additional tructure of the olution. For the ecific cae of three retailer tye we did not find any reult in the literature. We give a minimal lit of candidate contract for the otimal olution. The remainder i organied a follow. In Section 2 we reent an alternative model which how the hidden convexity and lead to an efficient olution method. We continue with tructural roertie of the contracting model in Section 3. In Section 4 we dicu the otimal menu of contract for two and three retailer tye, where we give examle of each occurring otimal menu. The derivation of the otimal contract for two and three tye are given in Aendice A and B. We end with a general dicuion of our reult in Section 5. 2 Efficient olution method In thi ection we how that the contracting roblem can be olved efficiently. Thi inight become aarent after a change of deciion variable of the contracting model. Before we give the detail, we rove that for ingle-dimenional retailer tye we can aume without lo of generality that the retailer holding cot i uncertain. Conequently, we can efficiently olve two kind of contracting model. 2. Equivalence when one cot arameter i uncertain Conider a contracting roblem where all retailer tye intead have the ame holding cot h, but different ordering cot f k. We can tranform any uch roblem to an equivalent contracting roblem where all tye have the ame ordering cot ˆf, but different holding cot ĥk. The tranformation i a follow. For arbitrary ˆd >0 and ˆ ˆd, define the following arameter: ˆω k = ω k, Ĥ = 2dF ) ˆˆd, ˆF = 2 H d )ˆd, ˆf = 2 h)ˆd, ĥ k = 2df k ). Thee arameter are well-defined and reult in a contracting roblem intance where all retailer tye have the ame ordering cot, intead of the ame holding cot. To ditinguih the intance, let Ŝ be the ulier and ˆ the retailer for the newly contructed roblem. We claim that both intance are equivalent, i.e., both have the ame otimal objective value and there i a bijection between the otimal olution. To how any equivalence between intance, the imortant exreion of the contracting model are: φ S, φ k, and φk. Conider any order quantity x k >0 and et ˆx k = /x k, leading to the exreion: φ S x k ) = df x k + 2 H d x k = 2 H d + df ˆx k = ˆx ˆd ˆF ˆx + ˆd k k 2Ĥ ˆ ˆx k = φŝˆx k ), φ k x k ) = df k + x k 2 hx k = df k ˆx k + 2 h ˆx = ˆd ˆf ˆx + ˆx k = φ kˆˆx k ), k k 2ĥk φ k = 2df k h = 2 ˆd ˆfĥk = φ k ˆ, where the equalitie follow by definition. Thu, any air x, z) i a feaible olution for the original intance if and only if /x, z) i feaible for the newly contructed intance. Moreover, the objective value of the two intance are equal. To conclude, the qualitative roertie of the contracting model with one uncertain cot arameter are irreective of which cot arameter ordering or holding cot) i uncertain. 6

7 Econometric Intitute eort EI Alternative convex model The contracting model i not convex, ince the IC contraint tate: df ) + x k x l 2 h k h l )x k + z l z k 0, k, l K. Here, the term /x l i not convex in the deciion variable. Non-convex otimiation roblem are generally difficult to olve, but we how that thi i not the cae for our roblem. We reveal a hidden convexity of our roblem by changing the erective from ide ayment to o-called information rent. An alternative contracting model can be obtained by recaling the ide ayment a follow. The individual rationality contraint imly that z k φ k x k) φ k 0. A uch, it i natural to interret the value φ k x k) φ k a the minimum ide ayment that alway ha to be aid to atify the I contraint. We introduce a new variable y k which denote the additional ide ayment required by the IC contraint: y k = z k φ k x k ) φ k ) 0. Thi variable i alo known a the information rent for tye k. Subtituting z k = y k + φ k x k) φ k in the default contracting model lead to: ) min ω k φ S x k ) + φ k x k ) + y k φ k, k K.t. y k 0, k K, 2.) y l y k + φ l x l ) φ k x l ) φ l φ k, k, l K, 2.2) x k 0, k K. So, 2.) are the I contraint and 2.2) are the IC contraint. We call the new model the alternative contracting model to differentiate it from the earlier defined default model. By definition of y k, there i a bijection between the feaible region of the alternative model and that of the default model. Furthermore, the correonding objective value are the ame. Hence, we can olve the default model by olving the alternative model and vice vera. Although both model are equivalent in the ene mentioned above, there i one