Balancing Expected and Worst-Case Utility in Contracting Models with Asymmetric Information and Pooling

Transcription

1 Balancing Expecte an Worst-Case Utility in Contracting Moels with Asymmetric Information an Pooling R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans Econometric Institute Report EI th January 2018 Abstract We consier a principal-agent contracting problem between a seller an a buyer, where the buyer has single-imensional private information. The buyer s type is assume to be continuously istribute on a close interval. The seller esigns a menu of finitely many contracts by pooling the buyer types a priori using a partition scheme. He maximises either his minimum utility, his expecte utility, or a combination of both a multi-objective approach). For each variation, we etermine tractable reformulations an the optimal menu of contracts uner certain conitions. These results are applie to a contracting problem with quaratic utilities. We show that the optimal objective value is completely etermine by the partition scheme, a single aggregate instance parameter, an a parameter encoing the seller s guarantee obtaine utility. This enables us to erive the optimal partition an exact performance guarantees. Our analysis shows that the seller shoul always offer at least two contracts in orer to have reasonable performance guarantees, resulting in at least 88% of the expecte utility compare to offering infinitely many contracts. By also optimising obtaine worst-case utility, he can potentially achieve only 64% of the maximum expecte utility. eywors: mechanism esign, asymmetric information, pooling of contracts, multi-objective optimisation 1 Introuction We consier a principal-agent problem where the principal is a seller of proucts an where the agent is a potential buyer. The seller has the initiative an market power to make a one-time offer to the buyer in which he presents a menu of contracts. We assume that this is a take-it-or-leave-it offer, i.e., we o not consier repeate offers or renegotiations. Each contract specifies an orer quantity x R 0 an a sie payment z R from the buyer to the seller. The buyer has the market power to accept or reject any contract from the menu. Furthermore, we assume that both the seller an the buyer act iniviually rationally an want to maximise their own utility. Thus, the buyer will accept an offere contract if this is most beneficial to himself. The buyer has private information that he oes not share with the seller. We consier the case where the buyer s private information can be encoe into a single-imensional parameter p, referre to as the buyer s type. We assume that the buyer s type can take on values in [ p, p] R with p > p an follows a continuous istribution with strictly positive ensity function ω : [ p, p] R >0. Although the buyer s type is private, this istribution is known to the seller. The seller has utility function φ S : R 0 R for an orer quantity an his net utility also inclues the sie payment. Thus, if the buyer accepts a contract x, z), the resulting seller s net utility is φ S x) + z. Likewise, a buyer with type p has utility function φ B p) : R 0 R. His net utility for contract x, z) is φ B x p) z. If the buyer rejects all contracts, we assume that his net utility is zero, which is also known as the buyer s reservation level. Due to the buyer s private information, the seller can an will use Mechanism Design to construct a menu of contracts for the buyer to choose from. For a general reference on Mechanism Design for contracting problems, see for example Laffont an Martimort 2002). We consier the case where the seller only offers a limite number of contracts, similar to the Robust Pooling approach in our earlier work erkkamp et al. 2017). That is, the seller first ecies how many contracts are offere, inicate by N 1. For notational convenience, let = {1,..., }. Secon, the seller partitions [ p, p] into subintervals [ p k, p k ] with p k > p k for k. Such a partition is calle a proper -partition. Thir, 1

2 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans the seller constructs a menu of contracts by solving a specific optimisation moel, given below, which epens on the chosen partition. Finally, this menu is offere to the buyer. From this point onwars, we refer to a contract by x k, z k ) with k an to a menu of contracts by x, z), where x = x 1,..., x ) an z = z 1,..., z ). The menu is esigne such that for each k it is for all types in [ p k, p k ] most beneficial to choose contract x k, z k ). We have the following constraints for the menu: φ B x k p k ) z k 0, p k [ p k, p k ], k, 1) φ B x k p k ) z k φ B x l p k ) z l, p k [ p k, p k ], k, l, 2) x k 0, k. 3) Constraints 3) simply enforce non-negative orer quantities. The other constraints 1) an 2) affect which contract the buyer will choose. Constraints 1) ensure iniviual rationality IR) for the buyer: for k an p k [ p k, p k ] contract x k, z k ) must not give a lower net utility than the buyer s reservation level. If 1) oes not hol, then type p k will never accept contract x k, z k ). We make the conventional assumption that if the buyer has multiple options which all maximise his net utility, then the seller can convince the buyer to choose from these the most beneficial option to the seller. Consequently, 1) guarantees that all types will choose a contract from the menu. Constraints 2) ensure for k that contract x k, z k ) has the highest net utility for all types in [ p k, p k ]. This is known as incentive compatibility IC). Thus, for a menu satisfying constraints 1)-3) contract x k, z k ) is chosen by all types [ p k, p k ]. In other wors, contract x k, z k ) is chosen by the buyer with probability ω k pk pk ωp)p k. 4) We consier two objective functions for the seller subject to constraints 1)-3): he maximises either his expecte net utility or his minimum net utility. With the above insight, the seller s expecte net utility is given by ω k φ S x k ) + z k ). 5) k This leas to the Maximise Expecte net utility ME) moel: max{5) : 1)-3)}. Similarly, the seller s minimum net utility is min φ Sx k ) + z k ), 6) k resulting in the Maximise Minimum net utility MM) moel: max{6) : 1)-3)}. It turns out that for a broa class of problems the MM moel has multiple optimal solutions, as we will show. The seller can therefore choose from these optimal solutions base on a secon criterion. In light of the seller s esire to maximise his utility, we consier the case that the seller selects the optimal MM solution with maximum expecte net utility. This can be interprete as a two-stage optimisation approach base on the MM an ME moels. In fact, we generalise this two-stage approach to a multi-objective approach by aing the constraint min k φ S x k ) + z k ) M, or equivalently φ S x k ) + z k M, k, 7) to the ME moel for some parameter M R. We note that 7) is also known as the seller s iniviual rationality constraint, where M is the seller s reservation level. We call the resulting moel the Multi- Objective MO) moel: max{5) : 7), 1)-3)}. Notice that by choosing M sufficiently small/negative 7) is non-restrictive an the MO moel becomes the ME moel. Likewise, by setting M to the optimal MM objective value, the MO moel has the above escribe two-stage interpretation an fins the optimal MM solution with maximum expecte net utility. Hence, the parameter M allows the seller to analyse the trae-off between maximising expecte or worst-case net utility in a multi-objective perspective. We shall refer to the MM, ME, an MO moels as pooling moels in general. Our goal is to analyse these pooling moels, etermine the optimal solutions analytically, an apply the results to a concrete contracting problem. In particular, we want to analytically quantify the effect of pooling the buyer types, the chosen partition scheme, an the buyer s reservation level M. 2

