Optimal design of sales contracts under

INFORMS

Vol. 00, No. 0, Xxxxx 0000, pp. 000–000 issn 0000-0000 | eissn 0000-0000 | 00 | 0000 | 0001

doi 10.1287/xxxx.0000.0000 c 0000 INFORMS °

Optimal design of sales contracts under information asymmetry Ying-Ju Chen Stern School of Business, New York University, [email protected]

Sridhar Seshadri Stern School of Business, New York University, [email protected]

Milind Sohoni Indian School of Business, Gachibowli, Hyderabad 500032, India, milind [email protected]

We consider a supply chain where one manufacturer sells products to local markets through dealers. Each dealer faces a geographically dispersed market with uncertain demand, and is able to observe demand signal and exert effort to enhance sales. Both signal and effort affect the sales outcomes probabilistically, and cannot be observed by the manufacturer. Furthermore, dealers’ signals are correlated across markets. We characterize sufficient conditions for the manufacturer to extract full surplus from dealers in this scenario. Surprisingly, when there are multiple dealers and signals are sufficiently informative, full surplus extraction can be implemented by a simple combination of linear contracts and lotteries. The linear contracts induce the dealers to exert the optimal effort, and the lotteries eliminate their incentives to misreport their signals. A linear contract plus lottery and nonlinear sales-contingent contracts can be used alternately to ensure full surplus extraction. Additionally, we show that using just linear contracts can leave large surplus for dealers. When the conditions for full surplus extraction fail to hold, we show that only a limited number of types receive information rent. We provide a two-stage procedure to compute the payment scheme. This scheme, as well, can be implemented using a linear payment scheme and a lottery. Key words : sales effort, surplus extraction, mechanism design, correlated types

1.

Introduction

Many researchers in economics and operations management have studied the design of salesforce incentives. A typical setting is one in which a manufacturer sells his products in different local markets through dealers. Based on past experience or local knowledge, a dealer (or salesperson) privately observes the demand signal in her local market. She then chooses her sales effort that further enhances the sales in her region as a function of the demand signal and her payoff. Both the demand signal and sales effort affect the sales outcome probabilistically, and are usually unobservable to the manufacturer. Thus, the inherent information asymmetry creates challenges for designing appropriate compensation (incentive structure) schemes. The optimal compensation scheme has to elicit the best sales effort and allow the manufacturer to extract the maximum possible surplus, but 1

Chen, Seshadri, and Sohoni: Optimal design of sales contracts c 0000 INFORMS 00(0), pp. 000–000, °

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this contract design problem is exposed to both adverse selection (hidden information) and moral hazard (hidden action) problems since both signal and effort are unobservable. A number of compensation schemes have been proposed and compared in the literature, including stair-step sales incentives, linear contracts, forecast scheme, and returns/rebates contracts. However, most researchers assume that demand signals of different salespersons are independent and hence propose individual contracts. This simplification cannot be applied to scenarios in which demands are correlated, especially if the surplus left on the table for dealers is large. Such correlation is inevitable when local demands are commonly affected by changes in general economic conditions (i.e., business cycle, seasonality, terrorism) or marketing factors (customers’ preference shift, introduction of competing products). Researchers have modeled this correlation and proposed to solve the problem via sales contests, but they have typically focused on ad hoc mechanisms. They also impose specific assumptions on demand distribution, payoff, and the cost of effort. In this paper we attempt to study the optimal contracting mechanism under general correlation structure of demand signals. How can the manufacturer use the information revealed from sales outcomes and dealers’ reports effectively? Should the manufacturer include all dealers’ reports in the compensation scheme? How and when can the manufacturer extract all surplus from dealers because of signal correlation? Can this be achieved using practical contracts? These are the research questions of this paper. We first consider the contract design problem when there is only one dealer (or equivalently, when the manufacturer uses only individual contracts). Even in this case, the manufacturer may be able to extract full surplus from dealers, i.e., the dealer is willing to disclose her demand signal and exert the optimal effort without receiving compensation in excess of the cost of effort. This is labeled as “full surplus extraction”. This occurs when the sales outcomes provide sufficient information regarding the demand signal as well as the dealer’s effort. We show that the possibility of full surplus extraction increases with the number of discrete sales outcomes. When there are multiple dealers, full surplus extraction can be achieved when other dealers’ signals are sufficiently informative, even if the manufacturer cannot distinguish dealers’ true signals by observing the sales outcomes. In other words, the demand signals are sufficiently informative. This condition coincides with the result of Cremer and McLean (1988) in the context of auction design. It is more likely to hold when more dealers are involved, because there are many more possible combinations of other dealers’ signals than the number of possible signals of a dealer. We show that the implementation of full surplus extraction is surprisingly simple. The manufacturer offers a menu of compensation schemes for dealers to self-select. Each compensation is composed


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of a linear contract over sales outcomes and a lottery that depends on other dealers’ reports. The linear contract provides incentive for the dealers to exert the optimal effort, as long as she gets a large enough share of the sales proceeds. The lotteries are used to eliminate dealers’ incentives to misreport their signals. In contrast, in the single-dealer problem, linear contracts might not prevent the dealer from either misreporting her signal or choosing a suboptimal effort. Thus, the manufacturer is able to use simple contracts based on signal correlation. With a slightly stronger condition–full rank condition–on signal correlation, we provide an alternative compensation scheme in which payments do not depend on a dealer’s own report. This further simplifies the contract design problem since the manufacturer can offer a single contract (rather than the aforementioned menu of contracts). It is worth noting that even greater simplification of the contract is possible as discussed in Section 5. Additionally, in this section we derive new conditions for full surplus extraction. These conditions use a combination of the probability distributions over sales outcomes and reports of signals by other dealers. They are therefore different from the ones stated above. When the sufficient conditions fail, we show that the manufacturer may have to leave information rent but only for a limited number of dealers. We provide a two-stage procedure to characterize the optimal compensation scheme. In the first stage, the manufacturer designs the sales-contingent payment schemes, and in the second stage he incorporates other dealers’ reports to implement these payments and provide incentive for dealers not to deviate. We show that the information about demand signals allows us to collapse the state space of sales outcomes. Thus, the manufacturer can use sales outcomes and signal correlation separately to implement the contracts. The rest of this paper is organized as follows. We review relevant literature in Section 2, and describe our model in Section 3. In Section 4, we demonstrate our main idea in the base model with two types and two effort levels. In Section 5 we carry out the equilibrium analysis for the general model. We draw our conclusion in Section 6, and include all the proofs in the Appendix.

2.

Related literature

A typical example of an incentive structure is the stair-step (threshold) sales incentive used in the automotive industry. Under a stair-step incentive, the dealer is paid on a per unit basis when the total sales exceeds a threshold value; a fixed bonus may also be offered (Sohoni et al. (2006)). Gonik (1978) proposed a scheme to extract maximum effort from the salespeople and induce them to forecast accurately. Under Gonik’s scheme, the firm asks each salesperson to provide a forecast of the territorial sales volume, and the salesperson compensation is jointly determined by the realized sales volume and the initial forecast. The compensation is piecewise linear in the realized sales.


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Other related papers in operations include the influential work by Porteus and Whang (1991), where they study coordination problems between one manufacturing manager (MM) and several product managers (PM). The PMs make sales effort decisions while the MM makes effort for capacity realization and decides inventory levels for different products. They develop incentive plans that induce managers to maximize the firm value (e.g., the expected return to the owner of the firm). Plambeck and Zenios (2000) develop a dynamic principal-agent model and identify an incentive payment scheme. Chen (2000, 2005) relates the salesforce compensation to the firm’s production and inventory costs, and compares the popular Gonik’s scheme and linear contracts. Taylor (2002) compares linear and target rebates with regards to the coordination issue when the dealer’s effort is made before the demand is realized. Krishnan et al. (2004) discuss the concern of contract-induced moral hazard, which arises when the dealer’s effort decision is made after observing initial sales. Cachon and Lariviere (2005) discuss situations where revenue sharing contracts do not coordinate a supply chain and develop a variation of revenue sharing that performs better. Taylor and Xiao (2006) investigate whether returns and rebates contracts can induce salespeople to invest on forecasting market demand. Surprisingly, they show that returns contracts (which intuitively provide insurance rather than lottery) intensify the salesperson’s incentives. Many researchers in economics have also investigated the possibility of extracting more rent by utilizing publicly observable signals that are correlated to an agent’s private information. The relevant signal can be either the ex-post public information, or the reports by other agents when their types are correlated. Mirrlees (1974) and Myerson (1981) independently illustrate this idea via specific settings/examples. The two seminal papers (Cremer and McLean (1985, 1988)), where they study auctions, formally identify necessary and sufficient conditions for full surplus extraction for all instances of agents’ utilities. McAfee and Reny (1992) extend the discussion to incorporate continuous uni-dimensional type spaces. Extensions of Cremer and McLean (1988)’s idea are dispersed in various environments. Mezzetti (2005) considers an interdependent-value setting (i.e., agents’ true valuations depend on other agents’ private information). Due to the interdependence of valuations, the payoffs are correlated. Hence, a two-stage mechanism that requires agents to report their types as well as their payoffs can be adopted to achieve the full surplus extraction. Obara (2006) allows the agents to exert effort that affects the probability distribution over types. He shows that conditions similar to Cremer and McLean (1988) continue to be valid in environment with moral hazard followed by adverse selection. In particular, it goes through for a special case when agents can spend effort in learning true states, namely, the “endogenous adverse selection” problem. Johnson et al. (1990) investigate


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whether it can be achieved among a group of agents whose actions generate externality for others. Riordan and Sappington (1988) study the possibility for contract design problems. The debate on whether full surplus extraction is possible continues to date. Robert (1991) shows that full surplus extraction is prone to risk aversion and limited liability. He establishes that when the agents’ types are almost independent, the principal must leave information rent for agents. Laffont and Martimort (2000) demonstrate that full surplus extraction may fail due to the collusive coalition of users in the context of public goods provision. Brusco (1998) argues that multiplicity of equilibria in Cremer and McLean (1988) may lead to implementation problems, and provides examples where there always exists another equilibrium that Pareto dominates the one with full surplus extraction from the agents’ perspective. He proceeds to propose a two-stage sequential mechanism to ensure the uniqueness of the dominant strategy equilibrium. Heifetz and Neeman (2006) uses a measure-theoretic argument to show that in the universal type space of common priors, full surplus extraction is unlikely to be achieved. Bose and Zhao (2006) argue that in some scenarios Cremer and McLean’s conditions do not hold (e.g., when there are fewer public signals than agents’ types). They solve some special cases where full surplus extraction cannot be achieved. To our knowledge, our paper is the first attempt to study the possibility of full surplus extraction in scenarios in which adverse selection is followed by moral hazard. We adopt risk neutrality of agents and do not consider the limited liability following the convention of this literature. With these two characteristics, our paper appears to be a fairly general framework to investigate the optimal contract design when agents’ types are correlated. We show that the implementation of full surplus extraction is surprisingly simple, and the information revealed from the sales outcomes and other agents’ reports can be used separately even when full surplus extraction fails.

