Unit VII J

Truthful multi-unit double auction with transaction costs and sellers’ changing marginal costs

Jiantao Guo a, Juliang Zhang a,*, T.C.E. Cheng b

a Department of Logistics Management, School of Economics and Management, Beijing Jiaotong University, Beijing, 100044, China b Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

A R T I C L E I N F O

Keywords: Procurement mechanism Double auction Platform economy Changing marginal cost Transaction cost

A B S T R A C T

It is well-known that lots of sellers offer discounts for large orders because of decreasing marginal costs and large productive capacities. Then some procurement platforms provide ways for small buyers to cooperate with their procurements and get discounts from these sellers. Moreover, these transactions will incur pair-related trans- action costs. Generally, buyers values and sellers’ costs are private information. Then a critical problem faced by the platforms is to induce the buyers (sellers) to reveal their values (costs) truthfully and to match the supplies and demands efficiently. To solve this problem, we design a multi-unit double auction mechanism based on the padding method where the buyers bid their values and the sellers submit their marginal costs of different quantities, and the platform selects the winners, allocates the transactions, and sets the transaction prices for both sides. We show that the mechanism is budget-balanced, individually rational, incentive-compatible, and asymptotically efficient. For the special case with zero transaction costs, we further simplify the mechanism and get some new findings. We further conduct numerical studies to compare our mechanism with four commonly used mechanisms to demonstrate its advantages and examine the impacts of some parameters on the perfor- mance of our mechanism. Finally, we consider the cases with general marginal cost structure as well as private quantity information, respectively.

1. Introduction

The platform economy has developed fast recently. By the beginning of 2020, there were 74 platform-based companies with a global market value of over $10 billion, a total value of $8.98 trillion, up 41.8% year- on-year (Smart Research Consulting, 2022). These platforms attract a significant number of buyers and sellers to trade their products or ser- vices because of efficient and fast transactions, and rich resources. However, these platforms also raise fierce competition among sellers (Evans et al., 2011; Li et al., 2021; Xiao et al., 2021). To attract buyers and increase sale volume, many sellers are willing to offer discounts for large orders (e.g., 56.1688.com; www.jd.com). Munson and Rosenblatt (1998) pointed out that the main reason for sellers to offer quantity discounts is their decreasing marginal costs brought about by scale economies. The scale economies are common in various industries, e.g., transport (Gansterer et al., 2020), industrial products (Ebadi and Madani, 2006), high-tech manufacturing (Hsu and Li, 2009), and con- sumer goods (Andrade et al., 2021), etc. The other reason is that many sellers have large productive capacities, therefore they want to sell more

by offering discounts (Munson and Rosenblatt, 1998). Generally, smaller and medium-sized buyers cannot benefit from such price concessions because of their small demands. To circumvent this problem, some on- line platforms provide ways for these buyers to cooperate with their procurements, i.e., to combine their demands to form a large order and then get discounts from the sellers. For example, the less-than-truckload (LTL) transport platform, 56.1688.com, increases its bargaining power by aggregating small member shippers’ demands to get lower quotes from carriers. Some e-commerce platforms (e.g., groupon.com; PDD platform) offer consumers group-buying opportunities to get lower seller discounts (Zhao et al., 2012). These platforms can help buyers lower their procurement costs and sellers sell more, which contributes to a win-win scenario for both sides and improve the social welfare (Liang et al., 2020).

In these bilateral markets, the procurement platforms need to match demands and supplies efficiently and set proper transaction prices. Generally, the buyers’ value information and the sellers’ cost informa- tion are private. Under incomplete information, double auctions are often adopted for the platforms to make these decisions (Myerson and

* Corresponding author. E-mail address: zhangjl@bjtu.edu.cn (J. Zhang).

Contents lists available at ScienceDirect

International Journal of Production Economics

journal homepage: www.elsevier.com/locate/ijpe

https://doi.org/10.1016/j.ijpe.2024.109430 Received 15 November 2023; Received in revised form 22 August 2024; Accepted 2 October 2024

Int. J. Production Economics 278 (2024) 109430

Available online 5 October 2024 0925-5273/© 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Satterthwaite, 1983; McAfee, 1992; Chen and Tseng, 2010). There are some studies investigating the double auction mechanisms considering the sellers’ decreasing costs in various bilateral markets such as e-commerce markets (Zhao et al., 2012), cloud computing markets (Sun et al., 2014), B2B procurement markets (Chen and Huang, 2018), and transport service procurement e-marketplaces (e.g., Liang et al., 2020). However, they do not consider transaction costs and neglect the impacts of the transaction costs on the mechanism design. As we know, trans- action costs are common in the transactions in these marketplaces. For example, on 56.1688.com, a carrier posts the same price for shippers for the LTL transport service on a fixed transport lane. When a transaction is executed, the carrier has to pick up the goods from the shipper’s location or the shipper has to deliver the goods to the carrier’s location, incurring an additional cost for the transaction. This cost can be regarded as the transaction cost in transport service procurements (Guo et al., 2022). In addition, the transactions of goods on online procurement platforms may incur transaction costs related to shipping costs from sellers to buyers (Babaioff et al., 2004; Chu and Shen, 2006).

Transaction costs should be considered when designing a transaction mechanism. If we ignore them, the mechanism efficiency will be reduced and the transaction cost bearers may be impaired. We can use a simple example to illustrate this. Consider a bilateral procurement market with three buyers, three sellers, and a platform. Buyers 1, 2, and 3 each want to buy one unit of the item with the values of 6, 5, and 4.5, respectively. Sellers 1, 2, and 3 can each provide one unit at the costs of 3, 3.5, and 4, respectively. When a buyer buys the item from seller 1, the platform has to cover 1.4 of the transaction costs associated with the delivery. From the classic McAfee auction mechanism (McAfee, 1992), the deal with the least profit is eliminated from the optimal allocation, and the buying and selling prices are determined based on the elimi- nated bid and ask, respectively. Under this mechanism, if the transaction costs are ignored, buyers 1 and 2 will buy one unit at 4.5, and sellers 1 and 2 will sell one unit at 4, respectively. Then the social welfare real- ized is 3.1 and the platform will lose 0.4. If we consider the transaction costs, buyers 1 and 2 can also buy one unit at 4.5 while sellers 2 and 3 will each sell one unit at 4.4. Then the realized social welfare increases to 3.5 and the platform can get a profit of 0.2, no longer subsidizing the transactions. This shows that the transaction costs play an important role on the auction mechanism design.

Motivated by the gap between the theory and practice mentioned above, we focus on the double auction design for a procurement plat- form (as a market broker) where multiple buyers and sellers (both called agents) want to trade multiple units of the item. The buyers have the same value for each unit of the item and the sellers have changing marginal cost structures (we first focus on the decreasing marginal costs and then discuss the general marginal cost structure). The values and costs are the agents’ private information but the demand and supply quantities are public information. When a buyer buys the item from a seller, a pair-related transaction cost is incurred. The transaction cost includes the costs associated with order handling, transport, and the buyer-seller relationship, which are absorbed by the platform1 (Chu and Shen, 2008; Guo et al., 2022). The platform pools the buyers’ demands to form a cooperative procurement and facilitates the transactions be- tween the buyers and the sellers. Each buyer submits a sealed multi-unit bid and each seller submits a sealed multi-unit ask that the price is related to the transaction quantity (called discriminatory bidding) to the

platform. The problem faced by the platform is to design a double auction mechanism to select the winning agents, allocate the trans- actions, and set the transaction prices.

It is well-known that a good mechanism should have the following properties (Chu, 2009): (i) Efficient,2 (ii) Incentive-compatible (IC), (iii) (Ex post) Individually rational (IR), and (iv) (Ex-post weakly) Budget-balanced (BB). Individual rationality is a participation constraint for agents. The budget balance means that the platform can make a profit from the transactions, which would incentivize the plat- form to run the mechanism (Chu and Shen, 2008; Chu, 2009). Note that no bilateral exchange mechanisms can be efficient, IC, IR, and BB simultaneously (Myerson and Satterthwaite, 1983). In this paper we aim to design a double auction mechanism that is BB, IR, IC, and asymp- totically efficient (AsE). By AsE, we mean that the social welfare con- verges to the maximum one as the number of bidders approaches infinity (Chu and Shen, 2008). Designing such a double auction mechanism for our problem presents the following challenges. First, when the sellers have changing marginal costs, they need to bid their price vectors related to their transaction quantities. It is very difficult to induce them to tell the truth because this may result in complicated strategic mis- reporting of bids (Huang et al., 2002). Given that the payment rules should be the same for different buyers (sellers) with different values (costs), it is also difficult to develop a BB and IR payment rule to balance the profit of the platform and the agents’ utility. Second, when there exist pair-related transaction costs, even if two sellers offer the same price for the same product, they may be heterogeneous to a buyer. This distinguishes the mechanism from those for the sale of multiple, but identical items. The mechanism should not only match quantities but also determine specific transactional relationships. This makes the effi- cient allocation more difficult than in the case with zero transaction costs. In addition, the payment rule should also take into account the impacts of the transaction costs to ensure budget balance.

Even though the classical Vickery-Clarke-Groves (VCG) mechanism can provide feasible IR and IC payment rules, it is not BB even if the agents trade one unit of item. A straightforward way to ensure the budget balance is that the platform charges a fixed percentage of the market clearing price, which is commonly used in some two-sided platforms (e.g., Zhang et al., 2022; McHardy, 2024). This way needs the assumption that there is an explicit functional relationship between the buying (selling) price and the demand (supply). In our situation, however, this relationship is implicit and not clearly known by the platform. If this way is used directly in the auction, the agents will consider the impacts of the charging percentage when submitting their bids. Then it is hard for the mechanism to be incentive-compatible. As we know, there are two popular approaches for designing the IC and BB multi-unit double auction mechanisms: the “padding” method (Chu, 2009) and the trade reduction approach (McAfee, 1992; Huang and Xu, 2013). The “padding” method turns the original auction into another one where the supply exceeds the demand by introducing a “virtual buyer” with an unlimited budget. Then the mechanism can ensure incentive compatibility and budget balance when the sellers have con- stant marginal costs. However, the traditional padding mechanism is not BB when the sellers have decreasing marginal costs. This is because, under its payment rules, the winning buyers pay the lowest marginal

1 Note that even if the transaction costs are shared by the agents, the platform can set the transaction prices that account for the transaction costs and then subsidies the realized ones to the agents.

2 As we know, most of the studies on the double auction aim to design effi- cient mechanisms to maximize the social welfare rather than to maximize the platform’s profit. There are three main reasons for this: First, as pointed out by Wise and Morrison (2000), the larger the market efficiency is, the more long-term profits an auction mechanism can generate for the platform. Second, an efficient auction mechanism focuses more on the agents’ utilities, which can attract more potential buyers and sellers and retain them, thus enhancing the competitiveness of the platform in a competitive market. Finally, a welfare-maximizing platform is preferred and supported by governments and regulators (Fershtman and Pavan, 2022).

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cost component of the eliminated seller while the winning sellers sell at the highest marginal cost component of the eliminated seller. As for the trade reduction approach, Liang et al. (2020) adopted it to propose a trade reduction mechanism with a quantity discount (TR-QD) for the case the sellers have decreasing average unit costs. However, their mechanism cannot be applied in the case with the pair-related trans- action costs because it is very difficult to set proper transaction prices. In addition, their mechanism only matches the transaction quantities but does not determine the buyer-seller transactional relationship, which will incur high transaction costs and thus reduce the efficiency. There- fore, it is a challenging problem to design a double auction mechanism that is BB, IR, IC, and AsE in our situation.

To address the challenge, we propose theMulti-unit Double Auction mechanism with Discriminatory Bidding (we call it the MDA-DB mechanism) by adopting the padding method. We introduce the different supply-side padding vectors related to the transaction costs rather than a demand-side padding and design the different pricing rules. These can overcome the shortcomings of the traditional padding method in the case with transaction costs and the sellers’ changing marginal costs. We show that the mechanism is IC, BB, IR, and AsE. We also show that the platform can manipulate the padding to maximize the social welfare or its profit. To get better theoretical properties and more findings, we further simplify the MDA-DB mechanism for the special case with sellers’ changing marginal costs and zero transaction costs. We show that the simplified mechanism can reduce the computational burden and further give bounds on the agents’ payment transfers and efficiency loss. We further examine how the transaction cost and the cost structure affect the performance of the MDA-DB mechanism, respec- tively. To see the advantages and disadvantages of the mechanism, we conduct numerical studies to compare our mechanism with the other four commonly used mechanisms, namely the double VCG mechanism, catalog mechanism, MSDA-DB mechanism (based on the multi-stage design approach), and TR-QD mechanism in the case with zero trans- action costs. At last, we consider the cases with general marginal cost assumption and private quantity information assumption, respectively.

Compared with the existing literature, this study has the following contributions.

(1) Although there are some studies on the double auction mecha- nisms considering the sellers’ changing marginal costs, they do not consider transaction costs. We consider the case with pair- related transaction costs in a bilateral procurement market where the sellers have changing marginal costs. We propose a multi-unit double auction mechanism with good properties (IC, BB, IR, and AsE) for it. The mechanism outperforms the catalog mechanism, usually used in practice, in terms of the social wel- fare, trading volume, and sellers’ utility.

(2) We extend the padding method to the case with transaction costs and the sellers’ changing marginal costs by introducing different supply-side padding vectors related to the transaction costs and designing different pricing rules for it. In addition, the mecha- nism with further modification can apply to the case with private quantity information.

(3) For the case with sellers’ decreasing marginal costs and zero transaction costs, we simplify the mechanism to reduce its computational burden. The mechanism adopts a differential payment rule for the buyers, which can facilitate cooperative procurement. In addition, the numerical studies show that our mechanism has more advantages in the efficiency and trading volume than the TR-QD mechanism proposed by Liang et al. (2020) in this case.

We organize the remaining sections as follows: Section 2 reviews the pertinent literature to illustrate the research gap and position this work. We discuss a bilateral procurement market in Section 3. We present the MDA-DB mechanism in Section 4. In Section 5 we simplify the

mechanism for the case with zero transaction costs. The numerical studies are conducted in Section 6. We discuss the relaxation of two key assumptions in Section 7. The conclusion and some future topics are given in Section 8. The proofs are presented in Appendix A.

2. Literature review

In this paper, we study the problem of designing a truthful multi-unit double auction mechanism considering transaction costs and sellers’ changing marginal costs. There are three streams of literature related to this study: the auctions with transaction costs, the auction mechanisms with sellers’ changing marginal costs, and the multi-unit double auction mechanism design.

2.1. Auctions with transaction costs

The first stream related to this study is on the auctions with trans- action costs because we consider the transaction costs between the buyers and the sellers in the auction. Noussair et al. (1998) showed that the transaction costs would reduce the transaction volume and market efficiency in auction markets. Some works examined one-sided auctions (forward auction and procurement auction) considering transaction costs. Nie (2012) studied an auction market for mobility credits where the buyers need to pay the transaction costs in addition to their bid prices. Zhang et al. (2013) incorporated the transaction costs (com- missions) into the forward uniform-price auction and discriminatory-price auction for water right transactions, and found that the transaction costs would reduce the seller’s revenue under both mechanisms. Xu et al. (2015) proposed two procurement combinatorial auction mechanisms for B2B e-commerce platforms to minimize the sum of intermodal service costs and transaction costs. Li et al. (2024) designed an IR and BB Critical Neighborhood Diffusion Auction mech- anism for intermediated markets, where the transactions are processed by intermediaries and incur transaction costs such as commissions and transportation costs.

Some literature investigated double auctions with transaction costs. Guo et al. (2022) designedmulti-unit double auctionmechanisms for the cases with homogeneous and heterogeneous transaction costs, respec- tively, where homogeneous transaction costs mean those that are un- related to the transactional relationship while the heterogeneous transaction costs refer to the pair-related transaction costs between buyers and sellers. Considering the homogeneous transaction costs associated with the transaction quantity, Fang et al. (2012) introduced the normalized prices in the double auction design, which integrates the bidding prices with the transaction costs. Sun et al. (2019) designed double auction mechanisms for intermodal transport service pro- curements with the homogeneous transaction costs related to transport lanes.

