Optimal contract design with a common agency in last-mile logistics

Transportation Research Part E 139 (2020) 101956

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Optimal contract design with a common agency in last-mile logistics

T

Xiang Chua, Jun Liub, Long Renc, , Daqing Gongd ⁎

a

School of Maritime Economics and Management, Dalian Maritime University, Dalian 116024, China International Business School, Beijing Foreign Studies University, Beijing 100089, China School of Information Technology and Management, University of International Business and Economics, Beijing 100029, China d School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China b c

ARTICLE INFO

ABSTRACT

Keywords: Common agency Last-mile logistics Moral hazard Profit-sharing contract Incentive payment

Cooperative package collection is challenging given the lack of contract design to guarantee that the shared last-mile logistics (LML) provider divides the workload among all partners in a fair way, as it may influence the customer’s carrier choice when sending a package. However, the agency’s fairness may be influenced by principals’ individual incentives. To solve this issue, we establish a theoretical model with multiple principals (express companies) and a common agency (LML provider). We find that the optimal ex ante contract among collusive principals is a profitsharing contract, which ensures that the principals have no motivation to take hidden actions to influence the agency’s workload division. We also provide the optimal incentive contract between the coalition and the agency.

1. Introduction In 2018, global package delivery volume neared 100 billion units, which translates to a 5% year-on-year increase (Apex Insight, 2019). Last-mile logistics (LML), which accounts for approximately 28% of the total cost (Zenezini et al., 2018), is one of the most expensive segments of the delivery process. Furthermore, LML also generates traffic noise, air pollution, and other social issues. To alleviate the negative impacts of the LML and fully exploit economies of scale, the concept of cooperative urban logistics (Ranieri et al., 2018) has been proposed based on the sharing of resources, infrastructure, transport, and revenue among participants within an LML system. In practice, a common agency model (Bernheim and Whinston, 1986) is used to implement the concept of cooperative urban logistics. The agency, the LML service provider, in the common agency model is jointly delegated by several express companies. For instance, UPS and FedEx transport packages to the destination city, then outsource the last leg of delivery to the United States Postal Service (USPS). In the cooperation among UPS, FedEx and USPS, the agency (USPS) only conducts package delivery, while the principals (UPS and FedEx) collect the package directly from the customer. USPS has delivered an average of approximately 2 million packages to homes and businesses per day for FedEx in recent years (Solomon, 2019), thereby reducing operating costs through economies of scale and offering neighborhoods pro-environmental outcomes by easing traffic congestion. To integrate first-mile pickup and last-mile delivery (Bergmann et al., 2020) and achieve further economies of scale, the express companies could also outsource package collection to the common agency. The customer may visit the agency’s collection-delivery point and choose a courier to send the package. However, the common agency model’s performance is typically far from satisfactory ⁎

Corresponding author. E-mail addresses: [email protected] (X. Chu), [email protected] (J. Liu), [email protected] (L. Ren), [email protected] (D. Gong).

https://doi.org/10.1016/j.tre.2020.101956 Received 27 August 2019; Received in revised form 14 March 2020; Accepted 18 April 2020 1366-5545/ © 2020 Elsevier Ltd. All rights reserved.

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during the package collection process compared with package delivery. In recent years, some competing express companies have attempted to co-operate with a common but independent agency to collect last-mile packages in practice. For example, City 100 logistics has served as an independent and common LML service provider for several express companies in Beijing, China since 2011. However, City 100 logistics receives limited orders from the express companies, far below expectations. As stated by Guoqing Wang, CEO of City 100 Logistics, “the fairness concern from express companies is the biggest challenge for common agency business, since the common agency might be suspected to influence the end customer’s decision on express choice when sending out a package”. Poor performance also exists in the practice of Cainiao Guoguo, which is an LML service provider established by Ali Group, the largest e-commerce giant in China. The common agency is believed to have illicit agreements with certain couriers since it unfairly provides customers with a default courier option.1 In anticipation of the aforementioned fairness concern, express companies usually prefer to hire an exclusive agency instead of sharing a common agency with their rivals to collect packages, as theoretically discussed by Martimort (1996). The fairness concern of express companies during the package receipt process has prohibited the implementation of cooperative urban logistics. Cooperation is established on the basis of mutual benefit, and alleviating the conflicts of interest between the express companies and the agency vital not only for industrial managers but also for academia. The most challenging problems for these companies are how to design a contract to ensure fairness and how to appropriately reward members in the LML system (Gunasekaran et al., 2015). Recent years have witnessed another form of cooperation among these participants. In 2016, EMS, YUNDA Express, TTK Express, BEST Express, and STO Express established a joint venture named FeiMa Express in Pei County, Jiangsu Province, China. The joint venture took over the package receipt transactions for its parent companies and handled goods with a total value of approximately 60 million RMB in 2017,2 which stands in contrast with the poor performance of City 100 logistics and Cainiao Guoguo. A survey of the existing literature shows that there is a significant body of research on allocating benefits and costs within the field of collaborative logistics (Cleophas et al., 2019; Jin et al., 2019; Kimms and Kozeletskyi, 2016; Anily, 2018). However, few studies have been conducted to guarantee the implementation of the proposed allocation scheme, especially when there is an ex post moral hazard issue. Motivated by business practice and the aforementioned research gap, we plan to address several research questions: (1) How should one design a contract to coordinate multiple competing express companies and a common LML provider by alleviating the fairness concern during the package collection process? (2) What are the impacts of important factors, e.g., package receipt volume, competitive intensity, market uncertainty, and the agency’s reservation level, on the optimal contract design? To investigate these research questions, we establish a common agency model with multiple principals and one common agency. The principals are express companies that delegate package receipt tasks to a common agency (the local LML service provider). The common agency performs package collection under authority delegated by the principals. The exact working time that the agency exerts is unobservable by the principals. Nevertheless, the higher the return on collecting a package for a specific principal, the more likely the agency is to spend more time on that principal, according to the traditional assumption of a monotonic likelihood ratio property (MLRP). To fully exploit the economies of scale of the common agency model and alleviate the impacts of such fairness concerns, principals need to form a stable coalition. The alliance makes ex ante rules regarding the members’ incentives with the common agency. For instance, the payment contracts need to ensure that all members have no motivation to take hidden actions to encourage the agency to exert greater effort on a specific express company. Following the rules agreed by the alliance, each member seeks to maximize its profits by optimizing a payment contract. We derive the optimal payment contract that can fully coordinate the common agency system. To the best of our knowledge, this paper is the first analytical study exploring the ex post moral hazard of the common agency in LML. Our findings enable us to make the following threefold contribution. First, unlike existing research on the contract between a common retailer and rival manufacturers that produce partially substitutable products (Chen et al., 2019b), we find that a coalition among principals (express companies) is possible in LML even if their products are completely substitutable and the optimal payment contract within the coalition is a profit-sharing contract. Second, extending the moral hazard agency literature (Poblete and Spulber, 2012), in this paper, principals might also take hidden actions. This requires simultaneously considering the subcontract price for the agent and the collaboration agreement among express companies, while both contracts are interactive. The proposed contracts guarantee that principals have no motivation to defect from the coalition and influence the agency’s effort via an individual contract in the LML business. Third, we generate some important managerial insights. Our proposed method is applicable to complex scenarios in which the scale, express price and service level of express companies are heterogeneous. It makes the common agency model feasible and creates the potential for tremendous economic and environmental benefits, which is important for operations managers and policymakers. The remainder of this paper is structured as follows. Section 2 briefly reviews the related literature. Section 3 introduces the assumptions and building blocks of our model. Section 4 investigates mechanisms to coordinate the LML system with a common agency. Section 5 provides numerical analyses to demonstrate the effects of the model parameters in the optimal contract design. We provide concluding remarks in Section 6.

