Combinatorial Auction of Used Cars Considering Pro-Environment Attribute: A Social Welfare Perspective

Abstract

Air pollution is becoming more and more serious as the number of vehicles increases. To address such problems, many cities have implemented many measures, including the circular economy mode, in which used cars with low carbon emission are becoming important in the sustainable transportation and carbon abatement. Considering multi-attribute demand, this study designed combinatorial auction mechanism for the bidders of automobile enterprises on an online used-car platform to achieve social welfare maximization. Two kinds of attributes were considered, namely, price attribute and non-price attributes; the latter particularly included the pro-environment attribute based on an analysis of complementarity and substitutability. Moreover, the mechanism was proved to satisfy individual rational condition and incentive compatibility condition. Numerical application showed that preference for the pro-environment attribute can better realize social welfare and respond to national energy conservation and emission reduction targets. As a result, from the social welfare perspective, the multi-attribute combinatorial auction can provide a reference for more fair and effective allocation of used cars to bidders and can promote both buyer’s utility and seller’s income.

1. Introduction

The number of motor vehicles in China exceeded 400 million by March 2022, thus bringing road congestion, environmental pollution, and other problems to residents’ health and lives, causing great harm and inconvenience [1,2]. To address the above problems, a number of departments issued relevant policy documents, including emission standards, policies related to the number of vehicles, and other aspects of restrictions [3]. For example, in response to the national “dual carbon” goal, many cities improved automobile emission standards according to local conditions, actively promoted the trading and circulation of low-emission vehicles, and restricted the production and circulation of high-emission vehicles. Relevant documents on promoting automobile consumption issued by the Ministry of Commerce mentioned that strengthening the creation of convenient conditions for the used-car market and accelerating the transaction and circulation of used cars are necessary to promote the recycling of motor vehicle resources. Therefore, focusing on the used-car market and considering the environmental properties of used cars are important tasks.

With the development of online used-car auctions, many platforms have recently emerged [4,5]. According to released data, China’s used-car market transaction volume increased by 8.76% in 2019, with online transaction volume accounting for nearly 20% of the total, reaching 30.26 million vehicles (data from China’s used-car Market Report 2020). On 9 February 2021, the Ministry of Commerce issued relevant documents on promoting automobile consumption, which mentioned that the used car market needs to strengthen the creation of convenient conditions, indicating that used-car trading has become an important part of the automobile trading market, and the country’s attention on used-car consumption is also increasing. Among these online used-car platforms, B2B or C2B modes are popular approaches, in which used car dealers (car companies) act as buyers [6]. This type of transaction and process is shown in Figure 1. Therefore, the auction mechanisms that are suitable for used car dealers need to be explored. With the heterogeneity of used cars taken into consideration, each car has various characteristics [7], which leads to the issue of how multiple used cars can be allocated effectively according to the multi-attribute demand of bidders.

Combinatorial auction is often used in these settings because most used-car dealers need to purchase more than one used car [8], and they always receive privileges or discounts as regular customers. In other words, more time and effort are required if a used-car dealer bids on a single car with individual buyers, without obvious advantages. As an effective way to solve the problem of multi-item auction, combinatorial auction is a hot topic in the field of auction research [9]. Vries and Vohra [10] reviewed the application of combinatorial auctions in various fields, focusing on how auctioneers establish mathematical programming models confronting allocation to maximize their own revenue. For example, Lee et al. [11] and Caplice [12] found that combinatorial auctions also have a significant advantage over single-route auctions in transportation service procurement auctions. This method manifests in the form of bidders bidding on various combinations of items.

Researchers have proved that the combinatorial auction is more efficient than the traditional auction method in allocating a variety of commodities or resources [13,14]. Its main advantage is that bidders can more freely show their own preference for their desired combination of auction items and improve the efficiency of resource allocation. As for the preference, that is, whether complementarity or substitutability exists between heterogeneous used cars, different buyers have different preferences under their different needs. Therefore, the relationship among items in combinational auctions needs to be considered. Komal et al. [15] analyzed the combinatorial auction model in which bidders’ preferences and payments for a bundle of items do not need to be quasi-linear. Schellhorn [16] considered the substitutability of commodities in a double-sided multi-unit combinatorial auction and proposed an algorithm for cleaning up the market. This algorithm is particularly effective when the number of traders is large and the commodities are divisible. Gao et al. [17] studied the optimal bidding strategy for two heterogeneous and complementary items in a combinatorial auction, considering the regretful psychological behavior of global bidders. However, to better reveal the preferences of bidders (car dealers) for different used cars, we emphasize that a crucial task is to uncover their need for multiple attributes hidden in their preferences, which was neglected in these studies.