In this study, we design a double auction mechanism for the case with pair-related heterogeneous transaction costs. Considering the het- erogeneous transaction costs due to different shipping costs between markets, Babaioff et al. (2004) proposed a single-unit trade-reduction mechanism. Chu and Shen (2006) considered a similar problem and designed two agent competition double auction mechanisms based on the multi-stage design approach. Chu and Shen (2008) further studied this problem and proposed two double auction mechanisms for the bilateral procurement market where the buyers want to procure a bundle of items while the sellers supply a single unit of the item. The above works assume that the sellers have a single unit of supply or multi-unit supplies with constant marginal costs. Different from them, we consider the case that the sellers have changing marginal costs in the supply quantity and can submit discriminatory bids in the auction.

2.2. Auction mechanisms with sellers’ changing marginal costs

As we mentioned before, we shall design a double auction for the

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

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case that the sellers have changingmarginal costs. The second streamwe reviewed is on the auction mechanisms with sellers’ changing marginal costs. Burke et al. (2008) and Bichler et al. (2011) investigated the winner determination problem in procurement auctions with quantity discounts. They focus on the algorithm design under the assumption that the sellers bid their cost structures truthfully. Ahlqvist et al. (2022) gave a review of auction-based pricing mechanisms for electricity markets to handle producers’ non-convex costs consisting of marginal costs, start-up, and no-load costs.

Many works study the double auction mechanism design problems for the bilateral markets with sellers’ changing marginal costs. They can be divided into two substreams: one for the cases where the sellers have increasing marginal costs, and another considers the cases with coop- erative procurement or group buying where the sellers have decreasing marginal costs or decreasing average unit costs. In the first substream, Segal-Halevi et al. (2018) designed a direct multi-unit double auction mechanism, and Loertscher and Mezzetti (2021) proposed an indirect double clock auction mechanism. Zhang and Kong (2022) proposed an auction-bargaining mechanism for emergency material procurement, whereby the demand-supply matches are determined by the double auction, and the transaction prices and quantities are determined through bargaining.

In the second substream, Zhao et al. (2012) proposed two double auction mechanisms for group buying where the sellers have unlimited supplies and the buyers have indivisible demands. However their mechanisms are only buyer-truthful or seller-truthful. Considering that each seller has a limited supply, Sun et al. (2014) developed a buyer-truthful combinatorial double auction mechanism for cloud resource group buying where the sellers can provide price discounts according to group size. For the case with the indivisible demands and limited supplies, Chen and Huang (2018) proposed a truthful multi-unit double auction mechanism with quantity discounts. Liang et al. (2020) relaxed the assumption of Chen and Huang (2018) that the buyers’ de- mands are indivisible and proposed a truthful multi-unit double auction mechanism for cooperative procurement of transport services. Milgrom and Watt (2022) designed two mechanisms for the case that the sellers can submit non-convex costs. They showed that they are BB, IR, and AsE, but only truthful in a very large market.

Our study is relevant to the second substream. We consider the case where the sellers have decreasing marginal costs and design a truthful, BB, IR, and AsE mechanism for it. Different from them, we also consider positive pair-related transaction costs between the buyers and the sellers while the above works assume zero transaction costs. Furthermore, we show that our mechanism is still truthful for the case where the sellers have a general marginal cost structure.

2.3. Multi-unit double auction mechanism design

The third stream of literature related to this study is on the multi-unit double auction mechanism design, which provides the methodology for us. It is well-known that there are three classical approaches for double auction mechanism design, i.e., the trade reduction approach (McAfee, 1992), the multi-stage design approach (Chu and Shen, 2007, 2008), and the padding method (Chu, 2009). Note that the mechanisms based on the multi-stage design approach are usually budget-imbalanced in the multi-unit cases (Chu, 2009; Cheng et al., 2016). Therefore we re- view here only the mechanisms based on the trade reduction approach and padding method.

McAfee (1992) first proposed the truthful McAfee mechanism using the trade reduction approach for a single-unit bilateral exchange envi- ronment. Huang et al. (2002) extended this approach to the case with multiple “divisible” units of an item. Huang and Xu (2013) designed three multi-unit trade reduction mechanisms for the logistics e-marketplace to determine the optimal exchange relationships. Xu et al. (2016) and Sun et al. (2019) designed double auction mechanisms based on this approach for the transport markets where the shippers purchase

multiple units (a bundle) of transportation services and the carriers provide bundles (multiple units) of transportation services. But their mechanisms are not truthful for the side with multi-unit demand or supply. Based on the trade reduction, Liang et al. (2020) proposed the TR-QD mechanism for the case where the sellers have decreasing average unit costs. Kong et al. (2021) proposed an online multi-unit double auction mechanism based on this approach for on-demand lo- gistics trading. Yu et al. (2022) extended this approach to the multi-attribute multi-unit case.

The padding method is proposed by Chu (2009), which introduces a demand-side padding vector to ensure the budget balance and truth- fulness for the multi-unit double auction mechanism. Chu (2009) pointed out that the mechanisms based on it have more advantages in social efficiency and individual payoff than that based on the trade reduction approach even in the case that each seller supplies a single unit of one item. Cheng et al. (2016) extended this approach to the multi-attribute multi-unit case. Considering parking slots are spatially heterogeneous, Cheng et al. (2022) proposed two padding-based VCG double auction mechanisms for the single-unit parking slot assignment problem.

In this study, we adopt the padding method to design the multi-unit double auction mechanism for the case with transaction costs and sellers’ changing marginal costs. Note that the traditional padding mechanism is not BB in a similar case even without transaction costs. We introduce different supply-side padding vectors related to the trans- action costs and design different pricing rules to ensure that the mech- anism is IR, IC, BB, and AsE. We also compare our mechanism to the TR- QD mechanism proposed by Liang et al. (2020) based on the trade reduction in the special case with zero transaction costs in Section 6.

3. The model

Consider a set I of buyers and a set J of sellers who procure and supply an identical commodity or service (item) on a procurement platform (we refer to a buyer as “she” and a seller as “he” in this paper). We refer to the minimum trade unit of the item as ‘‘one-unit item’’. Buyer i ∈ I wants to procure Xi units of the item and seller j ∈ J can supply Yj units of the item. Suppose that both supply quantity Yj and demand quantity Xi are public information. This is a common assump- tion in the literature on double auctions (e.g., Chu, 2009; Huang and Xu, 2013; Xu et al., 2016; Liang et al., 2020). We shall relax this assumption in Subsection 7.2. Buyer i ∈ I has a value vi for each unit of the item. Seller j ∈ J has a cost structure that the marginal cost is changing in his supply quantity. Let cqj be seller j’s marginal cost for the q-th unit of supply. We consider that the marginal cost of a seller is non-increasing in his supply quantity, i.e., c1j ≥ c2j ≥ ... ≥ cYj for each j ∈ J, which rep- resents economies of scale under the efficient scale. We shall discuss the case with the general marginal cost structure in Subsection 7.1. cqj and vi are the agents’ private information. When buyer i procures the item from seller j, a transaction cost dij is incurred. The transaction costs include the costs associated with order handling, transport, the buyer-seller relationship, etc., which are borne by the platform. Suppose that the transaction costs are public information (Chu and Shen, 2008; Guo et al., 2022).

The platform adopts a double auction to allocate the transactions and determine the transaction prices for both sides. We refer to “bid” and “ask” as the declarations of the buyers and the sellers, respectively. Buyer i submits a sealed bid {bid price (v̂i)} to the platform and seller j

submits a sealed discriminatory ask {ask price vector ( ( ĉqj

)

q∈{1,2,...,Yj} )}

to the platform. Assume that the bids and the asks are “divisible”, i.e., the demand of a buyer can be partially met and met by more than one seller and a seller can sell to more than one buyer with no more items than his supply capacity. This assumption is commonly used in the literature on the multi-unit double auctions, e.g., Huang and Xu (2013),

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Liang et al. (2020), Guo et al. (2022), and Yu et al. (2022). We also assume that there is no collusion among the agents.

Suppose that the self-interested agents submit their bids/asks to maximize their utilities. If an agent trades, its utility is the difference between the value (revenue) and the payment (cost); and if it does not trade, its utility is zero. The platform’s profit is the overall money received from the buyers minus the overall money paid to the sellers. The social welfare is the summation of the platform’s profit and the agents’ utilities.

What the platform needs to do is to design a double auction mech- anism (i.e., specify the bidding, winner determination, allocation, and payment rules) to maximize the social welfare, incentivize the agents to participate (IR), tell the truth (IC), and earn a profit (BB).

4. The MDA-DB mechanism for the case with pair-related transaction costs

4.1. Finding the maximal social welfare

If the agents reveal their information truthfully, we can solve the following integer program IP1(I, J) to find the maximal social welfare V(I,J):

IP1(I, J) : max(xi ,yj ,ηij)V(I, J)= ∑

i∈I xi v̂i −

∑

j∈J

∑Yj

q=1 yqj ĉqj −

∑

i∈I

∑

j∈J ηijdij

s.t. ∑Yj

q=1 yqj =

∑

i∈I ηij, j ∈ J, (1)

xi= ∑

j∈J ηij, i ∈ I, (2)

y(q− 1)j ≥ yqj , q∈ { 2, ...,Yj

} , j ∈ J, (3)

yqj ∈{0,1}, q∈ { 1,2, ...,Yj

} , j ∈ J, (4)

xi ∈{0,1, 2, ...,Xi}, i ∈ I, (5)

ηij ∈R, i ∈ I, j ∈ J, (6)

where xi denotes the trading quantity for buyer i, yqj denotes whether seller j sells the q-th item, and ηij denotes the quantity of the item pro-

cured by buyer i from seller j. Let yj = ( y1j , y2j , ...,yYj

) . Constraints (1)

and (2) guarantee the balance between the supply and the demand. Constraint (3) means that if seller j sells the q-th item, he must sell the n-th (n= 1, 2, ..., q − 1) item. Constraints (4)-(6) specify the types of de- cision variables.

Remark 1. In IP1(I, J), the platform can combine the buyers’ demands to form a large order for a supplier. From this, the buyers realize cooperative procurement and get the scale advantage. For example, there are two buyers with one unit of demand and one seller with two units of supply. Buyer 1’s value is higher than the seller’s marginal cost for the first unit and buyer 2’s value is lower than the marginal cost for the first unit but higher than the second unit. If they procure separately, buyer 1 can get one unit of the item while buyer 2 does not. In the allocation of IP1(I, J), both buyers can get the item.

Noting that there could be several optimum solutions to IP1(I, J), we employ Babaioff and Walsh’s (2005) perturbation method to pick just one. In this approach, each agent’s bid is adjusted for a perturbation factor. We consider their bids as v̂i + εi (ĉqj + εj) instead of v̂i (ĉqj ) for the buyers (sellers), where 0≪

( εj )

j∈J≪(εi)i∈I≪1. Then IP1(I, J) specified by the perturbation method will have a unique optimum solution, which is also the optimum solution to initial IP1(I, J) since ε≪ 1.

If the agents bid truthfully, the platform can solve IP1(I, J) to find the maximal social welfare. However, such centralized allocation schemes are often unusable because the agents have an incentive to misrepresent their private information. Therefore we propose the MDA-DB mecha- nism below to induce the agents to bid truthfully.

4.2. The mechanism

The double VCGmechanism is well-known to be efficient, IC, and IR. However, it is not BB even if the agents trade only one unit of the item. To ensure budget balance, we adopt the “padding” method proposed by Chu(2009) to design the MDA-DB mechanism. Different from Chu (2009), we create the gap between the demand and the supply by a group of padding vectors (Sk)k∈J which can be viewed as a group of “virtual sellers” with a supply of S at zero costs and different transaction costs. To make it easy to understand, we use a single-unit bilateral trading market with zero transaction costs to illustrate the idea, which is depicted in Fig. 1.

The original equilibrium transaction price and quantity (Q*) are determined by the intersection of the original demand and supply curves as the small black dot in Fig. 1. Padding S will increase the total supply and cause a rightward shift in the supply curve. It results in a new lower equilibrium price and a new larger equilibrium transaction quantity (Qʹ́ ). If we set the selling prices at the new equilibrium price, the transaction supply will decrease to Qʹ because the virtual seller does not trade. Keeping the supply and demand in balance, the buying price is correspondingly raised to a price higher than the original equilibrium price. From this, padding can make the platform get some surplus.

Specifically, the procedure of the MDA-DB mechanism is as follows.

Step 1. Collect bids and asks from the agents.

Step 2. Let the padding be S = H ( (Xi)i∈I,

( Yj )

j∈J

) , which is a non-

negative function of supplies and demands. Let Sk = ( Sjk )

k∈J denote

the supply padding vector for seller k, where Sjk = S if j = k, and otherwise Sjk = 0. The padding vector Sk can be viewed as a “virtual seller” with a supply of S at zero cost and the same transaction costs as seller k.

Step 3. The MDA-DB mechanism specifies whether and how much seller k ∈ J can sell by solving the following integer program IP2(I,J,Sk):

IP2(I, J,Sk) : max(xi ,yj ,ηij)V(I, J, Sk)= ∑

i∈I xi v̂i −

∑

j∈J

∑Yj

q=1 yqj ĉqj −

∑

i∈I

∑

j∈J ηijdij

Fig. 1. Supply padding in a single-unit bilateral trading market with zero transaction costs.

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s.t. ∑Yj

q=1 yqj + Sjk ≥

∑

i∈I ηij, j ∈ J, (7)

xi= ∑

j∈J ηij, i ∈ I, (8)

y(q− 1)j ≥ yqj , q∈ { 2, ...,Yj

} , j ∈ J,

yqj ∈{0,1}, q∈ { 1,2, ...,Yj

} , j ∈ J,

xi ∈{0,1, 2, ...,Xi}, i ∈ I,

ηij ∈R, i ∈ I, j ∈ J,

where constraints (7) and (8) balance the demand and supply with the padding vector Sk. Note that constraint (7) is an inequality rather than an equality because we allow

∑ i∈IXi < S in IP2(I,J,Sk). When

∑ i∈IXi ≥

S, ∑

i∈Iηij ≤ ∑Yj

q=1yqj + Sjk is equivalent to ∑

i∈Iηij = ∑Yj

q=1yqj + Sjk.

Because V(I, J,Sk) increases in ∑

i∈Ixi but decreases in ∑Yj

q=1yqj , the

optimal solution ( ( xki )

i∈I; ( ykj )

j∈J ; (

ηkij )

i∈I,j∈J ) to IP2(I, J,Sk) must satisfy

that ∑

i∈Ixki = ∑Yj

q=1ykj + Sjk even if we use an inequality constraint.

Step 4. Denote the winning seller set J̃ as the set of the sellers with ∑Yk

q=1ykqk > 0 in the optimal solution to IP2(I, J, Sk). In the next alloca-

tion, winning seller k ∈ J̃ sells either all of their supply qk = ∑Yk

q=1ykqk at

the total asking price Ĉqk = ∑qk

q=1 ĉqk or nothing.

Step 5. Allocate the transactions among the initial buyers i ∈ I and the winning sellers j ∈ J̃ by solving the following linear program LP1(I, J̃):

LP1(I, J̃) : max( xi ,zqj ,ηij

)Ṽ(I, J̃) = ∑

i∈I xi v̂i −

∑

j∈J̃

zqj Ĉqj − ∑

i∈I

∑

j∈J̃

ηijdij

s.t. zqj qj = ∑

i∈I ηij, j ∈ J , (9)

xi= ∑

j∈J̃

ηij, i ∈ I, (10)

0≤ xi ≤ Xi, j ∈ J̃,

0≤ zqj ≤ 1, j ∈ J̃,

ηij ≥0, i ∈ I, j ∈ J̃.

Let ((x̃i)i∈I; ( z̃qj

)

j∈J̃ ; ( η̃ij )

i∈I,j∈J̃) be the optimal solution to LP1(I, J̃).

Denote the winning buyer set Ĩ as the set of the buyers with x̃i > 0. Buyer i ∈ Ĩ procures η̃ij units of the item from seller j ∈ J̃.

Step 6. The payment P̃i of winning buyer i is her VCG payment in LP1(I, J̃) and the revenue R̃j of winning seller j is his VCG payoff in IP2

( I, J, Sj

) that takes into account the padding vector Sj.

The following proposition characterizes the structure of the optimal solution to LP1(I, J̃).

Proposition 1. In the optimal solution to LP1(I, J̃),

(i) z̃qj = 1 for each j ∈ J̃; (ii) (x̃i)i∈I and

( η̃ij )

i∈I,j∈J̃ are integers.

From Proposition 1, each winning seller k ∈ J̃will sell qk units under the allocation of LP1(I, J̃), the same as under the allocation of IP2(I, J,

Sk). This is because the supply in LP1(I, J̃), without the padding vector Sk, is less than that in IP2(I,J,Sk). So seller k will certainly sell qk units. Proposition 1 means that the optimal allocation is an integer although LP1(I, J̃) is a linear program. So the MDA-DB mechanism can develop a feasible transaction allocation between the buyers and sellers by solving a linear programming model, which can reduce the computation burden of the mechanism.