3 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans 1.1 Literature In the MM moel the seller maximises his minimum net utility, which is often calle having a maximin objective or being ambiguity averse in the literature. Here, the minimum is taken over all offere contracts or, equivalently, over all possible realisations of the buyer s type. Only the support [ p, p] of the istribution ω is neee. This is an extreme case of recent Robust Optimisation approaches to Mechanism Design, see for example Bergemann an Schlag 2011) an Pınar an ızılkale 2016). In these moels the minimum is taken over an uncertainty set for ω, e.g., the istribution ω cannot iffer too much from a reference istribution. The resulting robust moel is typically less conservative than the classical maximim moel. For further references on using Robust Optimisation, see Aghassi an Bertsimas 2006), Ben-Tal et al. 2009) an Bergemann an Morris 2005). Our pooling approach can be viewe as a ifferent application of Robust Optimisation. Consier the classical iscrete variant of the contracting problem see for example Laffont an Martimort 2002)). Here, the buyer s type lies in the set {p 1,..., p } an follows a iscrete istribution. If we associate an uncertainty set [ p k, p k ] to type p k, then our pooling moel is the Robust Optimisation variant for the iscrete moel. By consiering a continuum of types [ p, p] an using a partition scheme, we are in fact restricting the uncertainty sets [ p k, p k ] to form a partition of [ p, p]. A property of the pooling approach is to offer finitely many contracts to a continuum of buyer types. There are to our knowlege two papers in the literature that are strongly connecte to this approach: Bergemann et al. 2011) an Wong 2014). Bergemann et al. 2011) consier a linear-quaratic moel with limite communication between the seller an the buyer base on Mussa an Rosen 1978). The seller wants to maximise his expecte net utility. The limite communication restricts the seller to using a menu with finitely many contracts. In contrast to our pooling approach, they o not partition the types a priori. Instea, their menu maps each buyer type to one of the contracts without any restrictions. By reformulating the problem into a mean square minimisation problem an applying Quantisation theory, they are able to etermine the optimal menu of contracts an the corresponing optimal mapping of buyer types to contracts. In particular, their results show that the restriction to contracts leas to a loss in performance of the orer Θ1/ 2 ) compare to offering infinitely many contracts. Wong 2014) uses the same moelling approach as Bergemann et al. 2011), but analyses a more general non-linear pricing problem again maximising the seller s expecte net utility). His analysis focusses on the loss in performance when restricting to contracts instea of infinitely many contracts. In particular, he erives the same Θ1/ 2 ) loss in performance as Bergemann et al. 2011), but uner a more general setting. For the ME moel we have shown in erkkamp et al. 2017) that the pooling approach an those of Bergemann et al. 2011) an Wong 2014) are equivalent provie that we use the -partition of [ p, p] that maximises the seller s expecte net utility. That is, both approaches lea to partitioning the buyer types. However, as argue in erkkamp et al. 2017) an Wong 2014) etermining the optimal -partition is ifficult in general. In case the optimal partition cannot be erive, the benefit of our pooling approach is that it allows for heuristic partition schemes in a simple an controlle way. As we will show, the complexity of the pooling moels for a given partition is similar to classical iscrete contracting moels. The previously iscusse papers o not consier multi-objective optimisation. In terms of using a multi-objective approach, Zheng et al. 2015) has to our knowlege the strongest connection to our work. Zheng et al. 2015) consier a continuum of types [ p, p] an the seller offers a menu with infinitely many contracts, i.e., there is no a priori pooling. They suppose that the seller is not confient about the probability istribution ω an moel this by a so-calle ɛ-contamination: with probability 0 ɛ 1 the istribution ω is incorrect an the worst-case outcome of ω occurs. Consequently, the objective function is the weighte sum of the seller s expecte net utility an the seller s minimum net utility. The assume utility functions are φ S x) = cx an φ B x p) = pχx), where χ is a strictly increasing, positive, an continuously ifferentiable concave function. Furthermore, the istribution ω has a non-ecreasing hazar rate. They show that for 0 < ɛ < 1 the optimal menu effectively pools the types [ p, p ɛ)] for some < p p ɛ) < p an offers those types the same contract. For the types p ɛ), p] infinitely many contracts are offere. To compare their multi-objective approach to ours, we translate the moel of Zheng et al. 2015) to our pooling setting. For given 0 ɛ 1 the resulting moel is to maximize 1 ɛ) k ω k φ S x k ) + z k ) + ɛ min k φ Sx k ) + z k ) 3