1 2

https://www.v2ex.com/t/539195. http://www.px.gov.cn/px/zhxw/201808/2a686c69dc8445879aa7f921edebd92c.shtml. 2

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2. Relevant literature Our work is related to three streams of literature: business models in LML, common agency contracts, and horizontal collaboration. In this section, we provide a review of the relevant research in each stream. We begin with an overview of the first stream, focused on innovative business models in LML. 2.1. Business models in LML Collaborative and environmentally friendly urban transportation could alleviate the negative impacts of urban transportation. However, it faces organizational challenges in collaboration (see Cleophas et al. (2019) for a review). In recent years, several innovative business models based on collaboration and cooperation have been introduced in LML. Ranieri et al. (2018) comprehensively review these innovative models in LML, focusing on the reduction of costs associated with externalities. First, the collection-delivery point (CDP) or proximity station strategy uses a depot station where goods can be stored following home delivery failures (Liu et al., 2019). This improves the efficiency of LML, avoiding the risk of the unsuccessful delivery. Second, competing carriers form a horizontal coalition. Several studies (Dai and Chen, 2012; Kimms and Kozeletskyi, 2016) consider allocation schemes for the coalition. In these studies, the Shapley value and the core concept are frequently used to determine whether potential partners should join the coalition, which new partner or partners to accept, and how to design a mechanism to share costs and benefits. Third, new organizational models similar to urban consolidation centers (UCC) are used in LML (Heeswijk et al., 2019; Amaya et al., 2020). The UCC concept was proposed in the 1970s and suggests consolidating goods sorted by destination address in a depot before their delivery to the individual retailers. This strategy means that economies of scale become possible in LML. For the use of the UCC service, Handoko et al. (2014) propose a profit-maximizing auction mechanism, formulating the winner determination problem as a mixed-integer problem. Akeba et al. (2018) propose a crowdsourcing-based method to collect and deliver packages from the UCC using neighbors. Zhang et al. (2019) propose a last-mile split delivery approach to reduce delivery trip by consolidating shipments to the same customers in the station. The aforementioned studies focus either upstream or downstream of the UCC. However, there is scant academic research on how upstream third-party logistics (3PL) service providers outsource package collection jobs to the shared downstream LML provider, especially considering incentive mechanisms from a principal-agency perspective. This is within the scope of common agency theory, which is discussed next. 2.2. Common agency contract Under the common agency model, the interests of several principals conflict with the actions of a particular agency (Bernheim and Whinston, 1986). The theory of common agency is well-developed (Bernheim and Whinston, 1986; Grossman and Helpman, 1994; Martimort and Stole, 2009; Dutta et al., 2018) and has been widely used in studying political influence, industrial organization, public goods provision, and tourism (see Mallard, 2014; Mallard, 2019) since the 1980s. Information asymmetry is an important issue in a common agency context, just as it is for a single principal-agent problem. In common agency adverse-selection games, the revelation principle does not work in the equilibrium of multi-principal games (Martimort and Stole, 2002). Moreover, principals cannot observe the agency’s effort allocation among different principals. Reducing moral hazard via incentive contracts is one of the most important topics in this field. Poblete and Spulber (2012) introduce a critical ratio that measures the expected return to providing incentives in each state. According to the value of the critical ratio, the principal could design the optimal contract to efficiently motivate the agency’s effort. In the context of multiple principals and agencies, exclusive agencies have few incentives to cheat when exposed to effective competition and correspondingly charge lower informational rents if the products of the principals are substitutable. Although exclusive dealing lacks downstream coordination, it usually dominates due to noncooperative behavior among principals under common agency (Martimort, 1996). To overcome the substitution effect and realize cooperation among principals in the multiple-task agency problem, Sinclair-Desgagné (1999) propose changing the efforts required for the various tasks from substitutive to complementary in the agency’s reward function via a common agency contract. In this paper, we attempt to incorporate the notion of common contract agency into the LML problem. Moreover, unlike the existing moral hazard agency literature, principals might take hidden actions in our model. 2.3. Horizontal collaboration Complementarities are neither a necessary nor a sufficient condition for the formation of coalitions to increase efficiency (Cabral and Almeida, 2019). Although the mutual agency’s assortment strategy may intensify the competitive pressure facing coalition members (Heese and de Albéniz, 2018), the design of resource exchange enables horizontal competitors to achieve collaborative agreement (Chun et al., 2017). Even a firm can undertake part or all of the production activities for its competitors by horizontal subcontracting (Li and Jiang, 2018). However, the rules of the coalition are critical for its stability and members’ profitability. Firms’ selection of a coopetition strategy is affected by internal, relationship-specific, and external drivers (Chen et al., 2019b). Sim et al. (2019) explore the comprehensive effects of market structures on economic competitiveness and environmental sustainability. Crama et al. (2017) show how control rights, options, payment terms, and timing influence the value of collaboration. Caro et al. (2018) 3