Remark 2. Chu (2009) introduced the padding for the demand side where the sellers’ marginal costs are constant and no transaction costs are incurred. Here we introduce the padding for the supply side because the sellers’ marginal costs change with their sale quantities. If the demand-side padding is introduced, the winning buyers’ payments may be equal to the sellers’ lowest marginal costs under the VCG payment rule, which could be very low. Then budget balance cannot be guaran- teed. Cheng et al. (2022) also considered the supply-side padding and proposed the supplier competition padding-based-VCG (SC-PV) mech- anism for the case with the single-unit item and uniform valuation. Our mechanism has distinct differences from theirs. First, we formulate a different integer program model to allocate transactions for the ex- change environment with multi-unit discriminatory bidding and pair-related transaction costs. Second, to overcome the allocation chal- lenges posed by transaction costs, we introduce a series of supply padding vectors

( Sj )

j∈J instead of one. Each seller’s winning and pricing determinations are based on different integer program models IP2

( I, J,

Sj ) . Third, after the first allocation, we specify not only the winning

sellers, as in the SC-PV mechanism, but also the quantity that each winning seller can trade. This can prevent the sellers’ revenues from being set too high in the case with non-increasing marginal costs. Fourth, in terms of the final allocation and the buyers’ payment rule, we solve a linear program rather than the integer program in Cheng et al. (2022), which can reduce the computational burden.

4.3. An illustration example

We use an example to illustrate the implementation of the MDA-DB mechanism. Table 1 lists the bids and asks from three sellers and seven buyers, respectively and Table 2 lists the transaction costs. By simple calculation, the maximum social welfare is 29.

After collecting these bids and asks, we set the padding S = max{XI, YJ} = 6. Then we specify Sj for each seller and decide how much he can sell by solving IP2

( I,J,Sj

) . Specifically, for seller 1, S1 = {6,0,0}. He can

sell three units from the optimal solution to IP2(I, J,S1). By similar ar- guments, sellers 2 and 3 sell zero and four units, respectively. So sellers 1 and 3 are the winning sellers. Then LP1(I, J̃) is solved. Buyers 1 and 3 buy two units from seller 3, respectively and buyer 2 buys three units from seller 1. From the payment rule, buyers 1, 2, and 3 pay 186, 278, and 186, and sellers 1 and 3 receive 276 and 367, respectively. Then the three buyers’ total utility is 96× 2+ 95× 3+ 94× 2 − 186× 2 − 278 =

15, and the two sellers’ total utility is 276+ 367 − 91.5× 2 − 91 − 92× 2 − 90× 2 = 5. The platform’s profit is 278+ 186× 2 − 276 − 367 − 0 = 7. The efficiency of the mechanism is (15 + 5 + 7)/29 = 93.1%.

Table 1 The agents’ bids and asks.

Buyer: {v̂i, Xi} Seller: { ( ĉqj

)

q∈{1,2,...,Yj} }

1: {96, 2} 2: {95, 3} 3: {94, 2} 4: {93, 2} 5: {92, 4} 6: {91, 4} 7: {87, 3}

1: {ĉq1 =

{ 91.5, 1 ≤ q ≤ 2 91,3 ≤ q ≤ 4 }

2: {ĉq2 =

⎧ ⎨

⎩

96,1 ≤ q ≤ 2 94,3 ≤ q ≤ 4 91,5 ≤ q ≤ 6

}

3: {ĉq3 =

{ 92,1 ≤ q ≤ 2 90,3 ≤ q ≤ 4 }

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6

4.4. The properties

From now on, we show the MDA-DB mechanism has nice properties. Note that the sellers will ask the prices related to their transaction quantities, so we modify the definition of incentive compatibility as follows.

Definition 1. When sellers can submit discriminatory bids, a mecha- nism is incentive-compatible for the sellers if seller j’s (j ∈ J) weakly dominant strategy is to ask that ĉqj = cqj for each q ∈

{ 1,2, ...,Yj

} .

Next, we shall show that the MDA-DB mechanism is BB, IR, and IC when the padding is large enough. Let XI = max{Xi|i∈ I} and YJ =

max { Yj ⃒ ⃒j ∈ J

} .

Theorem 1. The MDA-DB mechanism is BB, IR, and IC if S ≥ max{XI, YJ}.

From Theorem 1, the BB property means that the platform can make a profit by implementing this mechanism. This would incentivize the platform to hold the auction. In addition, the IR property means that the buyers and the sellers will get non-negative utilities. So they are willing to participate in the auction. Moreover, the agents will submit their marginal costs or values truthfully from the IC property. Truthful bid- ding can reduce the computation burdens for the bidders to get the optimal bidding strategies and obtain information about others. So- phisticated players could not reap more utilities than novices by manipulating their bids, which promotes fairness in transactions (Yu et al., 2022).

We shall further show that the MDA-DB mechanism is AsE. Accord- ing to Chu and Shen (2008), we can regard the transaction cost dij as the cost that is proportional to the distance between buyer i and seller j. Suppose the agents are independently located from a continuous dis- tribution U over some compact area A. That means that the transaction cost function is continuous with respect to the agents’ locations.

Theorem 2. With a bounded value distribution and continuous transaction costs, the MDA-DB mechanism is AsE for a fixed padding S.

Theorem 2 shows that the MDA-DB mechanism can achieve the market efficiency of 100% as the number of agents approaches infinity. In real-life applications, the higher the market efficiency is, the more conducive the auction mechanism is to the long-term development of a platform (Wise and Morrison, 2000). Therefore, the platform could adopt our MDA-DB mechanism to realize high market efficiency in the long term.

Remark 3. Under the MDA-DB mechanism, the platform can choose the padding, which can affect the mechanism efficiency and the plat- form’s profit. Intuitively, a larger padding means a lower efficiency due to more removed transactions and fewer transaction volumes. This also has a negative effect on the platform’s profit. On the other hand, a large padding will exacerbate the gap between the buying and selling prices, which increases the platform’s profit (a positive effect). Therefore, the platform can select a proper padding to maximize the profit by con- trolling the two conflicting effects. It can also manipulate the padding to balance the social welfare and profit. We shall conduct numerical studies to investigate this problem further in Appendix A.10.

5. The special case with zero transaction costs

In Section 4, we propose the MDA-DB mechanism for the case with sellers’ decreasing marginal costs and pair-related transaction costs and show that the mechanism has good properties. To deeply understand the performance of this mechanism, in this section, we consider the special case with zero transaction costs. We show that the MDA-DB mechanism can be further simplified, which can reduce the computational burden largely. In addition, we find it has some new properties in this case. From this, we can show that our mechanism has some advantages over the existing ones (Zhao et al., 2012; Sun et al., 2014; Chen and Huang, 2018; Liang et al., 2020) because they only consider this special case.

5.1. Finding the maximal social welfare

In this case, IP1(I, J) can be simplified as follows:

IP3(I, J) : max(xi ,yj)V(I, J) = ∑

i∈I xi v̂i −

∑

j∈J

∑Yj

q=1 yqj ĉqj

s.t. ∑

j∈J

∑Yj

q=1 yqj =

∑

i∈I xi, (11)

y(q− 1)j ≥ yqj , q ∈ { 2, ...,Yj

} , j ∈ J,

yqj ∈ {0,1}, q ∈ { 1, 2, ...,Yj

} , j ∈ J,

xi ∈ {0,1, 2, ...,Xi}, i ∈ I,

where constraint (11) guarantees the supply and the demand balance. Note that the agents are only concerned with their transaction quantities but not with whom they are trading in this case. So the decision variable ηij can be omitted in IP3(I, J).

Proposition 2 characterizes the properties of the optimal solution

( ( x*i )

i∈I; ( y*j )

j∈J ) to IP3(I, J).

Proposition 2. In the optimal solution to IP3(I, J),

(i) there is at most one j’ with 0 < ∑Yj’

q=1y*qj’ < Yj’ ;

(ii) for any j ∕= j’, either y*Yj = 1 or y*1j = 0;

(iii) let v̂1 = min { v̂i ⃒ ⃒x*i > 0, i ∈ I

} and v̂2 = max

{ v̂i ⃒ ⃒x*i = 0, i ∈ I

} .

Then v̂1 ≥ v̂2; (iv) there is at most one i with 0 < x*i < Xi.

From Proposition 2 (i)-(ii), if a seller wins in the efficient allocation, his supply must be allocated as much as possible. This is because the sellers’ marginal cost decreases for each additional unit sold. Due to the limitation of the demand, at most one supplier will sell his supply partially. In other words, in the efficient allocation, it is impossible that there are two sellers partially selling their supplies. If this occurred, it would be more efficient to reduce one seller’s transaction volume but increase another one to his capacity. From Proposition 2 (iii)-(iv), the demands will be met sequentially from the highest to the lowest bid prices, because a higher bid price (values) leads to a higher increase in the social welfare.

5.2. The simplified MDA-DB mechanism

If the transaction costs are zero, the MDA-DB mechanism can be simplified as follows.

Step 1. Collect bids and asks from the agents.

Step 2. Determine the padding S = H ( (Xi)i∈I,

( Yj )

j∈J

) .

Table 2 The transaction costs between buyers and sellers.

Buyer 1

Buyer 2

Buyer 3

Buyer 4

Buyer 5

Buyer 6

Buyer 7

Seller 1

1 0 0 0 0 1 2

Seller 2

0 2 1 0 0 2 0

Seller 3

0 1 0 0 0 0 0

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Step 3. Allocate the transactions among the initial buyers i ∈ I and sellers j ∈ J with padding S by solving the following IP4(I,J,S):

IP4(I, J, S) : max(xi ,yj)V(I, J, S)= ∑

i∈I xi v̂i −

∑

j∈J

∑Yj

q=1 yqj ĉqj

s.t. ∑

i∈I xi ≤

∑

j∈J

∑Yj

q=1 yqj + S , (12)

y(q− 1)j ≥ yqj , q∈ { 2, ...,Yj

} , j ∈ J,

yqj ∈{0,1}, q∈ { 1,2, ...,Yj

} , j ∈ J,

xi ∈{0,1, 2, ...,Xi}, i ∈ I,

where constraint (12) balances the demand and supply with padding S.

Step 4. Let ((xiʹ)i∈I; ( yjʹ )

j∈J ) be the optimal solution to IP4(I,J,S). The

sellers with ∑Yj

q=1yqjʹ > 0 will be the winning sellers. In the next allo-

cation, winning seller j ∈ J̃ sells either all of their supply qj = ∑Yj

q=1yqjʹ at

the total asking price Ĉqj = ∑qj

q=1 ĉqj or nothing.

Step 5. The transactions are conducted based on the optimal solution

((x̂i)i∈I; ( ẑqj

)

j∈J̃ ) to the following linear program LP2(I, J̃):

LP2(I, J̃) : max( xi ,zqj

)Ṽ(I, J̃)= ∑

i∈I xi v̂i −

∑

j∈J̃

zqj Ĉqj

s.t. ∑

j∈J̃

zqj qj = ∑

i∈I xi,

0≤ zqj ≤ 1, j ∈ J̃,

0≤ xi ≤ Xi, i ∈ I.

Each buyer i ∈ I gets ẽi = x̂i units of the item and each seller j ∈ J̃ sells q̃j = ẑqj qj units of the item.

Step 6. The payment P̃i of winning buyer i is her VCG payment in LP2(I, J̃) and the revenue R̃j of winning seller j is his VCG payoff in IP4(I, J, S) with the padding S.

The following proposition characterizes the structure of the optimal solution to LP2(I, J̃).

Proposition 3. In the optimal solution to LP2(I, J̃),

(i) ẑqj = 1 for each j ∈ J̃;

(ii) x̂i is an integer for each i ∈ I; (iii) let ṽ1 = min{v̂i|x̂i > 0, i ∈ I} and ṽ2 = max{v̂i|x̂i = 0, i ∈ I}.

Then ṽ1 ≥ ṽ2；

(iv) there is at most one i with 0 < x̂i < Xi.

Proposition 3 shows that in the optimal solution to LP2(I, J̃), each winning seller k ∈ J̃ will eventually sell qk units, the same as under the allocation of IP4(I,J,S). This is because that the supply with the padding S in IP4(I, J, S) is more than that in LP2(I, J̃). In addition, the buyers are sequentially selected to be the winners from the highest to the lowest bid prices. If a buyer wins, her demand will be allocated as much as possible. These can help us further reduce the computation of the transaction

allocation, and we shall show the simplified allocation rule in Remark 5.

Let p[h] be the bid price of the h-th highest unit of the item.3 We can find that the payment rule for the buyers under the simplified MDA-DB mechanism has the following characteristics.

Observation 1. When S ≥ XI,

(i) winning buyer i with ẽi = Xi will get his Xi units one-by-one at non-increasing prices from p[∑

j∈J̃ q̃j+1

], p[∑ j∈J̃ q̃j+2

] , …, to

p[∑ j∈J̃ q̃j+Xi

];

(ii) winning buyer í with ẽí < Xí will get his ẽí units one-by-one at non-increasing prices from p[∑

j∈J̃ q̃j+Xí − ẽí +1

], p[∑ j∈J̃ q̃j+Xí − ẽí +2

] ,

…, to p[∑ j∈J̃ q̃j+Xí

].

Remark 4. From Observation 1, we find that (i) the more a buyer buys, the lower the unit buying price is; (ii) the buyers whose demands are fully met will pay the same for the same demands; (iii) a buyer whose demand is partially met will pay no more than a buyer whose demand is equal to the former’s purchase quantity and fully met (for the same q, the payment of buyer i with q = Xi is

∑q h=1p[∑

j∈J̃ q̃j+h

], that is no less

than ∑q

h=1p[∑ j∈J̃ q̃j+Xí − q+h

], that of buyer í with q < Xí ). Schotanus et al.

(2008) found that the uniform price allocation method leads to dissat- isfaction with the gain allocation among the members of the purchasing group. Our differential payment rule for buyers can facilitate coopera- tive procurement.

Remark 5. From Proposition 3 and Observation 1, the simplified MDA-DB mechanism can adopt a further simplified allocation rule and buyer payment rule to reduce the computational burden.

Simplified allocation rule.

(i) Seller j sells q̃j = qj units of the item; (ii) Sort the buyers’ bids in a decreasing order. Without loss of gen-

erality, we assume that v̂1 > v̂2 > .... > v̂|I|;

(iii) Buyer 1 gets ẽ1 = min { X1,

∑ j∈J̃ q̃j

} units of the item and buyer i ∕=

1 gets ẽi = min { Xi,

∑ j∈J̃ q̃j −

∑i− 1 n=1ẽn

} units of the item.

Simplified payment rule for the winning buyers.

(i) Winning buyer i with ẽi = Xi pays ∑ẽi

h=1p[∑ j∈J̃ q̃j+h

];

(ii) Winning buyer í with ẽí < Xí pays ∑ẽí

h=1p[∑ j∈J̃ q̃j+Xi − ẽí +h

].

In addition, the padding vector Sk can be reduced to padding S because of the zero transaction costs. So the simplified mechanism solves the integer program IP4(I, J, S) only once to determine all the winning sellers, whereas the original mechanism needs to solve the integer program once for each seller to determine whether he can trade. This can dramatically reduce the computational burden.

3 For example, consider that buyer 1 procures one unit at 4 and buyer 2 procures two units at 5. Then, p[1] = p[2] = v̂2 = 5 and p[3] = v̂1 = 4.

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

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5.3. The properties

In this subsection, we show the simplified MDA-DB mechanism still possesses the desired properties. By similar proofs of Theorems 1 and 2, we can show that the simplified mechanism is still IR, IC, and AsE.

Theorem 3. The simplified MDA-DB mechanism is

(i) IR and IC; (ii) AsE with a bounded value distribution and a fixed padding S.

From Theorem 3, we know that the simplified mechanism can ensure that the agents get non-negative utilities from participating in the auction and induce the agents to submit their marginal costs or values truthfully. Also, the simplified MDA-DB mechanism can realize the maximal social welfare as the number of agents approaches to infinity.

Now we show that the simplified mechanism is still BB when the padding is large enough. First we give bounds on the agents’ total payment transfers.