4 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans subject to 1)-3) with variables x an z. This is equivalent to maximising 1 ɛ) k ω k φ S x k ) + z k ) + ɛm subject to 7) an 1)-3) with variables x, z, an M. We refer to this moel as the weighte objective moel. Although this moel is very similar to our MO moel, there are ifferences. The most obvious ifference is that for ɛ = 1 the weighte objective moel is our MM moel, not our MO moel with correctly corresponing M). In this case, our MO moel is a two-stage optimisation moel which etermines the optimal MM solution that maximises the seller s expecte net utility. Typically, the MM moel has multiple optimal solutions, whereas the MO moel has just one as we will show later). Furthermore, the parameter M in the MO moel has a natural interpretation, namely the seller s reservation level. A similar interpretation of ɛ only follows inirectly after solving the weighte objective moel an observing the corresponing optimal M. We will iscuss further similarities an ifferences in more etail uring our analysis. Besies solving the pooling moels for given number of contracts, partition scheme, an seller s reservation level M, we want to quantify the effect of these choices on the corresponing optimal objective values. Due to the complexity we focus on a concrete contracting problem for such an analysis: the Linear-Quaratic-Uniform LQU) problem aapte from Wong 2014). For the ME moel variant of the LQU problem we also refer to erkkamp et al. 2017), where we complete the analysis of Wong 2014). Since the MO moel generalises the ME moel, our results in this paper supersee the mentione analyses of the LQU problem. Furthermore, we are able to relate results for the LQU problem to Zheng et al. 2015). 1.2 Contribution We analyse a contracting problem where the seller offers a menu of finitely many contracts to a buyer with a continuum [ p, p] of types. Here, the seller uses a partition scheme to pool the types a priori. Moreover, we consier a multi-objective approach for the seller s objective function which balances expecte an worst-case minimum) net utility. Compare to the literature, we exten relate work by either consiering a multi-objective approach Bergemann et al. 2011) an Wong 2014)) or by pooling the buyer types with a partition scheme Zheng et al. 2015)). Furthermore, there are ifferences in the moelling approaches as iscusse in Section 1.1. Uner commonly use assumptions, we erive tractable reformulations for the pooling moels an etermine the optimal menu of contracts. The optimal menus all turn out to be the maxima of certain moifie joint net utility functions. We apply an exten these results to a concrete contracting problem, namely the LQU problem. In particular, we erive the optimal partition scheme for the LQU problem an the corresponing optimal objective values. Consequently, we can analyse various performance measures that quantify the effect of the number of contracts offere an of the seller s reservation level M. This leas to performance guarantees that give insight into the trae-off between maximising expecte or worst-case net utility. All results are analytical an expresse in close-form formulas. The remainer of this paper starts with the general analysis of the pooling moels in Section 2, followe by the application to the LQU problem in Section 3. We conclue our finings in Section 4. 2 General analysis In this section we analyse the three pooling moels in etail. First, we present the essential etails of the setting an the three moels in Section 2.1. In Section 2.2 we erive tractable reformulations for the moels uner a common assumption on the buyer s utility function. Finally, we etermine the optimal solution of the moels for a broa class of problems in Section 2.3. All corresponing proofs are given in Appenix A. 2.1 The moels As introuce in Section 1, we consier a seller with utility function φ S : R 0 R an a buyer with utility function φ B p) : R 0 R for type p [ p, p] R. The joint utility function is enote by φ J p) φ S ) + φ B p). The buyer s type follows a continuous istribution with strictly positive ensity function ω : [ p, p] R >0. The seller offers the buyer a menu with a limite number of contracts 4

5 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans by pooling the buyer types a priori. In this menu, the k-th contract x k, z k ) specifies the orer quantity x k R 0 an the sie payment z k R from the buyer to the seller. First, the seller chooses the number of contracts N 1 to offer. Secon, the seller partitions [ p, p] into subintervals [ p k, p k ] with p k > k for k = {1,..., }, leaing to aggregate probabilities ω k for k given by 4). Given this partition/pooling p scheme, the seller constructs a menu of contracts by solving one of our pooling moels: the MM moel max{6) : 1)-3)}, the ME moel max{5) : 1)-3)}, or the MO moel max{5) : 7), 1)-3)}. We focus on the MO moel: max x,z ω k φ S x k ) + z k ), k s.t. φ S x k ) + z k M, k, 7) φ B x k p k ) z k 0, p k [ p k, p k ], k, 1) φ B x k p k ) z k φ B x l p k ) z l, p k [ p k, p k ], k, l, 2) x k 0, k. 3) The MO moel maximises the seller s expecte net utility uner iniviual rationality constraints for the seller 7) an the buyer 1), an uner incentive compatibility constraints 2). Note that the buyer s reservation level in 1) is assume to be zero, whereas the seller s reservation level M R in 7) is set by the seller. With the parameter M the seller can balance his expecte net utility with his minimum net utility. In particular, by an appropriate choice of M we can solve the ME or MM moel with the MO moel. That is, if we increase M then the optimal solution transitions from an optimal ME solution to an optimal MM solution. As a final note, if M is too large, then the MO moel is infeasible. These insights will be mae more concrete in the following analysis. 2.2 Tractable reformulation In orer to obtain tractable reformulations of our moels an the results to come, we nee to assume aitional structure on the buyer s utility function. Assumption 1 states that φ B is non-ecreasing in the buyer s type an satisfies the strictly increasing ifferences property. Assumption 1. The buyer s utility function φ B satisfies the following properties: φ B x λ) φ B x µ) λ µ R, x 0, 8) φ B x λ) φ B x λ) < φ B x µ) φ B x µ) λ < µ R, 0 x < x. 9) Note that we implicitly assume that φ B λ) is efine for all λ R, not just for [ p, p]. However, it follows from the proofs that we only nee to consier 2 instance-epenent values for λ. Assumption 1, or the stronger Single-Crossing Conition, is common in the Mechanism Design literature an leas to non-ecreasingness in the orer quantities with respect to the buyer s type see also Elin an Shannon 1998), Laffont an Martimort 2002) an Schottmüller 2015)). This is also the case for the pooling moels, as shown in Lemma 1. Lemma 1. Uner Assumption 1, any x satisfies 1)-3) if an only if 0 x 1 x. Furthermore, for fixe orer quantities x, the IR an IC constraints 1) an 2) imply a ual shortest path problem structure on the sie payments z. For non-pooling moels this has been ientifie before, see for example Rochet an Stole 2003) an Vohra 2012). For our pooling moels, the sie payments are even more restricte, leaing to the optimal formulas for z given in Lemma 2. In fact, we only nee to assume 8) for this result, but we have chosen to merge certain assumptions for reaability. Lemma 2. Consier the ME, MM, or MO moel uner Assumption 1. It is necessary an sufficient for optimality to set k 1 z k = φ B x k p k ) φb x i p i ) φ B x i p i ) ) k. 10) i=1 With Lemma 2 we can eliminate the sie payments from our moels. Using Lemma 1 we can then simplify the constraints from infinitely many to linear constraints. This results in the tractable reformulations as shown in Theorem 3. Recall that φ J p) is the joint utility function with respect to type p. 5