ignificant difference. Notice that the non-linear term in 2.2) cancel out if we exand the cot function: y l y k + 2 h l h k )x l = y l y k + φ l x l ) φ k x l ) φ l φ k. Thu, all contraint of the alternative model are linear in the deciion variable. Since the objective function i convex, we conclude that the alternative model i convex. Moreover, the feaible olution x k = x k and y k = ɛ >0 for all k K i a Slater oint, i.e., trictly feaible. It i well-known that a convex model with differentiable function and Slater oint can be olved efficiently uing calable method uch a interior-oint or cutting-lane method ee Berteka 205; Boyd and Vandenberghe 2004). Thi concluion i tated in Theorem 2.. Theorem 2.. The contracting model can be olved efficiently via the alternative model. Proof. The roof i given in the above dicuion. 7

8 Econometric Intitute eort EI206-8 emark 2.2. ecalling the reult from Section 2., we note that the contracting model with ingledimenional tye can be olved efficiently. If both the ordering cot f and the holding cot h are rivate information, we have two-dimenional retailer tye ecified by cot arameter f k, h k ). In thi cae, both the default model and the alternative model fall in the category of Difference of Convex function DC) rogramming. In the literature, there exit good numerical method to find local otima of DC model, ee Hort et al. 99; Pham Dinh and Le Thi 204. However, to guarantee global otimality uch method need to be incororated into for examle a Branch-and- Bound rocedure. To conclude, in ractice we can determine otimal olution of our roblem numerically. We have imlemented a cutting-lane algorithm uing Gurobi a Linear Programming olver. Tyical comutational time are le than a econd for one hundred tye on a tandard dekto comuter. However, it i worthwhile to further analye the model theoretically. In the following ection we determine qualitative roertie of the otimal menu of contract and in ome cae even rovide cloed-form olution. The ued model default or alternative) ha no ignificant effect on the reult. Hence, we reent all reult uing the default model and lace remark where needed for the alternative model. 3 Structural roertie We continue with additional roertie of the contracting model and it otimal olution. Thee reult hold for a general number of retailer tye. In articular, the model i connected to a one-to-all hortet ath roblem in a certain directed grah. Thi allow u to ue the theory of the hortet ath roblem and have a different view of the contracting model. Furthermore, we ue the well-known Karuh-Kuhn-Tucker condition to determine tructure in the otimal olution. In the end, we derive a ufficient condition to guarantee unique contract in the otimal olution. Moreover, the analyi lead to a minimal lit of menu of contract for two and three retailer tye which contain the otimal olution. Thee are dicued in Section Shortet ath interretation A cloer look into the tructure of the I and IC contraint how a connection with a dual hortet ath interretation. For given fixed quantitie x k, contraint.5) and.6) can be een a the dual contraint of a hortet ath roblem. To be ecific, for given x k the whole model i equivalent to the dual of a ecific minimum cot flow formulation for the one-to-all hortet ath roblem. A imilar connection to hortet ath ha been decribed in Vohra 202. Conider the directed grah G = V, A) with node V = {} K and directed arc A = {, k) : k K} {k, l) : k, l K, k l}. That i, G i the comlete grah of K retailer node with a ource added. See Figure for an examle. We call uch a grah an IIC grah, which tand for Individual ationality and Incentive Comatibility grah for reaon to become aarent. The length or cot) of the arc are: arc, k) with k K ha length φ k φk x k), arc k, l) with k, l K, k l, ha length φ l x k) φ l x l). Finally, node ha uly k K ω k and each retailer node k K ha demand ω k. There are no caacity retriction on the arc. Conequently, flow will be ent along hortet ath in the otimal olution of the flow formulation. Hence, we ee thi flow formulation a a one-to-all hortet ath rereentation. 8