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consolidate the coalition through joint and shared audits with a collective penalty. Liu and Wang (2019) propose a combined contract between the carrier coalition and the port to coordinate a maritime transport chain. The existence of multiple principals in a coalition introduces the issue of how the benefits and costs are divided among members. In the context of cooperative game theory, Cleophas et al. (2019) identify all potential sharing mechanisms, such as the Shapley value, the nucleolus, separable and non-separable costs, shadow prices and volume weights. Many allocation methods implement the Shapley value and the core concept from cooperative game theory (Dai and Chen, 2012; Kimms and Kozeletskyi, 2016; Anily, 2018; Chen et al., 2019a; Jin et al., 2019). Instead of benefit and cost allocation fairness, this paper focuses on the fairness of the agency’s effort allocation. In view of the existing research, we attempt to investigate the principal-agency problem in a context where multiple 3PL service providers delegate package receipt jobs to a downstream common LML provider. Distinct from the literature, the principals in our model may take hidden actions. Moreover, unlike the existing research considering the contract between one common retailer and rival manufacturers, a coalition between the upstream 3PL providers is more likely due to common delivery, even if their services are completely substitutable. We introduce the building blocks of our model in the next section. 3. Common agency model Consider the following problem. There are m express companies i = 1, …, m , which form a coalition and delegate package receipt jobs to a common agency, e.g., a local logistics service provider. All players can observe the performance of the agency, which is labeled i . For instance, i could be i’s revenue. However, only the agency knows exactly how much effort, ei 0 , is spent on collecting the packages of express company i. Herein, effort ei is defined as the percentage of time the agency spends on acquiring m customers and processing orders for i, i.e., i = 1 ei = 1. We take the following example to illustrate the effort allocation mechanism. Assume that a customer chooses express company i to send a package. If the agency encourages the customer to choose i or even does not influence the customer’s decision, all effort of the agency is spent on the execution of i’s job in this transaction. If the agency i and 1 j m , the effort on acquiring customers is for i and encourages the choice of i but the customer ultimately chooses j, j the effort on processing orders is for j. A high input ei raises the probability of a high output i . As principals, express companies infer ei imperfectly from i . Following the classic work (Conlon, 2009; Poblete and Spulber, 2012), the relationship between the agency’s effort and the performance (revenue) of principal i is i

=

i ( i,

(1)

ei ),

where i depends on the state of nature i and the agency’s effort ei . Since companies are heterogeneous, the function expression i of company i may be different from that of other companies. The uncertainty of express company i’s revenue is reflected in the state i , which is a random variable with probability density function f ( i ) . At the beginning of a contract period, the agency chooses an effort allocation scheme e = (e1, …, em) before the realization of state vector = ( 1, …, m) . In the contract period, for instance, a negative event may produce a shock for the principal with respect to number of packages received, in addition to the effort of the LML provider. At the end of the contract period, the principals are able to observe = ( 1, …, m) and the realization of but cannot observe the agency’s effort e . We assume that all principals and the agency are risk-neutral. Equivalently, all principals and the agency seek to maximize their expected profits. The relationships between the principals and the agency are illustrated in Fig. 1. Under the coalition agreement, each principal i attempts to influence the agency’s effort allocation via an ex ante incentive contract wi , which entails a transfer payment from i to the agency. Thus, principal i’s expected payoff is

Vi (wi (·), ei ) =

[

i ( i,

ei)

wi (·)] f ( i ) d i.

(2)

i ), this will create an ex post moral hazard If wi (·) depends only on i regardless of other principals’ performance j , (j problem for the principals. Specifically, each principal thus intends to motivate the agency to spend more working time on its own package collection process for a positive marginal return via an individual contract, as shown in Fig. 1. Finally, in equilibrium, the agency obtains excess profit from the competition among principals. In anticipation of this cutthroat competition, principals will not agree to join the coalition and will instead hire an independent agency, and hence, the common agency model fails. Therefore, the coalition must overcome the aforementioned moral hazard issue by sharing the incentive cost among multiple principals, i.e.,

Fig. 1. The relationships among players. 4

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wi (·) = wi (

1,

m) ,

…,

the linear form of which is as follows: m

wi (

1,

…,

m)

=

aij

j,

(3)

j =1

where aij is a ratio coefficient. From Eq. (3), we find that express company i also pays the agency with aij other companies’ performance. Thus, the total incentive payment received by the agency is m

w(

1,

…,

m)

=

j,

which is a function of

m

Ai

i,

where

Ai =

i=1

aji.

(4)

j=1

Here, Ai is the incentive payment rate of i offered to the common agency. In other words, the coalition pays Ai i to the agency for his effort devoted on behalf of express company i, and aij (j = 1, …, m) determine how the payment Ai i is shared among coalition members. The following constraint assures each individual principal that there are disincentives to take ex post hidden actions to unilaterally motivate the agency:

Vi (wi ( 1, …, ej

m ),

ei)

=

Vi (wi ( 1, …, ek

m),

ei )

,

i, j, k = 1, …, m.