6 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans Theorem 3. Uner Assumption 1, the ME moel is equivalent to max ω k φ J x k p k ) φ B x k p k ) φ B x k p k ) ) ω i 0 x 1 x ω k k i=k+1 ), 11) the MM moel to k 1 max min φ J x k p k ) φb x i p i ) φ B x i p i ) )), 12) 0 x 1 x k i=1 an the MO moel to s.t. max x ω k φ J x k p k ) φ B x k p k ) φ B x k p k ) ) ω k k i=k+1 k 1 ω i ), 13) φ J x k p k ) φb x i p i ) φ B x i p i ) ) M, k, 14) i=1 x x ) From the reformulations it is clear that the complexity of solving our pooling moels epens on the shape of φ J p k ) an of φ B p k ) φ B p k ) for k. For example, if φ J p k ) is ifferentiable an concave, an if φ B p k ) φ B p k ) is linear, then all three moels are concave optimisation problems with ifferential functions, which can be solve numerically in an efficient way. We focus on classifying problems for which the optimal solutions can be escribe in a unifie way. 2.3 Optimal solutions The next assumption exclues situations where the seller coul potentially achieve infinite utility from the menu of contracts, see Assumption 2. Assumption 2. For any λ R the joint utility function φ J λ) has a maximum on R 0 an on any close subinterval of R 0. In Assumption 2, the existence of a maximum on any close subinterval of R 0 is neee because of a technicality see the proof of Lemma 4 to come). In particular, Assumption 2 is satisfie if φ S an φ B are continuous functions in the orer quantity. The maximum of φ J λ) for specific values of λ has a central role in the optimal solutions for our moels. Therefore, we have an intermeiate result on the maximisers of φ J λ), see Lemma 4. Lemma 4. Uner Assumptions 1-2, there exists a non-ecreasing function M : R R such that M λ) max x 0 {φ Jx λ)}. For M M p), there exists a non-ecreasing function x M) : R R 0 such that x λ M) min argmax{φ J x λ)}, x x M where x M R 0 is given by x M min { x 0 : φ J x p) M }. Recall that the seller s reservation level M only affects the MO moel. For the MM an ME moels, we can implicitly use an non-restrictive value for M, namely M =. For M = we have x M = 0, hence x λ M) optimises over the entire omain x 0. In this case we simplify our notation an use x λ) instea of x λ ). We can now express the optimal solution for the MM moel, see Theorem 5. Theorem 5. Uner Assumptions 1-2, the optimal objective value for the MM moel is M p), which can be attaine by offering a menu with a single contract with orer quantity x p). Note that this oes not epen on the partition of [ p, p]. Another optimal solution is x k = x p k ) for k, which oes epen on the partition. 6

7 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans A consequence of Theorem 5 is that the complexity of solving the MM moel is completely etermine by the complexity of maximising φ J x p) over x 0. For a similar result for the ME an MO moels, we nee the assumption that each k-th term in the objective function can be written as φ J x k λ k ) for some λ k non-ecreasing in k. This assumption is formalise in Assumption 3. Assumption 3. The ensity function ω an the buyer s utility function φ B are such that there exist parameters π k R for k satisfying π 1 π an φ B x π k ) = φ B x p k ) φ B x p k ) φ B x p k ) ) i=k+1 ω i ω k x 0, k. 16) The parameters π k are strongly relate to the virtual valuation of the buyer types see e.g. Laffont an Martimort 2002)). Uner Assumptions 1 an 3 it is trivial to show that π k < p k for all k \ {} an π p see the proof of Theorem 7 to come). In Appenix B we present an example problem class for which we prove that it satisfies Assumption 3 an provie a close-form expression for π k. In the example, the buyer s utility function is φ B x p) = ψx) + pχx) for some functions ψ an χ, where χ is strictly increasing an non-negative. Furthermore, ω is a continuous istribution with a non-ecreasing hazar rate, e.g., the uniform istribution. Uner the aitional assumption, we can erive the optimal solution for the ME moel as shown in Theorem 6. Theorem 6. Uner Assumptions 1-3, an optimal solution for the ME moel is x k = x π k ) for k. Compare to the MM moel, where an optimal solution is given by the maximisers of φ J p k ) for k, an optimal ME solution is specifie by the maximisers of φ J π k ). In other wors, we nee to shift the buyer types ownwars from p k to π k. Last but not least, we have the optimal solution for the MO moel. The MO moel maximises the seller s expecte net utility uner the constraint that the seller s minimum net utility is at least his reservation level M. From Theorem 5 we know that the minimum net utility is at most M p), being the optimal objective value of the MM moel. Therefore, any seller s reservation level M M p) can be satisfie an any M > M p) is infeasible. Theorem 7 states the optimal MO solution. Theorem 7. Uner Assumptions 1-3, the MO moel is feasible if an only if M M p), an an optimal solution for the MO moel is x k = x π k M) for k. In particular, if the seller sets M = M p), then the MO moel is a two-stage optimisation which maximises first the seller s minimum net utility an secon the seller s expecte net utility. Hence, uner Assumptions 1-3 we have ientifie a thir optimal MM solution. This leas to the next straightforwar corollary. Corollary 8. Uner Assumptions 1-3, if for each k the function k 1 φ J x k p k ) φb x i p i ) φ B x i p i ) ) is concave on x x 1 0, then any convex combination of is an optimal solution for the MM moel. i=1 x k = x p) k, x k = x p k ) k, x k = x π k M p)) k, Returning to Theorem 7, if φ J λ) is concave, then the optimal MO solution can be foun in two steps as follows. First, etermine the optimal ME solution by using the shifte buyer types π k, resulting in x k = x π k ) for k. Secon, set any x k < x M to the threshol orer quantity x M in orer to guarantee the seller s reservation level M. Zheng et al. 2015) erive a similar solution structure for their concave setting with infinitely many contracts, where the types [ p, p ] for some p < p < p are offere the same contract like x M in our case). Returning to our result, the threshol x M coul lea to aitional pooling of types as multiple contracts can specify the orer quantity x M. This implies that the original partition of [ p, p] can be improve to increase the seller s expecte net utility. 7