9 Econometric Intitute eort EI Figure : IIC grah for K = 4 retailer tye. It i ueful to mention ome well-known roertie of the dual flow formulation, ee alo Ahuja et al Conider the otimal olution x, z) of the contracting model. The value z k i equal to the length of the hortet, k)-ath. Moreover, Strong Duality imlie that the IIC grah contain a negative cycle if and only if the dual i infeaible. In uch cae there exit no ide ayment that will atify the IC contraint for the conidered order quantitie x k. Thu, the IC contraint can be atified if and only if the correonding IIC grah ha no negative cycle. In the non-degenerate cae, the et of all ued arc in the otimal hortet ath from to the other node form a anning tree in the IIC grah. In the degenerate cae, thi doe not hold, but the otimal olution can be modified uch that the ued arc form a anning tree again. In articular, if the et contain cycle, thee cycle mut have length 0. From the comlementary lackne condition it follow that if arc i, j) i in the anning tree, then the correonding contraint in the dual i atified with equality. For examle, if arc, k) i art of the hortet ath tree, then the I contraint for tye k i tight. If arc k, l) i ued, with k, l K, then tye l want to retend to be tye k. That i, the IC contraint φ l x l) z l φ l x k) z k i atified with equality. Due to the bijection between retailer tye and retailer tye node, and the bijection between arc and the I and IC contraint, we often interchange interretation and terminology. For examle, we can refer to outgoing arc out of a retailer tye, referring to the outgoing arc of the correonding node in the grah. Thee inight exlain why we call the grah the IIC grah. emark 3.. We note that the ame reult hold for the alternative model, with the excetion that the arc length are given by: arc, k) with k K ha length 0, arc k, l) with k, l K, k l, ha length φ l x k) φ k x k) + φ k φl. emark 3.2. Due to eronal tate, one can refer a longet ath formulation intead. The arc length are omewhat eaier to remember directly from the model by rewriting the I and IC contraint to: ) z k φ k x k ) φ k, z }{{} k z l + arc length, k) φ k x k ) φ k x l ) }{{} arc length l, k) In the otimal olution, z k i the length of the longet ath from node to node k when uing thee arc length. Naturally, the longet and hortet ath formulation are equivalent.. 9

10 Econometric Intitute eort EI Adjacent retailer tye Since the tye are ordered uch that h < h 2 < < h K, there i a ene of adjacent or neighbouring tye. We define the neighbour of tye k K to be the tye k and k +, where tye and K have only one neighbour. The adjacency of tye lay an imortant role a we will ee. Intuitively, one would exect that in an otimal olution a tye with higher holding cot get offered a lower order quantity i.e., more frequent ordering) to revent too high inventory cot. Lemma 3.3 how that thi intuition i mathematically correct. Lemma 3.3. Any feaible menu of contract atifie x x 2 x K. Proof. Conider a feaible menu of contract x, z). From the hortet ath interretation in Section 3. we know that no negative cycle exit in the correonding IIC grah. In articular, any 2-cycle in the IIC grah ha non-negative length. Without lo of generality, conider i, j K with h i < h j and conider the length of the 2-cycle between node i and j, which mut be non-negative: φ i x j ) φ i x i ) ) ) + φ j x i) φ j x j) 0 2 h j h i )x i x j ) 0 h i <h j ) x i x j 0. Hence, x i x j mut hold in any feaible olution. The ordering or monotonicity) in the order quantitie i a common roerty of contracting model, ee for examle Laffont and Martimort 2002; Vohra 202. However, there i no monotonicity in the ide ayment ee Section 4 for examle). A conequence of Lemma 3.3 i that adjacent retailer tye follow both from the holding cot and from the feaible) order quantitie. In fact, uing thi reult we can retrict the incentive comatibility contraint to take only the neighbouring tye into account, without changing the feaible region. See Lemma 3.4 for the reult. We call thee contraint the adjacent IC contraint. Lemma 3.4. The adjacent incentive comatibility contraint are ufficient to enure general incentive comatibility. Proof. Let x, z) be the otimal menu of contract when we only ue the adjacent IC contraint, intead of all general IC contraint. Conider a cycle C = i,..., i C ) of unique retailer node in the IIC grah correonding to x, z). We rove that any uch cycle ha non-negative length, imlying that all general IC contraint are atified. The roof i by induction on the cardinality of C. If C = 2, then the adjacent IC contraint enforce that the cycle length i non-negative. Therefore, let C > 2 and without lo of generality, aume that tye i C ha the greatet holding cot. By induction, the cycle i,..., i C ) ha non-negative length. We comare the difference in length between the two cycle, ee alo Figure 2: φ i C x ic ) φ i C x ic ) ) + φ i x ic ) φ i x i ) ) ) = φ i C x ic ) φ i C x ic ) + φ i x ic ) φ i x ic ) = 2 h i C h i )x ic x ic ) 0. ) φ i x ic ) φ i x i ) The inequality follow from our aumtion on the holding cot h ic > h i ) and Lemma 3.3. Thu, C mut have non-negative length a well. Conequently, all IC contraint hold without 0