(5)

The economic implication behind Eq. (5) is that each unit of working time brings the same marginal revenue for the principals. Thus, a principal has no incentive to motivate the agency to allocate more working time to its own package collection process. m Given contract w (·) = i = 1 wi (·) , the agency chooses an effort allocation scheme e = (e1, …, em) to maximize his utility function, m

U (w (·), e) =

wi (

1 ( 1,

e1), …,

m ( m,

em)) f ( i ) d i.

(6)

i=1

Note that the cost of the agency’s effort is not included in the above utility function for the following reasons. First, this paper discusses effort allocation fairness rather than effort induction. Second, package delivery and receipt share the same logistics network, meaning that most of the cost of receipt has been covered by delivery jobs. The principal coalition’s problem of choosing an optimal incentive contract w (·) , i.e., incentive payment rate vector A = {A1 , …, Am } , subject to feasibility restrictions can be stated as follows: m

Max A

(1

Ai )

i ( i,

ei ) f ( i ) d i ,

(7)

i=1

subject to

e = (e1, …, em)maximizesU (w (·), e),

(8)

U (w (·), e)

(9)

Vi (wi ( 1, …, ej

U0, m ),

ei)

=

Vi (wi ( 1, …, ek

m),

ei )

,

(10)

m

ei

0,

ei = 1,

(11)

i=1

(12)

i , j, k = 1, …, m.

The objective function (7) maximizes the total profit of all express companies; the agency is paid at a rate of Ai from sales revenue of company i = 1, …, m . Eq. (8) is the incentive compatibility (IC) constraint: given the payment scheme provided by the principals, the agency always selects an effort allocation scheme that maximizes his expected utility. Eq. (9), the participation constraint (PC) of the agency, ensures that the agency in total earns no less than U0 , where U0 is the agency’s reservation utility level representing the original utility of the agency with a single principal. The economies of scale and existence of a suitable contract within the coalition ensure that each principal also does not earn less. As mentioned above, each unit of working time brings the same marginal revenue for the principals. Thus, Eq. (10) eliminates the ex post moral hazard for the principals. Eq. (11) denotes the time (effort) constraint of the agency, where ei , i = 1, …, m denotes the percentage of time the agency spends on express company i. Obviously, each allocated working time (effort) is non-negative. The Mirrlees-Rogerson condition is extensively adopted in principal-agency problems (Jewitt, 1988; Sinclair-Desgagné, 1994). It includes the monotone likelihood ratio property (MLRP) and the convex distribution function condition (CDFC). Under these conditions, i exhibits diminishing marginal returns in effort ei as follows. Assumption 1. The performance

i ( i,

ei ) of express company i is increasing and weakly concave in ei for all

i,

i = 1, …, m .

Assumption 1 allows us to use the first-order approach to replace the IC constraint (8) with the agency’s first-order condition. Next, to avoid triviality, we make the following assumption as commonly employed in the literature. Assumption 2. There exist at least one feasible contract w and an effort allocation e such that constraints 8, 9,11 are satisfied. 5

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Table 1 Notation summary. Notation

Meaning

i ei

Notations of express companies, i = 1, 2, …, m . Working time spent by the agency on express company i, where i = 1, 2, …, m . A random variable indicating the state of nature with probability density function f ( i) . The random state vector = ( 1, …, m) . Revenue function of express company i, where i = 1, 2, …, m . Incentive payment rate i offered to the common agency. Incentive payment rate vector A = {A1 , …, Am } . The transfer payment from company i paid to the common agency in the ex ante contract. Expected profit of express company i. The agency’s utility function. Reservation level of the agency’s utility function.

i i ( i , ei )

Ai A wi Vi (wi, U U0

i)

The above assumption is frequently adopted in the principal-agency literature, e.g., Poblete and Spulber (2012). It holds whenever the scale effect of sharing a common agency is sufficiently large compared with the agency’s reservation level U0 . If the assumption does not hold, the common agency model does not save money compared with the exclusive agency model. This may occur when the common agency model leads to high organizing and management costs. Our work is based on the common belief of industries, i.e., a common agency saves money, which supports Assumption 2. The notations of our paper are summarized in Table 1 as follows: 4. The optimal contract This section explores the optimal incentive contract between the coalition and the agency, together with the cooperative agreement among the principals. First, Section 4.1 analyzes a centralized system that serves as the benchmark. Next, Section 4.2 discusses a decentralized model in which all players strive to maximize their own profits. We provide a complete characterization of the optimal contracts and demonstrate the existence of the unique solution. Finally, Section 4.3 provides the steps for identifying the optimal contracts in real-world LML practice. 4.1. Centrally coordinated system In a centrally coordinated system, all m + 1 members (including m principals and 1 agency) are integrated. The effort scheme is determined to maximize the profit of all members as follows: m

Max e

C

=

i ( i,

ei ) f ( i ) d i ,

(13)

i=1

where the superscript “C ” denotes the centralized model. Lemma 1 describes the optimal allocation scheme in a centralized system. Lemma 1. The centrally coordinated system has a unique optimal effort allocation scheme, i.e., e = ( e 1 , …, e m) , which is jointly determined by the first-order conditions. Proof. The Hessian matrix of 2

1

e12

H (e1, e2, …, em) =

C

f ( 1) 0

Because eigenvalues

i f ei2

0 2

2

e22

…

0

f ( 2)

.