Lemma 1. Under the simplified MDA-DB mechanism,

(i) ∑

k∈J̃ ∑q̃k − 1

h=0 p[∑ j∈J̃ q̃j+S− h

] is an upper bound of the winning

sellers’ total payoff. (ii)

∑ i∈I\{í |x̂í <Xí ,í ∈I}

∑Xi h=1 p[∑

j∈J̃ q̃j+h

] +

∑ i∈{í |x̂í <Xí ,í ∈I}

∑x̂i h=1 p[∑

j∈J̃ q̃j+Xi − x̂i+h

] is a lower bound of the

winning buyers’ total payment when S ≥ XI.

Theorem 4. The simplified MDA-DB mechanism is BB if S ≥ max{XI, YJ}.

From Theorem 4, we know that the platform can make a profit by implementing the simplified MDA-DB mechanism.

When the transaction costs are zero, we can further asseremoass the efficiency of the simplified MDA-DB mechanism. To this end, we need the following assumptions. Suppose that all buyers’ values are drawn independently from a distribution F with continuous positive density f on [v, v] and all sellers’ average costs c̃j =

∑Yj q=1cqj/Yj are drawn inde-

pendently from a distribution G with continuous positive density g on [c, c]. In addition, we assume that the demand quantities and supply quantities are drawn independently from bounded discrete distributions respectively. These assumptions do not affect the implementation of the simplified mechanism and the validity of Theorems 3 and 4.

Theorem 5. The efficiency loss of the simplified MDA-DB mechanism

is bounded by O (

max { S K,

S (L− 1)

})

, where K and L are the element

numbers of sets { i ⃒ ⃒x*i > 0, i ∈ I

} and

{ j ⃒ ⃒ ⃒y*1j = 1, j ∈ J

} , respectively.

Theorem 5 gives a precise estimation of the efficiency loss of the simplified MDA-DB mechanism. It says that the efficiency loss is related to the padding and the number of the winning agents. In terms of the padding, if S is set to 0, the MDA-DB mechanism is equivalent to the efficient double VCG mechanism and the efficiency loss is zero. How- ever, the platform may face a budget deficit. To ensure that the platform does not have to subsidize transactions, S should be set to at least max {XI, YJ} by Theorem 4. However, if we set the padding S too large, there is a large loss in the efficiency of the mechanism. On the other hand, the efficiency loss decreases as the total number of winning agents increases. Summarizing above, if the platform wants to improve the efficiency and get more profit, it should attract more agents to participate in the auction.

Remark 6. Zhao et al. (2012), Sun et al. (2014), Chen and Huang (2018), and Liang et al. (2020) have proposed the double mechanisms

for this special case. However, the mechanisms proposed by Zhao et al. (2012) and Sun et al. (2014) are buyer-truthful or seller-truthful. Our simplified mechanism can be always truthful for both sides. Chen and Huang (2018) proposed a truthful double auction mechanism for the case where the buyers have indivisible demands. But their mechanism cannot ensure incentive compatibility in our divisible demand case. Liang et al. (2020) propose the IC, IR, BB, and AsE TR-QDmechanism for this special case based on the trade reduction approach. We further compare the performance of this mechanism with ours in the case with zero transaction costs through the experimental studies (Subsection 6.2).

6. Numerical studies

In this section we compare our mechanism’s performance with that of the double VCG (D-VCG) mechanism, the catalog mechanism, also known as the posted-price mechanism (Jin and Yu, 2015), the MSDA-DB mechanism, and the TR-QD mechanism in Liang et al. (2020), to demonstrate the advantages of our mechanism in the case with zero transaction costs. We further investigate how the transaction costs and cost structure affect our mechanisms in terms of the social welfare, trading volume, platform profit, and agents’ utilities.

6.1. The experimental data

Consider the Cainiao Transport Market (56.1688.com), an online logistics platform in China, on which many small e-commerce com- panies (transport service buyers) buy LTL transport services from some large 3 PL companies (transport service sellers). The experimental data is set based on the “Shanghai-Nanjing, Jiangsu province” submarket in this platform.

The LTL transport market has more buyers than sellers, and the ratio of the numbers of buyers to sellers is 5:1. We set the basic market size to be (100, 20). But the total demand is no more than the total supply. The average freight demand is about 100 kg and the minimum trading unit is 10 kg. From this, the buyers’ demands (kg) are generated from the discrete uniform distribution UD[80, 120]. The sellers’ maximal supply quantities are set to be 500 kg.

Because the buyers’ values and the sellers’ costs are private infor- mation, these data are generated according to the sellers’ posted prices on the Cainiao Transport Market. Specifically, these prices without discount are in [500, 1200] ¥/ton. The sellers have decreasing step cost structures that a seller divides the transport supplies into K intervals on average and the marginal cost varies in different intervals. In the ex- periments, K is picked randomly from {2,3, 4}with equal probability. In practice, the sellers will offer a posted price that is higher than the cost to make a profit. Thus, assume that the marginal cost c1 for the quantity in the first interval is generated from the continuous uniform distribution UD(450,1080), which is 10% lower than the posted prices on the plat- form. We use a discount factor σ to measure the sellers’ decreasing marginal cost structure, where the marginal cost in the k-th interval is set to be σk− 1c1. The smaller σ is, the faster the marginal cost decreases, which implies a stronger scale effect. Different sellers have different σ that are generated from the continuous uniform distribution UD(0.6,1). To cover the cost, we assume that the buyers’ values are generated from the continuous uniform distribution UD(500,1200) that is the same as the posted prices. For ease of calculation, we use the pick-up costs as transaction costs. The transaction cost between the buyer i and seller j is proportional to the distance, trading volume, and transaction cost rate. We assume that the distances (km) are generated from the continuous uniform distribution UD(20,50). We set the transaction cost rate to be 2 ¥/ton-km, which is about 70% of the average posted price. We repeat the auctions 200 times and report the averages.

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

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6.2. Comparison with the mechanisms based on other design approaches

In this subsection, we compare the performance of the MDA-DB mechanism with that of the D-VCG mechanism, catalog mechanism, MSDA-DB mechanism, and TR-QD mechanism at different market sizes. The catalog mechanism is currently used by the Cainiao Transport Market for non-member shippers. Under the catalog mechanism, sellers first post their prices. Then buyers arrive at the platform sequentially, select the sellers with low prices, and buy the service at their posted prices. We show the detailed procedure of the MSDA-DB mechanism in Appendix A.9. The padding of the MDA-DB mechanism is set to be the maximal supply quantity of the sellers. Since the TR-QD and MSDA-DB mechanisms can only apply to the case with zero transaction costs, we set the transaction costs to zero to facilitate the comparison. We use the numbers of buyers and sellers (m, n) to measure the market size and set the values of (m, n) to increase from (50, 10) to (250, 50) in the step of (50, 10). Tables 3 and 4 summarize the results under the five mecha- nisms, respectively, “SW” stands for social welfare and “E” for efficiency.

From Tables 3 and 4, the D-VCG mechanism achieves the largest efficiency and trading volume, the MSDA-DB mechanism achieves the second, the MDA-DB mechanism achieves the third, the TR-QD mech- anism achieves the fourth and the catalog mechanism achieves the smallest ones. It is obvious that the D-VCG mechanism maximizes social welfare and delivers the optimum trading volume. As the market size increases, the efficiencies of the latter three mechanisms increase and tend to the optimum. Our mechanisms result in very small efficiency and welfare loss. For example, when the market size is (250,50), the effi- ciency loss decreases to 0.03% and the social welfare loss is only 3.

The MDA-DB mechanism can realize more social welfare and trading volumes than the catalog mechanism at any market size. Even in a small market size (50, 10), the efficiency gap is 10.63% and the trading vol- ume gap is 150 kg. This means that our mechanism outperforms the mechanism used in practice largely. Note that the four double auction mechanisms are more efficient than the catalog mechanism, especially in a large market. This is because the buyers cannot reap the procure- ment scale advantages in the catalog mechanism. The other mechanisms can coordinate the buyers’ procurements to secure lower buying prices. This can increase the scale of transactions and thus social welfare.

The MDA-DB mechanism is more efficient than the TR-QD mecha- nism. The reasons are as follows: first, the TR-QD mechanism may remove more efficient transactions than the MDA-DB mechanism. This can be found in Table 5, which records the number of times that the two mechanisms are efficient in 200 experiments, respectively. Second, the TR-QD mechanism conducts the final allocation according to bidding time. However, the MDA-DB mechanism efficiently allocates trans- actions among the initial buyers and remaining sellers, which is more efficient than the TR-QD mechanism.

Compared with the MSDA-DB mechanism, the MDA-DB mechanism realizes lower efficiency, which is consistent with Theorem EC.3 (shown in Appendix A.9). However, as the market increases, the effi- ciency difference between the two mechanisms decreases and ap- proaches zero. From this, we conclude that the MDA-DB mechanism performs better than the MSDA-DB mechanism because the latter may not be BB and/or IC in some situations.

Tables 6 and 7 show the division of social welfare. The percentage of

the platform’s profit over social welfare is also listed behind the platform profit.

From Tables 6 and 7, the MDA-DB mechanism can balance the di- vision of the social welfare among the buyers, the sellers, and the plat- form. Although D-VCG realizes the largest social welfare, the platform gets negative profit. Under the Catalog mechanism, the platform cannot profit from the transactions. So in practice, Cainiao needs to realize profitability in other ways, such as by providing value-added services. The MSDA-DB mechanism may not be BB in some experiments, but the platform’s average profit is positive. Among the five mechanisms, the platform can get the most profit under the MDA-DB mechanism. The reason is that the MDA-DB mechanism introduces the padding to the supply side, which induces the mechanism to set a higher buying price and a lower selling price compared to the other mechanisms. Moreover, as the market size increases, both the agents’ utilities and the platform’s profit increase, even though the share of the platform’s profit decreases. This means that the MDA-DB can attract potential buyers and sellers to participate in the platform and also give the platform more incentive to hold auctions.

It is worth mentioning that the buyers’ utility, sellers’ utility, and platform profit under the MDA-DB mechanism are all larger than those under the catalog mechanism when the market size is large. Even when the market size is very small (50, 10), the utility obtained by buyers under the MDA-DB mechanism is only 1 smaller than that under the catalog mechanism. Combined with the previous findings, compared to the catalog mechanism currently used by Cainiao, the MDA-DB mech- anism can improve social welfare, help buyers purchase at lower prices, help sellers profit by selling more transport services, and make the platform profitable. It gives important support for the application of our MDA-DB mechanism in practice.

6.3. The impacts of the transaction costs on the mechanisms’ performance

In this subsection, we study the impacts of the transaction costs on the social welfare, trading volume, platform profit, and the agents’ utilities under the D-VCG and MDA-DB mechanisms. Note that the TR- QD and MSDA-DB mechanisms cannot apply to this case and the

Table 3 The social welfare and efficiencies under the five mechanisms at different market sizes.

(m,n) D-VCG Catalog TR-QD MSDA-DB MDA-DB

E SW E SW E SW E SW E SW

(50, 10) 100.00% 1446 88.69% 1290 98.63% 1426 99.60% 1440 99.32% 1436 (100, 20) 100.00% 2927 90.86% 2663 99.61% 2916 99.90% 2924 99.86% 2923 (150, 30) 100.00% 4415 90.88% 4015 99.85% 4408 99.96% 4413 99.92% 4412 (200, 40) 100.00% 5910 91.31% 5400 99.91% 5905 99.97% 5909 99.95% 5908 (250, 50) 100.00% 7387 91.41% 6756 99.94% 7382 99.99% 7386 99.97% 7384

Table 4 The trading volumes under the five mechanisms at different market sizes.

(m,n) D-VCG Catalog TR-QD MSDA-DB MDA-DB

(50, 10) 3464 3042 3136 3262 3192 (100, 20) 6936 6182 6568 6728 6683 (150, 30) 10388 9373 10067 10188 10105 (200, 40) 13864 12566 13500 13673 13574 (250, 50) 17310 15679 16965 17132 17012

Table 5 The number of the efficient allocation under the TR-QD and MDA-DB mechanisms.

(m,n) (50, 10) (100, 20) (150, 30) (200, 40) (250, 50)

MDA-DB 81 95 83 76 69 TR-QD 64 50 63 46 51

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

10

catalog mechanism has no advantages at this market size (100, 20), so we do not consider them. We set the values of the transaction cost rate to increase from 0 ¥/ton-km to 4 ¥/ton-km in the step of 1 ¥/ton-km. The results are summarized in Table 8, where “TV” represents trading volumes.

From Table 8, the transaction costs would reduce the realized social welfare and trading volumes. This is consistent with the findings of previous works (e.g., Noussair et al., 1998; Sun et al., 2019; Guo et al., 2022). Moreover, as the transaction cost rate increases, the efficiency of the MDA-DB mechanism decreases. Although the transaction costs would reduce the efficiency of the MDA-DB mechanism, the efficiency loss is very small. For example, even at the transaction cost rates of 4 ¥/ton-km, the efficiency can reach 99.21%. These show that our mechanism can ensure a high efficiency even in the presence of trans- action costs. Table 9 reports the division of social welfare under the two mechanisms, respectively.

From Table 9, we find that the transaction costs are mainly borne by the platform under the D-VCG mechanism. Although the platform is the actual bearer of the transaction costs, the MDA-DB mechanism tends to transfer the transaction costs to the agents by raising the buying price and lowering the sale price, which results in a large price difference. As the transaction cost rate increases, the price difference increases and thus the platform can make more money. As a result, the platform will prefer to adopt the MDA-DB mechanism as the transaction costs increase.

6.4. The impacts of the cost structure on the mechanisms’ performance

In this subsection, we study the impacts of the cost structure on the social welfare, trading volume, platform profit, and agents’ utilities

under the D-VCG and MDA-DB mechanisms. We randomly generate the discount factor σ from the continuous uniform distributions UD(0.8,1), UD(0.7, 0.9), UD(0.6,0.8) and UD(0.5,0.7) to simulate different cost structures. A small σ implies a strong scale effect and the marginal cost decreases rapidly in the transaction quantity. The trading results are summarized in Table 10, where “TV” represents trading volumes.

From Table 10, the efficiency loss of the MDA-DB mechanism is very small. Moreover, as σ decreases, the social welfare realized by the mechanism increases but the efficiency loss decreases. For example, when σ is randomly generated from UD(0.5,0.7), there is only a 0.29% efficiency loss under the MDA-DB mechanism. This is because the wel- fare loss resulting from the removed transactions is smaller than the welfare increase brought about by the reduction in σ. These show that the MDA-DB mechanism is more advantageous for a market where the sellers have a strong scale effect. Table 11 reports the division of social welfare under the two mechanisms, respectively.

From Table 11, we find that the profit of the platform increases as the scale effect increases. This can give the platform more incentive to hold auctions. In addition, we find that the sellers’ scale effect benefits the buyers more than the platform and the sellers. That is, as σ decreases, the ratio of the sellers’ utility decreases, the ratio of the platform profit changes slightly, and the ratio of the buyers’ utility increases. This is because the agents’ utilities and the platform’s profit are related to two factors: the trading volume and transaction prices. There is no doubt that a stronger scale effect will increase the trading volume. However, it will pull down the buying price and the selling price. These two effects are beneficial to the buyers. Therefore the ratio of the buyers’ utility in- creases as the scale effect grows. For the sellers, the lower selling price brings negative utility. Then the ratio of the sellers’ utility may decrease as the scale effect grows. For the platform, the spread between the buying price and the selling price changes slightly with the scale effect, and accordingly the ratio of its profit varies slightly.

6.5. Recommendation for the mechanism selection

From the analysis in the above sections, we have the following findings:

The agents prefer the D-VCGmechanism because it can simplify their bidding and offer them the largest utility. However, it is not BB and the platform has to subsidize the operation of the auction.

The platform prefers the MDA-DB mechanism because it can get more profit under this mechanism. However, when the platform is in its infancy or competing with some rivals for agents, the MSDA-DB and TR- QD mechanisms may be better choices. Because these two mechanisms can give the agents more utilities while maintaining an expected budget balance for the platform.

From a system-wide perspective, governments and regulators would focus on high social welfare (efficiency) while ensuring the IR, IC, and BB properties. The proper mechanisms under different exchange envi- ronments are given in Table 12. Note that although our numerical studies only consider the LTL transport market, the findings and rec- ommendations are still applicable to other bilateral trading markets.

Table 6 The agents’ utilities under the five mechanisms at different market sizes.