3.

The Model

We consider a supply chain with one manufacturer and multiple dealers (indexed by k ∈ K). Each dealer faces a geographically dispersed market with uncertain demand, and is able to observe a demand signal and exert effort to further enhance sales in her local market. Each dealer receives a signal that conveys (noisy) information of her local demand. For ease of notation, we assume that dealers are symmetric, but our analysis carries over to the asymmetric case immediately; the reason will become clear when we study the manufacturer’s problem. Moreover, we discretize the state space since the discrete formulation shows the clear structure of the contract design problem. Throughout this paper, we refer to the dealer as the feminine pronoun “she” and the manufacturer as the masculine one “he” to avoid confusion.


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In each local market, the demand signal can take value from a finite state space N with associated probabilities {Pn , n ∈ N }. This signal is privately observed by the corresponding dealer before she signs the contract with the manufacturer. Thus, this private information also represents the dealer’s type. After receiving the signal n ∈ N , the dealer can choose a costly effort j ∈ J to further enhance her sales. Since the effectiveness of the sales effort depends on the potential market demand, we assume that the cost of effort is signal-dependent and denote it by {enj , n ∈ N, j ∈ J }. Neither the signal nor the effort can be observed by the manufacturer. The sales outcome of a local market, denoted by m, can take values from a finite set M. The probability distribution of sales can depend on both the demand signal and the dealer’s effort. Suppose that the dealer receives a signal n ∈ N and exerts effort level j ∈ J, then the states of sales in her local market occur with (conditional) probabilities {qnjm , m ∈ M }, respectively, and P qnjm = 1.1 We define Qnj = (qnjm )m∈M as the row vector of conditional probabilities. m∈M

Since potential demands of these local markets are commonly affected by changes of economic conditions or marketing factors, we assume that the signals received by these dealers are correlated. We model this correlation by conditional probability vectors: Given that dealer 1 receives signal n ∈ N, she determines the conditional probabilities {φni }, where i ∈ N |K|−1 denotes the signal profile received by other dealers, |K | is the cardinality (the number of elements) of set K, and N |K|−1 is the product space of |K | − 1 identical sets N . By symmetry this defines the conditional probabilities from other dealers’ perspective as well. We further define Φn ≡ (φni )i∈N |K|−1 , n ∈ N.2 The manufacturer’s goal is to design an appropriate compensation payment structure for each dealer such that the dealer chooses the optimal sales effort and the manufacturer is able to extract the maximum surplus. Such surplus extraction may be practically possible when the manufacturer has market power or the dealers sell the manufacturer’s products exclusively. Given a signal n, we denote j = jn as the effort level that maximizes the expected sales: Assumption 1. ∀n ∈ N,

P m∈M

qnjn m m >

P m∈M

qnjm m, and enjn > enj , j ∈ J \ {jn }.

Note that enjn > enj rules out dominated effort levels that result in lower expected sales with a higher cost. We assume that the manufacturer wants to implement {jn , n ∈ N }. 1

The probability distribution {qnjm } over the sales in a local market is unaffected by the signals received by other dealers as well as the effort exerted by other dealers. Thus, effort exertion has no spill-over effect across markets. Another observation is that different market signals and effort levels can result in the same sales albeit with different probabilities. Therefore, the manufacturer has no way of distinguishing the states based on sales outcomes. 2

If signals are independent, these conditional probability vectors will be identical, i.e., Φn = Φs , n, s ∈ N .


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With a single dealer the game proceeds as follows: The dealer observes the market demand signal prior to the announcement of payment structure. The manufacturer then announces the salescontingent payments {Πnm , m ∈ M } for the dealer to self-select. If a dealer selects the payment schedule Πn = (Πn1 , ..., Πn|M | ), she will receive the amount Πnm from the manufacturer when the sales is m ∈ M (which is observable to both contracting parties). Using the extended revelation principle (Laffont and Martimort 2002, Chapter 7), we restrict our attention to contracts wherein the manufacturer requests the dealer to report her signal (since our model involves hidden action as well). Thus, the subscript of Πn refers to the dealer’s report. Given this contract, if a type-n dealer reports the signal truthfully and exerts optimal effort, her P net payoff will be qnjn m Πnm − enjn , where the first component is the expected payment she m∈M

receives from the manufacturer, and enjn is her cost of exerting effort. Nevertheless, if she falsely P reports her type to be s ∈ N and exerts effort level j ∈ J, her net payoff becomes qnjm Πsm − enj . m∈M

When there are |K | ≥ 2 dealers, we assume that the manufacturer announces the payments {Πknmi } that are contingent on dealers’ identities, sales outcomes, and dealers’ reports. Each dealer

then reports her signal and decides her effort level. The sales are realized and the manufacturer awards Πknmi to the dealer if the report profile from other dealers is i ∈ N |K|−1 . The contingency on other dealers’ reports affords the manufacturer greater flexibility in designing compensation schemes, and hence it may further reduce the dealers’ surplus. We focus on pure-strategy equilibria throughout this paper, and adopt Bayesian Nash equilibrium as the solution concept. A Bayesian Nash equilibrium requires that ∀k ∈ K, each type of dealerk’s strategy is her best response given that all other “players” follow the equilibrium behavior (Fudenberg and Tirole (1994)). For ease of notation, we let XT denotes the transpose of arbitrary matrix X, and vectors and matrices are presented in bold fonts. 3.1. Discussion Our payment schemes {Πnm } and {Πknmi } are the most general form of dealers’ compensation, since the manufacturer uses all the available information. In reality firms may adopt simpler mechanisms that impose restrictions on these payments. For example, if the manufacturer cannot penalize dealers, then payments have to be nonnegative (i.e., Πknmi ≥ 0). A popular compensation scheme is the linear contract, in which each dealer is awarded a fixed unit payment depending on her sales outcome (i.e., Πknmi = akn m + bkn , where akn ≥ 0). We may also present the piecewise (stair-step) incentive: Suppose the threshold is T , then Πknmi = akn (m − T )+ + bkn 1{m ≥ T }, where akn ≥ 0 is the per unit payment, bkn ≥ 0 is the additional bonus, (m − T )+ = max(m − T, 0), and 1{m ≥ T } is the indicator function that the sales exceed the threshold. Under Gonik’s scheme, if a forecast Fnk is made, then the dealer is paid Πknmi = ckn m + dkn − akn (m − Fnk )+ − bkn (Fnk − m)+ .


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Note that our payment scheme does not cover sales contests, which includes all sales outcomes across different local markets in each dealer’s compensation. Our model mainly elaborates the possibility of using information from other dealers’ reports. As long as other dealers report (truthfully) their demand signals to the manufacturer, the information asymmetry between other dealers and the manufacturer regarding a specific dealer’s true type is completely eliminated. Hence, including realized sales in other markets does not provide extra information content in this model.3

4.

Base model

In this section, we present a version of the general model, which we have labelled as the base model. It illustrates the key concepts without additional complications due to state-contingent payoffs. The base model has two dealers, two types of demand signals, two effort levels, and three deterministic sales outcomes. We label the signals and effort levels as N = {H, L}, J = {1, 2}. The cost of effort does not depend on signal, i.e., enj = ej , n ∈ N, j ∈ {1, 2}, and e1 > e2 . When a dealer receives a high signal and exerts high effort, the realized sales is mH . When a low signal occurs and low effort is exerted, the sales outcome mX is very low and will be identified. Nevertheless, outcome mL occurs when either type-H dealer exerts low effort or type-L dealer exerts high effort. This stylized setting captures the problem of designing dealer’s compensation when both adverse selection and moral hazard are present.4 4.1.

Single-dealer case

Let us consider the simple supply chain when there is one manufacturer and one dealer. Since the sales outcome is deterministic, the manufacturer expects to see total sales of mH and mL corresponding to these demand signals. Moreover, with deterministic outcomes, it suffices to make the payments contingent on the sales outcomes, i.e., ΠH , ΠL , and ΠX . With some abuse of notation, the subscript denotes the label of sales outcome.5 To induce the high effort, the manufacturer solves the following problem: max PH (mH − ΠH ) + PL (mL − ΠL ) 3

Sales contests are proposed when demands of different markets are subject to common random shock, and hence they can amplify competition among salespeople to induce effort when market conditions are significantly positively correlated. They are more appropriate for other scenarios where rank orders are relatively easy to obtain but subjective performance measure is hard to justify in the court, or when the firms would like to avoid reneging by predetermining the prize structure (Prendergast (1999)). 4

In our notation, M = {mH , mL , mX }, and the probability distributions are respectively QH1 = [1, 0, 0], QH2 = QL1 = [0, 1, 0], and QL2 = [0, 0, 1]. 5

To see this, suppose that a type-H dealer reports H but exerts low effort, then the realized outcome is mL . The manufacturer knows immediately that either the report is false or low effort was exerted, and can severely punish the dealer to avoid such a deviation. Similarly, if she reports L but exerts high effort, the sales outcome mH is also inconsistent with her report. Likewise, a type-L dealer never reports H since the expected outcome mH cannot be achieved regardless of her effort.


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s.t. Πn − e1 ≥ 0, n = H, L,

(1)

ΠH − e1 ≥ ΠL − e2 ,

(2)

ΠH , ΠL unrestricted, where Eq. (1) guarantees that the dealer is willing to participate (the individual rationality constraints), and Eq. (2) ensures that type-H dealer does not deviate to exert low effort. The manufacturer can make ΠX a huge penalty to prevent type-L dealers from choosing low effort. The solution to this optimization problem is ΠL = e1 , ΠH = 2e1 − e2 . Hence, the manufacturer must pay an extra amount PH (e1 − e2 ) to the dealer. Therefore he is unable to extract all surplus. This is because the information disclosed from the realized sales is not sufficient for the manufacturer to infer the demand signal and the dealer’s effort. 4.2.