0 0

2

is as follows:

…

0

2

m

2 em

f(

m)

(14)

( i ) , i = 1, …, m , of the Hessian matrix H are all negative (due to the concavity assumption on

i ( i,

ei ) ),

H (e1, e2, …, em) is negative definite. This ensures that the profit function is strictly jointly concave in ei . Hence, the optimal effort allocation scheme e is unique, which is jointly determined by first-order conditions. □ C

e represents the state in which the common agency attempts to help the express companies earn as much as possible from customers, regardless of individual payoff. For simplicity and without loss of generality, we assume that e i > 0 for all i = 1, …, m . This assumption can be satisfied by removing all principals i with e i = 0 . Nevertheless, the corresponding principals {i | e i = 0} remain in the coalition due to their contributions to the coalition, e.g., they enhance the scale effect by sharing ingoing packages. They are subsidized once other express companies, i.e., {i | e i > 0} , are selected by the customer. Furthermore, we find the following property of e . 6

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Lemma 2. At e , the first-order conditions of the revenue functions equal one another, i.e., j

( i, e j ) f ( i) d

ej for all 1

i

k

=

ek

( k, ek ) f ( k) d k,

(15)

m.

j, k

Lemma 2 is easy to prove. We do so by contradiction. If Eq. (15) does not hold, the agency reallocates some working time from a specific principal’s package collection process with lower marginal revenue to another with higher marginal revenue, thereby constructing a more profitable solution than e . Lemma 2 implies that the agency receives the same marginal payment in effort by the coalition, regardless of which courier is chosen. Taking the centralized system as a benchmark, we then investigate the decentralized case. 4.2. Decentralized system We explore the decentralized common agency system in this subsection. This subsection is organized as follows. First, we show how to design an optimal contract between the coalition and the agency and prove that the optimal contract is unique. Then, we solve the profit redistribution problem within the coalition. Finally, we guarantee that the proposed contracts can improve the performance of the decentralized system to the centralized level. For the optimal contract design problem, we solve for the coalition’s and the agency’s decisions by backward induction. For a given set of incentive contracts, the agency decides on the working time allocated to each principal. Then, the coalition maximizes its profit by setting the parameters in the contracts. For a given incentive contract w offered by the coalition, the agency’s problem is shown in Eq. (8) restricted by Eq. (9) and Eq. (11). By the Karush-Kuhn-Tucker conditions, the optimal e should satisfy the following first-order conditions:

U = Ai ei for 1

i

ei

( i , ei ) f ( i ) d

i

+ µ = 0,

(16)

m , where µ is the Lagrange multiplier. Eq. (16) and Lemma 2 imply the following proposition.

i

Proposition 1. To make the agency incentive compatible, the coalition should set an equivalent incentive rate for all payment functions, i.e., Ai = Aj , for all 1 i, j m . Proposition 1 shows that the coalition, as a unified decision maker, should set the same incentive rate for all payment functions. However, intuitively, the proposed incentive rate in Proposition 1 appears unfair since it induces the agency’s effort on a particular principal with easier or more profitable express service. Nevertheless, it is incentive compatible and profitable for the coalition’s total payoff as a whole, which could be subsequently redistributed among members in the coalition. Moreover, due to the concavity of i ( i , ei ) in Assumption 1, more profitable jobs may become less profitable along with increasing effort, so that the agency will not spend all its effort on a particular principal. It is better for the agency to distribute effort following Proposition 1. The coalition anticipates the reaction of the agency regarding the choice of e . It is straightforward to show that the coalition’s profit is maximized when w (A , ) satisfies the following first-order conditions of the Lagrangian function:

(1

)

i ( i,

ei ) f ( i ) d

i

=0

(17)

for all 1 i m . The dual feasibility condition is (18)

0. The complementary slackness condition is

(U (w , e )

(19)

U0 ) = 0.

Since the agency’s effort generates a positive payoff, i.e., i ( i , ei ) > 0 , we have dition must bind. To that end, we have the following corollary.

> 0 from Eq. (17). Thus, the slackness con-

Corollary 1. The relationship between the parameters of the optimal incentive contract and the agency’s effort scheme is

Ai =

m

U0 i ( i,

ei ) f ( i ) d

,

for all i = 1, …, m.

i

(20)

i=1

Corollary 1 above provides an approach to calculate the optimal incentive rate. Let w ( ) = ensures that the calculated result is the unique optimal solution.

m i=1

Ai

i.

The following proposition

Proposition 2. In the decentralized system, the principals have a unique optimal payment scheme w ( ) , and the agency has a unique optimal effort allocation scheme e . 7

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Proof. Lemma 1 finds the agency’s unique optimal effort allocation scheme e in the centralized system. With e , we obtain the unique payment scheme between the coalition and the agency using backward induction in Proposition 1 and Corollary 1 for a decentralized system. Therefore, we can intuitively prove Proposition 2. □ Proposition 2 sheds light on how to coordinate the relationship between the coalition and the agency. From Proposition 2, we find that the agency’s effort allocation scheme in the decentralized system could be the same as that in the centralized system, as long as a compatible incentive is offered to the agency. Thus far, we have obtained the optimal profit for the coalition as a whole. Next, we need to redistribute the profit within the coalition. Derived from Eq. (10), we have

(1 for all 1

1

i

aii ) i, j

aii =

ei

( i , ei ) f ( i ) d

m and j

i

=

aij

j

ej

( j , ej ) f ( j ) d

j

i . Furthermore, by Lemma 2, we have (22)

aij .

Denoting ri = 1

(21)

aii =

aij and substituting aii and aij into Eq. (2), we have

m

Vi (wi , ei ) = ri

j ( j,

ej ) f ( j ) d j.

(23)