(m,n) D-VCG Catalog TR-QD MSDA-DB MDA-DB

Buyers Sellers Buyers Sellers Buyers Sellers Buyers Sellers Buyers Sellers

(50, 10) 839 671 722 567 798 571 751 598 721 568 (100, 20) 1697 1298 1480 1183 1654 1191 1600 1225 1582 1192 (150, 30) 2518 1960 2228 1787 2483 1853 2429 1888 2389 1854 (200, 40) 3383 2610 2999 2401 3336 2495 3282 2531 3236 2493 (250, 50) 4217 3241 3762 2994 4184 3114 4133 3171 4077 3126

Table 7 The platform’s profit under the five mechanisms at different market sizes.

(m,n) D-VCG Catalog TR-QD MSDA-DB MDA-DB

(50, 10) − 64 (− 4.46%)

0 (0%) 57 (4.02%)

92 (6.37%)

148 (10.27%)

(100, 20)

− 68 (− 2.33%)

0 (0%) 71 (2.43%)

99 (3.40%)

149 (5.10%)

(150, 30)

− 63 (− 1.43%)

0 (0%) 73 (1.65%)

96 (2.18%)

169 (3.82%)

(200, 40)

− 82 (− 1.39%)

0 (0%) 74 (1.25%)

95 (1.61%)

179 (3.03%)

(250, 50)

− 71 (− 0.97%)

0 (0%) 85 (1.14%)

83 (1.12%)

181 (2.45%)

Table 8 The trading results under the D-VCG and MDA-DB mechanisms with different transaction cost rates.

Transaction cost rate D-VCG MDA-DB

E SW TV E SW TV

0 ¥/ton-km 100.00% 2927 6936 99.86% 2923 6683 1 ¥/ton-km 100.00% 2775 6772 99.68% 2766 6422 2 ¥/ton-km 100.00% 2626 6609 99.56% 2615 6196 3 ¥/ton-km 100.00% 2480 6422 99.37% 2464 5957 4 ¥/ton-km 100.00% 2338 6253 99.21% 2320 5757

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

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7. Discussion

In the previous sections, we show that the mechanism has good properties under the assumptions that each seller has a non-increasing marginal cost structure and that the agents’ supply and demand quan- tities are public information. In this section, we discuss the implications of relaxing these two assumptions for our mechanism, respectively.

7.1. The case with the sellers’ general marginal cost structures

Note that the assumption that the sellers’ marginal costs are non- increasing in their supply quantities generally holds because of the economies of scale arising from mass production and joint sourcing of raw materials. However, some sellers may have more complex marginal cost structures in practice. If the demand exceeds a seller’s capacity and the seller has to work in shifts or source items from third-party suppliers, the costs will increase (Bichler et al., 2011). Compared to the case with non-increasing marginal costs, the sellers’ bidding behavior will be more complicated and it is more difficult to induce the sellers to tell the truth and balance the budget for the platform. In the following, we can show that the MDA-DB mechanism still has nice properties.

Theorem 6. In the case with the sellers’ general marginal cost struc- tures, the MDA-DB mechanism is

(i) IR, IC, and BB if padding S ≥ max{XI, YJ}; (ii) AsE with a bounded value distribution and a fixed padding S.

Theorem 6 shows that in the case that the sellers have general marginal cost structures, the MDA-DB mechanism still possesses the necessary properties (IR, IC, BB, and AsE) if we choose an appropriate padding. This means that the mechanism is scalable and can be used in more general scenarios.

7.2. The case with private quantity information

In the previous sections, we assume that the supply and demand quantities are public information. In reality, this information may be private, and the agents may benefit from lying about it. As pointed out by Huang and Xu (2013) and Guo et al. (2022), it is very difficult to design IC and BB double auctions for this case. In this subsection, we further consider that the buyers’ demands and the sellers’ supplies are private information and discuss the properties of our mechanism. In this case, buyer i submits a sealed bid {v̂i, X̂i} and seller j submits a sealed

discriminatory ask { ( ĉqj

)

q∈{1,2,...,Ŷj} , Ŷ j}, where X̂i (Ŷ j) is the demand

(supply) quantity submitted by buyer i (seller j). Theorem 7 states that the MDA-DB mechanism is still IC if the

padding is determined independently of the agents’ bids.

Theorem 7. Given a padding S, the MDA-DB mechanism has the following properties

(i) the sellers will bid their costs and supply quantities truthfully; (ii) the buyers will bid their values and demand quantities truthfully

when qj = Yj for each j ∈ J.

From Theorem 7, although the demand and supply quantity infor- mation is private, the sellers will still bid their costs and supply quan- tities truthfully under the MDA-DBmechanism. The buyers will bid their values and demand quantities truthfully in a short-supply market. This is because the supply shortage will increase the competition among the buyers, which is more likely to incentivize the buyers to tell the truth.

Note that the platform needs to decide the padding S before the agents bid but it does not know the demand and supply quantities in this case. Then how to decide it is a problem faced by the platform because

Table 9 The agents’ utilities and platform’s profits under the two mechanisms with different transaction cost rates.

Transaction cost rate D-VCG MDA-DB

Buyers Sellers Platform Buyers Sellers Platform

0 ¥/ton-km 1697 (57.98%) 1298 (44.35%) − 68 (− 2.33%) 1582 (54.12%) 1192 (40.78%) 149 (5.10%) 1 ¥/ton-km 1618 (58.30%) 1234 (44.46%) − 77 (− 2.76%) 1466 (53.00%) 1111 (40.17%) 189 (6.83%) 2 ¥/ton-km 1546 (58.89%) 1174 (44.72%) − 95 (− 3.61%) 1373 (52.50%) 1037 (39.65%) 205 (7.85%) 3 ¥/ton-km 1464 (59.03%) 1116 (45.02%) − 100 (− 4.05%) 1273 (51.68%) 966 (39.19%) 225 (9.13%) 4 ¥/ton-km 1394 (59.60%) 1060 (45.32%) − 115 (− 4.92%) 1187 (51.16%) 897 (38.68%) 236 (10.16%)

Table 10 The trading results under the D-VCG and MDA-DB mechanisms with different discount factors.

σ D-VCG MDA-DB

E SW TV E SW TV

UD(0.8,1) 100.00% 2205 5934 99.43% 2192 5545 UD(0.7,0.9) 100.00% 2616 6612 99.55% 2605 6212 UD(0.6,0.8) 100.00% 2999 7235 99.61% 2988 6845 UD(0.5,0.7) 100.00% 3418 7765 99.71% 3408 7390

Table 11 The agents’ utilities and platform’s profits under the two mechanisms with different discount factors.

σ D-VCG MDA-DB

Buyers Sellers Platform Buyers Sellers Platform

UD(0.8,1) 1250 (56.68%) 1033 (46.87%) − 78 (− 3.55%) 1096 (50.01%) 907 (41.39%) 189 (8.60%) UD(0.7,0.9) 1539 (58.81%) 1162 (44.40%) − 84 (− 3.21%) 1372 (52.68%) 1026 (39.40%) 206 (7.92%) UD(0.6,0.8) 1835 (61.20%) 1257 (41.92%) − 94 (− 3.12%) 1639 (54.87%) 1113 (37.27%) 235 (7.86%) UD(0.5,0.7) 2109 (61.70%) 1412 (41.30%) − 103 (− 3.00%) 1912 (56.11%) 1248 (36.61%) 248 (7.28%)

Table 12 The mechanism recommendation.

Exchange environments Mechanism selection

The case with single-unit demand and zero transaction costs

MSDA-DB mechanism

The case with multi-unit demand and zero transaction costs

Simplified MDA-DB mechanism

The case with multi-unit demand and transaction costs

MDA-DB mechanism

The case with private quantity information MDA-DB mechanism The case with general sellers’ marginal cost structures

MDA-DB mechanism

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

12

padding S should meet S ≥ max{X̂I, ŶJ} to ensure the budget surplus, where X̂I = max{X̂i|i∈ I} and ŶJ = max

{ Ŷ j ⃒ ⃒j∈ J

} . In practice, the

platform can determine appropriate padding in the following ways. For a mature market without newcomers, the platform can get an upper bound of the sellers’ supplies and the buyers’ demands by historical transaction records and specify it as the padding. Another way is that at the stage of announcing the auction rule, the platform can pick a proper upper bound of demand and supply quantities that agents cannot bid beyond and set it as the padding.

8. Conclusion

In this paper, we study the problem of designing the truthful multi- unit double auction mechanism considering transaction costs and sellers’ changing marginal costs. We propose the MDA-DB mechanism for this problem and show that the mechanism is BB, IR, IC, and AsE. For the special case with zero transaction costs, we further simplify the mechanism. We show that the simplified mechanism can reduce the computational burden and further give the bounds on the agents’ pay- ment transfers and efficiency loss. Moreover, we discuss the cases of relaxing the decreasing marginal cost assumption and private quantity information assumption, respectively, and show our mechanism still has nice properties.

We further conduct numerical studies to compare our mechanism’s performance with that of commonly used mechanisms and examine the impacts of the transaction cost and sellers’ cost structure on the mech- anism’s performance. We obtain the following findings: (i) The effi- ciency loss of our MDA-DBmechanism is very small, especially in a large market and/or with a strong scale effect; (ii) The MDA-DB mechanism can balance the division of the social welfare and the platformmakes the most money under it among these mechanisms; (iii) Comparing with the catalog mechanism used in practice, the MDA-DB mechanism can improve the social welfare, increase the trading volumes and agents’ utilities, and make the platform profitable; (iv) The transaction costs would slightly reduce the efficiency of the MDA-DB mechanism but in- crease the profits of the platform; and (v) A strong scale effect favors the buyers more than the platform and the sellers. From these findings, we can provide recommendations for a procurement platform to select the mechanism as follows: Generally, the platform should adopt the MDA- DB mechanism because of its good properties and high efficiency. If the buyer’s demand is one unit, it is better to choose the MSDA-DB

mechanism. When the platform is in its infancy or competing with some rivals for agents, the MSDA-DB and TR-QD mechanisms may be preferable choices. If a platform wants to attract a large number of buyers and sellers without considering making a profit, the D-VCG mechanism is a good choice.

Based on this work, some topics can be studied further. Our mech- anisms need to solve some integer programming models, which incur a computing burden for the platform, especially when the numbers of sellers and their supplies are enormous (Huang et al., 2002). However the linear relaxations of the integer program cannot ensure incentive compatibility. It is an interesting and challenging topic to design a computationally efficient double auction mechanism for our problem. In our study, the platform’s decisions are mainly made by the monetary bids of the agents. In reality, many firms place a large emphasis on non-monetary attributes such as scale, quality, lead time, reliability, etc. (e.g., Perrone et al., 2010; de la Boulaye et al., 2019). Then an attractive study topic is to take the non-monetary attributes into account in double auctions with discriminatory bidding.

CRediT authorship contribution statement

Jiantao Guo: Writing – original draft, Methodology, Conceptuali- zation. Juliang Zhang: Writing – review & editing, Supervision, Funding acquisition. T.C.E. Cheng: Writing – review & editing, Supervision.

Data availability

No data was used for the research described in the article.

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China under grant numbers 72471018 and 72171016, the Major Project of Humanities and Social Sciences of Ministry of Educa- tion of China (23JZD009), the Fundamental Research Funds for the Central Universities under grant numbers 2023JBWZD002 and 2022YJS035, and the Beijing Logistics Informatics Research Base. The authors also thank the editor and the referees for their valuable com- ments to help us improve the paper greatly.

Appendix

A.1. Proof of Proposition 1

To prove Proposition 1, we first give the following lemma. Lemma EC.1. For seller k ∈ J and j ∈ J\{k},

∑Yk q=1y

j qk ≥

∑Yk q=1ykqk in the optimal solution to IP2

( I, J,Sj

) .

Proof. In IP2(I, J, Sk), seller k faces more competition than in IP2 ( I, J,Sj

) for any j ∈ J\{k} because Skk ≥ Sjk. So if seller k sells

∑Yk q=1ykqk units in the

optimal solution to IP2(I, J,Sk), he can sell at least ∑Yk

q=1ykqk units in the optimal solution to IP2 ( I, J, Sj

) for j ∈ J\{k}. Then we prove this lemma. Q.E.

D.

In LP1 ( I, J̃

) , the competition for seller k is less than in IP2

( I, J,Sj

) for j ∈ J\{k} because there are fewer sellers. From Lemma EC.1, seller k can sell

∑Yk q=1ykqk units in the optimal solution to LP1

( I, J̃

) , i.e., z̃qk = 1. From constraint (9),

∑ i∈I η̃ij must be an integer for each seller j ∈ J̃. Given the supply-

demand balance constraints, ∑

j∈J̃ ∑Yj

q=1ỹqj (an integer) units will be procured. Because the buyer’s demand is divisible, we have that x̃i is an integer for

each i ∈ I. From constraint (10), ∑

j∈J̃η̃ij is also an integer for each seller j ∈ J̃. Then we can rewrite the linear program LP1 ( I, J̃

) as follows:

LP1 ( I, J̃

) : max(

xi ,zqj

)V ( I, J̃

) = −

∑

i∈I

∑

j∈J̃

ηijdij

s.t. ∑

i∈I ηij = z̃qj qj, j ∈ J ,

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

13

∑

j∈J̃

ηij = x̃i, i ∈ I,

ηij ≥ 0, i ∈ I, j ∈ J̃.

Next we introduce Lemma EC.2. Lemma EC.2. (Georgiou and Floudas, 1989; Guo et al., 2022). For a linear program LP,

LP max z = cTx

s.t. Ax = b ,

x ≥ 0.

If matrix A is a totally unimodular matrix and b is an integer vector, then the fundamental solution x̂ = (x̂1, x̂2, x̂3, ..., x̂m)T of LP is an integer vector.

By a similar proof in Guo et al. (2022), the coefficient matrix of LP1 ( I, J̃

) is a totally unimodular matrix. Therefore,

(

η̃ij )

i∈I,j∈J̃ are integers in the

optimal solution to LP1 ( I, J̃

) from Lemma EC.2.

In summary, we prove Proposition 1. Q.E.D.

A.2. Proof of Theorem 1

Budget balance. To prove that the MDA-DB mechanism is BB, we need the following lemmas. Lemma EC.3. When S ≥ XI, suppose that seller k ∈ J̃ asks Cqk ≤ Ĉqk ≤ R̃k. Then,

(i) the winning seller set J̃ and revenue R̃j for seller j ∈ J̃ do not change;

(ii) the original optimal solution to LP1 ( I, J̃

) is still optimal;

(iii) winning buyer set Ĩ and payment P̃i for buyer i ∈ Ĩ do not change.

Proof. Because revenue R̃k of seller k is equal to seller k’s VCG payoff in IP2 ( I, J,Sj

) , R̃k is the highest ask price for him to sell qk units. Even though

seller k asks Cqk ≤ Ĉqk ≤ R̃k, he can still trade qk units. From Lemma EC.1, seller k can trade ∑Yk

q=1y j qk ≥ qk in the optimal solution to IP2

( I, J,Sj

) where

j ∈ J\{k}. So if seller k does not ask higher than R̃k, the optimal solution to IP2 ( I, J,Sj

) for j ∈ J̃\{k} is unchanged. From the definition of J̃, J̃ does not

change with the unchanged optimal solution. Seller j ∈ J̃ will get the VCG payoff in IP2 ( I, J, Sj

) , which is determined by the buyers with positive

allocations or the sellers with zero allocations in the optimal solution to IP2 ( I, J, Sj

) . Then revenue R̃j for seller j ∈ J̃ does not change either. We prove

Lemma EC.3. (i).

With unchanged J̃, from Proposition 1. (i), the original optimal solution to LP1 ( I, J̃

) is still optimal and only the optimum will decrease by Ĉqk −

Cqk . We prove Lemma EC.3. (ii). From the definition of Ĩ, Ĩ does not change either with the unchanged optimal solution. From Observation 1, the payment of buyer i ∈ Ĩ is

∑ẽi h=1p[∑

j∈J̃ q̃j+h

] or ∑ẽi

h=1p[∑ j∈J̃ q̃j+Xi’ − ẽi+h

] when S ≥ XI, which is determined by the buyers with zero allocations in the optimal solution to LP1 ( I, J̃

) .

Then the payment P̃i for buyer i ∈ Ĩ does not change either. We prove Lemma EC.3. (iii). Q.E.D. Lemma EC.4. When S ≥ YJ and buyer k ∈ Ĩ bids P̃k/x̃k ≤ v̂k ≤ vk. Then,

(i) winning seller set J̃ and the revenue R̃j for seller j ∈ J̃ do not change;

(ii) the original optimal solution to LP1 ( I, J̃

) is still optimal;

(iii) winning buyer set Ĩ and the payment P̃i for buyer i ∈ Ĩ do not change.