Two-dealer case

Now consider the two-dealer case and assume that they are symmetric. Let Pni be the joint probability that dealer 1 receives signal n ∈ N and dealer 2 receives signal i ∈ {H, L}. A similar argument shows that the manufacturer can restrict his attention to payments that are independent of the dealer’s own report. However, we assume that the manufacturer still asks for their reports, since a dealer’s report is useful in knowing the other dealer’s true signal. Let {Πkmi } denote the payment received by dealers. Since a dealer’s compensation is independent of her own report, she has no incentive to lie regarding the signal. Under a Bayesian Nash equilibrium, the manufacturer solves the following optimization problem: min PHH (Θ1H + Θ2H ) + PHL (Θ1H + Θ2L ) + PLH (Θ1L + Θ2H ) + PLL (Θ1L + Θ2L ) s.t. φHH ΠkHH + φHL ΠkHL − e1 ≥ φHH ΠkLH + φHL ΠkLL − e2 , k = 1, 2, φHH ΠkHH + φHL ΠkHL − e1 − ΘkH = 0, k = 1, 2,

(3) (4)

φLH ΠkLH + φLL ΠkLL − e1 − ΘkL = 0, k = 1, 2, Θ1H , Θ1L , Θ2H , Θ2L ≥ 0, Πkni unrestricted, k = 1, 2, n, i = H, L, where φHi =

PHi ,n, i PHH +PHL

= H, L, and likewise for {φLi }. The decision variables ΘkH and ΘkL rep-

resent the excess payments to dealer k when the dealer’s type is H or L respectively. Eq. (3) guarantees that each player is willing to exert high effort. Eq. (4) is the individual rationality (IR) constraint. This, along with the nonnegativity of ΘkH and ΘkL , ensures that all players are willing to participate. If at optimality the objective function is 0, the manufacturer extracts full surplus. Note that the objective can be rearranged as (PH1 Θ1H + PL1 Θ1L ) + (PH2 Θ2H + PL2 Θ2L ), where Pn1 = PnH + PnL , and Pn2 = PHn + PLn ,n ∈ N , and there are no linking constraints (such as budget balance constraints). Thus, this problem is decomposable with respect to dealers’ identities, and we can


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focus on the optimization problem facing an individual dealer. We suppress the index of the dealer’s identity in the sequel. The manufacturer’s optimization problem facing dealer 1 then becomes: min PH ΘH + PL ΘL s.t.

(5)

φHH ΠHH + φHL ΠHL − e1 ≥ φHH ΠLH + φHL ΠLL − e2 , φnH ΠnH + φnL ΠnL − e1 − Θn = 0, n ∈ N,

(Y )

(wn )

(6) (7)

ΘH , ΘL ≥ 0, Πni unrestricted, n, i = H, L, where inside the parentheses are corresponding dual variables. From Eq. (7), we can replace the left-hand side of Eq. (6) by ΘH , which then becomes φHH ΠLH + φHL ΠLL − ΘH ≤ e2 . The dual program of Eq. (5) is max (wH + wL )e1 + Y e2 s.t. φHi wH = 0, i = H, L,

(8)

φnH Y + φnH wL = 0, n = H, L,

(9)

−(wH + Y ) ≤ PH , −wL ≤ PL ,

Y ≤ 0, wH , wL unrestricted. Eq. (8) implies that wH = 0. Eq. (9) leads to Y

¡φ

HH

φHL

¢

= −wL

¡φ

LH

φLL

¢ . If φHH 6= φLH , then Y = wL = 0,

in which case the objective value is zero. Thus, full surplus extraction can be achieved. When φHH 6= φLH , full surplus extraction can be implemented by the following compensation 2 scheme: Πmi = am+ Πi , where a = max( meH1 −e , e1 −e2 ), ΠH = −mL mL −mX

a(φLH mH −φHH mL ) φHH −φLH

a[(1−φHH )mL −(1−φLH )mH ] φHH −φLH

+ e1 , ΠL =

+ e1 . To verify that this compensation achieves full surplus extraction, let us first

examine whether type-H dealer is willing to exert high effort. Note that {Πi } is independent of her decision and hence can be excluded in the comparison. If she exerts high effort, her payoff −mL is amH − e1 = a(mH − mL ) + e2 − e1 + amL − e2 = −(e1 − e2 )[1 − a meH1 −e ] + amL − e2 . Since 2

a≥

e1 −e2 , mH −mL

−mL 1 − a meH1 −e ≤ 0 and we have amH − e1 ≥ amL − e2 . The last term corresponds to 2

the payoff when she exerts low effort, and therefore she is willing to exert high effort. Given this compensation scheme, a type-H dealer’s expected payoff is amH − e1 + φHH ΠH + φHL ΠL = mH −φHH mL ) L −(1−φLH )mH ] amH + φHH a[(1−φHHφ)m + (1 − φHH ) a(φLHφHH = 0. Likewise, the expected payoff −φLH HH −φLH

of type-L dealer is zero after some algebra. This implies that no excess payment is needed and dealers are willing to participate. We now show that by using linear contracts alone the manufacturer may leave huge excess payment for dealers. Suppose that Πknmi = akn m + ckn , n = H, L. Note that we further allow the


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payment to be contingent on the dealer’s own report. Since the compensation schemes for these two dealers are completely irrelevant in this case and they are symmetric, we suppress the identity index k and focus on dealer 1. From the definition of ΘL , we have cL = ΘL − aL mL + e1 . Moreover, since type-L dealer exerts high effort in equilibrium, we have aL mL + cL − e1 ≥ aL mX + cL − e2 , which implies aL ≥

e1 −e2 mL −mX

. Also, type-H dealer cannot benefit from pretending to be type-L, and

therefore ΘH = aH mH + cH − e1 ≥ aL mH + cL − e1 . Plugging in cL = ΘL − aL mL + e1 , we obtain ΘH ≥ aL mH +ΘL − aL mL +e1 − e1 = ΘL +aL (mH − mL ) ≥ ΘL +

mH − mL mH − mL (e1 − e2 ) ≥ (e1 − e2 ), mL − mX mL − mX

where the last inequality follows from ΘL ≥ 0. Therefore, the excess payment left for dealers is mH −mL lower bounded by PH ΘH + PL ΘL ≥ PH ΘH ≥ PH m (e1 − e2 ). If mL − mX → 0+ , the excess L −mX

payment approaches infinity. This happens when the dealer’s effort does not enhance sales much when demand signal is low.

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From the base model, we make the following observations: 1. Only one type of dealer receives rent. 2. Knowledge of sales outcomes may be used to prevent dealers from misrepresenting their signal and effort. 3. The manufacturer can extract full surplus when signals are sufficiently informative. 4. When full surplus extraction is possible, it can be achieved by simple contracts. 5. In the optimal compensation, the manufacturer uses the information revealed from sales outcomes and signal correlation separately. For the two-signal case, it can be shown that all these observations continue to hold, regardless of the number of effort levels, sales outcomes, and dealers.7 In the next section, we investigate whether these findings go through in more complicated scenarios with arbitrary numbers of signals, effort levels, sales outcomes, and dealers.

5.

Equilibrium analysis of general model

In this section, we return to the model described in Section 3 and investigate the generality of the above results. Unlike the base model, since any combination of demand signal and effort level leads to a probability distribution over sales outcomes, observing sales is not sufficient for identifying the signal. We first study the single-dealer problem and show that full surplus extraction may be achieved. We then investigate the contract design problem in the multi-dealer case. 6

The intuition is as follows. When the effort is not effective given a low signal, the manufacturer has to give away a large share of proceeds in order to induce the dealer to exert optimal effort when the signal is low. However, this large share also makes it attractive for the type-H dealer to pretend to be type-L. The manufacturer needs to compensate type-H dealer so that she will not deviate. 7

Details are provided in the online appendix: http://www.stern.nyu.edu/˜ychen0/css-appendix.pdf.


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5.1.

Single-dealer case

Recall that if the dealer reports her type to be s ∈ N and exerts effort level j ∈ J, her net payoff is P qnjm Πsm − enj . In equilibrium, truth-telling has to be the dealer’s best response, which leads m∈M

to the following truth-telling (TT) constraints: X m∈M X

qnjn m Πnm − enjn ≥ qnjn m Πnm − enjn ≥

m∈M

X m∈M X

qnjm Πnm − enj , j ∈ J \ {jn }, n ∈ N, qnjm Πsm − enj , j ∈ J, s 6= n, n, s ∈ N,

m∈M

where the first inequality ensures that when reporting the true signal n, the dealer is willing to exert optimal effort. The second inequality guarantees that the dealer cannot benefit from misreporting the signal, regardless of the effort level she chooses. These constraints are also known as the incentive compatibility constraints. In addition, the manufacturer also has to induce the dealer to participate, and therefore the contracts have to satisfy the dealer’s individual rationality: P qnjn m Πnm − enjn ≥ 0, n ∈ N. m∈M

The manufacturer’s optimization problem is as follows: X

min s.t. TT :

Pn Θn

n∈N X m∈M X

IR :

m∈M X

qnjn m Πnm − enjn ≥ qnjn m Πnm − enjn ≥

X m∈M X

qnjm Πnm − enj , j ∈ J \ {jn }, n ∈ N, qnjm Πsm − enj , j ∈ J, s 6= n, n, s ∈ N,

m∈M

qnjn m Πnm − enjn − Θn = 0, n ∈ N,

(10)

m∈M

Θn ≥ 0, Πnm unrestricted, n ∈ N, ∀m ∈ M. As defined in the base model, Θn is the excess payment made to type-n dealer. Eq. (10) and the nonnegativity of {Θn } guarantee that the dealers are willing to participate. From Eq. (10), we can replace the left-hand sides of those truth-telling constraints by {Θn }’s respectively. The manufacturer’s optimization problem then becomes (P − S) min s.t.

X

Pn Θn

n∈N X m∈M X m∈M X

qnjm Πnm − Θn ≤ enj , j ∈ J \ {jn }, n ∈ N, qnjm Πsm − Θn ≤ enj , j ∈ J,

(Ynsj )

qnjn m Πnm − Θn = enjn , n ∈ N,

(wn )

(βnj )

(11) (12) (13)

m∈M

Θn ≥ 0, Πnm unrestricted, n ∈ N, ∀m ∈ M, where the variables in parentheses adjacent to the constraints are the corresponding dual variables.