A multi-attribute auction generally focuses on reverse auctions or procurement [18,19]. Ran et al. [20] studied the equilibrium price bidding strategy of global suppliers in two stages by using the reverse recursion method and conducted a comparative static analysis. Koksalan et al. [21] developed an evolutionary algorithm for multi-attribute, multi-item reverse auctions that involves generating the whole Pareto front to solve the allocation problem. Ma et al. [22] proposed a method to transform a multi-attribute auction into a single attribute auction, with the aim of reducing the operation cost and complexity of multi-attribute auction. Karakaya et al. [23] designed an interactive approach for buyers and bidders with nonlinear preference functions in a bi-attribute, multi-item auction setting. As for procurement, Baranwal and Vidyarthi [24] considered non-price attributes and price attributes to determine the winning service providers by multi-attribute combinatorial reverse auction. The relationships of complementarity and substitutability are scarcely analyzed in the context of multi-attribute auction. In this paper, we analyzed price attribute and non-price attributes (risk attribute, pro-environment attribute, service attribute, quality attribute), in which complementarity and substitutability were considered.

However, the research on the pro-environment attribute in the field of used cars is still relatively lacking. To solve multi-attribute decision-making problems, Mehdi et al. [25] employed data envelopment analysis when facing heterogeneous attributes, which ignored the pro-environment attribute. Yu et al. [26] rated and sorted multi-attribute new energy vehicles to provide consumers with a more reliable and real reference for selection. However, they focused only on comments from a third party and did not consider the consumers’ own multi-attribute demands. Moreover, in a traditional multi-attribute auction, price is mainly considered, neglecting other important green attributes [27]. Therefore, to fill this gap, under the form of combinatorial auction, this paper designed the multi-attribute used-car auction mechanism considering the pro-environmental attribute. In other words, our purpose was to design one combinatorial auction considering the multi-attribute demand, especially the pro-environment attribute, from the perspective of social welfare. Moreover, complementarity or substitutability were analyzed before analyzing multi-attribute demand.

Therefore, our contributions focused on three main aspects: (1) Multi-attribute demand and preference relationship (complementarity and substitutability) of bidders were incorporated to explore the optimal revenue in a used-car combinatorial auction. This approach not only reflects the actual situation of the bidder and helps the seller achieve a higher revenue, but also assists in achieving social welfare. (2) The combinatorial auction mechanism considering multi-attribute demand was designed, which was proved to satisfy individual rational (IR) condition and incentive compatibility (IC) condition and achieve higher social welfare more than by only considering price by numerical application. (3) A variety of fuzzy number descriptions were given in case of different type of data and actual situations. In this way, the mechanism can support the accurate evaluation of multiple attributes and be closer to the real needs.

The remainder of this paper is organized as follows: The materials and methods used in this paper are introduced in Section 2, in which a multi-attribute combinatorial auction model is constructed based on an analysis of complementarity and substitutability. The numerical application and results are given in Section 3. Some discussions are shown in Section 5. Conclusions are given in Section 5.

2. Materials and Methods

2.1. Multi-Attribute Demand of Bidders

For the multiple attributes of used cars to be considered, environmental attributes need to be taken into account from the perspective of social welfare. This approach can promote the circulation of used-car resources and improve the environment. Therefore, except price attribute, we further considered non-price attributes, including risk attribute ($q_1$), pro-environment attribute ($q_2$), service attribute ($q_3$), and quality attribute ($q_4$), which is shown below (Figure 2).

Additionally, some notations used in this paper are listed in

Table 1. Notations and their descriptions.

Parameter	Descriptions
$\overline{w}$	non-price attribute bids
$r\left(\overline{w_j}\right)$	bidder’s demand rating for the j-th non-price attribute
${\overline{o}}_j$	assessment of j-th attribute’ importance, i.e., weight
$q_1$	risk attribute
$q_2$	pro-environment attribute
$q_3$	service attribute
$q_4$	quality attribute
$p_i$	price attribute
C	risk type attribute
B	benefit type attribute
n	quantity of bidders
$m$	quantity of indivisible heterogeneous objects
$M$	set of indivisible heterogeneous objects
$v_i$(S)	bidder i’s estimated value of item/combination S
$x_i$	allocation to bidder i
$s_i$	demand combination of bidder i
$e_i$	payment of bidder i
$\epsilon$	substitutability parameter
$\delta$	complementarity parameter
${\phi }_i$	true type of bidder i
${\phi }_i^{\prime }$	false type of bidder i
$u_i$	utility of bidder i
$U$	utility of seller
$W$	social welfare

for convenience.

Price attribute is the estimated value of the used car from the bidder. For the non-price attributes, the online platform would publish the scoring content of non-price attribute bids $\overline{w}=(\overline{w_1},\overline{w_2},\overline{w_3},\dots \overline{w_j})$, and the bidding functions

$$r\left(\overline{w}\right)={{\displaystyle \sum }}_{k=1}^j{\overline{o}}_{k}r\left({\overline{w}}_k\right)$$

(1)

where k = 1,2,3,…, j, is the sequence of non-price attributes. $r\left(\overline{w_j}\right)$ denotes the bidder’s demand rating for the j-th non-price attribute. ${\overline{o}}_j$ shows the assessment of j-th attribute’ importance, i.e., weight.

In the following

Table 2. Non-price multi-attribute indexes.