8 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans This brings us to one of the ecisions the seller has to make: the partition of [ p, p]. Base on our results, we have the following strategy for the seller. First, the seller must etermine the optimal MM objective value M p), which is inepenent of the partition. Secon, he must ecie on his reservation level M M p). Thir, he chooses the number of contracts offere. Finally, the seller selects a partition an uses the above results to etermine an optimal menu of contracts for the MO moel. Ieally, the seller optimises the partition such that his expecte net utility is maximise. Unfortunately, such optimisation appears to be ifficult in general. Given the complexity of the analysis, we focus on the so-calle Linear-Quaratic-Uniform LQU) problem aapte from Wong 2014). In Section 3 we erive the optimal partition an analyse performance guarantees for the LQU problem. 3 Application to the LQU problem In this section, we apply the results of Section 2 to a concrete contracting problem, calle the Linear- Quaratic-Uniform LQU) problem. We formalise the LQU problem in Section 3.1 an translate our general results from Section 2 to this setting in Section 3.2. In Sections we continue the analysis, erive the optimal partition, an etermine performance guarantees when using the optimal partition. All corresponing proofs are given in Appenix C. 3.1 The Linear-Quaratic-Uniform problem In the Linear-Quaratic-Uniform LQU) problem, the seller s utility function is linear in the orer quantity: φ S x) = P x, where P R >0 is the seller s utility per unit of sol prouct. The buyer s utility function is characterise by a saturation effect: the marginal utility of buying an aitional prouct ecreases linearly. That is, for orer quantity x R 0 the buyer s marginal utility of an aitional prouct is p rx. Here, p [ p, p] R >0 with p > is the buyer s private) type an r R >0 is a saturation rate parameter. Note that p is strictly positive. p The buyer s type is assume to have a uniform istribution, i.e., ωp) = 1/ p Consequently, the buyer s utility function is p). φ B x p) = x 0 p ru)u = px 1 2 rx2. Notice that for large orer quantities the buyer s utility is negative, which moels for example that excess proucts must be ispose of at a cost. Furthermore, orering no proucts leas to zero utility for the buyer, which is his reservation level. The pooling of contracts for the LQU problem has been analyse in Wong 2014) an uner the name DMU-1 in erkkamp et al. 2017), both with the goal to maximise the seller s expecte net utility the ME moel). As mentione in Section 1, we exten the analysis to the MO moel, with which we can balance the maximisation of the seller s worst-case an expecte net utility. 3.2 Optimal solutions It is straightforwar to verify that the LQU problem satisfies Assumptions 1-3 of Section 2. In particular, since ω is the uniform ensity function we have an therefore 16) of Assumption 3 simplifies to π k x 1 2 rx2 = p k x 1 2 rx2 p k p k )x Hence, the parameters π k are ω k = p k p k p p, i=k+1 p i p i p k p k = p k + p k p)x 1 2 rx2. π k = p k + p k p k, which satisfy π 1 π an π k p k for all k. In contrast to the buyer s type p the parameter π k can be negative for some k, epening on the instance parameters an the partition. 8

9 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans Since φ J x λ) = P + λ)x 1 2 rx2, the function M state in Lemma 4 is { 1 M λ) = 2r P + λ)2 if P + λ 0. 0 otherwise For the MO moel, we nee to realise that for a non-negative seller s reservation level M 0) the seller s IR constraint 14) is non-restrictive. This follows from Theorem 7 an the efinition of x M) in Lemma 4. More precisely, since φ J 0 p) = 0 we have x M = 0 for M 0 an hence the optimal MO solution is the optimal ME solution. To conclue, the only proper values for M for the MO moel are 0 M M p). Therefore, we only consier M = βm p) for some β [0, 1]. For notational convenience, we often switch from M to β. We efine x β x βm p) an x β) x βm p)), resulting in x β = min { x 0 : φ J x p) βm p) } = min { x 0 : P + p)x 1 2 rx2 β 1 2r P + p)2} = 1 r 1 1 β)p + [0, p) 1 r P + p)]. Thus, the function x state in Lemma 4 is x λ β) = 1 r {1 max } 1 β)p + P + λ. 17) p), In Corollary 9 we collect an translate the results of Theorems 5, 6, an 7 for the LQU problem. Corollary 9. For the MM-LQU moel, any convex combination of x k = 1 r P + k, p) x k = 1 r P + k, pk) x k = 1 r max { } P + P p + p k + k k, p, p is an optimal solution. For the ME-LQU moel, the only optimal solution is x k = 1 r max { 0, P p + p k + p k } k. 18) For the MO-LQU moel an M = βm p) for some β [0, 1], the only optimal solution is x k = 1 r max { 1 1 β)p + p), P p + p k + p k } k. 19) Note that the optimal solutions for the ME an MO moels are unique ue to the strict concavity of φ J λ) the etails are given in the proof of Corollary 9). Furthermore, for certain instances the optimal ME solution 18) results in no trae with a range of buyer types those for which P p + p k + k 0). Consequently, the seller s worst-case net utility is zero for such instances. It might be preferable p to always trae with a potential buyer to at least make some revenue, even if this results in a potentially lower expecte net utility. This is exactly what happens with the optimal MO solution 19) for β > 0: the optimal MO menu always instigates trae with the buyer if β > 0. A similar result is observe in Zheng et al. 2015) for their concave setting with infinitely many contracts. We can use Corollary 9 to illustrate a ifference between the approach of Zheng et al. 2015) an our MO moel. If we translate the results of Zheng et al. 2015) to our pooling setting, their multi-objective approach coul lea to a iscontinuity at β = 1 the equivalent to their ɛ = 1) in x k as function of β. This is ue to the fact that their approach leas to the MM moel if β = 1, which has multiple optimal solutions, as seen in Corollary 9. In contrast, the MO solution is always unique an continuous in β. Finally, for β = 0 the optimal MO solution is the optimal ME solution. Similarly, for β = 1 the optimal MO solution is the unique) optimal MM solution that also maximises the seller s expecte net utility as a two-stage optimisation process. For this reason, we focus completely on the MO moel in the results to come. 9