13

We first find that the above problem is always feasible. Moreover, truth-telling can always be implemented by a simple linear contract. Theorem 1. Suppose Assumption 1 holds. In the single-dealer problem, the following linear e

−e

njn nj P contract is feasible to (P-S): Πnm = am + c, where a ≥ maxn∈N maxj∈J\{jn } P qnj m qnjm m n m− m∈M m∈M P and c ≥ maxn∈N (enjn − a qnjn m m).

m∈M

The above theorem implies that the optimal effort level can always be implemented if the dealer’s compensation increases significantly with the sales outcome. This is a revenue-sharing scheme where a is the share of sales proceeds awarded to the dealer, and c is the fixed payment she receives when accepting the contract. The value of a has to be large enough so that the dealer is willing to exert the optimal effort, which corresponds to the idea of a “high-powered” compensation (see Laffont and Martimort (2002)). The fixed payment has to overcome the dealer’s reservation value. Therefore, the feasibility is always guaranteed.8 We next investigate the possibility of full surplus extraction, in which case the manufacturer is able to find payments {Πnm } such that Θn = 0, n ∈ N. Definition 1. A compensation scheme {Πnm } is said to achieve full surplus extraction in the single-dealer problem if {Θn = 0, n ∈ N } in (P-S). In other words, the dealer receives no surplus regardless of her signal. To this end, we find it convenient to work with the dual program of this optimization problem, as presented below. (D − S) max

X

wn enjn +

n∈N

s.t.

X X

XX n∈N j∈J

Ysnj qsjm +

s∈N \{n} j∈J

−(

X

X

Ynsj +

X

X

βnj enj

n∈N j∈J\{jn }

s∈N \{n} X βnj qnjm + wn qnjn m = 0, ∀m, n,

(14)

j∈J\{jn }

X X

βnj +

j∈J\{jn }

enj

Ynsj + wn ) ≤ Pn , n ∈ N,

(15)

s∈N \{n} j∈J

βnj ≤ 0, j ∈ J \ {jn }, Ynsj ≤ 0, wn unrestricted, j ∈ J, n, s ∈ N, where Eq. (14) corresponds to the coefficients of Πnm , and Eq. (15) follows from the coefficients of Θn . In the dual program, we can further combine those {qnjm }’s as column vectors {QTnj } and represent the dual program as max

X n∈N

s.t.

8

wn enjn +

XX

enj

X

Ynsj +

s∈N \{n} X X n∈N j∈J X T Ysnj Qsj + βnj QTnj j∈J\{jn } s∈N \{n} j∈J

X

X

βnj enj

n∈N j∈J\{jn } + wn QTnjn = 0, n

∈ N,

(16) (17)

We can assume that the manufacturer has an alternate objective function and derive all the results in this paper if we additionally assume that the single-dealer problem is feasible.


14

−(

X

βnj +

j∈J\{jn }

X X


s∈N \{n} j∈J

βnj ≤ 0, j ∈ J \ {jn }, Ynsj ≤ 0, wn unrestricted, j ∈ J, n, s ∈ N.

(18)

We next give the necessary and sufficient condition for the feasibility of this problem. Define Q = [Q1 , ..., Q|N | ]T as the collection of probability vectors Qn = [Qn1 , ..., Qn|J| ]T , ∀n ∈ N . Moreover, we let E ≡ [ET1 , ..., ET|N | ]T denote the column vectors of costs of effort, where En = [en1 , ..., en|J| ]T . Theorem 2. In the single-dealer case, full surplus extraction is feasible if and only if for each n ∈ N , the following system is infeasible: ¶ ¶ µ µ Q E T T > 0, yn ≤ 0, = 0, yn yn −Qnjn −enjn

(19)

where yn is an (|N | × |J | + 1)-element row vector. The proof applies theorems of the alternative. From the above theorem, we can provide sufficient conditions for full surplus extraction. Corollary 1. In the single-dealer case, full surplus extraction is achievable if: 1. For each n ∈ N , there exists no nonnegative solution {λnsj } to QTnjn = P P n T λsj Qsj .

P

j∈J\{jn }

λnnj QTnj +

s∈N \{n} j∈J

1a. The above condition is automatically true when Q has full row rank (|N | × |J |). 2. E ∈ span (Q). When sales outcomes are probabilistic, even in the single-dealer case the manufacturer may be able to extract full surplus from the dealer. This is because the probability matrix Q may convey enough information regarding the dealer’s private signal and her effort level, as predicted by Condition 1. This condition fails in the base model since the same sales outcome appears when a type-H dealer exerts low effort or when a type-L dealer exerts high effort. Full surplus extraction may also be achieved when the costs of effort belong to the span of the column of Q (as demonstrated in Condition 2). In this case, the optimal compensation can be obtained by letting Θn = 0, n ∈ N, and making all truth-telling constraints binding in (P-S), i.e., Eqs. (11) and (12) are satisfied as equalities.9 5.2. Multi-dealer case Now suppose that there are |K | ≥ 2 dealers. The contract proposed by the manufacturer again has to induce truth-telling and satisfy the dealers’ individual rationality. We first set up the manufacturer’s optimization problem, and then provide sufficient conditions for full surplus extraction and how the manufacturer can implement the payment scheme. In the end we discuss the general case. 9

In other words, all dealers are indifferent between truth-telling and any other combination of report and effort. Therefore, under this compensation scheme they are willing to reveal the information.


5.2.1.

15

The manufacturer’s problem. Assume that all other dealers follow their equilibrium

strategies. Suppose a type-n dealer reports that she receives a signal s, and exerts effort level j, P then her expected payoff will be qnjm φni Πsmi − enj . Note that the only difference from m∈M,i∈N |K|−1

the single-dealer case is that now the payments also depend on other dealers’ reports. Thus, in equilibrium the truth-telling and individual rationality constraints are respectively X

qnjn m φni Πnmi − enjn ≥

m∈M,i∈N |K|−1

X

qnjm φni Πnmi − enj , j ∈ J \ {jn }, n ∈ N,

m∈M,i∈N |K|−1

qnjn m φni Πnmi − enjn ≥

m∈M,i∈N |K|−1

X

X X

qnjm φni Πsmi − enj , ∀j ∈ J, s 6= n, n, s ∈ N,

m∈M,i∈N |K|−1

qnjn m φni Πnmi − enjn ≥ 0, n ∈ N.

m∈M,i∈N |K|−1

We now consider the manufacturer’s optimization problem. Following the argument similar to that in Section 4, the problem is decomposable with respect to dealers’ identities since there is no linking constraints across dealers.10 Therefore, we only present the manufacturer’s optimization problem for an individual dealer, say dealer 1: min s.t. TT :

X

Pn Θn n∈N X

qnjn m φni Πnmi − enjn ≥ qnjn m φni Πnmi − enjn ≥

X

X

qnjm φni Πsmi − enj , j ∈ J, s 6= n, n, s ∈ N,

m∈M,i∈N |K|−1

m∈M,i∈N |K|−1

IR:

qnjm φni Πnmi − enj , j ∈ J \ {jn }, n ∈ N,

m∈M,i∈N |K|−1

m∈M,i∈N |K|−1

X

X

qnjn m φni Πnmi − enjn − Θn = 0, n ∈ N,

(20)

m∈M,i∈N |K|−1

Θn ≥ 0, Πnmi unrestricted, n ∈ N, ∀m ∈ M, ∀i ∈ N |K|−1 , where Eq. (20) and the nonnegativity of {Θn }’s guarantee that the dealers are willing to participate. From Eq. (20), we can replace the left-hand sides of (TT) conditions by {Θn }’s. The optimization problem then becomes (P − NE) min

X n∈N

s.t.

Pn Θn X

qnjm φni Πnmi − Θn ≤ enj , j ∈ J \ {jn }, n ∈ N,

(βnj )

m∈M,i∈N |K|−1

X

qnjm φni Πsmi − Θn ≤ enj , j ∈ J, s 6= n, n, s ∈ N,

m∈M,i∈N |K|−1

X

qnjn m φni Πnmi − Θn = enjn , n ∈ N,

(wn )

m∈M,i∈N |K|−1

Θn ≥ 0, Πnmi unrestricted, n ∈ N, ∀m ∈ M, ∀i ∈ N |K|−1 , 10

This suggests that our analysis applies to scenarios with asymmetric dealers as well.

(Ynsj )


16

where {βnj }’s, {Ynj }’s, and {wn }’s are the corresponding dual variables.11 We can again define the case for “full surplus extraction”. Definition 2. A compensation scheme {Πnmi } is said to achieve full surplus extraction in the multi-dealer problem if {Θn ≥ 0, n ∈ N } in (P-NE). From (P-NE), we can write down its dual program as follows. (D − NE) max

X

wn enjn +

n∈N

X X

s.t.

XX n∈N j∈J

X

j∈J\{jn }

X

βnj +

Ynsj +

s∈N \{n}

X

Ysnj qsjm φsi +

s∈N \{n} j∈J

−(

enj

X

X

βnj enj

n∈N j∈J\{jn }

βnj qnjm φni + wn qnjn m φni = 0,

(21)

j∈J\{jn }

X X

∀m ∈ M, ∀i ∈ N |K|−1 , ∀n ∈ N,


(22)

s∈N \{n} j∈J

βnj ≤ 0, j ∈ J \ {jn }, Ynsj ≤ 0, wn unrestricted, j ∈ J, n, s ∈ N.

(23)

This dual program provides structural properties of the optimal solution. It allows us to characterize sufficient conditions for full surplus extraction and how the manufacturer implements it. It also provides a systematic procedure to solve the compensation design problem for general cases when full surplus extraction may not be achievable. These issues are described next. 5.2.2.

Full surplus extraction using conditions on {Φn }. We next discuss several special

cases that lead to full surplus extraction and its implementation. We find that if each Φn cannot be represented as a convex combination of other Φn ’s, then full surplus extraction can be achieved. Proposition 1. Suppose Assumption 1 holds. In the multi-dealer game, full surplus extraction P is achievable if ∀n ∈ N, there exists no nonnegative solution {ξsn } to ΦTn = ξsn ΦTs . s∈N \{n}

The condition on {Φn } formalizes the idea that other dealers’ signals are sufficiently informative. P If we take conv(Φ) as the convex hull of {Φn } (conv(Φ) = {x : ∃{ξn ≥ 0}, ξn = 1, s.t. x = n∈N P ξn Φn } ), this condition means that each vector is an extreme point of this set.12 This condition n∈N

coincides with the one in the seminal work by Cremer and McLean (1988) done in the context of auction design. The condition does not imply demand signals are either positively or negatively correlated. Informativeness of signals is based on how distinguishable they are rather than whether they form a specific pattern. We next show that the implementation of full surplus extraction is surprisingly simple. 11

This is similar to (P-S) in the single-dealer problem. It has the same number of constraints (and hence the same dual variables). The only difference is that now the payments could depend on other dealers’ reports. 12

Note that there are |N | such vectors and each of them has |N | × (|K| − 1) elements, and therefore this condition is quite likely to hold. Suppose that a convex combination exists. Any slight perturbation of this system breaks down the combination and hence this condition is not very restrictive.