j=1

Eq. (23) shows that each express company earns a fixed proportion ri , i = 1, …, m , from the revenue of the coalition in a stable common agency model. Thus, we have the following proposition. Proposition 3. The optimal contract among principals in the coalition is a profit-sharing contract. Proposition 3 illustrates that a profit-sharing contract can facilitate the redistribution of profits among members in the coalition. In practice, the principal coalition announces an ex ante profit-sharing contract, the parameters of which, ri , i = 1, …, m , can be determined through bargaining. In bargaining, the equilibrium choices of the related parameters depend on the relative importance of the participating members. Extensive studies (Shapley, 1953; Cleophas et al., 2019) provide approaches to derive these choices in the context of a coalition. Assuming that these parameters are given, this paper focuses on the implementation of the agreement. During contract execution, a principal is likely to take hidden action, i.e., offer an ex post incentive contract demonstrated in Fig. 1, to influence the agency’s effort allocation by offering a higher marginal profit. The profit-sharing contract is able to prevent such ex post moral hazard for principals. Proposition 3 also suggests that express companies should form a coalition in an output-observable way, e.g., creating a joint venture such as FeiMa Express. Note that the common agency also earns a fixed proportion Ai of the coalition’s revenue. In light of Proposition 2 and Proposition 3, we conclude: Proposition 4. The profit-sharing contract and incentive payment scheme w ( ) can effectively coordinate the decentralized common agency system. Proof. Proposition 2 finds that there exists a unique payment scheme w ( ) inducing the agency’s effort allocation scheme e and total expected profit of all players that are identical to those in the centrally coordinated system. Moreover, Proposition 3 indicates that w ( ) requires a profit-sharing contract among principals. Therefore, we can intuitively prove Proposition 4. □ Existing studies on the contract between a common agency and competing principals assume that the goods of principals are partially substitutable (Chen et al., 2019b). In contrast, Proposition 4 shows that a coalition among principals (express companies) is possible in LML even if their products are completely substitutable. Moreover, members of the coalition can be coordinated with a profit-sharing contract. Proposition 4 provides clear managerial implications for cooperation between express companies and local LML providers in practice. It also vindicates business evidence provided in the introduction regarding why City 100 Logistics, which is an independent local LML provider, only receives limited collection packages from its partners, while FeiMa Express, which is a joint venture of express companies, outperforms. The following subsection demonstrates how to implement the above findings in industrial practice. 4.3. The application of the optimal contract In the practical implementation of the common agency model, contract design is required both within and outside the coalition. Based on the above findings, we propose the following steps to construct the optimal incentive contracts. Step 1: Estimate the revenue functions of express companies related to the agency’s effort, i.e., i ( i , ei ) for i = 1, …, m . These functions can be constructed by either using expert knowledge or fitting the curve to historical data. Step 2: Find e . According to Lemma 1 and 2, the optimal effort scheme in the centrally coordinated system, e , is uniquely determined by the following set of equations:

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e i = 1, i=1 j

ej

( j , ej ) f ( j ) d

j

=

k

ek

( k, ek ) f ( k) d k,

for all 1

j, k

m.

(24)

Step 3: Determine the optimal incentive payment rate Ai , i = 1, …, m , offered to the agency, according to Corollary 1. Before this, the agency’s reservation utility U0 for the common agency model needs to be determined. It depends on his original payoff and the savings resulting from the common agency model. Section 5 provides an example of calculating U0 . Step 4: Coalition members negotiate on how to distribute the profit after paying the common agency. The profit distribution scheme is determined by the profit-sharing contract’s parameters ri , i = 1, …, m , which depend on the express companies’ relative market power. A widely accepted real-world approach at present is to divide profits among express companies simply by their volumes of packages to be delivered. However, other company characteristics, such as reputation, logistics networks, and marketing, also affect the receipt volume and could be considered in a sophisticated negotiation. 5. Experimental study Our theoretical results in Section 3 and Section 4 provide an approach to design incentive contracts to coordinate the coalition and the agency, as well as redistribute the remaining earnings among coalition members in a general form. In this section, we employ practice-related data to investigate the impacts of other important factors on the optimal contract design. Actual industrial practice in China has authenticated the benefits of the common agency model in LML. In Pei County, the model has reduced occurrences of delivery delay, package loss and consumer complaint by 80%, 96%, and 60%, respectively. In addition, it saves 0.4 million dollars per year in terms of venue rent, labor and vehicles.3 In Huizhou, the efficiency and salary of couriers have increased by 35% and 25%, due to the 50% increase in packages delivered.4 Thus, our numerical analyses in this section are designed to understand the effects of the model parameters on incentive contract design to generate further managerial insights instead of discussing how much benefit the model can deliver. For this reason, we first describe real-world data from practice and design the setting of the numerical study. Next, we conduct sensitivity analysis and present managerial insights in greater detail. 5.1. Data The following data and information are collected either via the Internet or interviews with employees of related companies. 5.1.1. Components of express fees According to the data shown in Fig. 2 from National Business Daily of China (NBD), the average express fee charged to the customer is approximately 10 yuan. From package receipt, the agency earns 3 yuan, and the other 1.9 yuan is spent on the express agent’s commission and the cost of shipping packages to the distribution center. Regarding package delivery, 1.5 yuan per package is paid to the agency at the package’s destination. The express company earns the remaining 3.6 yuan for transportation and distribution management, with a cost of 1.92 yuan.5 Therefore, the following experiments set the revenue per receipt of express companies at 4.68 = 3 + 3.6 1.92 , which includes the incentive paid to the agency during the package receipt process. 5.1.2. The Pei County case Five express companies focusing on nationwide transportation and distribution of parcels jointly outsource LML in Pei County to FeiMa Express, which is a joint venture and a shared LML service provider. The five express companies collect 30 thousand units of packages from customers on a daily basis. After they hired the shared agency for package delivery and receipt, the common agency model saved 10 thousand yuan for the shared agency on vehicles, labor and rent per day. This real-world case sheds light on the following experimental design and setting. 5.2. Experiment setting Consider an area with Q thousand units of packages to be collected on a daily basis. In the past, m express companies exclusively hired individual LML service providers. The agencies earned P yuan per package receipt, a total of PQ thousand yuan from the express companies in the context of exclusive agencies. On the other hand, the express company earns 4.68 yuan per package received from a customer on average. Now, the m express companies consider hiring a shared agency to make full use of economies of scale. If hired, the shared agency will save S thousand yuan on package delivery and receipt. 3

http://www.px.gov.cn/px/zhxw/201808/2a686c69dc8445879aa7f921edebd92c.shtml. https://www.iyiou.com/p/68233.html. 5 The cost is calculated by NBD according to the 2017 annual report of YTO Express. 4

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Fig. 2. The components of express fee.