Proof. Buyer k’s VCG payment in IP2 ( I, J,Sj

) is no higher than that in LP1

( I, J̃

) , because there are more supplies in IP2

( I, J,Sj

) . Payment P̃k of

buyer k is equal to buyer k’s VCG payment in LP1 ( I, J̃

) . So as long as buyer k does not bid lower than P̃k, she can get the same allocation in the optimal

solution to IP2 ( I, J,Sj

) as she bids truthfully. Then the original optimal solution is still optimal. By similar arguments as in the proof of Lemma EC.3.

(i), J̃ does not change. Payment P̃k of buyer k is equal to buyer k’s VCG payment in LP1 ( I, J̃

) , so P̃k/x̃k is the lowest bid price for her to get x̃k units.

Because J̃ does not change, even though buyer k ∈ Ĩ bids P̃k/x̃k ≤ v̂k ≤ vk, he can still trade x̃k units. Then the optimal solution to LP1 ( I, J̃

) is un-

changed. We prove Lemma EC.4. (ii).

Seller j ∈ J̃ will get the VCG payoff in IP2 ( I, J, Sj

) , which is determined by the buyers with zero allocations in the optimal solution to LP1

( I, J̃

) or

the sellers with zero allocations in the optimal solution to IP2 ( I, J, Sj

) because S ≥ YJ. Because the two optimal solutions do not change, revenue R̃j for

each seller j ∈ J̃ does not change either. We prove Lemma EC.4. (i).

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

14

From the definition of Ĩ, Ĩ does not change either with the unchanged optimal solution. Buyer i ∈ Ĩ pays the VCG payment in LP1 ( I, J̃

) , which is

determined by the sellers with positive allocations or the buyers with zero allocations in the optimal solution to LP1 ( I, J̃

) . With the unchanged

optimal solution, payment P̃i for buyer i ∈ Ĩ does not change either. We prove Lemma EC.4. (iii). Q.E.D.

The optimum of LP1 ( I, J̃

) must be non-negative, i.e.,

∑ i∈I x̃i v̂i −

∑ j∈J̃ z̃qj Ĉqj −

∑ i∈I ∑

j∈J̃η̃ijdij ≥ 0 because zero is a feasible solution. By Lemma

EC.3, if we let all the winning sellers raise ask price to R̃j one by one, the optimal solution to LP1 ( I, J̃

) is unchanged. Then, by Lemma EC.4, if we let

all the winning buyers decrease the bid price to P̃i one by one, J̃ and optimal solution to LP1 ( I, J̃

) are also unchanged. So, the optimum of LP1

( I, J̃

)

after the agents adjust their prices is still non-negative. Then we have that ∑

i∈Ĩ P̃i − ∑

j∈J̃R̃j − ∑

i∈Ĩ ∑

j∈J̃η̃ijdij ≥ 0, i.e., the MDA-DB mechanism is BB when S ≥ XI.

Incentive compatibility for sellers. If seller j has no contribution to IP2 ( I, J,Sj

) , he will sell nothing; if seller j has a positive contribution, that is,

IP2 ( I, J, Sj

) > IP2

( I, J\{j}, Sj

) , he will sell q̃j units of the item. Moreover, his payoff is his VCG payoff in IP2

( I, J,Sj

) . It is determined by agent set I ∪

J\{j} and Sj, which is not affected by seller j’s ask. Therefore, seller j ∈ J will ask ĉqj = cqj for each q ∈ { 1, 2, ...,Yj

} under the MDA-DB mechanism.

Incentive compatibility for buyers. The payment of winning buyer l is her VCG payment in LP1 ( I, J̃

) , which is determined by agent set I\{l} ∪ J̃.

Lemma EC.5 will show that a buyer does not manipulate her bid price to affect the winning seller set J̃. Lemma EC.5. A buyer does not manipulate her bid price to change J̃ under the MDA-DB mechanism. Proof. We first discuss buyer l with xkl = 0, who does not trade in the optimal solution to IP2(I, J, Sk) for each k∈J. If she decreases her bid prices

(v̂l < vl), the optimal solution to IP2(I, J, Sk) is not changed and J̃ keeps unchanged. Next, we show that if she increases bid prices (v̂l > vl) such that ∑

j∈J̃ q̃j increases and she finally trades h > 0 units, her utility will be negative. If buyer l increases her bid prices such that she can eventually trade, she

needs to pay pVCGl

( I, J̃

) , her VCG payment in LP1

( I, J̃

) . In addition, we have that pVCGl (I, J,Sk) ≤ pVCGl

( I, J̃

) where pVCGl (I, J, Sk) is buyer l’s VCG

payment in IP2(I, J, Sk). If buyer l bids truthfully, the VCG payment pVCGl (I, J, Sk) is vlh. Then we have that vlh = pVCGl (I, J, Sk) ≤ pVCGl

( I, J̃

) . If she

increases her bid prices such that she can finally trade, she will get a negative utility. So buyer lwith xkl = 0will not manipulate her bid price to affect J̃.

Next, we discuss buyer l with 0 < x̃l = max { xjl|j ∈ J

} . Let k = argmax

{ xjl|j ∈ J

} . If buyer l bids vl + δl (δl >0) instead of vl, the optimal solution to

IP2(I, J,Sk) is unchanged but the objective value increases by δlx̃l. Then J̃ remains the same. If she decreases bid prices (v̂l ≤ vl) such that ∑

j∈J̃ q̃j decreases, she must trade fewer units than truthful bidding. Under the VCG payment rule, she loses utility from the units not obtained and pays the same money for the units traded as bidding truthfully. Then her utility will decrease. So buyer l with 0 < x̃l = xkl will not manipulate her bid prices to affect J̃.

Last, we discuss buyer l with x̃l < xkl . If she increases her price to increase ∑

j∈J̃ q̃j, her utility would decrease, which is similar to that buyer l with xkl = 0 increases her bid price. If she decreases the price to decrease

∑ j∈J̃ q̃j, her utility will also decrease, which is similar to the case that buyer l with

0 < x̃l = xkl decreases her bid price. So buyer l with x̃l < xkl will not manipulate her bid price to affect J̃. Q.E.D. From Lemma EC.5, I\{l} ∪ J̃ is not affected by the bid price v̂l. Thus the buyers will bid truthfully.

Individual rationality. The payment of a winning buyer is her VCG payment in LP1 ( I, J̃

) and the payoff of a winning seller is his VCG payoff in

IP2 ( I, J, Sj

) . By reporting their true values and marginal costs, the agents secure non-negative utility. Then the MDA-DB mechanism is IR. Q.E.D.

A.3. Proof of Theorem 2

Given r, we can divide the compact area A into a finite number (k) of regions (i.e., the radius of each partition is less than r). For each region, the auction is conducted for the agents that are in this region. When the radius r is small enough (this makes sense in practice because the transaction cost is generally much less than the value of the item in the cooperative procurement. Even if the difference is too large, we can divide them into different partitions), up to S units of possible transactions are removed in each partition. For a finite number (k) of partitions, the social welfare loss is bounded above by kvS, where v is the upper bound of the buyers’ values. Because the cost of the “virtual seller” introduced in the mechanism is zero, the upper bound on the social welfare loss is a constant for a fixed padding S.

As |I| and |J| approach infinity, the maximum social welfare approaches infinity and the efficiency loss approaches zero. i.e.,

lim |I|→∞,|J|→∞

social welfare loss V(I, J)

≤ lim |I|→∞,|J|→∞

kvS V(I, J)

=0%.

So the MDA-DB mechanism is asymptotically efficient. Q.E.D.

A.4. Proof of Proposition 2

Suppose there are two sellers j1 and j2 meeting 0 < q̃js = ∑Yjs

q=1y*qjs < Yjs (s = 1, 2) in the optimal solution to IP3(I, J). So ∑Yj1

q=1 ĉqj1 + ∑q̃j2 − Yj1+q̃j1

q=1 ĉqj2 > ∑q̃j1

q=1 ĉqj1 + ∑q̃j2

q=1 ĉqj2 and ∑q̃j1 − Yj2+q̃j2

q=1 ĉqj1 + ∑Yj2

q=1 ĉqj2 > ∑q̃j1

q=1 ĉqj1 + ∑q̃j2

q=1 ĉqj2 . We have ∑Yj1

q=q̃j1+1 ĉqj1 >

∑q̃j2 q=q̃j2 − Yj1+q̃j1+1

ĉqj2 and

∑Yj2 q=q̃j2+1

ĉqj2 > ∑q̃j1

q=q̃j1 − Yj2+q̃j2+1 ĉqj1 . Since the marginal cost ĉqj decreases in the quantity, we have

∑q̃j2 q=q̃j2 − Yj1+q̃j1+1

ĉqj2 ≥ ∑Yj2

q=q̃j2+1 ĉqj2 . Then, we get

∑Yj1 q=q̃j1+1

ĉqj1 > ∑q̃j1

q=q̃j1 − Yj2+q̃j2+1 ĉqj1 , which contradicts the assumption that the marginal cost decreases with the transaction quantity. Therefore, there

is at most one j́ such that 0 < ∑Yj́

q=1y*qj́ < Yj́ . For other j ∕= j́ , we have either y*Yj = 1 or y*1j = 0. We prove Proposition 2. (i) and (ii).

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

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Because the buyers’ demands are divisible, the total of ∑

j∈J ∑Yj

q=1y*qj demand units will be fulfilled sequentially from the highest to the lowest bid

prices. Then, there is at most one i satisfying 0 < x*i < Xi and v̂ 1 > v̂2. Then we prove Proposition 2. (iii) and (iv).

A.5. Proof of Proposition 3

Suppose that there is a seller k ∈ J̃ such that 0 ≤ ẑqk < 1 in the optimal solution to LP2(I, J̃). Then we have ∑

j∈J ∑Yj

q=1yqjʹ = ∑

j∈J̃\{k} ẑqj qj + qk

> ∑

j∈J̃\{k} ẑqj qj+ ẑqk qk. Note Ĉq̃k/q̃k ≤ p[∑ j∈J

∑Yj q=1

yqj ʹ+1

]. Ṽ(I, J̃) can increase by

∑⌊(1− ẑqk )qk⌋− 1 h=0 p[∑

j∈J̃ qj − h

] + ( ( 1 − ẑqk

) qk −

⌊( 1 − ẑqk

) qk ⌋) p[∑

j∈J̃ qj −

⌊

(1− ẑqk )qk

⌋] − ( 1 − ẑqk

) Ĉqk through setting ẑqk as 1. So ẑqj = 1 for each j ∈ J̃ and we

prove Proposition 3. (i). From Proposition 3. (i), LP2(I, J̃) can be rewritten as follows:

LP2(I, J̃) : max(xi)Ṽ(I, J̃)= ∑

i∈I xi v̂i

s.t. ∑

i∈I xi =

∑

j∈J̃

qj ,

0≤ xi ≤ Xi, i ∈ I.

Because ∑

j∈J̃qj and Xi are an integer, the total of ∑

j∈J̃qj demand units with the highest bid prices are met one by one in the optimal solution.

Therefore, (x̂i)i∈I, the optimal solution to LP2(I, Ĵ) will be an integer vector. Then we prove Proposition 3. (ii)-(iv). Q.E.D.

A.6. Proof of Lemma 1

Since the payoff of winning seller k is his VCG payoff in IP4(I, J,S), his payoff is no more than ∑q̃k − 1

h=0 p[∑ j∈J̃ q̃j+S− h

]. The an upper bound on the

winning sellers’ payoffs is ∑

k∈J̃ ∑q̃k − 1

h=0 p[∑ j∈J̃ q̃j+S− h

].

The payment of winning buyer l (x̂l = Xl) is ∑x̂l

h=1p[∑ j∈J̃ q̃j+h

]. There is at most one buyer ĺ such that 0 < x̂ ĺ <Xĺ from Proposition 3. (iv) and her

payment is ∑x̂ĺ

h=1p[∑ j∈J̃ q̃j+Xĺ − x̃ĺ +h

]. Then a lower bound on the winning buyers’ payments is ∑

l∈Ĩ,l∕=ĺ ∑x̂l

h=1p[∑ j∈J̃ q̃j+h

] + ∑x̂ĺ

h=1p[∑ j∈J̃ q̃j+Xĺ − x̂ĺ +h

].

A.7. Proof of Theorem 4

From Lemma 1. (i), we havemore low-priced terms p[∑ j∈J̃ q̃j+S− h1

] than high-priced terms p[∑ j∈J̃ q̃j+S− h2

] in the summation ∑

k∈J̃ ∑q̃k − 1

h=0 p[∑ j∈J̃ q̃j+S− h

]

(h1 < h2) and the highest price term is at most p[∑ j∈J̃ q̃j+1

] when S≥ YJ. Let α1S+ β1 = ∑

j∈J̃ q̃j = ∑

i∈I x̂i, where α1 and β1 are integers and β1 < S. Then

an upper bound on the total payoffs can be α1 ∑S

h=1p[∑ j∈J̃ q̃j+h

]+ ∑S

h=S− β1+1p [∑

j∈J̃ q̃j+h

].

From Lemma 1. (ii), we have more terms at the higher price p[∑ j∈J̃ q̃j+h1

] than at the lower price p[∑ j∈J̃ q̃j+h2

] in the summation

∑ l∈Ĩ,l∕=ĺ

∑x̂l h=1p[∑

j∈J̃ q̃j+h

] (h1 < h2) and the lowest price term is at less p[∑ j∈J̃ q̃j+S

] when S ≥ XI. Let α2S + β2 = ∑

l∈J,l∕=ĺ x̂ l, where α2 and β2 are integers

and β2 < S. Then a lower bound on the overall payments is ∑β2

h=1p[∑ j∈J̃ q̃j+h

]+ ∑S

h=S− x̃ĺ +1 p[∑

j∈J̃ q̃j+h

]+ α2 ∑S

h=1p[∑ j∈J̃ q̃j+h

].

From the constraint on the balance between supply and demand, we have α1S + β1 = α2S + β2 + x̂ĺ . There are two cases (i) α1 = α2 and β1 = β2 + x̂ ĺ , and (ii) α1 = α2 +1 and β1 + S = β2 + x̂ĺ .

For case (i), we have α2 ∑S

h=1p[∑ j∈J̃ q̃j+h

] = α1 ∑S

h=1p[∑ j∈J̃ q̃j+h

] and ∑S

h=S− x̂ĺ +1 p[∑

j∈J̃ q̃j+h

] + ∑β2

h=1p[∑ j∈J̃ q̃j+h

] ≥ ∑S

h=S− x̂ĺ +1 p[∑

j∈J̃ q̃j+h

] +

∑S− x̂ĺ +1 h=S− β1+1

p[∑ j∈J̃ q̃j+h

] = ∑S

h=S− β1+1p [∑

j∈J̃ q̃j+h

].

For case (ii), it is clear that β2 >β1 and x̂ ĺ >β1. Furthermore, we have (α2 +1) ∑S− 1

h=S− β2p [∑

j∈J̃ q̃j+S− h

] + α2 ∑S− 1− β2

h=0 p[∑ j∈J̃ q̃j+S− h

] +

∑x̂ĺ − 1 h=0 p[∑

j∈J̃ qj+S− h

] = (α2 +1) ∑S

h=1p[∑ j∈J̃ q̃j+h

]− ∑β2

h=1p[∑ j∈J̃ q̃j+h

] + ∑S

h=S− x̂ĺ +1 p[∑

j∈J̃ q̃j+h

]. Then α1 ∑S

h=1p[∑ j∈J̃ q̃j+h

] = (α2 + 1) ∑S

h=1p[∑ j∈J̃ q̃j+h

],

∑S h=S− x̂ĺ +1

p[∑ j∈J̃ q̃j+h

] = ∑S

h=S− β1+1p [∑

j∈J̃ q̃j+h

] + ∑S− β1

h=S− x̂ĺ +1 p[∑

j∈J̃ q̃j+h

] and ∑S− β1

h=S− x̂ĺ +1 p[∑

j∈J̃ q̃j+h

] − ∑S

h=β2+1p [∑

j∈J̃ q̃j+h

] ≥ 0.

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

16

Summarizing the above, we have (α2 + 1) ∑S− 1

h=S− β2p [∑

j∈J̃ q̃j+S− h

]+ α2 ∑S− 1− β2

h=0 p[∑ j∈J̃ q̃j+S− h

]+ ∑x̂ĺ − 1

h=0 p[∑ j∈J̃ q̃j+S− h

] ≥ α1 ∑S

h=1p[∑ j∈J̃ q̃j+h

]+

∑S h=S− β1+1p

[∑

j∈J̃ q̃j+h

]. That is, the lower bound on the overall payment is not less than the upper bound on the overall payoff. The profit of the

platform is non-negative. Therefore, when S ≥ max{XI,YJ}, the MDA-DB mechanism is BB. Q.E.D.