11 Econometric Intitute eort EI206-8 exlicitly incororating the correonding IC contraint in the otimiation model. To conclude, x, z) i alo otimal for the comlete contracting model with all general IC contraint i i C i i i C i C i i C i C i C i C i C a) Smaller cycle i,..., i C ). b) Larger cycle C = i,..., i C ). Figure 2: elevant arc in the induction roof of Lemma 3.4. We can ue Lemma 3.4 to rove that order quantitie atifying x x 2 x K > 0 can alway be extended to a feaible menu of contract x, z), ee Corollary 3.5. Therefore, we ometime call uch order quantitie feaible for the contracting model. Corollary 3.5. For given order quantitie atifying x x 2 x K > 0, it i feaible and otimal to determine the ide ayment via the hortet ath interretation. Proof. From Lemma 3.4 it follow that for feaibility we only need to determine ide ayment uch that the adjacent IC contraint are atified. From the hortet ath interretation, we know that ide ayment atifying the adjacent IC contraint exit if and only if 2-cycle in the correonding grah have non-negative length. Now conider arbitrary i, j K with h i < h j. The roof of Lemma 3.3 how that the 2-cycle between i and j ha non-negative length if and only if x i x j, which hold by aumtion. Hence, we can determine feaible ide ayment by olving a one-to-all hortet ath roblem a decribed in Section 3.. Furthermore, thi lead to the bet oible feaible ide ayment with reect to the given order quantitie. 3.3 KKT condition Since the contracting model conit of continuouly differentiable function with a continuou domain, there are well-known neceary condition for otimality and even ufficient otimality condition in certain cae. Uing thee condition we can deign candidate olution for further inection. Thi allow u to analytically invetigate roertie of the otimal menu of contract. In the following ection we ue the Karuh-Kuhn-Tucker KKT) otimality condition to do o. Firt of all, we oint out a ubtle iue regarding KKT condition. The default contracting model i non-convex. A uch, the general KKT condition, alo known a Fritz-John condition, ee Brinkhui and Tikhomirov 2005; John 948), are neceary for the otimal olution. We need regularity condition to be able to ue the tandard KKT condition Karuh 939; Kuhn and Tucker 95), uch a the Mangaarian-Fromovitz contraint qualification. However, with a light detour we can ignore thi iue. We have an equivalent convex model with a Slater oint, namely the alternative contracting model of Section 2.2. Thu, the tandard KKT condition are neceary and ufficient for the alternative model. Both model lead to the ame general KKT condition, from which we conclude that the tandard KKT condition are alo neceary and ufficient for the default model.

12 Econometric Intitute eort EI206-8 The KKT condition lead to a large) et of candidate olution. Thee olution will be called KKT menu and their contract KKT contract. The otimal olution of our model i the bet KKT menu. However, in general determining thi et will be intractable due to it ize. Therefore, we analye our roblem to exclude certain KKT menu. Unfortunately, we do not end u with a tractable olution aroach for a general number of retailer tye. Hence, thi KKT aroach eem unucceful to rovide a generaliable olution method, but the analyi will neverthele rovide additional inight in otimal menu of contract. With the above mentioned remark in mind, we determine the general KKT condition for the contracting model. Uing Lemma 3.4 we only incororate the adjacent IC contraint in our model. The Lagrangian function with Lagrange multilier κ 0, λ, ν K 0, and µ 2K 2 0 i given by: Lx, z, κ, λ, µ, ν) = κ ω k φ S x k ) + z k ) + ) λ k φ k x k ) z k φ k k K k K + ) µ k,k φ k x k ) z k φ k x k ) + z k k K\{} + k K\{K} + k K ν k x k ). µ k+,k φ k x k ) z k φ k x k+ ) + z k+ ) We deliberately chooe thi order of the indice of µ and will exlain in Section 3.4 why thi notation i ueful. The KKT condition conit of rimal and dual feaibility, comlementary