10 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans 3.3 Normalising the objective function To construct an optimal menu of contracts, the seller has to ecie on his reservation level M or equivalently β), the number of contracts offere, an the partition of the buyer types. To quantify the effect of these ecisions, we nee to express the optimal objective function value in terms of the state ecisions. We can simply substitute 19) in the objective function 13), but it turns out to be useful to normalise various parameters as follows. First, we reefine the partition as pk = p + δ k 1 p p) an p k = p + δ k p p), where δ 0 = 0, δ k [0, 1] for k = 1,..., 1, an δ = 1. For a proper -partition, we have 0 = δ 0 < < δ = 1. Consequently, ω k = δ k δ k 1. Secon, we introuce the normalisation factor ν an the aggregate instance parameter α: ν = 2r p p) 2 > 0 an α = p p P + > 0. p As we will show, the relative) performance measures of interest can be expresse completely in terms of α, β, an δ k k ). The normalisation factor ν is use to simplify the expressions an cancels out in relative measures. Let Γ be the optimal MO objective value when using a proper -partition, i.e., using 19). We can express the normalise optimal MO objective value νγ in terms of the introuce normalise/aggregate parameters, see Lemma 10. Lemma 10. For any proper -partition the normalise optimal MO-LQU objective value is given by νγ = k β k=1 + δ k δ k 1 ) k=k β +1 β α + 2δ 2 k + δ k 1 1)1 ) 1 β) 1 α δ k δ k 1 ) 1 α + δ k + δ k 1 1 ) 2, 20) where k β is the largest inex affecte by the seller s reservation level: { { }} k β = max 0, max k : δ k + δ k 1 < 1 1 α 1 β. Notice that k β <, since δ = 1. Furthermore, for instances with α 0, 1 β] 0, 1] we have k β = 0, implying that the seller s reservation level is non-restrictive for all contracts. Two extreme cases are ν = α 2 an the limit of νγ for using any sensible partition such as δ k = k/): νγ = δ β where δ β correspons to the limit of k β : Hence, we get 0 β α + 4δ 2)1 ) 1 1 β) 1 2 α δ + 1 α + 2δ 1) 2 δ, δ β δ β = max { 0, } 1 α 1 β). { 1 α + 1 νγ = 2 3 if α 1 β β α α 1 β) 3 if α > 1 β. 21) Notice that Γ is inepenent of the partition, as shoul be the case since we are effectively offering infinitely many contracts. Trivially, we have Γ Γ for any α, β,, an partition. Hence, we can use Γ as a benchmark to evaluate the performance of the chosen partition scheme. Recalling the objective of the MO moel, a natural choice for the partition is the one which maximises the seller s expecte net utility Γ. For the LQU problem we are able to etermine this optimal partition, as we will show in the next section. 10

11 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans 3.4 Optimal partition The goal of this section is to etermine close-form formulas for the partition that maximises Γ or equivalently νγ ) for a given α, β, an. As mentione in erkkamp et al. 2017) an Wong 2014), it seems to be ifficult to etermine such close-form formulas in general, either ue to complex system of equations neee to be solve or ue to the existence of multiple local optima. However, the structure of the LQU problem allows us to etermine close-form formulas for the optimal partition. First, we prove that offering the same contract multiple times is suboptimal an that we shoul use all available contracts. This is intuitively clear, but formalise in Lemma 11. Lemma 11. The optimal MO-LQU partition satisfies k β {0, 1} an 0 = δ 0 < δ 1 < < δ 1 < δ = 1. Lemma 11 greatly restricts the value of k β when etermining the optimal partition, making the analysis manageable. We can now erive the optimal MO partition, see Theorem 12. Theorem 12. For 0 < α equiistant partition: For α > 1 1 β the optimal MO-LQU partition satisfies k β = 0 an is the δ opt k = k k. 1 1 β the optimal MO-LQU partition satisfies k β = 1 an is δ opt k = 1 k α 1 β) k. In particular, for β = 1 the equiistant partition is suboptimal for all α > 0. Sketch of the proof. Since k β {0, 1} by Lemma 11 we only nee to consier two variants for the formula of νγ. For each variant we set the graient to zero, leaing to systems of linear equations after simplification, an etermine the corresponing maximiser. The maximiser must be in line with the consiere value of k β, resulting in a specification of a vali range of instances: either 0 < α 1 1 β or α > 1 1 β. These ranges are isjoint an capture all instances α > 0, which completes the proof. The result in Theorem 12 an its proof) is a generalisation of the erive optimal partition in erkkamp et al. 2017). A remarkable property is that the equiistant partition is optimal for a range of instances, as specifie by the relation between α, β, an. This range of instances increases if β ecreases by lowering the seller s reservation level) or if ecreases by offering less contracts). Moreover, if the equiistant partition is not optimal, then the optimal partition can be foun by increasing δ 1 an partitioning the remaining subinterval [δ 1, 1] equiistantly for δ 2,..., δ 1. When using the optimal partition the expression for νγ 20) can be simplifie to a similar expression as 21). This is shown in Corollary 13. Corollary 13. For the optimal partition the normalise optimal MO-LQU objective value 20) is ν = { 1 ) α if α 1 1 β 2 β α + 2 1) ) α 1 β) 3 if α >. 22) 1 1 β Notice that 22) clearly converges to 21) if, as shoul be the case. Also, on certain intervals ν an νγ either iffer by a constant 1/3 2 ) or by a factor 4 1)/2 1) 2 in the cubic term. In the next section, we analyse the relative ifference between an Γ in more etail to obtain performance guarantees. 3.5 Performance guarantees In this section we analyse the performance of the optimal menu of contracts when using the optimal partition. We consier two performance measures: the pooling performance Section 3.5.1) an the reservation level performance Section 3.5.2). Both measure the effectiveness of pooling the buyer types compare to offering infinitely many contracts in their own way. In particular, the first measure is useful when the seller s reservation level cannot be ajuste, whereas the secon is insightful when the seller s reservation level is a ecision variable. 11