17

Theorem 3. In the multi-dealer problem, if the conditions in Proposition 1 hold, then the manufacturer can extract full surplus by using the following compensation: Πnmi = an m + cn + Πni , P enjn −enj P where an ≥ maxj6=jn P qnj m , cn = −an qnjn m m + enjn , and the lottery Πni satisqnjm m n m− m∈M m∈M P Pm∈M fies φni Πni = 0 and φsi Πni is sufficiently small, ∀s ∈ N \ {n}. i∈N |K|−1

i∈N |K|−1

The contract is composed of three parts: a fixed payment, the linear contract based on sales outcomes, and a lottery that depends on other dealers’ reports. The share an of sales proceeds has to be large enough to align the dealer’s incentive of choosing optimal effort, and the fixed compensation cn ensures that the dealer does not receive any surplus. The lottery Πni depends only on other dealers’ reports, and it yields zero expected payment if the dealer reports her signal truthfully. However, this lottery imposes a severe penalty on those dealers who attempt to pretend as other types (this construction is also used in Bose and Zhao (2006)). This lottery rules out the possibility of misreporting the signals, and therefore the sales-contingent payment can be designed taking into account only the type-n dealer’s effort decision. It is worth noting that in the singledealer problem, the linear contract may not be incentive compatible. Thus, when more dealers are involved and the manufacturer utilizes the information provided from other dealers, the contract may actually be significantly simplified. The “reports” correspond to dealers’ sales forecasts of their local markets. Therefore, under the optimal scheme, a dealer’s compensation is determined by three elements: (1) her own forecast, (2) the sales outcome of her local market, and (3) other dealers’ forecasts.13 The first two elements are consistent with what Chen (2005) and Gonik (1978) propose: They show that sales forecasts can be used as a powerful tool to elicit dealers’ demand information. The compensation scheme that involves sales forecasts has been adopted in practice, e.g., by IBM (Chen (2005)). Under our optimal scheme, the sales-contingent payment is a linear contract, which is a special (degenerate) case of stair-step incentive or Gonik’s scheme (see Section 3). The idea of using other dealers’ forecasts to further extract dealer’s private information is a new and important addition to the existing literature. Our result shows that other dealers’ forecasts can allow the manufacturer to reduce the excess payment for dealers.14 13

The implementation of this scheme should not be an issue, since in practice the manufacturer collects his sales forecasts. To implement the optimal compensation scheme, he simply has to put these forecasts all together and modify the payment schemes accordingly. 14

One way to to implement this compensation scheme would be to maintain an escrow account for each dealer. Additionally, each dealer is provided with a computational tool that computes her remaining escrow balance. Over several periods the dealer will observe that the escrow balances goes to zero if she truthfully reports her type; however, it may become negative if she chooses to lie. This ensures that dealers are willing to report truthfully.


18

Moreover, if we select an = maxj6=jn

P m∈M

enjn −enj P , qnjn m m− qnjm m

and its corresponding fixed payment

m∈M

cn , the manufacturer uses the minimum coefficients among linear contracts that induce truthtelling. Thus, such a contract minimizes the dependence on the sales outcomes. Corollary 2. For any Πnmi = an m + cn + Πni , an = maxj6=jn

P m∈M

enjn −enj P qnjn m m− qnjm m

is the con-

m∈M

tract with minimum share of sales proceeds that induces truth-telling from dealers. A special case of the condition in Proposition 1 is when {Φn } has full row rank, i.e., they are linearly independent. This is because no nontrivial linear combination exists for {Φn }. Proposition 2. Suppose Assumption 1 holds. In the multi-dealer problem, if {Φn , n ∈ N } has full rank, full surplus extraction can be achieved. The full rank condition is also known as the spanning condition (e.g., Cremer and McLean (1988)). It is more likely to hold when more dealers are involved, because the state space of i expands (N |K|−1 ) but that of n remains unchanged (N ). According to Cremer and McLean (1988), this condition holds in “nearly all” economic environments when the number of types is fixed. In this case, Proposition 1, Theorem 3, and Corollary 2 continue to hold. However, the manufacturer can implement full surplus extraction by a further simplified scheme where a dealer’s payment does not depend on her own report. Theorem 4. Suppose Assumption 1 holds and {Φn , n ∈ N } has full rank in the multi-dealer problem. The manufacturer can extract full surplus by using the following compensation: Πnmi = P enjn −enj P , and { Π φni Πi = am + Πi , where a ≥ maxn∈N maxj6=jn P qnj m i } solves q m m− njm n |K|−1 m∈M m∈M i∈N P −a qnjn m m + enjn , n ∈ N . m∈M

Since Πnmi = am + Πi does not depend on a dealer’s own report, each dealer has no incentive to misreport a wrong signal. Moreover, the manufacturer can simply offer a single contract, as opposed to a menu of contracts for dealers to self select (e.g., Πnmi = an m + cn + Πni ). Given this payment scheme, the dealer has only two decisions to make: (1) whether to accept the compensation or not; (2) if she accepts, which effort level to choose. The expected payment from Πi is independent of a dealer’s effort decision. Thus, when the dealer gets a sufficient share of the sales proceeds (a large enough), she is willing to choose the optimal effort even if the manufacturer cannot observe her action. The choice of {Πi } guarantees that each dealer is willing to participate and receives no information rent. Full rank condition on {Φn } gives the manufacturer enough flexibility to construct such a lottery.


19

The exact choice of {Πi } can be obtained from, e.g., Gaussian elimination. In fact, the manufacturer simply needs to make the payments contingent on any minimum spanning subset of i for {Φn }, because arbitrary such subset of N |K|−1 has full rank. Thus, at most |N | of them are needed to construct such lotteries by Cramer’s rule.15 Therefore, the contract design problem may be surprisingly simple when there are many dealers. We now show that in some cases using linear contracts alone (i.e., {Πnmi = an m + cn , n ∈ N }) may be extremely costly for the manufacturer. 6 jn such Theorem 5. Suppose that: (1) There exist two types n, s and an effort level j = P P that qnjm m > qsjs m m and enj ≤ esjs . (2) There exists another effort level l 6= js but P m∈M P m∈M P P e −e qslm m > qsjs m m − ². Then Θn > sjs² sl ( qnjm m − qsjs m ). m∈M

m∈M

m∈M

m∈M

Condition (1) may hold when signal n leads to a much better market condition than s and therefore the expected sales under the suboptimal effort j is still higher than the maximum expected sales given the signal s. Condition (2) implies that given signal s, effort level l is suboptimal, but the difference of expected sales is very small. The lower bound of information rent approaches infinity if ² is arbitrarily small. In this case, a linear contract leads to an unbounded rent. On the contrary, full surplus extraction is achieved when the manufacturer uses both linear contracts and lotteries. 5.2.3.

Full surplus extraction using joint conditions on {Qnj } and {Φn }. We can also

characterize sufficient conditions for full surplus extraction that comprise of two parts: one that is imposed using the probability distributions over sales outcomes, and the other using signals of other dealers. These conditions appear to be new to the literature. Theorem 6. Suppose Assumption 1 holds. In the multi-dealer game, full surplus extraction can be achieved if there exists a set N1 ⊆ N such that

P • ∀n ∈ N1 , there exists no nonnegative solution {λnsj } to QTnjn = λnnj QTnj + j∈J\{jn } P P n T λsj Qsj . s∈N \{n} j∈J P • ∀n ∈ N \ N1 , there exists no nonnegative solution {ξsn } to ΦTn = ξsn ΦTs . s∈N \{n}

Compared to Proposition 1, we impose a stronger condition on probability distributions over sales outcomes and relax the condition for type correlation. The condition on {Qnj } ensures that 15

Moreover, as long as {−a

P m∈M

qnjn m m + enjn , n ∈ N } is inside the span of {Φn }, the same idea may apply even

though {Φn } fails to have full rank.


20

demand signals are sufficiently distinguishable with respect to sales outcome. We might expect this to hold when different demand signals lead to significantly different distributions over sales. According to this condition, we can leverage the informativeness of sales outcomes and other dealers’ signals with regard to a dealer’s true type. Since the manufacturer can infer a dealer’s true signal from both the realized sales and other dealers’ reports, he should make the best use of them. We next propose a compensation scheme that implements full surplus extraction. Theorem 7. In the multi-dealer problem, suppose the conditions in Theorem 6 hold. The manufacturer can extract full surplus using the following compensation: • For n ∈ N \ N1 , the compensation Πnmi = an m + cn + Πni in Theorem 3 is adopted. • For n ∈ N1 , there exists a payment scheme {Πnm } that does not involve other dealers’ reports

and leads to no deviation. In this case, the manufacturer uses a combination of compensations. If a dealer reports that she receives a signal n ∈ N \ N1 , she is offered a linear contract plus a lottery aforementioned. This compensation provides the right incentive for the dealer to choose optimal effort and prevents other types of dealers to pretend as a type n ∈ N \ N1 . When the dealer reports as a type-n ∈ N1 , her compensation is independent of other dealers’ reports. This (potentially nonlinear) payment scheme ensures that the dealer is willing to choose the optimal effort and no other type of dealer intends to take this scheme if she is not type-n ∈ N1 . 5.2.4.

General case. When the conditions in Propositions 1, 2, and Theorem 6 do not hold,

the manufacturer may leave some rent for dealers. With some abuse of notation, let N s denote an arbitrary minimum spanning subset of {Φn }, i.e., span(N s ) = span(Φ), where span(Φ) = {x : P ∃{µn }, s.t. x = µn Φn }.16 Given N s , a type n is called a spanning type if n ∈ N s . We first show n∈N

that generically at most one spanning type can receive rent. Theorem 8. Suppose Assumption 1 holds. In the multi-dealer problem, at most one spanning type of dealer receives rent. The above theorem sets a limit on the number of types that receive rent in generic cases (cf: the base model). This phenomenon is also observed by Bose and Zhao (2006), where they study the mechanism design problem with only hidden information. They focus on two special cases when the rank of {Φn } is |N | − 1 or |N | − 2. We do not impose any rank condition. 16

N s are the sets of types than probability vectors {Φn }. For each n ∈ N , there exists a unique representation P rather µn {µt } such that Φn = t Φt . t∈N s


21

We now demonstrate a two-stage procedure to solve the manufacturer’s problem for general cases. Let N e denote the set of extreme points of conv(Φ). The dual program of multi-dealer problem is equivalent to the following. Lemma 1. (D-NE) is equivalent to the following: max

X

wn enjn +

n∈N

s.t.