lackne, and tationarity contraint. The dual feaibility contraint require all multilier to be non-negative with the additional condition that not all multilier are zero. The comlementary lackne contraint are: ) λ k φ k x k ) z k φ k = 0, k K, ) µ k,k φ k x k ) z k φ k x k ) + z k = 0, k K \ {}, ) µ k+,k φ k x k ) z k φ k x k+ ) + z k+ = 0, k K \ {K}, ν k x k = 0, k K. 3.) Since x k = 0 i never otimal, it follow from 3.) that ν k = 0 for all k K. Therefore, we et ν k = 0 and ignore related term comletely. Likewie, we argued above that the tandard KKT condition hold, imlying that κ =. Thu, we ignore thi multilier a well. For each k K, the tationarity contraint with reect to x k are: and with reect to z k : ω k dφ S dx x k) + λ k dφ k dx x k) + µ k,k + µ k+,k ) dφk dx x k) dφ k µ k,k dx x dφ k+ k) µ k,k+ dx x k) = 0, 3.2) ω k λ k µ k,k + µ k+,k ) + µ k,k + µ k,k+ ) = 0, 3.3) 2

13 Econometric Intitute eort EI206-8 where all ill-defined multilier with out of bound indice are et to zero. We can imlify the tationarity contraint by ubtituting 3.3) in 3.2): ω k df + F ) x 2 + h k + H d k 2 ) ) + 2 µ k,k h k h k ) + 2 µ k,k+h k h k+ ) = ) To conclude, the KKT condition conit of the rimal and dual feaibility contraint, comlementary lackne contraint, and tationarity contraint 3.3) and 3.4). emark 3.6. The KKT condition for the alternative model directly give 3.3) and 3.4). 3.4 KKT grah A mentioned before, we only ue adjacent IC contraint Lemma 3.4). The hortet ath interretation of Section 3. till hold and the correonding Adjacent IIC grah i hown in Figure 3. Notice that the order of indice of µ correond nicely to the Adjacent IIC grah. If µ lk > 0, then the equality φ k x k) z k = φ k x l) z l mut hold by the KKT comlementary lackne condition. Hence, arc l, k) i ued by the hortet ath, a dicued in Section 3.. The ame hold for λ k, contraint φ k x k) z k φ k, and arc, k). Conequently, we have bijection between multilier λ or µ), the I or IC) contraint, and certain arc in the Adjacent IIC grah. A uch, we can refer to the multilier of an arc in the Adjacent IIC grah. Thu, the trictly oitive multilier indicate which arc are for certain art of hortet ath in the IIC grah. Unfortunately, there could be arc in a hortet ath for which the multilier i zero, a degenerate cae may occur Figure 3: Adjacent IIC grah for K = 4 tye. Each KKT menu can be identified by the ubet of multilier which are trictly oitive. We can viualie the contract in the Adjacent IIC grah by only conidering the arc for which the correonding multilier are trictly oitive. That i, we have a directed grah Ĝ = V, Â) with V = {} K and arc, k) with k K if λ k > 0, k, k ) with k K \ {} if µ k,k > 0, k, k + ) with k K \ {K} if µ k,k+ > 0. We call thi grah the KKT grah. In the reult to come, we often ue the term connected comonent of the KKT grah. To avoid confuion, a ubet S V i a connected comonent if between each air of node in S there exit an undirected ath in the grah. Furthermore, a node i called iolated if it ha no in- or outgoing) arc. 3

14 Econometric Intitute eort EI206-8 The KKT grah allow for eay-to-draw name of KKT menu. We call arc, k) the U arc for retailer tye k K, arc k, k + ) the ight arc, and arc k, k ) the Left arc. The name of a KKT menu i imly a lit of the U, ight, and Left arc of the correonding KKT grah. For examle, KKT menu ight2uleft3uleftight4x i hown in Figure Figure 4: KKT grah for ight2uleft3uleftight4x. 3.5 Proertie of otimal contract The reult that only adjacent IC contraint need to be taken into account greatly reduce the number of oible KKT menu to conider. We continue to analye which cae can alo be excluded from conideration, i.e., which combination of trictly oitive multilier or which KKT grah) can occur. We tart with Lemma 3.7, which how an exlicit connection to hortet ath. Lemma 3.7. Every retailer node k K mut be reachable from ource node in the KKT grah. Proof. Firt, uoe k K ha no ingoing arc, i.e., λ k = µ k,k = µ k+,k = 0. From 3.3) we have: ω k + µ k,k + µ k,k+ = 0 = µ k,k + µ k,k+ < 0. Thi contradict the fact that all multilier are non-negative. Hence, any node in K mut have an ingoing arc. Second, let S = {i, i +,..., j, j} K be an arbitrary maximal connected ubet of retailer node with λ k = 0 for all k S. That i, no node in S i directly reachable from node. Adding u 3.3) for all k S reult in: ω k µ i,i µ j+,j + µ i,i + µ j,j+ = ) k S Notice that all internal arc of S cancel