12 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans Pooling performance The pooling performance /Γ is the fraction of the seller s expecte net utility attaine by offering contracts instea of infinitely many contracts. Here, the seller uses the optimal menu of contracts an the optimal partition, as erive in Sections 3.2 an 3.4. In other wors, it measures how much is lost ue to the pooling of the buyer types by offering a limite number of contracts. Note that the seller s reservation level M or β) must always be satisfie by both menus corresponing to an Γ. In Figure 1 we illustrate the attaine pooling performance for two example instances in terms of β an. Here, we use α = 2 an α = We have chosen for α = 2 because this is the threshol value in 22) for = 2 an β = 0. The reason for the other chosen instance will be given in the next section. We observe in Figure 1 that for β = 0 the pooling performances are 88% = 2) an 96% = 3) for both instances. As β increases the pooling performance increases. However, the rate of increase iffers significantly between = 2 an = 3, an between the two instances. Finally, notice that for fixe {2, 3} an for any 0 < β 1 the pooling performance is higher for α = 2 than for the other instance. Figure 1: Pooling performance for the MO-LQU moel with the optimal partition, where α is fixe to α 1 = 2 or α 2 = By analysing expressions 21) an 22) for Γ an, respectively, we can generalise the above observations. Furthermore, we are able to erive guarantees for the pooling performance, see Theorem 14. Theorem 14. For the optimal partition the MO-LQU pooling performance /Γ is continuous an non-increasing in α, an continuous an non-ecreasing in β. In particular, we have the tight pooling performance guarantee 4 1) Γ 2 1) 2 α > 0, 0 β 1. 23) Sketch of the proof. The iea is to consier all cases that occur base on 21) an 22). Analysing the close-form an manageable formula for each case leas to the following insights. For = 1 we have For > 1 we have if β = 0: Γ < 0 for all α > 0. Γ < 0 for 0 < α < 1 an Γ if 0 < β 1: opt Γ < 0 for all α > 0. We have β Γ = 0 for 0 < α 1 β an β Γ = 0 for α 1, Γ > 0 for α > 1 β. Hence, for any 0 β 1 we obtain a tight pooling performance guarantee by taking the limit α, resulting in 23). The full proof is given in Appenix C. 12

13 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans In Figure 2 we show the attaine pooling performance for various choices of α, β, an. The aggregate instance parameter α increases when the seller s utility per unit of sol prouct P ecreases or when the buyer s type interval [ p, p] wiens uner certain conitions). We can interpret the first case ecreasing P ) as a higher investment risk in proucts, since a prouct provies less utility. The secon case wiening [ p, p]) can be interprete as an increase in uncertainty on the buyer s ientity. Hence, Theorem 14 implies that an increase in investment risk or in the uncertainty on the buyer s ientity has a negative effect on the pooling performance. In contrast, increasing the seller s reservation level encoe in β) has a positive effect on the pooling performance, provie that the seller s reservation level is restrictive α > 1 β). However, if we want to give a guarantee for the pooling performance that hols for any instance, then this positive effect has no influence. In fact, the seller s reservation level oes not affect the pooling performance guarantee. This follows from the proof of Theorem 14, where we show that the guarantee 23) is tight for any 0 β 1. Table 1 shows the values of this guarantee for = 1,..., 5. From Figure 2 an Table 1 we can conclue that the seller shoul not offer a single contract ue to poor pooling performance in general. In contrast, offering two contracts alreay leas to a pooling performance guarantee of 88% an offering three contracts results in 96%. Recall that for β = 0 the MO moel is equivalent to the ME moel. Therefore, the bouns for β = 0 correspon to the results in erkkamp et al. 2017) an Wong 2014). Our analysis shows that the same bouns hol for the MO moel for any 0 β 1. Thus, the pooling of buyer types leas to a simpler menu of contracts an can be one with a controllable loss in the seller s expecte net utility. Tight lower boun for /Γ Figure 2: Attaine pooling performance /Γ for the MO-LQU moel with the optimal partition. Table 1: Pooling performance guarantees for the MO-LQU moel with the optimal partition Reservation level performance The reservation level performance /Γβ=0 is similar to the pooling performance, except that the use benchmark isregars the seller s reservation level. That is, the numerator epens on β as before, but the enominator always uses β = 0. In particular, is the seller s maximum attainable expecte net utility over all β an. We can use the reservation level performance to quantify how much expecte net utility is lost by the seller s reservation level, again in terms of the number of contracts offere. We first consier the attaine reservation level performance for two example instances, see Figure 3. As before, we use α = 2 an α = Realise that for β = 0 the reservation level performance an the pooling performance are the same. In contrast to the pooling performance, the reservation level performance ecreases when β increases, as seen in Figure 3. We observe that for α = 2 the performance is less sensitive to changes in β for low values of β compare to the other instance. For both instances there is a steep ecrease in performance when β approaches 1, i.e., when the seller fully consiers his worst-case utility. 13

14 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans Figure 3: Reservation level performance for the MO-LQU moel with the optimal partition, where α is fixe to α 1 = 2 or α 2 = Similar to Theorem 14, we are able to generalise the above observations an etermine guarantees for the reservation level performance. These results are shown in Theorem 15, where the term unimoal refers to being non-increasing at first an then non-ecreasing. Theorem 15. For the optimal partition the MO-LQU reservation level performance /Γβ=0 is continuous an unimoal in α, an continuous an non-increasing in β. In particular, we have the tight reservation level performance guarantee 8 1) 4 1) + 2 1) ) 6 1) + 2 1) ) 2 α > 0, 0 β 1. 24) Sketch of the proof. We nee to consier all cases that occur base on 21) an 22). By analysing each case, we obtain the following results. For = 1 we have For > 1 we have if β = 0: if 0 < β 1: α > α, < 0 an β = 0 for all α > 0. < 0 for 0 < α < 1 an β = 0 for 0 < α 1 < 0 for 0 < α < α, = 0 for α 1, = 0 for α = α, an 1 β an β < 0 for α > Γ 1 1 β. β=0 > 0 for Here, the minimiser α is efine for > 1 an 0 < β 1 by 2 1) + 1)β + 2 1) β 2 1)1 1 β) + β ) α = )1, 25) 1 β) which satisfies α > 1 an α > 1 1 β if it exists. Therefore, the reservation level performance guarantee is zero for = 1 an for > 1 it follows by taking β = 1 an evaluating the performance for α = α. The full proof is given in Appenix C. Note that in Figures 1 an 3 the example instance with α = correspons to the minimiser 25) for = 2 an β = 1. The attaine reservation level performance for various choices of α, β, an is epicte in Figure 4. For β = 0 the reservation level performance an the pooling performance are equivalent see also Figure 2). For 0 < β 1 there is a unique minimiser for the reservation level performance, namely α state in 25). Hence, for any given β we can easily etermine the minimum reservation level performance. We omit the verbose exact expressions, simply use 21), 22), an 25). 14