XX

enj

X

Ynsj +

X

X

βnj enj

n∈N j∈J

n∈N j∈J\{jn } s∈N \{n} X X X n s βnj qnjm + wn qnjn m ) Ysnj qsjm + µt ( µt j∈J j∈J\{jn } s∈N \{n} X βnj qnjm + wn qnjn m = 0, n ∈ N e , (Πnm ) j∈J\{jn }

−(

X

βnj +

j∈J\{jn }

X X

= 0, ∀m ∈ M, ∀t ∈ N s , ∀n ∈ N \ N e , (Πtnm )

Ynsj + wn ) ≤ Pn , n ∈ N, (Θn )

s∈N \{n} j∈J

βnj ≤ 0, j ∈ J \ {jn }, Ysnj ≤ 0, s ∈ N \ {n}, wn unrestricted, j ∈ J, n ∈ N \ N e . where the variables {Πtnm } and {Πnm } correspond to these constraints. From the equivalent formulation, we can write down its corresponding primal as follows: (P − NE − Reduced) min s.t.

X

Pn Θn

n∈N X m∈M X m∈M X m∈M X

qnjm qsjm

X t∈N Xs

µnt Πtnm − Θn ≤ enj , j ∈ J \ {jn }, n ∈ N \ N e , µst Πtnm − Θs ≤ esj , j ∈ J, s 6= n, n ∈ N \ N e ,

t∈N s

qnjm Πnm − Θn ≤ enj , j ∈ J, n ∈ N e , qnjn m Πnm − Θn = enjn , n ∈ N,

m∈M

Θn ≥ 0, Πnm , Πtnm unrestricted, t ∈ N s , n ∈ N, ∀m ∈ M,

The above primal program does not involve any {φni } and other dealers’ reports (they are included implicitly via the linear combination {µnt }). The information about {Φn } allows us to collapse the state space of sales outcomes (using the µ’s) and to represent the original problem as a simple single-dealer problem. Based on this formulation, we can solve the manufacturer’s problem in a two-stage procedure. Theorem 9. In the multi-dealer problem, the optimal compensation scheme can be solved in the following two-stage procedure: 1. Design sales-contingent payments: Solve (P-NE-Reduced) and obtain {Π∗nm , n ∈ N e }, {Πt∗ nm , t ∈ N s , n ∈ N \ N e }, and {Θ∗n }. 2. Incorporate other dealers’ reports:


22

P

• For n ∈ N e , find {Πni } such that ∀s ∈ N \ {n}. Let Πnmi = Π∗nm + Πni .

i∈N |K|−1

• For n ∈ N \ N e , find {Πnmi } such that

P

φni Πni = 0 and P

i∈N |K|−1

φsi Πni sufficiently small,

i∈N |K|−1

s φti Πnmi = Πt∗ nm , ∀t ∈ N .

In the first stage, we remove all truth-telling constraints associated with misreporting as typen ∈ N e . Therefore, the compensation simply has to ensure that this dealer exerts the optimal effort. For those types that dealers might pretend to be, the manufacturer designs the optimal payment scheme using the payments for the spanning types (i.e., n ∈ N s ). The effective payment scheme seen by other types (n ∈ N \ N e ) can be represented as an affine combination of these basic payment schemes. This represents a state space collapse due to information! In the second stage, we find the appropriate payment schemes that implement these basic schemes for types that dealers might pretend to be. For payment schemes in N e , the manufacturer designs appropriate lotteries such that all other types will be severely punished if they pretend to be types in N e . Nevertheless, from the true type’s perspective, the lotteries yield a zero expected value and therefore have no net impact. The above procedure demonstrates that the compensation design problem can be separated. The information disclosed via the sales outcomes is used first, and then the information conveyed by other dealers’ reports are patched as a second tool to further strengthen dealers’ incentives. Not withstanding the common belief, the contract design problem gets complicated when full surplus extraction is not possible. The complication mainly results from the payments for those types that dealers may pretend to be. The payment schemes for types in N e can be constructed following the approach in Section 5.2.2. In particular, it can be implemented by a linear contract and a lottery. Corollary 3. Consider the multi-dealer problem. For n ∈ N e , the optimal compensation scheme can be achieved by Π∗nm = an m + cn , where an ≥ maxl6=jn P = −an qnjn m m + enjn + Θ∗n .

P m∈M

enjn −enl P qnjn m m− qnlm m

and cn

m∈M

m∈M

6.

Conclusion

This paper studies the optimal design for salesforce compensation when demand signals are correlated across local markets. We characterize sufficient conditions for the manufacturer to extract the full surplus. We show that the manufacturer can capitalize on the information revealed from the realization of sales outcomes as well as the reports of other dealers. Surprisingly, full surplus extraction with multiple dealers can be implemented using simple contracts. Under mild conditions


23

on signal correlation, we propose a simple menu of linear contracts and lotteries to extract full surplus. The linear contracts induce the dealers to exert the optimal effort, and the lotteries eliminate their incentives to misreport their signals. Omitting the lottery can result in large surplus for the dealers. We further discuss situations where the manufacturer can fully extract dealers’ surplus by offering a single contract instead of a menu, and show that the manufacturer can leverage the informativeness of sales outcomes and other dealers’ reports. When these conditions fail, only a limited number of types receive information rent. The contract design problem can be solved in a two-stage procedure wherein the information regarding sales and signals are used separately. Other contracting issues can be studied using our approach. For example, when should the manufacturer use individual contracts even if demand signals are correlated? This helps identify the situations where optimal compensation schemes need not include other dealers’ reports. Moreover, if dealers are concerned that other dealers may behave irrationally or make mistakes accidentally, solution concepts stronger than Bayesian Nash equilibrium may make a better sense. An example is the dominant strategy equilibrium, in which a dealer’s equilibrium strategy is the best response for all strategy profiles of other dealers. Adopting stronger solution concepts to examine the robustness of mechanism design problem is therefore of practical importance. The idea of capitalizing on type correlation can be applied in other scenarios. In particular, in the procurement auction context where suppliers’ production costs are correlated, the retailer may be able to extract more surplus from these suppliers. Another possibility is to use the correlation of retailers’ demands when the supplier has to allocate limited capacity to different regions. These deserve further investigation.

Appendix: Proofs. Proof of Theorem 1. Given this compensation, when a type-n dealer reports that she receives signal and chooses effort P P level j, her expected payoff is qnjm Πsm − enj = qnjm (am + c) − enj = m∈M,i∈N |K|−1 m∈M,i∈N |K|−1 P a qnjn m m − enj + c. The dealer’s report does not change her compensation, and therefore she m∈M

has no incentive to lie regarding the signal. Therefore, a

X

qnjn m m − enj + c = a(

m∈M

X

m∈M

qnjm m −

X

qnjn m m) − enj + enjn + a

m∈M

X m∈M

qnjn m m + c − enjn

X X X euju − eul P ( qnjm m − qnjn m m) + enjn − enj + a qnjn m m + c − enjn u∈N l6=jn quju m m − qulm m m∈M m∈M m∈M m∈M m∈M P P   qnjn m m − qnjm m euju − eul m∈M m∈M  P max max P = (enjn − enj ) 1 − u∈N l6 = j enjn − enj quju m m − qulm m n ≤ max max P

m∈M

m∈M


24

+a

X

qnjn m m + c − enjn ,

m∈M

P P where the inequality follows from that qnjm m > qnjn m m. Since enjn > enj , we obtain m∈M P P Pm∈M a qnjn m m − enj + c ≤ a qnjn m m + c − enjn = qnjn m (am + c) − enjn . Therefore, when a m∈M

m∈M

m∈M

dealer accepts the contract, she is willing to tell the truth and exert the optimal effort. Moreover, P P the dealer is willing to participate since qnjn m (am + c) − enjn = a qnjn m m − enjn + c ≥ 0. ¤ Proof of Theorem 2.

m∈M

m∈M

Since the seller receives no surplus, we can replace Θn by 0 and write down (TT) and (IR) conditions P P as qnjm Πnm − enj ≤ 0, ∀j ∈ J, ∀n ∈ N, and qnjn m Πnm − enjn = 0, ∀n ∈ N. Decomposing Eq. m∈M m∈MP P (13) into qnjn m Πnm − enjn ≥ 0, ∀n ∈ N, and qnjn m Πnm − enjn ≤ 0, ∀n ∈ N, and removing m∈M m∈M P the duplicate equations, we can rewrite the linear system as qnjm Πnm ≤ enj , ∀j ∈ J, ∀n ∈ N, m∈M P and − qnjn m Πnm ≤ −enjn , ∀n ∈ N. m∈M

Now let us rearrange the sequence of these inequalities and regroup all inequalities associated P P with a specific report n. That is, ∀n ∈ N, qsjm Πnm ≤ esj , ∀s ∈ N, ∀j ∈ J, and − qnjm Πnm ≤ m∈M

m∈M

−enjn . Note that when the true signal is s, it is suboptimal to report as if the dealer receives signal

n, regardless of her effort level. After such regrouping the problem is decomposed according to {Πnm } rather than types. If there is no solution to a particular report n (with associated payment

schedule {Πnm }), then the entire problem has no solution. This is because any choice of {Πsm }s6=n will not affect the constraints associated with {Πnm }. Thus, the problem is fully decomposable, and we in the following focus on a particular report n. For a given n ∈ N , we can represent it in matrix form as follows:     q111 q112 ... q11|M | e11    ..      ...  . Πn1      q1|J|1 q1|J|2 ... q1|J||M |   .   e1|J|    .  ≤   .    ..  . ..    .  .   Πn|M |    q|N ||J|1 q|N ||J|2 ... q|N ||J||M |   e|N ||J|  −qnjn 1 −qnjn 2 ... −qnjn |M | −enjn Recalling the definitions of {Qnj } and {Qn }, the linear system can be expressed as       ET1¦ Q1 Πn1  ..   .   .   ..   ..    .  ≤  T .  Q|N |   E|N |¦  Πn|M | −Qnjn −enjn The µ

above equation can be suppressed ¶ µ further ¶ Q E [Πn1 , ..., Πn|M | ]T ≤ − , ∀n ∈ N. −Qnjn −ejn

as

the

following

matrix-expression:


25

Recall that Gale’s theorem of the alternative: for any pair of matrices A and b (with appropriate dimensions), a linear system (Ax ≤ b) is feasible if and only if the alternative system ³ ´ T yT A = 0, y b >0, y ≤ 0 has no solution (Mangasarian (1969)). Therefore, if we let An = µ ¶ µ ¶ Q E , bn = − , ∀n ∈ N, the alternative system is as stated in the theorem. ¤ −Qnjn −enjn Proof of Corollary 1. The idea is to characterize sufficient conditions such that the alternative system (i.e., Eq. (19)) is infeasible. Suppose that the only solution to (ynT An = 0, yn ≤ 0) is yn = 0. Then ynT bn < 0 cannot be satisfied for ¶any bn . Both conditions (1) and (1a) are sufficient for this case. Any µ P P P Q βnj QTnj + Ysnj QTsj . = 0 must satisfy (wn − βnjn )QTnjn = solution to ynT −Qnjn j∈J\{jn } s∈N \{n} j∈J Since βnj , Ysnj ≤ 0, ∀n, s ∈ N, ∀j ∈ J, wn − βnjn ≥ 0 as well. Therefore, when either condition (1)¶ or µ E condition (1a) holds, all dual variables {βnj }, {Ysnj } are zero. We then obtain that ynT = −enjn (βnjn − wn )enjn = 0, which implies that the n-th alternative system is infeasible, ∀n ∈ N . By Theorem 2, full surplus extraction is achievable under these cases. We now suppose that condition (3) holds. Since if there exists a solution to An [Πn1 , ..., Πn|M | ]T = bn , it is also feasible to An [Πn1 , ..., Πn|M | ]T ≥ bn . Note that the last row is redundant. Eliminating the last row, the matrix on the left-hand side becomes Q and the right-hand side is E , independent of n. Therefore, if E ∈ span (Q), then the original system has a solution. ¤ Proof of Proposition 1. From Eq. (21), we obtain ∀n ∈ N, X X s∈N \{n} j∈J

⇔

X

φsi [

Ysnj qsjm φsi +

X

βnj qnjm φni + wn qnjn m φni = 0, ∀m ∈ M, ∀i ∈ N |K|−1 ,

j∈J\{jn } X (−Ysnj )qsjm ] = φni [wn qnjn m +

j∈J X T Φs [ (−Ysnj )qsjm ] j∈J s∈N \{n} s∈N \{n}

⇔

X

X

βnj qnjm ], ∀m ∈ M, ∀i ∈ N |K|−1 ,

j∈J\{jn }

=

ΦTn [wn qnjn m

+

X

βnj qnjm ], ∀m ∈ M.

(24)

j∈J\{jn }

P P P Recall that qsjm = 1,∀j ∈ J, s ∈ N, we obtain that ΦTs [ (−Ysnj )] = ΦTn [wn + m∈M j∈J s∈N \{n} P βnj ], ∀n ∈ N. Since Ysnj ≤ 0, ∀j ∈ J, ∀n, s ∈ N , the above equation implies that ΦTn can be j∈J\{jn }

rewritten as a convex combination of {ΦTs , s ∈ N \ {n}}. The condition aforementioned forces all P P the coefficients to be zero, i.e., Ysnj = 0, ∀j ∈ J, ∀n, s ∈ N, and wn + βnj = 0, ∀n ∈ N. This j∈J

j∈J\{jn }

further implies that Ysnj = 0, ∀j ∈ J, ∀n, s ∈ N since they are all nonpositive. From Eq. (24), we P then have wn QTnjn + βnj QTnj = 0, ∀n ∈ N. j∈J\{jn }

Now we prove that Assumption 1 implies that the above equation forces {wn }, {βnj } = 0. Since {βnj }’s are nonpositive, this is equivalent to that Qnjn cannot be written as a convex combination


26

of {Qnj , j 6= jn }. Suppose that such convex combination exists for a type n, i.e., there exists {ρnj ≥ P P 0} such that QTnjn = ρnj QTnj . Therefore, qnjn m = ρnj qnjm , ∀m ∈ M. We can then j∈J\{j j∈J\{j P P n} P P P n} P P obtain qnjn m m = ρnj qnjm m = ρnj qnjm m < ρnj qnjn m m = m∈M m∈M j∈J\{jn } m∈M m∈M j∈J\{jn } j∈J\{j P P n} qnjn m m, where we have applied Assumption 1 in the inequality and ρnj = 1 in the last m∈M

step. This leads to a contradiction. Thus, all dual variables are zero. ¤ Proof of Theorem 3.

j∈J\{jn }

Consider first the incentive of type-n dealer to choose a wrong effort level. If she chooses effort level j 6= jn , her expected payoff will be X m∈M,i∈N |K|−1

X

= =

qnjm φni (an m + cn + Πni ) − enj

m∈M,i∈N |K|−1

X

qnjm φni (an m + cn ) − enj +

m∈M,i∈N |K|−1

X

X

qnjm φni Πnmi − enj =

m∈M

m∈M,i∈N |K|−1

X

qnjm (an m + cn ) − enj +

qnjm φni Πni

φni Πni = an

X

qnjm m + cn − enj ,

m∈M

i∈N |K|−1

where the last equality follows from the choice of {Πni }. Note that an

X

qnjm m + cn − enj = an (

m∈M

X

qnjm m −

m∈M

X

X

qnjn m m) − enj + enjn + an

m∈M

qnjn m m + cn − enjn

m∈M

X X X enjn − enl P ( qnjm m − qnjn m m) + enjn − enj + an qnjn m m + cn − enjn l6=jn qnjn m m − qnlm m m∈M m∈M m∈M m∈M Pm∈M P   qnjn m m − qnjm m enjn − enl m∈M m∈M  P = (enjn − enj ) 1 − max P l6=jn enjn − enj qnjn m m − qnlm m m∈M m∈M X +an qnjn m m + cn − enjn , ≤ max P

m∈M

where the inequality follows from the choice of an . Because enjn > enj and by Assumption 1, we obtain that X X an qnjm m + cn − enj ≤ an qnjn m m + cn − enjn m∈M m∈M X X X = qnjn m (an m + cn ) − enjn + φni Πni = m∈M

i∈N |K|−1

P m∈M

qnjn m m >

P

qnjm m

m∈M

qnjn m φni (an m + cn + Πni ) − enjn .

m∈M,i∈N |K|−1

Since the last term corresponds to the expected payoff if the type-n dealer chooses effort level jn , she has no incentive to choose any other effort level. Moreover, she is willing to participate because P P by the choice of cn , qnjn m φni (an m + cn + Πni ) − enjn = an qnjn m m + cn − enjn = 0. m∈M

m∈M,i∈N |K|−1

Now let us consider whether other types of dealers may report as type-n. Suppose a type-s dealer pretends to be type-n and chooses effort level j. Her expected payoff will be X m∈M,i∈N |K|−1

qsjm φsi (an m + cn + Πni ) − esj =

X m∈M

qsjm (an m + cn ) − esj +

X i∈N |K|−1

φsi Πni .


27

P Recall that ∀n ∈ N, there exists no nonnegative solution {ξsn } to ΦTn = ξsn ΦTs . Farkas’ lemma s∈N \{n} P P then implies that for each n ∈ N , there exists {fni } such that φni fni = 0, and φsi fni < i∈N |K|−1 i∈N |K|−1 P 0, ∀s ∈ N \ {n}. Let Πni = tn fni , ∀i ∈ N |K|−1 . This payment scheme satisfies φni Πni = 0. i∈N |K|−1 P Since qsjm (an m + cn ) − esj is finite, we must be able to find a sufficiently large tn so that P m∈M P qsjm (an m + cn ) − esj + φsi Πni < 0, ∀s ∈ N \ {n}, j ∈ J. This implies that such deviation m∈M

i∈N |K|−1

is never optimal for type-s dealer (who can at least get zero by not participating). Combining all above for each n ∈ N , no dealer can benefit from misreporting her signal, and choosing any other effort level yields a strictly negative expected payoff. Thus, this compensation scheme implements full surplus extraction. ¤ Proof of Corollary 2. The proof is by contradiction. Suppose the claim is false. Then there must exist another compensation scheme {Πnmi = an m + cn + Πni } such that there exists s ∈ N and as < maxl6=js

P m∈M

esjs −esl P . qsjs m m− qslm m

Let η = arg maxl6=js

m∈M

m∈M

effort η, her expected payoff will be X X

X

qsηm φsi Πsmi − esη =

qsηm (as m + cs ) − esη +

X

esjs −esl P . qsjs m m− qslm m

If type-s dealer chooses

m∈M

qsηm φsi (as m + cs + Πsi ) − esη

m∈M,i∈N |K|−1

m∈M,i∈N |K|−1

=

P

φsi Πsi = as

X

qsηm m + cs − esη +

X

φsi Πsi

|K|−1 m∈M i∈NX i∈N |K|−1 X φsi Πsi qsjs m m) + esjs − esη + as qsjs m m + cs − esjs + |K|−1 m∈M m∈M m∈M i∈N P P   qsjs m m − qsηm m X X m∈M m∈M  + as = (esjs − esη ) 1 − as φsi Πsi . qsjs m m + cs − esjs + esjs − esη |K|−1 m∈M

m∈M

= as (

X

qsηm m −

X

i∈N

Since as
s . qsjs m m − qslm m ² m∈M

Moreover, since type-n dealer does not benefit from misreporting as type-s and choosing effort j, X qnjn m (an m + cn ) − enjn ≥ qnjm (as m + cs ) − enj m∈MX m∈M X X X X = as ( qnjm m − qsjs m ) + as qsjs m m + cs − enj = as ( qnjm m − qsjs m ) + Θs + esjs − enj ,