out. Furthermore, by maximality of the ubet, all remaining multilier in 3.5) mut be zero. Thi lead to a contradiction, ince ω k > 0 for all k K. To conclude, every maximal connected comonent i reachable from. Finally, by iteratively uing that each node ha an ingoing arc we can conclude that every node mut be reachable from node. Notice that thi i a tronger roerty than the fact that the ide ayment follow from hortet ath. Shortet ath imly that each node i reachable from uing only arc for which the correonding contraint i tight. A weak comlementary lackne may hold, tightne doe not automatically imly that the correonding multilier i trictly oitive. However, a trictly oitive multilier doe imly tightne of the contraint. Thi reult allow u to dicard certain combination of multilier, ignificantly reducing the number of otion. The next lemma decribe a general attern a T-attern ) that will never occur in the otimal olution. 4

15 Econometric Intitute eort EI206-8 Lemma 3.8. There exit no k K \ {, K} uch that the contraint correonding to arc, k), k, k ), and k, k + ) are atified with equality. Proof. Let i, j, k K, i < k < j, be uch that the contraint correonding to arc, k), k, i), and k, j) are atified with equality. Conequently, we have: φ k x k ) z k = φ k, φ i x i ) z i = φ i x k ) z k, φ j x j) z j = φ j x k) z k, φ i x i ) z i φ i, Combining thee relation lead to the following: ewriting thee reult give: φ j x j) z j φ j. φ i φ i x i ) z i = φ i x k ) z k = φ i x k ) φ k x k ) + φ k = 2 h i h k )x k + φ k, φ j φj x j) z j = φ j x k) z k = φ j x k) φ k x k ) + φ k = 2 h j h k )x k + φ k. φ i φk h i h k 2 x k φj φk. h j h k ecall that φ l = 2dfh l for all l K. Thu, we arrive at the following inequality: hk h i h k h i hj h k h j h k hk + h i hj + h k hi h j. The firt inequality comare two loe between three oint on the quare root curve. Such an inequality never hold for h i < h k < h j, a the equivalent inequality how. Corollary 3.9. A retailer node directly connected to node in the KKT grah ha at mot one outgoing arc. Proof. Suoe a node k K directly connected to ha more outgoing arc. The direct connection to node imlie λ k > 0. Furthermore, the outgoing arc mut be k, k ) and k, k + ), o µ k,k, µ k,k+ > 0. The KKT comlementary lackne condition imly that the correonding contraint are tight, violating Lemma 3.8. Corollary 3.9 imlie that the grah in Figure 4 i never a valid KKT grah, ince node 3 ha U, Left, and ight arc violating the corollary) Cycle are retrictive A a reult of Lemma 3.4, the only cycle of interet are 2-cycle between adjacent node. The next lemma and corollary how that 2-cycle lead to having the ame contract, alo called bunching in the literature. Lemma 3.0. Both incentive comatibility contraint between adjacent) retailer tye i and j are tight if and only if x i = x j. Furthermore, if x i = x j then z i = z j mut hold. 5

16 Econometric Intitute eort EI206-8 Proof. Firt, uoe the order quantitie for tye i, j K are the ame. The incentive comatibility contraint tate: φ i x i ) z i φ i x j ) z j = φ i x i ) z j z i z j, φ j x j) z j φ j x i) z i = φ j x j) z i z j z i. Thu, z i = z j mut hold and both contract are the ame. Conequently, ubtituting z i = z j how that both incentive comatibility contraint are tight. Second, uoe both incentive comatibility contraint between i and j are tight: φ i x i ) z i = φ i x j ) z j, φ j x j) z j = φ j x i) z i. Combining both equalitie lead to: φ i x i ) φ j x i) = φ i x j ) φ j x j) 2 h i h j )x i = 2 h i h j )x j x i = x j. The firt equivalence follow from having the ame ordering cot f and the lat equivalence from h i h j. A roved above, x i = x j imlie that z i = z j. Thu, the contract for tye i and j are the ame. Corollary 3.. Tye art of a 2-cycle in the KKT grah have the ame contract. Proof. Being art of a 2-cycle in the KKT grah mean that µ kl, µ lk > 0 for ome adjacent k, l K. From comlementary lackne it follow that both incentive comatibility contraint between tye k and l mut be tight. Lemma 3.0 imlie that x k = x l and z k = z l. The KKT condition become more retrictive if certain tye have the ame order quantity, a it introduce additional deendency between the deciion variable. Uing thi fact, we can exclude more cae from conideration, ee Lemma 3.2. Lemma 3.2. In the KKT grah, a maximal ubet of retailer node connected with only 2-cycle mut have at leat one ingoing arc oibly from node ) and exactly one outgoing