A.8. Proof of Theorem 5

For IP3(I,J), the maximal social welfare is V(K,L) = ∑

i∈Ivix*i − ∑

j∈J ∑Yj

q=1cqj y*qj . Without loss of generality, we assume that c̃1 ≤ c̃2 ≤ .... ≤ c̃|J| and v1 ≥ v2 ≥ .... ≥ v|I|. If there are L sellers and K buyers who eventually trade in the optimal solution to IP3(I,J), from Proposition 2,

V(K, L) = ∑K

i=1 vix*i −

∑L

j=1

∑Yj

q=1 cqj y

* qj =

∑K− 1

i=1 (vi − vK)Xi +

∑L− 1

j=1

(

c̃L − c̃j )

Yj + (

vK − c̃L ) ∑K

i=1 x*i

= (v1 − v2)X1 + (v2 − v3)(X1 + X2) + ...+ (vK− 1 − vK) ∑K− 1

i=1 Xi

+

(

c̃2 − c̃1 )

Y1 +

(

c̃3 − c̃2 )

(Y1 + Y2) + ...+ (cL − cL− 1) ∑L− 1

j=1 Yj +

(

vK − c̃L ) ∑K

i=1 x*i .

The welfare loss wl(K, L) is caused by the efficient transaction between the sellers and the buyers removed due to padding S. Then we have

wl(K, L, S)≤ ∑K

i=K* vi −

∑L

j=L*

∑Yj q=1

c̃jy*qj ≤(v1 − vK)S+ (̃cL+1 − c̃L)S+ (̃cL − c̃1)S,

where K* = max { í | ∑K

i=í x*i ≥ S, í ∈ I } and L* = max

{ j́ | ∑L

j=j́ ∑Yj

q=1y*qj ≥ S, j́ ∈ J } . Define el(K, L, S) = E[wl(K,L,S)]

E[V(K,L)] to be the efficiency loss.

To prove the theorem, we introduce the following lemma that states the gap of the values/costs between two consecutive buyers/sellers is always bounded.

Lemma EC.6. (Huang et al., 2002). For all i = 1, 2, ..., |I| − 1, 1 φ(|I|+1) ≤ E[vi − vi+1] ≤ 1

ϕ(|I|+1); and for all j = 1, 2, ..., |J| − 1,

1 δ(|J|+1) ≤ E

[

c̃j+1 − c̃j ]

≤ 1 γ(|J|+1), where ϕ, γ, φ, and δ are defined as ϕ = min{f(x) : v≤ x≤ v}, γ = min{g(x) : c≤ x≤ c}, φ = max{f(x) : v≤ x≤ v} and

δ = max{g(x) : c≤ x≤ c}. By Lemma EC.6, we have that

E[V(K, L)] ≥ E [

X1 + (X1 + X2) + ...+ ∑K− 1

i=1 Xi ]

φ(|I| + 1) +

E

[

Y1 + (Y1 + Y2) + ...+ ∑L− 1

j=1 Yj

]

δ(|J| + 1)

= K(K − 1)E[X] 2φ(|I| + 1) ⏟̅̅̅̅̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅̅̅̅̅⏟

A

+ L(L − 1)E[Y] 2δ(|J| + 1) ⏟̅̅̅̅̅̅̅̅̅ ⏞⏞̅̅̅̅̅̅̅̅̅ ⏟

B

and

E[wl(K, L, S)] ≤ E[v1 − vK]S+ E[̃cL+1 − c̃L]S+ E[̃cL − c̃1]S,

≤ (K − 1)S ϕ(|I| + 1)

+ S

γ(|J| + 1) +

(L − 1)S γ(|J| + 1)

= LS

γ(|J| + 1) ⏟̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅⏟

C

+ (K − 1)S ϕ(|I| + 1) ⏟̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅ ⏟

D

.

Note that C+DA+B ≤ max ( C B，

D A

)

, we have

el(K, L, S)≤max (

S (L − 1)

2δ γE[Y]

, S K

2φ ϕE[X]

)

=O (

max {

S (L − 1)

, S K

})

. Q.E.D.

A.9. Proof of Theorem 6

The difference between this case and Section 4 is that the sellers’ marginal costs are not necessarily decreasing in their transaction quantities. We first discuss incentive compatibility for the sellers. If seller j’s ask satisfies IP2

( I,J,Sj

) = IP2

( I,J\{j},Sj

) , he will sell nothing; if seller j’s ask satisfies

IP2 ( I,J,Sj

) > IP2

( I,J\{j},Sj

) , he will sell q̃j units of the item. Moreover, his payoff is his VCG payoff in IP2

( I,J,Sj

) . Moreover, his payoff is his VCG

payoff in IP2 ( I,J,Sj

) , which is determined by agent set I ∪ J\{j} and Sj but seller j’s ask. Therefore, seller j ∈ J will ask truthfully.

Then, by similar arguments as in the proof of Theorems 1 and 2, we can easily show that the MDA-DBmechanism is BB, IR, IC for buyers, and AsE. Q.E.D.

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

17

A.10. Proof of Theorem 7

We first consider the seller side. Since S is given beforehand, each seller cannot manipulate S. Note that seller j’s marginal cost vector ( cqj

)

q∈{1,2,...,Yj} can be written as {c1j , c2j , …, cYj , ∞, ∞,…}. Bidding Ŷ j > Yj is equivalent to bidding {c1j , c2j , …, cYj , ĉYj+1, …, ĉŶj , ∞,…} (ĉŶj ∕= ∞) and

bidding Ŷ j < Yj is equivalent to bidding {c1j , c2j , …, cŶj , ∞, ∞,…}. From Theorems 1 and 6, the sellers will ask truthfully.

Then we show consider the buyer side. Similarly, the buyers cannot manipulate S. Since buyers pay VCG payments in LP1(I,J̃), buyer lwith xkl = 0 will not trade by misrepresenting her demand alone. From the proof of Theorem 1, she has no incentive to increase her bid price. Buyer l will bid truthfully.

For buyer lwho will trade x̃l = max { xjl ⃒ ⃒ ⃒j∈ J

} > 0 when she bids truthfully, she can trade X̂l units if she bids X̂l < Xl. Under the VCG payment rule,

she loses utility from units not obtained and pays the same money for units traded as bidding truthfully. Then her utility will decrease. If she bids X̂l > Xl such that she is allocated X̂l units, she would pay the same money for Xl units as bidding truthfully but have to pay for the extra X̂l− Xl units that are zero values.

Finally, we discuss buyer l with x̃l < xkl . When qj = Yj for each j ∈ J, she cannot get more if she only increases her bid demand. If she decreases her bid demand and trades less than x̃l. By similar arguments as in the case that buyer l with 0 < x̃l = xkl bids X̂i < Xi, we can show that her utility will decrease. So buyer l with x̃l < xkl will bid her value and demand truthfully.

Summarizing above, we prove Theorem 7. Q.E.D.

A.10. The impacts of the padding on the MDA-DB mechanism’s performance

In this subsection, we study the impacts of the padding on the social welfare, efficiency, trading volume, and platform profit under the MDA-DB mechanism. We set the market size to be (100, 20). We repeat the auctions 200 times and report the average values. The results are summarized in Table A.1.

From Tab. A.1, we find that the impacts of the padding on the MDA-DB mechanism’s performances are consistent with Remark 3. First, the larger the padding is, the fewer the final transactions are. This unquestionably will reduce the efficiency of the mechanism. Second, the padding brings a negative effect (decreasing the transactions) and a positive effect (increasing the buy-sell spreads) to the platform’s profit. When S ≤ 600, the positive effect outweighs the negative one, so the platform’s profit increases as the padding gets larger. When S > 600, the negative effect from the increasing padding prevails. Therefore, the platform’s profit decreases. The platform can set the padding at around 600 to maximize the profits as much as possible. In addition, the efficiency of the mechanism is robust with the change of the padding sizes. For example, if we set S = 250, 5 times max{XI, YJ}, the MDA-DB mechanism can still achieve an efficiency of 95%. Then the platform’s profit increases by about 743. This gives the platform some leeway to determine the padding to balance the efficiency with the platform’s profitability.

A.11. The mechanism based on the multi-stage design approach

Chu and Shen (2007) pointed out that the mechanisms based on the multi-stage design approach have some advantages in social efficiency and individual payoff. In this subsection, we design amechanism (MSDA-DBmechanism) using this multi-stage approach for the case with zero transaction costs and compare it with the MDA-DB mechanism to show the advantages and disadvantages of these two approaches.

Now we state the MSDA-DB mechanism in detail. Step 1: Collect bids and asks from the agents. Step 2: Allocate the transactions among the buyers i ∈ I and the sellers j ∈ J by solving IP3(I, J). The sellers whose allocations are zero will be

removed and the others will remain. Step 3: Further eliminate some sellers from the remaining sellers. Let ĉ1 be the highest average asking price among the remaining sellers, i.e., ĉ1 =

max { Ĉqj /qj

⃒ ⃒ ⃒qj> 0,j∈ J

} , where qj =

∑Yj q=1y*qj . If ĉ

1 ≤ v̂2, there are no sellers to be eliminated. Otherwise, the seller with the highest average ask price

ĉ1 will be eliminated further. After this, the remaining sellers are winning suppliers (j ∈ J̃). Winning seller j will sell either qj units of the item at the asking price Ĉqj or nothing.

Step 4: Allocate the transactions among the initial buyers i ∈ I and the winning sellers j ∈ J̃ by solving LP2(I, J̃).

Step 5: Winning buyer i pays her VCG payment in LP2(I, J̃). Winning seller j gets the payoff min { qj v̂

2 , Ĉqj +Ṽ(I, J) − Ṽ(I, J\{j})

} if ĉ1 ≤ v̂2, and

min { qj ĉ

1 , Ĉqj +Ṽ(I, J) − Ṽ(I, J\{j})

} otherwise.

Note that Proposition 3 still holds under the MSDA-DBmechanism by similar arguments, which characterizes the structure of the optimal solution to LP2(I, J̃).

Theorems EC.1 and EC.2 show the properties of the MSDA-DB mechanism. Theorem EC.1. The MSDA-DB mechanism is.

(i) IC and IR; (ii) BB if each buyer’s demand is a single unit.

Proof. Incentive compatibility for sellers. The seller’s payoff is min { qj v̂

2 or qj ĉ 1 , Ĉqj + V(I, J) − V(I, J\{j})

} . The sellers eliminated in the first

allocation will not manipulate their asking prices since their payments are at most the VCG payoff. We next consider left sellers. If ĉ1 > v̂2, all sellers

with Ĉqj/qj < ĉ1 finally trade. Winning seller j’s payoff ismin {

qj ĉ 1 , Ĉqj + V(I,J) − V(I,J\{j})

} , which is independent of his ask. If his ask satisfies Ĉqj/

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

18

qj < ĉ1, he can trade at min {

qj ĉ 1 , Ĉqj + V(I,J) − V(I,J\{j})

} . If his ask satisfies Ĉqj/qj ≥ ĉ1, he will be eliminated. For a seller with Ĉqj/ qj = ĉ1, if he

lowers his price and trades, he will get a negative utility. When ĉ1 ≤ v̂2, winning seller j’s payoff is min { qj v̂

2 , Ĉqj + V(I, J) − V(I, J\{j})

} , which is

independent of his ask. If his ask satisfies Ĉqj/qj < v̂2, he can trade at min { qj v̂

2 , Ĉqj + V(I,J) − V(I,J\{j})

} . If his ask satisfies Ĉqj/ qj > v̂2, he will be

eliminated. In either case, a seller cannot obtain a higher utility by manipulating his asking price. Therefore, seller j ∈ J will ask ĉqj = cqj for each q ∈ { 1, 2, ...,Yj

} under the MSDA-DB mechanism.

Incentive compatibility for buyers. The buyers’ payments are their VCG payments in LP2(I, J̃). We first show that the buyers do not manipulate their bid prices to affect the trading seller set J̃.

Lemma EC.7. The buyers do not manipulate their bid prices to change J̃ under the MSDA-DB mechanism. Proof. We first consider buyer lwith x*l = Xl. If she increases her bid price vl to vl + δl (δl >0), V(I, J)will increase by δlXl and the optimal solution to

IP3(I, J) is unchanged. If she lowers her bid price (v̂l < vl) such that ∑

j∈J̃ ∑Yj

q=1y*qj is smaller, she would trade fewer than truthful bidding. Under the VCG payment rule, she will lose utility from unsatisfied demands and pay the same money for the units traded as bidding truthfully. Therefore, buyer l with x*l = Xl will not manipulate her bid to change J̃.

Then we consider buyer l with Xl > x*l > 0. We first show that she does not increase her bid prices (v̂l > vl) to make ∑

j∈J̃ ∑Yj

q=1y*qj increase. Let

( ( x*i,l

)

i∈I ; ( y*qj ,l

)

j∈J ) be the optimal solution to IP3(I,J|v̂l> vl), where buyer l submits v̂l instead of vl. Then

∑ j∈J̃

∑Yj q=1y*qj ,l >

∑ j∈J̃

∑Yj q=1y*qj and buyer l

would get x*l,l − x*l more demands. But these extra trading units incur a negative utility for buyer l because her values corresponding to these demands

are lower than the costs corresponding to the supplies. By similar arguments as above, buyer l will not lower her bid prices to make ∑

j∈J̃ ∑Yj

q=1y*qj decrease either. Therefore, buyer l will not manipulate her bid prices to change J̃.

Finally, we consider buyer l with x*l = 0. If she lowers her bid price, she will still not trade either. If she increases her bid price vl to make ∑

j∈J̃ ∑Yj

q=1y*qj increase, by similar arguments as above, we can show that she will get less utility. Therefore, buyer lwill not manipulate her bid price to

change J̃. Summarizing the above, we prove Lemma EC.7. Q.E.D. From Lemma EC.7, we know that the buyers do not manipulate their bid prices to change J̃. Moreover, they pay their VCG payments in LP2(I, J̃).

Therefore, the buyers will bid truthfully.

Individual rationality. Seller j’s total payoff ismin { qj v̂

2 or qj ĉ 1 , Ĉqj + V(I,J) − V(I,J\{j})

} . If his payoff is qj v̂

2 or qj ĉ 1, we have that v̂2 > ĉ1 ≥ Ĉqj/

qj and Ĉqj/qj < ĉ1 for any seller j ∈ J̃. By reporting his true marginal cost, the utility of winning seller j is non-negative. If his payoff is Ĉqj + V(I,J) −

V(I,J\{j}), he also can secure a non-negative utility by reporting his true marginal costs. Winning buyer i pays her VCG payment in LP2(I, J̃). So we have V(I, J̃) ≥ V(I\{i}, J̃) for each i ∈ I. By reporting the true value, each buyer gets a non-negative utility. Therefore, the MSDA-DB mechanism is IR.

Budget balance for the case with a single-unit demand. We first consider the case where ĉ1 < v̂2. For this case, the overall payment for winning

buyers is ∑

i∈Ĩ ẽi v̂ 2. The overall payoff for winning sellers is

∑ j∈J̃ min

{ qj v̂

2 , Ĉqj + V(I,J) − V(I,J\{j})

} . Note that

∑ i∈Ĩ ẽi =

∑ j∈J̃qj, we have

∑ i∈Ĩ ẽi v̂

2 ≥

∑ j∈J̃ min

{ qj v̂

2 , Ĉqj + V(I,J) − V(I,J\{j})

} . The budget is balanced for this case.

Then we consider the case ĉ1 ≥ v̂2, where a remaining seller k is removed. The total payment of the winning buyers is at least ∑

i∈Ĩ ẽi ĉ 1. The total

payoff of the winning sellers is ∑

j∈J̃ min {

qj ĉ 1 , Ĉqj + V(I,J) − V(I,J\{j})

} . Noting that

∑ i∈Ĩ ẽi =

∑ j∈J̃qj, we have

∑ i∈Ĩ ẽi ĉ

1 ≥

∑ j∈J̃ min

{

qj ĉ 1 , Ĉqj + V(I,

J) − V(I,J\{j}) } . The budget is also balanced for this case. Q.E.D.

Now we show that the MSDA-DB mechanism is AsE.