Θn =

X

m∈M

m∈M

m∈M

m∈M

m∈M

where we have applied the definition of information rent Θs in the last equality. Since Θs ≥ 0 and P P P P e −e enj ≤ esjs , we obtain Θn ≥ as ( qnjm m − qsjs m ) > sjs² sl ( qnjm m − qsjs m ). ¤ m∈M

m∈M

m∈M

m∈M

Proof of Theorem 6. P P P Summing over i ∈ N |K|−1 in Eq. (21) for n ∈ N1 , we have Ysnj qsjm + βnj qnjm + j∈J s∈N \{n} j∈J\{j } n P wn qnjn m = 0, ∀m ∈ M, where we have applied φsi = 1,∀s ∈ N . Since {βnj }’s, {Ysnj }’s are all i∈N |K|−1

nonpositive, the first part of the condition implies that all coefficients involved in these equations P P P Ysnj qsjm φsi + are zero. The residual dual variables have to satisfy βnj qnjm φni + s∈N \{n} j∈J

j∈J\{jn }

wn qnjn m φni = 0, ∀m ∈ M, ∀i ∈ N |K|−1 , ∀n ∈ N \ N1 . The second part guarantees that all variables are zero following an argument similar to Proposition 1. Thus, the corresponding objective value is zero, i.e., full surplus extraction is possible. ¤ Proof of Theorem 7. The argument of Proposition 1 works for each n ∈ N \ N1 , and hence no dealer can benefit from misreporting as type-n ∈ N \ N1 if such compensation is offered. Thus, we in the sequel focus P λnnj QTnj + on the case n ∈ N1 . Since there exists no nonnegative solution {λnsj } to QTnjn = j∈J\{jn }


P

P

s∈N P\{n} j∈J m∈M

29

λnsj QTsj , .Farkas’ lemma implies that there exists {fnm } such that

P m∈M

qnjn m fnm = 0, and

qsjm fnm < 0,∀(s, j) 6= (n, jn ). Let Πnmi = xn fnm + enjn , where xn is a scalar to be chosen.Given

this payment scheme, if a type-n dealer chooses the optimal effort, her expected payoff will be X m∈M,i∈N |K|−1

X

=

X

qnjn m φni Πnmi − enjn =

qnjn m φni (xn fnm + enjn ) − enjn

m∈M,i∈N |K|−1

qnjn m (xn fnm + enjn ) − enjn = xn

m∈M

X

qnjn m fnm + enjn − enjn = 0.

m∈M

P However, if she chooses another effort j, her expected payoff will be qnjm φni Πnmi − |K|−1 m∈M,i∈N P P enj = xn qnjm fnm + enjn − enj . Similarly, a type-s gets an expected payoff xn qsjm fnm + m∈M

m∈M

enjn − esj if she chooses effort j. We can make xn sufficiently large such that all these expected payoffs are negative. Thus, no dealer has an incentive to deviate. ¤ Proof of Theorem 8. From (D-NE), we can replace {φni } by the unique representation φni = N . Eq. (21) then becomes ∀m ∈ M, ∀i ∈ N |K|−1 , ∀n ∈ N, X X

Ysnj qsjm

s∈N \{n} j∈J

X

⇔

[

X

µst

t∈N s s∈N \{n}

X

µst φti + (

X

t∈N s

βnj qnjm + wn qnjn m )

t∈N s

j∈J\{jn } X n Ysnj qsjm + µt ( βnj qnjm j∈J j∈J\{jn }

X

P

µnt φti , ∀i ∈ N |K|−1 , ∀n ∈

X

µnt φti = 0,

t∈N s

+ wn qnjn m )]φti = 0.

P P P P Summing over ∀m ∈ M , we obtain [ µst Ysnj + µnt ( βnj + wn )]φti = 0, which t∈N s s∈N \{n} j∈J j∈J\{jn } P P P P implies [ µst Ysnj + µnt ( βnj + wn )]ΦTt = 0. Since N s is a minimum spant∈N s s∈N \{n}

j∈J

j∈J\{jn }

ning subset, {Φt } ’s are linearly independent. Therefore, µst = 0, ∀s 6= t, s, t ∈ N s . Moreover, P P P µst Ysnj + µnt ( βnj + wn ) = 0, ∀t ∈ N s , ∀n ∈ N, according to the definition of N s . j∈J

s∈N \{n}

j∈J\{jn }

The above equation can be separated into three cases. Fix a t ∈ N s . If we take n = t, we have P P P P P P µst Ystj + ( βtj + wt ) = 0. If n 6= t, n ∈ N s , Ytnj + µst Ysnj = 0. Finally, s j∈J j∈J j∈J s∈N \N s j∈J\{j } s∈N \N n P P P P when n ∈ N \ N s , Ytnj + µst Ysnj + µnt ( βnj + wn ) = 0. j∈J

s∈N \N s

j∈J

j∈J\{jn }

Now consider the inequality constraint of (D-NE), i.e., Eq. (22). For a particular t ∈ N s , we can represent the left-hand side as follows: X X X X X Ytnj + wt ) = −( βtj + wt ) − Ytnj − Ytnj s \{t} j∈J j∈J\{jn } n∈N \N s j∈J n∈N j∈J\{jn } n∈N \{t} j∈J   X X X X X X X X X  µst µst Ystj + Ysnj + µnt ( βnj + wn ) + µst Ysnj

−(

=

X

s∈N \N s

=

X s∈N \N s

βtj +

j∈J

µst (

XX

n∈N j∈J

X X

n∈N \N s

Ysnj ) +

s∈N \N s

X n∈N \N s

j∈J

µnt (

X

j∈J\{jn }

j∈J\{jn }

βnj + wn ) =

X s∈N \N s



µst 

n∈N s \{t} s∈N \N s

XX

n∈N j∈J

Ysnj +

X j∈J\{jn }

j∈J



βsj + ws  ,


30

which is equivalent to a combination {−µst } of left-hand sides of Eq. (22) for types s ∈ N \ N s . The above argument works for each type in N s . Suppose a left-hand side of Eq. (22) for N \ N s is given. Unless Pt , t ∈ N s is chosen to match these ratios, generically only one of the constraints for t ∈ N s is binding, i.e., at most one spanning type receives rent. ¤ Proof of Lemma 1. If Φn is an extreme point of conv(Φ) , the dual variables {Ysnj }’s associated with Πnmi , where n ∈ \N e , are zero following the argument of Theorem 1. Therefore, they are eliminated, and we are left with those constraints that prevent dealers from misreporting as a type n ∈ N \ N e . We can now represent the {Φn }’s as a combination of the minimum spanning subset N s : max

X

wn enjn +

n∈N

s.t.

X X

XX n∈N j∈J

Ysnj qsjm

s∈N \{n} j∈J

X

X

X

s∈N \{n} µst φti + (

t∈N s

X

Ynsj + X

X

βnj enj

n∈N j∈J\{jn }

βnj qnjm + wn qnjn m )

X

µnt φti = 0,

(25)

t∈N s

j∈J\{jn }

∀m ∈ M, ∀i ∈ N |K|−1 , ∀n ∈ N \ N e ,

βnj qnjm + wn qnjn m = 0, n ∈ N e ,

j∈J\{jn }

−(

enj

X

j∈J\{jn }

βnj +

X X


s∈N \{n} j∈J

βnj ≤ 0, j ∈ J \ {jn }, Ysnj ≤ 0, s ∈ N \ {n}, wn unrestricted, j ∈ J, n ∈ N \ N e . In Eq. (25), we can collect " terms associated with φti together and find that# ∀m ∈ M, ∀i ∈ P P P P N |K|−1 , ∀n ∈ N \ N e , µst Ysnj qsjm + µnt ( βnj qnjm + wn qnjn m ) φti = 0. Since t∈N s s∈N \{n} j∈J j∈J\{jn } P P {Φt , t ∈ N s }0 s are linearly independent, the above equation implies that µst Ysnj qsjm + j∈J s∈N \{n} P µnt ( βnj qnjm + wn qnjn m ) = 0, ∀m ∈ M, ∀n ∈ N \ N e . ¤ j∈J\{jn }

Proof of Theorem 9. Since (D-NE) is equivalent to the dual program in Lemma 1, the optimal value of (P-NE-Reduced) is equal to that of the original problem. It then suffices to find the appropriate compensation schemes that achieve the optimality. When the manufacturer solves (P-NE-Reduced), he obtains the optimal {Θ∗n }, {Π∗nm , n ∈ N e }, s e and {Πt∗ nm , t ∈ N , n ∈ N \ N }. The next step is to find Πnmi such that other types will not pretend

as type-n in N e , and are implemented if the dealer reports n ∈ N \ N e . For n ∈ N e , we can follow the approach of Proposition 1 to construct the lottery {Πni } that yields a zero expected value for the true type but punishes other types severely. Therefore, we can define Πnmi = Π∗nm + Πni to ensure that the truth-telling constraints are satisfied. Now consider n ∈ N \ N e . Our goal is to find {Πnmi } such that

P i∈N |K|−1

s φti Πnmi = Πt∗ nm , ∀t ∈ N .

Since {Φt , t ∈ N s }0 s are linearly independent, the existence of solution is guaranteed and can be


31

obtained by Cramer’s rule. For type-s dealer that does not belong to N s but reports as a type-n, P P P s her expected payment when sales outcome m appears is φsi Πnmi = µt φti Πnmi = |K|−1 |K|−1 t∈N s i∈N i∈N P s P P s t∗ φti Πnmi = µt µt Πnm . Thus, {Πt∗ nm } is indeed implemented by such a choice. ¤

t∈N s

t∈N s

i∈N |K|−1

Proof of Corollary 3. P Fix n ∈ N e . In (P-NE-Reduced), the only constraints associated with Πnm are qnjm Πnm − m∈M P Θn ≤ enj , j ∈ J, n ∈ N e , and qnjn m Πnm − Θn = enjn , n ∈ N e . When Θ∗n is given, it suffices to m∈M

verify that these constraints are satisfied under the proposed contracts. The equality constraint is automatically true from the construction of an and cn . Moreover, ∀j ∈ J, X X qnjm m + cn − Θ∗n qnjm (an m + cn ) − Θ∗n = an qnjm Πnm − Θ∗n = m∈M m∈M X X X X m∈M qnjm m) + enjn − enj + enj . qnjn m m − = an qnjm m − an qnjn m m + enjn = −an ( X

m∈M

m∈M

Therefore,

P m∈M

m∈M P

qnjm Πnm − Θ∗n = (enjn − enj )[1 − an m∈M

inequality follows from an ≥ maxl6=jn

m∈M qnjn m m−

m∈M

qnjm m

enjn −enj

enjn −enl P P . qnlm m qnjn m m−

m∈M

P

] + enj ≤ enj , where the

Thus, the optimality is achieved. ¤

m∈M

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