5.2.1. The performance function i The well-known Cobb-Douglas production function is adopted to characterize the relationship between the agency’s effort and principals’ revenue, i.e., i ( i , ei ) = i Pi Ci ei , 1 i m . To characterize the heterogeneity among principals, we incorporate Pi and Ci in i . Specifically, the profitability parameter Pi is the earnings of the agency and principals for each package express company i receives. Ci measures how difficult it is for the agency to obtain a customer’s package for express company i. It depends on express company i’s advertising, service, and promotion, among other factors. The power of ei is 0.5, indicating that the production function has decreasing returns to scale. In addition, a random variable i uniformly distributed in the range [0, 1], i.e., i ~U [0, 1], is included to represent the uncertainty of the outcome. For example, a negative event (a macroeconomic downturn) could decrease the realization of i . 5.2.2. Other parameters It is important to know how the optimal incentive contract varies in different scenarios that consider the total volume of receipts in the area, the structure of the coalition (including the number of express companies m and the degree of member heterogeneity), and the agency’s reservation utility (which depends on P and S). Experiments are designed by adjusting the values of these parameters as follows. (a) Coalition Structure Cases 1 and 2, which consider homogeneous members, and Cases 3 and 4, which consider heterogeneous members, are explored to determine the effect of coalition structure. We take Pi and Ci as parameters to characterize the heterogeneity among members in the coalition. In all cases, it is necessary to preset values of Ci related to the expected total receipt volume Q as follows: m

E [ i] Ci ei = Q.

(25)

i=1

Parameter settings are listed in Table 2. In the homogeneous cases, Pi is set at 4.68 provided in Section 5.1.1, and Ci is determined as Ci = 2Q / m from E [ i ] = 0.5 and ei = 1/ m . In the heterogeneous cases, we proxy for the heterogeneity of express companies using the sales volumes of ZTO Express (8.5 billion packages), YUNDA Express (6.8 billion packages), YTO Express (6.7 billion packages), BEST Express (5.5 billion packages) and STO Express (5.1 billion packages) in 2018. After normalizing, we obtain the following: C1 = 1.15Q , C2 = 0.92Q , C3 = 0.9Q , C4 = 0.74Q , and C5 = 0.69Q . Case 3 still sets all Pi at 4.68, while Case 4 sets Pi differently under the condition that the average per receipt is 4.68. (b) Receipt Volume The total receipt volume Q is critical for the feasibility of the common agency model in a given area. The model mainly benefits the agency in the delivery stage, and the agency may have distinct attitudes towards the incentive provided by the coalition under different levels of demand. Therefore, we use a set of values of Q from 10 to 50 to determine its effect on the incentive contract. Cases with low or high receipt volumes are not included because contract design is not necessary for moral hazard in a nonreceipt scenario (Q = 0 ). Moreover, a high-volume case, i.e., Q > 50 case, is also not considered since the common agency model does not work in areas with high receipt volume. For example, Yiwu City of Zhejiang, China, which is called China’s largest small commodities market, exports daily use products to more than 200 countries and areas. In these areas, express companies have Table 2 Parameter settings in Cases 1–4. Parameters

Case 1

Case 2

Case 3

Case 4

m Pi ei Ci/Q

2 4.68 0.5

5 4.68 0.2

5 4.68 0.33, 0.21, 0.20, 0.14, 0.12

5 4.78, 4.5, 4.6, 4.7, 4.8 0.34, 0.19, 0.20, 0.14, 0.12

1.41

0.94

1.15, 0.92, 0.90, 0.74, 0.69

1.15, 0.92, 0.90, 0.74, 0.69

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Fig. 3. The effect of receipt volume and coalition structure.

limited market power. Instead, the local agency controlling customer resources (a small commodity seller) has more bargaining power. The common agency model does not work in such a case. (c) Reservation Utility The agency’s reservation utility is U0 = W0 Q S thousand yuan in our common agency model. Only if the agency obtains an offer in excess of the reservation level U0 is he willing to accept the contract and thus participate in the common agency model. Thus, both W0 and S affect the agency’s adoption. Section 5.1.1 shows an average receipt fee, i.e., 3 yuan, while the exact amount W0 varies from agency to agency in an exclusive delegation scenario. Similarly, savings S may be different from those in Pei County, i.e., 10 thousand yuan, given in Section 5.1.2. Therefore, we vary W0 from 2 to 4 and S from 0 to 20 to investigate the effect. 5.3. Results and managerial insights After executing the experiments designed in Section 5.2.2, we report the sensitivity analysis results in Figs. 3 and 4. They demonstrate the effects of receipt volume, coalition structure, and the agency’s reservation utility on the optimal contract design. Moreover, we discuss the impact of the randomness of . 5.3.1. The effect of receipt volume In Fig. 3(a), the x -axis denotes the total receipt volume of m express companies, and the left y -axis and right y -axis represent the incentive rate and amount the coalition pays to the agency for each receipt, respectively. The results show the following: Remark 1. The express companies need to offer the common agency a higher unit payment if the total receipt volume Q increases. The insight is that the common agency model benefits the agency enough in the delivery activity such that the agency would be willing to compromise earnings from receipt activity. Moreover, an increase in Q makes the agency less willing to compromise and raises the unit payment. This coincides with the views of practitioners we collected in the survey. The common agency model is easier to adopt in areas with high-volume delivery and low-volume receipt.