Lemma EC.8. The efficiency loss of the MSDA-DB mechanism is bounded by O (

max {

1 K,

1 L(L− 1)

})

.

Proof. Obviously, the MSDA-DB mechanism can realize the efficient allocation when ĉ1 ≤ v̂2. So we need only consider the bound on the efficiency loss when ĉ1 > v̂2. From the proof of Lemma 1, we have

E[V(K, L)]≥ K(K − 1)E[X] 2φ(|I| + 1) ⏟̅̅̅̅̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅̅̅̅̅⏟

A

+ L(L − 1)E[Y] 2δ(|J| + 1) ⏟̅̅̅̅̅̅̅̅̅ ⏞⏞̅̅̅̅̅̅̅̅̅ ⏟

B

.

The welfare loss is caused by the lost transactions between the seller L and the buyers. Then the total welfare loss is

wl(K, L)= ∑K

i=K* vi −

∑YL q=1

cqL y * qL ≤(v1 − vK)YL + (̃cL+1 − c̃L)YL,

where K* = max { í | ∑K

i=í x*i ≥ ∑YL

q=1y*qL , í ∈ I } . By Lemma EC.6, we know that

E[wl(K, L)] ≤ E[̃cL+1 − c̃L]E[YL] + E[v1 − vK]E[YL],

≤ E[YL]

γ(|J| + 1) ⏟̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅⏟

E

+ (K − 1)E[YL]

ϕ(|I| + 1) ⏟̅̅̅̅̅̅̅̅̅ ⏞⏞̅̅̅̅̅̅̅̅̅ ⏟

F

.

Note that E+F A+B ≤ max

( F A,

E B

)

, therefore we have

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

19

el(K, L)≤max ( 1 K 2φE[Y] ϕE[X]

, 1

L(L − 1) 2δ γ

)

=O (

max { 1 K ,

1 L(L − 1)

})

. Q.E.D.

Theorem EC.2. With the bounded value distribution, the MSDA-DB mechanism is AsE. Proof. As |I| and |J| approach to infinity, K and L approach to infinity. Then, the MSDA-DB mechanism’s efficiency loss approaches to zero, i.e.,

limK→∞,L→∞el(K,L) ≤ limK→∞,L→∞O(max{1 /(L(L − 1)),1 /K}) = 0. So the MSDA-DB mechanism is AsE. Q.E.D. Next, we compare the properties of the simplified MDA-DB and MSDA-DB mechanisms. Theorem EC.3 shows that the MSDA-DB mechanism can

achieve larger efficiency than the MDA-DB mechanism. Theorem EC.3. For any S ≥ max{XI, YJ}, the MSDA-DB mechanism is more efficient than the simplified MDA-DB mechanism. Proof. Because the cost of the “virtual seller” in the MDA-DB mechanism is zero, there are two relationships between the optimal solutions to

IP3(I, J) and IP4(I,J,S): (i) ∑

j∈J ∑Yj

q=1y*qj = ∑

i∈Ix*i = ∑

j∈J ∑Yj

q=1yqjʹ + S = ∑

i∈Ixiʹ and (ii) ∑

j∈J ∑Yj

q=1y*qj = ∑

i∈Ix*i < ∑

j∈J ∑Yj

q=1yqjʹ+ S = ∑

i∈Ixiʹ. In case (i), at most one efficient seller is removed under the MSDA-DB mechanism. Because S ≥ YJ, the more efficient allocations are obviously

removed under the MDA-DB mechanism. Thus the efficiency loss of the MDA-DB mechanism is greater than that of the MSDA-DB mechanism. In case (ii), no efficient sellers are removed and the MSDA-DB realizes efficient allocation. The MDA-DB mechanism can realize the efficient allocation only when

∑ j∈J

∑Yj q=1yqjʹ =

∑ j∈J

∑Yj q=1y*qj . For either case, the MSDA-DB mechanism can realize an efficiency no less than that of the simplified MDA-DB

mechanism when S ≥ max{XI, YJ}. Q.E.D. From Theorem EC.3, we know that the MSDA-DB mechanism has an advantage over the MDA-DB mechanism in terms of the efficiency. However,

the MSDA-DB mechanism cannot guarantee budget balance in the general case with multi-unit demands from Theorem EC.1. This means that the MSDA-DB mechanism sheds the platform’s profit to increase the social welfare. Furthermore, we can show that the MSDA-DB mechanism is not IC in the case with private quantity information and the case with the sellers’ general marginal cost structures.

Consider there are two sellers j1 and j2 with non-zero allocations in the efficient allocation. Seller j1 sells qj1 (qj1 ≤ Yj1 ) units of the item. If Ĉqj1 /

qj1 < Ĉqj2 /qj2 = ĉ1 > v̂2, seller j1’s total payoff is qj1 ĉ 1 = qj1 Ĉqj2 /qj2 if he asks truthfully. However, if he bids ĉ(q+1)j1 (ĉ(q+1)j1 < ĉqj2 ), he can sell qj1+ 1

units while seller j2 will sell qj2 − 1 units. Then seller j1’s total payoff will be ( qj1 + 1

) Ĉqj2 − 1/

( qj2 − 1

) , which is larger than qj1 ĉ

1 = qj1 Ĉqj2 / qj2 . Then

seller j1 will lie and the MSDA-DB mechanism is not IC. In the case with private quantity information, a buyer can overstate her demand to increase |J̃| such that she can finally trade when ĉ1 > v̂2 under

the MSDA-DBmechanism. A seller can also lie and understate his supply to increase ĉ1 and v̂2, which may result in a greater total utility for him even if he sells fewer quantities. The key reason is that the MSDA-DBmechanism determines ĉ1, v̂2 and J̃ based on the optimal solution to IP3(I,J). The agents canmisrepresent their bid quantities to manipulate this optimal solution so that ĉ1, v̂2 and J̃ to favor them. So theMSDA-DBmechanism is not IC in this case.

In summary, the MSDA-DB mechanism performs better than the MDA-DB mechanism for the case with a single unit of demand, because both mechanisms are IC, IR, BB, and AsE but the former is more efficient. For the cases where the buyers’ demands are multiple units, the sellers’ marginal cost structures are general or the demand/supply quantity information is private, the MDA-DB mechanism performs better than the MSDA-DB mechanism because the latter is not BB in the first case and not IC in the latter two cases.

Table A.1 The performance of the MDA-DB mechanism under the different padding sizes.

S (unit) Social welfare Efficiency Trading volume Platform profit

50 2923 99.86% 6683 149 100 2907 99.32% 6380 362 150 2881 98.41% 6080 562 200 2846 97.20% 5795 739 250 2800 95.61% 5502 892 300 2741 93.59% 5196 1036 350 2677 91.38% 4920 1149 400 2598 88.72% 4622 1245 450 2509 85.63% 4322 1314 500 2416 82.47% 4044 1367 550 2311 78.86% 3756 1398 600 2183 74.45% 3440 1412 650 2033 69.40% 3104 1394 700 1855 63.41% 2738 1347 750 1639 56.07% 2334 1260 800 1386 47.51% 1903 1124

References

Ahlqvist, V., Holmberg, P., Tangerås, T., 2022. A survey comparing centralized and decentralized electricity markets. Energy Strategy Rev. 40, 100812.

Andrade, X., Guimarães, L., Figueira, G., 2021. Product line selection of fast-moving consumer goods. Omega 102, 102389.

Babaioff, M., Nisan, N., Pavlov, E., 2004. Mechanisms for a spatially distributed market. In: Proceedings of the 5th ACM Conference on Electronic Commerce, pp. 9–20.

Babaioff, M., Walsh, W.E., 2005. Incentive-compatible, budget balanced, yet highly efficient auctions for supply chain formation. Decis. Support Syst. 39 (1), 123–149.

Bichler, M., Schneider, S., Guler, K., Sayal, M., 2011. Compact bidding languages and supplier selection for markets with economies of scale and scope. Eur. J. Oper. Res. 214 (1), 67–77.

Burke, G.J., Carrillo, J., Vakharia, A.J., 2008. Heuristics for sourcing from multiple suppliers with alternative quantity discounts. Eur. J. Oper. Res. 186 (1), 317–329.

Cheng, M., Ning, Y., Xu, S.X., Wang, Z., 2022. Novel double auctions for spatially distributed parking slot assignment with externalities. IISE Transactions 1–13.

Cheng, M., Xu, S.X., Huang, G.Q., 2016. Truthful multi-unit multi-attribute double auctions for perishable supply chain trading. Transport. Res. E Logist. Transport. Rev. 93, 21–37.

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

20

Chen, S., Tseng, M.M., 2010. A Negotiation-Credit-Auction mechanism for procuring customized products. Int. J. Prod. Econ. 127 (1), 203–210.

Chen, S., Huang, M., 2018. Truthful multi-unit double auctions with quantity discount. In: 2018 Chinese Control and Decision Conference, pp. 428–432.

Chu, L.Y., 2009. Truthful bundle/multiunit double auctions. Manag. Sci. 55 (7), 1184–1198.

Chu, L.Y., Shen, Z.J.M., 2006. Agent competition double-auction mechanism. Manag. Sci. 52 (8), 1215–1222.

Chu, L.Y., Shen, Z.J.M., 2007. Trade reduction vs. multi-stage: a comparison of double- auction design approaches. Eur. J. Oper. Res. 180 (2), 677–691.

Chu, L.Y., Shen, Z.J.M., 2008. Truthful double-auction mechanisms. Oper. Res. 56 (1), 102–120.

de la Boulaye, P., Erriquez, M., Gener Bago, M., Jiménez Iribarren, A., Russo, F., 2019. How B2B online marketplaces could transform indirect procurement. McKinsey & Company. Available at: https://www.mckinsey.com/business-functions/operations /our-insights/how-b2b-online-marketplaces-could-transform-indirect-procurement.

Ebadi, J., Madani, S.M., 2006. The economies of scale in Iran manufacturing establishments. Iran. Econ. Rev. 11 (1), 143–170.

Evans, D.S., Schmalensee, R., Noel, M.D., Chang, H.H., Garcia-Swartz, D.D., 2011. Platform Economics: Essays on Multi-Sided Businesses. Competition Policy International. Available at: SSRN: https://ssrn.com/abstract=1974020.

Fang, D., Wu, J., Tang, D., 2012. A double auction model for competitive generators and large consumers considering power transmission cost. Int. J. Electr. Power Energy Syst. 43 (1), 880–888.

Fershtman, D., Pavan, A., 2022. Matching auctions. Rand J. Econ. 53 (1), 32–62. Gansterer, M., Hartl, R.F., Savelsbergh, M., 2020. The value of information in auction-

based carrier collaborations. Int. J. Prod. Econ. 221, 107485. Georgiou, A., Floudas, C.A., 1989. Optimization model for generic rank determination of

structural matrices. Int. J. Con. 49 (5), 1633–1644. Guo, J., Zhang, J., Cheng, T.C.E., Zhao, S., 2022. Truthful double auction mechanisms for

online freight platforms with transaction costs. Transp. Res. Part B Methodol. 158, 164–186.

Huang, G.Q., Xu, S.X., 2013. Truthful multi-unit transportation procurement auctions for logistics e-marketplaces. Transp. Res. Part B Methodol. 47 (1), 127–148.

Huang, P., Scheller-Wolf, A., Sycara, K., 2002. Design of a multi-unit double auction e- market. Comput. Intell. 18 (4), 596–617.

Hsu, C.I., Li, H.C., 2009. An integrated plant capacity and production planning model for high-tech manufacturing firms with economies of scale. Int. J. Prod. Econ. 118 (2), 486–500.

Jin, M., Yu, A.J., 2015. Procurement auctions and supply chain performance. Int. J. Prod. Econ. 162, 192–200.

Kong, X.T., Kang, K., Zhong, R.Y., Luo, H., Xu, S.X., 2021. Cyber physical system-enabled on-demand logistics trading. Int. J. Prod. Econ. 233, 108005.

Liang, R., Wang, J., Huang, M., Jiang, Z.Z., 2020. Truthful auctions for e-market logistics services procurement with quantity discounts. Transp. Res. Part B Methodol. 133, 165–180.

Li, B., Hao, D., Zhao, D., 2024. Diffusion auction design with transaction costs. Aut. Agents Multi-Agent Syst. 38 (1), 2.

Li, Z., Zhou, X., Huang, S., 2021. Managing skill certification in online outsourcing platforms: a perspective of buyer-determined reverse auctions. Int. J. Prod. Econ. 238, 108166.

Loertscher, S., Mezzetti, C., 2021. A dominant strategy double clock auction with estimation-based tâtonnement. Theor. Econ. 16 (3), 943–978.

McAfee, R., 1992. A dominant strategy double auction. J. Econ. Theor. 56, 434–450. McHardy, J., 2024. Platform business models and strategic price interaction. Transp. Res.

Part B Methodol. 182, 102920. Milgrom, P., Watt, M., 2022. Walrasian Mechanisms for Non-convex Economies and the

Bound-form First Welfare Theorem. Working paper, Stanford University. Available at: https://mitchwatt.github.io/files/PricingMechanismsNonConvex.pdf.

Munson, C.L., Rosenblatt, M.J., 1998. Theories and realities of quantity discounts: an exploratory study. Prod. Oper. Manag. 7 (4), 352–369.

Myerson, R.B., Satterthwaite, M.A., 1983. Efficient mechanisms for bilateral trading. J. Econ. Theor. 29 (2), 265–281.

Nie, Y.M., 2012. Transaction costs and tradable mobility credits. Transp. Res. Part B Methodol. 46 (1), 189–203.

Noussair, C., Robin, S., Ruffieux, B., 1998. The effect of transaction costs on double auction markets. J. Econ. Behav. Organ. 36 (2), 221–233.

Perrone, G., Roma, P., Nigro, G.L., 2010. Designing multi-attribute auctions for engineering services procurement in new product development in the automotive context. Int. J. Prod. Econ. 124 (1), 20–31.

Schotanus, F., Telgen, J., de Boer, L., 2008. Unfair allocation of gains under the Equal Price allocation method in purchasing groups. Eur. J. Oper. Res. 187 (1), 162–176.

Segal-Halevi, E., Hassidim, A., Aumann, Y., 2018. MUDA: a truthful multi-unit double- auction mechanism. In: Proc. AAAI Conf. Artif. Intell. 32 (1).

Smart Research Consulting, 2022. https://www.chyxx.com/research/202010/899277. html, 08/01/2022.

Sun, J., Li, G., Xu, S.X., Dai, W., 2019. Intermodal transportation service procurement with transaction costs under belt and road initiative. Transport. Res. E Logist. Transport. Rev. 127, 31–48.

Sun, Z., Zhu, Z., Chen, L., Xu, H., Huang, L., 2014. A combinatorial double auction mechanism for cloud resource group-buying. In: 2014 IEEE 33rd International Performance Computing and Communications Conference, pp. 1–8.

Wise, R., Morrison, D., 2000. Beyond the exchange–the future of B2B. Harv. Bus. Rev. 78 (6), 86–96.

Xiao, F., Wang, H., Guo, S., Guan, X., Liu, B., 2021. Efficient and truthful multi-attribute auctions for crowdsourced delivery. Int. J. Prod. Econ. 240, 108233.

Xu, S.X., Cheng, M., Huang, G.Q., 2015. Efficient intermodal transportation auctions for B2B e-commerce logistics with transaction costs. Transp. Res. Part B Methodol. 80, 322–337.

Xu, S.X., Huang, G.Q., Cheng, M., 2016. Truthful, budget balanced bundle double auctions for supplier collaboration. Transport. Sci. 51 (4), 1365–1386.

Yu, H., Huang, M., Chao, X., Yue, X., 2022. Truthful multi-attribute multi-unit double auctions for B2B e-commerce logistics service transactions. Transport. Res. E Logist. Transport. Rev. 164, 102814.

Zhang, C., Chen, J., Raghunathan, S., 2022. Two-sided platform competition in a sharing economy. Manag. Sci. 68 (12), 8909–8932.

Zhao, D., Zhang, D., Perrussel, L., 2012. Multi-unit double auction under group buying. In: Proceedings of the 20th European Conference on Artificial Intelligence, pp. 882–887.

Zhang, L.H., Jia, S.F., Leung, C.K., Guo, L.P., 2013. An analysis on the transaction costs of water markets under DPA and UPA auctions. Water Resour. Manag. 27, 475–484.

Zhang, M., Kong, Z., 2022. A multi-attribute double auction and bargaining model for emergency material procurement. Int. J. Prod. Econ. 254, 108635.

J. Guo et al. International Journal of Production Economics 278 (2024) 109430

21