arc. Proof. The tatement that at leat one ingoing arc exit follow directly from Lemma 3.7. We rove the tatement for the outgoing arc by contradiction. Let S = {i, i +,..., j, j} K be uch a maximal ubet and uoe that S a a whole ha no outgoing arc. By Corollary 3., all k S get the ame contract, ay order quantity x. The tationarity condition tate that ω k df + F ) x h k + H ) ) d + 2 µ k,k h k h k ) }{{} >0 + 2 µ k,k+h k h k+ ) = 0. }{{} <0 By aumtion, µ i,i = 0 or non-exitent if i = ). Likewie, µ j,j+ = 0 or non-exitent if j = K). The tationarity contraint for tye i require that: df + F ) x 2 + h i + H d 2 ) ) > 0. Thi imlie that the imilar term in the tationarity contraint for j i alo trictly oitive, a h i < h j. However, the reulting contraint only contain trictly oitive term, which i infeaible. Thu, S a a whole ha at leat one outgoing arc. Suoe S a a whole ha two outgoing arc, i.e., µ i,i, µ j,j+ > 0. By Lemma 3.7, all node in S mut be reachable from via arc with trictly oitive multilier. Corollary 3.9 imlie that λ i,..., λ j = 0, otherwie the olution i infeaible. Therefore, µ i,i > 0 and/or µ j+,j > 0 mut hold, but thi contradict the maximality of S. To conclude, S ha exactly one outgoing arc. 6

17 Econometric Intitute eort EI206-8 For examle, the grah in Figure 4 i not a valid KKT grah, ince node and 2 form a 2-cycle but do not have an outgoing arc. A direct conequence i that for two retailer tye KKT grah with 2-cycle are not valid KKT grah. For more than two retailer tye 2-cycle in the otimal olution can actually occur, ee Section 4. Thi imlie that tye can get the ame contract in the otimal olution. We return to thi iue in Section The joint order quantity If a retailer node k K ha no outgoing arc in the KKT grah it i traightforward to determine that x k = x k J mut hold. The next lemma how that thi i an if-and-only-if relation. Lemma 3.3. In the otimal olution, x k = x k J the KKT grah. if and only if node k K ha no outgoing arc in Proof. Firt, uoe node k K ha no outgoing arc in the KKT grah, i.e., µ k,k = µ k,k+ = 0. The KKT tationarity condition 3.4) require that: ω k df + F ) x 2 + ) ) h k + H d k 2 = 0 = x k = x k J, which rove one direction of the lemma. Second, uoe that x k = x k J. Again uing the KKT tationarity condition 3.4), we get: µ k,k h k h k ) + µ k,k+ h k h k+ ) = 0. Since h k < h k < h k+, either µ k,k, µ k,k+ > 0 node k ha two outgoing arc) or µ k,k = µ k,k+ = 0 node k ha no outgoing arc). In the latter cae we are done. Therefore, uoe µ k,k, µ k,k+ > 0. We dicern two cae. Cae I: node k i not art of a 2-cycle, i.e., µ k,k = µ k+,k = 0. By Lemma 3.7 node k mut be reachable from node, imlying λ k > 0. Thi cae i infeaible, ee Corollary 3.9. Cae II: node k i art of a 2-cycle. Let k be art of the maximal ubet S = {i, i +,..., k,..., j, j} K of retailer node connected with 2-cycle. ecall that from Corollary 3. we know that all tye in S have the ame contract. Furthermore, by Lemma 3.2 either µ i,i > 0 or µ j,j+ > 0 but not both). Conider the cae that µ i,i > 0, and thu µ j,j+ = 0 and j > k. The KKT tationarity condition tate: ω j df + F ) x 2 + j 2 h j + H d ) ) + 2 µ j,j h j h j ) = 0. Since µ j,j h j h j ) > 0, it mut hold that x j < x j J. Since x k = x j, we have the required contradiction: x k J = x k = x j < x j J < xk J. The other cae, µ i,i = 0 and µ j,j+ > 0, i imilar and i omitted. To conclude, if x k = x k J node k mut have no outgoing arc in the KKT grah, which comlete the roof. Our lat reult for thi ection, Lemma 3.4, tate that at leat one tye i aigned the joint order quantity in the otimal olution. 7

18 Econometric Intitute eort EI206-8 Lemma 3.4. In the otimal olution the order quantity for at leat one retailer tye i the joint order quantity. Moreover, the total cot for at leat one retailer tye equal it default cot. Proof. The reult that there exit a retailer tye with the ame total cot a it default otion follow directly from Lemma 3.7. That i, there exit a k K uch that λ k > 0. Hence, φ k x k) z k = φ k by comlementary lackne. By combining Lemma 3.2 and 3.3, we can rove the other claim a follow. Suoe each retailer node ha an outgoing arc in the KKT grah. Thu, arc, 2) from tye to tye 2 exit in the grah. If arc 2, 3) i in the grah, we continue to tye 3. If tye 2 ha only outgoing arc 2, ), then tye 2 form a 2-cycle with tye. By Lemma 3.2 arc 2, 3) mut exit a well, a contradiction. eeat thi argument until we reach tye K. Since node K alo ha an outgoing arc, a 2-cycle with tye K i formed. Again by Lemma 3.2, thi cycle mut have an outgoing arc, namely arc K, K 2). eeat thi argument until we reach tye. Hence, all retailer node are art of 2-cycle which contradict Lemma 3.2. Thu, there exit at leat one tye with no outgoing arc. Lemma 3.3 tate that thi retailer tye i aigned the joint order quantity in the otimal olution. 