Fig. 4. The effect of the agency’s reservation utility. 11

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5.3.2. The effect of coalition structure We also find that Cases 1–4 share the same incentive rate from Fig. 3(a). This implies the following: Remark 2. The structure of the coalition does not change the profit distribution between the coalition and the agency. Recall from Corollary 1 that our finding holds only if the coalition’s expected total profit remains the same. Although the heterogeneity of express companies does not affect the incentive scheme, it diversifies individual income as shown in Fig. 3(b). Compared with Case 2, income in Case 3 is re-distributed among the coalition members according to the profit-sharing contract stated in Proposition 3. From Fig. 3(b), we find that income is increasing in the market share of the express company. 5.3.3. The effect of reservation utility In addition to total receipt volume, the agency’s reservation utility also depends on original unit payment P of the exclusive agency and cost-savings S brought by the common agency model, which both linearly impact the incentive scheme as shown in Fig. 4. The results imply the following: Remark 3. The cost-savings generated by the common delivery model are the critical motivation for the agency, so that less unit payment is offered per receipt than in the scenario with exclusive agencies. Via cost savings, the agency’s delivery volume assigned by the coalition indirectly affects the incentive contract. The more packages are delivered, the more saving is possible. This paper derives the lowest incentive level the agency can accept according to his reservation utility. However, in practice, to maintain sustainable cooperation, the coalition would be willing to pay more after two-sided negotiation. 5.3.4. Discussion on All experiments above consider the incentive scheme when forming a coalition; consequently, the coalition and the agency know the expectation of , not its realization. In the execution of the contract, the realization inevitably deviates from the expectation. As a result, some express companies may receive more orders than expected in a contract period, and some may receive less. Hence, we discuss the effect of deviations in ’s realization here. Recall that the combined contracts we proposed in which the payoffs of all players depend on the realization of the coalition’s total receipt volume rather than ’s realization for individual express companies. If ’s realization makes the total receipt volume exceed (less than) the estimated, the agency is paid less (more) than it deserves since a higher incentive rate should have been offered according to the results in Fig. 3(a). Although a good realization of makes all players better off, a high (low) realization makes the agency underpaid (overpaid). This reduces the agency’s motivation to help express companies achieve a better realization of . An additional incentive contract proposed by Sinclair-Desgagné (1999) can resolve this issue and induce the agency to devote more effort on behalf of the coalition. 6. Conclusion 6.1. Concluding remarks and managerial implications In this paper, we establish a theoretical model to investigate the common agency model in last-mile cooperative urban logistics. In the considered system, express companies are modeled as principals, and a local logistics service provider is the common agency. Principals need to form a coalition to fully exploit the economies of scale offered by the common agency model. However, because some principals may take hidden actions to motivate the agency to devote more working time to that principal’s package collection jobs, contracts should be designed to guarantee the sustainable operation of the coalition. We explore the incentive contract between the coalition and the agency and the cooperation agreement among coalition members. Some interesting findings, insights and managerial implications are summarized below. First, we find that the optimal contract among express companies in the coalition takes the form of profit-sharing, which is the only way to address their fairness concern. This finding is an important implication for operations managers of express companies when choosing a mode of cooperation. It also vindicates our conjecture for why City 100 Logistics, which is an independent LML service provider, receives limited delegated jobs from third-party express companies, while FeiMa Express, which is a joint venture of express companies, performs well. Second, we find that the coalition should set equivalent incentive rates for all jobs delegated by express companies to the shared agency. The optimal incentive rate depends on the agency’s reservation utility and the total revenue of the coalition. Our finding suggests that principals do not need to worry about the agency’s preference for a particularly easy or profitable express service. It is incentive compatible and maximizes the coalition’s total expected in total. Our profit redistribution mechanism allows all express companies to be better off. Third, we find that our proposed method is applicable to scenarios with heterogeneous express companies in terms of package volume, express price, and service level, among other factors. Operations managers could effectively coordinate various common agency systems using the incentive payment scheme and profit-sharing contract. The potential economic value of this research can be inferred from the rapid development of the recent parcel locker business, which is another form of CDP in LML. For instance, the parcel delivery locker pilot in China, Hive Box, was founded in 2015. By 2018, it had delivered an average of 9 million packages for express companies per day, with more than 120,000 branches. Parcel lockers eliminate the fairness concern of express companies by offering customers human-machine interaction (HMI) rather than human12

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human interaction (HHI) for courier selection. However, the HHI mode is more flexible than the HMI mode. It can grow faster as long as the moral hazard problem is resolved. 6.2. Limitations and future studies Our study is the first that analytically examines the use of contract design models to reduce the ex post moral hazard in last-mile logistics. Nevertheless, our model has several limitations and extensions that could be addressed in future research. First, we investigate the optimal ex ante contract design, where the level of incentives is determined at the beginning of a contract period. However, the realized receipt volume is inevitably different from the original estimation, and as a result the agency is not paid the appropriate amount. In that setting, a more extensive computational experiment is needed to fully explore the dynamics of the system. Second, we assume that the performance function is independent of the agency’s effort cost. However, express companies want an agency to invest more in delegated jobs. It would be worthwhile to investigate how to induce greater effort by the agency in a fair context. Third, we assume that the agency’s cost savings resulting from common delivery are public information. In reality, the agency would benefit from reporting lower savings. Finally, we assume that multiple principals and only one agency co-exist. An extension into multiple competing agencies would be interesting and challenging. How will the competition among agencies impact the optimal contract? We will address these research questions in the future. CRediT authorship contribution statement Xiang Chu: Conceptualization, Methodology, Writing - original draft. Jun Liu: Visualization, Writing - review & editing. Long Ren: Formal analysis, Writing - original draft, Writing - review & editing. Daqing Gong: Resources, Writing - review & editing, Validation. Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 71802037); Fundamental Funds for the Ministry of Education (MOE) Humanities and Social Sciences Research Youth Foundation, China (No. 19YJC630137); the Fundamental Research Funds for the Central Universities (No. CXTD10-05, CXTD11-04, 3132020231). Finally, we thank the anonymous reviewers for the constructive comments. References Akeba, H., Moncef, B., Durand, B., 2018. Building a collaborative solution in dense urban city settings to enhance parcel delivery: An effective crowd model in paris. Transp. Res. Part E 119, 223–233. Amaya, J., Arellana, J., Delgado-Lindeman, M., 2020. Stakeholders perceptions to sustainable urban freight policies in emerging markets. Transp. Res. Part A 132, 329–348. Anily, S., 2018. Full characterization of the nonnegative core of some cooperative games. Naval Res. Logist. 65 (4), 303–316. Apex Insight, 2019. 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