15 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans Instea, we epict the resulting minima in Figure 5, which are tight reservation level performance guarantees for each 0 β 1. In Figure 5 the values for β = 0 correspon to 23) an the values for β = 1 to 24). As seen in Figure 5, the reservation level performance guarantee ecreases more rapily for larger values of β. These guarantees are also given in Table 2 for certain choices of an β. For example, increasing β from 0 to 3 4 leas to approximately the same percentage point ecrease in the guarantee as increasing β from 9 10 to 1. Furthermore, notice that even with infinitely many contracts = ) it is not always possible to obtain full reservation level performance. If we let, then the boun in 24) is In other wors, with infinitely many contracts the seller obtains at least 70% of the maximum expecte net utility if he first maximises his worst-case net utility β = 1) an this boun can be attaine epening on the instance. Similarly, when using two three) contracts, the seller achieves at least 64% 68%) when first maximising his worst-case net utility, an these bouns can be attaine see Table 2). Lowering the seller s reservation level raises these guarantees. For example, for β = 1 2 these are 83%, 89%, an 92% for equal to 2, 3, an, respectively. Overall, we conclue that the seller s reservation level has a significant impact on the seller s expecte net utility, irrespective of the number of contracts offere. In terms of pooling performance, increasing the seller s reservation level has a positive effect, whereas it has a negative effect in terms of the reservation level performance. In any case, the seller shoul always offer at least two contracts in orer to have a reasonable reservation level performance guarantee. Figure 4: Attaine reservation level performance /Γβ=0 for the MO-LQU moel with the optimal partition. Figure 5: Tight reservation level performance guarantees in terms of β for the MO-LQU moel with the optimal partition. Tight lower boun for /Γβ=0 β = 0 β = 1 2 β = 3 4 β = 9 10 β = Table 2: Reservation level performance guarantees for the MO-LQU moel with the optimal partition. 15

16 R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans 4 Concluing remarks When face with a continuum [ p, p] of buyer types, the seller can pool the buyer types to obtain a simpler menu of finitely many contracts. We analyse a pooling approach where the seller partitions the set of types [ p, p] a priori into N 1 subintervals [ p k, p k ] for k {1,..., }. The resulting menu consists of contracts an is esigne such that the types in the k-th subinterval [ p k, p k ] choose the k-th contract in the menu. In aition to pooling, we consiere multiple objective functions for the seller: he maximises either his minimum net utility, his expecte net utility, or a combination of both resulting in a multi-objective approach). We moelle the multi-objective approach by maximising the seller s expecte net utility uner the aitional constraint that his minimum net utility must be at least his reservation level. Here, the seller s reservation level is an aitional moel parameter ecie by the seller. Our analysis shows that uner commonly use assumptions the three consiere pooling moels have tractable reformulations an that the optimal menus are maxima of certain moifie joint net utility functions. In particular, the maximum obtainable minimum net utility is the maximum joint net utility with respect to the lowest buyer type p. Using this property, the seller can fine-tune his reservation level in the multi-objective pooling moel to balance his expecte an worst-case net utility. Effectively, the multi-objective moel encompasses the other moels. With this moel the seller can, for example, first maximise his minimum net utility, followe by his expecte net utility as a two-stage approach). The consiere pooling moels epen on the chosen partition scheme an the seller s reservation level. We consiere a contracting problem with quaratic utilities to quantify the effect of these ecisions mae by the seller. For this problem we first erive the optimal partition scheme, which then le to manageable formulas for the corresponing optimal objective value. In turn, these formulas can be use to etermine various performance measures. We note that these results are analytical/exact an hol for any number of contracts. We focusse on two performance measures. The first is the pooling performance, which is the obtaine expecte net utility by offering menus compare to infinitely many contracts. It quantifies purely the effect of pooling the buyer types. The secon is the reservation level performance, which is the obtaine expecte net utility compare to ignoring the seller s reservation level an using infinitely many contracts. Here, the benchmark is the highest attainable expecte net utility over all seller s reservation levels an all number of contracts. Both measures have been fully analyse, which resulte in performance guarantees lower bouns) in terms of the number of contracts offere. For example, offering a single contract has poor performance in both measures) an is ill-avise. In contrast, offering two, three, or infinitely many contracts leas to a pooling performance guarantee of 88%, 96%, an 100%, respectively. The corresponing reservation level performance guarantees are 64%, 68%, an 70%, respectively. Note that the latter guarantees are overall bouns an can be mae more specific for a fixe seller s reservation level. In particular, a reservation level near the maximum feasible value is costly in performance. All mentione bouns can be attaine for certain instances an are therefore tight. From our analysis, we conclue that pooling of buyer types results in a simpler menu of contracts an any loss in performance can be controlle by the number of contracts offere. High performance can alreay be achieve with up to five contracts. Furthermore, a multi-objective optimisation approach can be performe by incluing the seller s reservation level as a ecision parameter. The seller s reservation level has a significant impact on the seller s expecte net utility, irrespective of the number of contracts offere. Increasing the reservation level has a positive effect on the pooling performance, but a negative effect on the reservation level performance. Therefore, the seller has to balance his expecte an worstcase net utility an can use the state performance measures to justify his choices. References Aghassi, M. an D. Bertsimas 2006). Robust game theory. Mathematical Programming 107.1, pp Ben-Tal, A., L. El Ghaoui an A. Nemirovski 2009). Robust optimization. Princeton Series in Applie Mathematics. Princeton University Press. Bergemann, D. an S. Morris 2005). Robust mechanism esign. Econometrica 73.6, pp Bergemann, D. an. Schlag 2011). Robust monopoly pricing. Journal of Economic Theory 146.6, pp Bergemann, D., J. Shen, Y. Xu an E. Yeh 2011). Mechanism esign with limite information: the case of nonlinear pricing. 2n International ICST Conference on Game Theory for Networks. 16