3.6 Uniquene of contract Additional aumtion are needed in order to guarantee that the otimal menu of contract uniquely identifie each tye, i.e., that each contract in the menu i unique. Suoe that the ulier only ha bound for the retailer holding cot, h [h LB, h UB ], and due to the lack of information aume a uniform ditribution for h. We dicretie the uniform ditribution uing K N equiditant oint: h {h,..., h K } with h k+ = h k + δ for ome aroriate te δ >0. Furthermore, the uniform ditribution imlie that all weight are equal, ω k = ω for all k K. The aumtion lead to the following KKT tationarity condition for k K: 2df + F ) ) x 2 + H d + h k + δµ k,k µ k,k+ ) = 0, 3.6) k λ k µ k,k µ k+,k + µ k,k + µ k,k+ = 0, 3.7) where all ill-defined multilier with out of bound indice are et to zero. Notice that without lo of generality we et ω k = in the KKT condition by uniformly recaling all multilier. It turn out that uniformity on tye and equiditant holding cot i ufficient to guarantee a riori to obtain an otimal menu with unique contract, ee Theorem 3.5. Be aware that the excluion of 2-cycle of arc with trictly oitive multilier doe not automatically imly that all contract are unique, at leat not without imroving the reult of Corollary 3.. Theorem 3.5. Aume uniformity on tye and equiditant holding cot. In the otimal olution, all contract are unique. Proof. For all tye k K, let ω k = ω and h k+ = h k + δ for ome δ >0. Firt, realie that if x k = x l for ome l > k +, then all intermediate tye alo have the ame order quantity: x k = x k+ = = x l = x l. Thi follow from the ordering of the order quantitie Lemma 3.3). Second, if x k = x l then automatically z k = z l mut hold to be feaible Lemma 3.0). So, both contract are exactly the ame. Aume that there are contract for retailer tye that are the ame, ele there i nothing to rove. Let S = {i, i +,..., j, j} K with i < j be a maximal et of tye with the ame contract. Note that Lemma 3.8 imlie that λ i+,..., λ j = 0. We have to ditinguih two cae baed on the KKT multilier. 8

19 Econometric Intitute eort EI206-8 Cae I: µ i,i > 0. We have three direct imlication: λ i = 0 by Lemma 3.8), µ i,i = 0 by maximality of S and Corollary 3.), and thu µ i+,i > 0 by Lemma 3.7). Furthermore, ince all node in S mut be reachable Lemma 3.7), we have that µ i+,i,..., µ j,j > 0. Finally, we can conclude that µ j,j+ = 0 with a imle argument by contradiction uing the maximality of S or Lemma 3.7 and 3.8. Now we can derive two contradictory equation. The firt equation i a follow. Since λ i,..., λ j = 0, the um of the correonding KKT condition 3.7) from i to j i equal to j i ) + µ i,i µ j,j + µ j,j = 0. For the econd equation we need to conider the KKT condition 3.6) for tye i and j. A x i = x j both condition have a common art, hence the difference mut be equal: h i + δµ i,i µ i,i+ ) = h j + δµ j,j µ i,i µ i,i+ = j i) + µ j,j. Finally, both equation combined tate that µ i,i+ = 2j i) + µ j,j + µ j,j > 0. Thi contradict that µ i+,i i non-negative. Cae II: µ i,i = 0. The KKT tationarity for tye i imlifie to 2df + F ) ) x 2 + H d + h i = δµ i,i+ 0. i Therefore, for any k S, k > i, it mut hold that µ k,k+ > 0, ince x k = x i and from the above inequality: 0 = 2df + F ) ) x 2 + H d + h k + δµ k,k µ k,k+ ) h k h i ) + δµ k,k µ k,k+ ). k Conequently, λ j = 0 by Lemma 3.8, and µ j+,j = 0 by maximality of S and Corollary 3.. A in Cae I, we derive two contradictory equation. The um of the correonding KKT condition 3.7) from i + to j i equal to j i ) + µ i+,i µ i,i+ + µ j,j+ = 0. The KKT condition 3.6) for tye i and j lead to: h i δµ i,i+ = h j + δµ j,j µ j,j+ ) µ i,i+ = j i) + µ j,j µ j,j+. Thee two equation give a contradiction: 0 = 2j i) + µ j,j + µ i+,i. To conclude, a menu with non-unique contract between retailer tye i never otimal, irreective of the actual value of δ. Thi how that uniformity of tye and equiditant holding cot are ufficient for unique contract. Corollary 3.6. If we aume uniformity on tye and equiditant holding cot, the KKT grah ha no cycle. 9