Attribute	Description	Format	Type
Risk ($q_1$)	Fluctuation of deal price	Triangular fuzzy number	C
Pro-environment ($q_2$)	Environment requirement: low, inferior, medium, superior, high	Fuzzy semantic	B
Service ($q_3$)	Service level: 1, 2, 3, 4, 5.	Determinate number	B
Quality ($q_4$)	Vehicle quality rating from 1 to 5	Interval number	B

, the descriptions of multiple non-price attributes are given, including risk attribute ($q_1$), environment attribute ($q_2$), service attribute ($q_3$), and quality attribute ($q_4$), whose details can be seen in Table 1. For the risk attribute, a small value is better, whereas for the other attributes, a large value is better. The former was named risk type attribute (C), and the latter was named the benefit type attribute (B).

These four categories of attributes were normalized according to fuzzy descriptions in this paper to quantify the above attribute requirements uniformly.

(1) For interval numbers, the normalizing formula is as follows [28]:

$$w_{i,j}=\left\{\begin{array}{cc}\left[\frac{q_{i,j}^L}{K_1},\frac{q_{i,j}^H}{K_1}\right],\hfill & i\in N,j\in J\cap j\in C\hfill \\ \left[\frac{1}{K_2{q}_{i,j}^H},\frac{1}{K_2{q}_{i,j}^L}\right],\hfill & i\in N,j\in J\cap j\in B\hfill \end{array}\right.,$$

(2)

where $q_{i,j}^L$ is the lower bound of bidder $i$ on attribute $j$, and $q_{i,j}^H$ is the higher bound. We set $K_1=\sqrt{\sum _{i=1}^n[{\left(q_{i,j}^L\right)}^2+{\left(q_{i,j}^H\right)}^2]}$, $K_2=\sqrt{\sum _{i=1}^n[{\left(1/q_{i,j}^L\right)}^2+{\left({1/q}_{i,j}^H\right)}^2]}$ and obtain the demand level

$${\overline{w}}_{i,j}=\left\{\begin{array}{cc}\left(\frac{q_{i,j}^L}{K_1}+\frac{q_{i,j}^H}{K_1}\right)/2,\hfill & i\in N,j\in J\cap j\in C\hfill \\ \left(\frac{1}{K_2{q}_{i,j}^H}+\frac{1}{K_2{q}_{i,j}^L}\right)/2,\hfill & i\in N,j\in J\cap j\in B\hfill \end{array}\right.$$

(3)

(2) For the triangular fuzzy number, the normalizing formula is as follows [29]:

$$w_{i,j}=\left\{\begin{array}{cc}\left[\frac{q_{i,j}^L}{K_1},\frac{q_{i,j}^M}{K_1},\frac{q_{i,j}^H}{K_1}\right],\hfill & i\in N,j\in J\cap j\in C\hfill \\ \left[\frac{1}{K_2{q}_{i,j}^H},\frac{1}{K_2{q}_{i,j}^M},\frac{1}{K_2{q}_{i,j}^L}\right],\hfill & i\in N,j\in J\cap j\in B\hfill \end{array}\right.$$

(4)

where $q_{i,j}^L$ is the first number among the triangular fuzzy numbers of bidder $i$ on attribute $j$, $q_{i,j}^M$ is the second number, and $q_{i,j}^H$ is the third number. We set $K_1=\sqrt{\sum _{i=1}^n[{\left(q_{i,j}^L\right)}^2+{\left(q_{i,j}^M\right)}^2+{\left(q_{i,j}^H\right)}^2]}$, $K_2=\sqrt{\sum _{i=1}^n[{\left(1/q_{i,j}^L\right)}^2+{\left({1/q}_{i,j}^M\right)}^2+{\left({1/q}_{i,j}^H\right)}^2]}$ and obtain the demand level

$${\overline{w}}_{i,j}=\left\{\begin{array}{cc}\left(\frac{q_{i,j}^L}{K_1}+4\frac{q_{i,j}^M}{K_1}+\frac{q_{i,j}^H}{K_1}\right)/6,\hfill & i\in N,j\in J\cap j\in C\hfill \\ \left(\frac{1}{K_2{q}_{i,j}^H}+4\frac{1}{K_2{q}_{i,j}^M}+\frac{1}{K_2{q}_{i,j}^L}\right)/6,\hfill & i\in N,j\in J\cap j\in B\hfill \end{array}\right..$$

(5)

(3) The fuzzy semantic is generally transformed into triangular fuzzy numbers [30].

In accordance with the descriptions and definitions of researchers [31], we set $L$ as a set of fuzzy semantic terms, $L=\left\{\left.L_k\right|k=\mathrm{0,1},2,\dots ,T\right\}$. If $T=4$, then the semantic terms correspond to low, inferior, medium, superior, and high, respectively. Calculation is facilitated generally by converting the fuzzy semantic into a triangular fuzzy number [32]

$$e=\left\{\max \left\{\frac{k-1}{T},0\right\},\frac{k}{T},\mathrm{m}\mathrm{i}\mathrm{n}\{\frac{k+1}{T},1\}\right\}$$

(6)

(4) For the determinate numbers, the normalizing formula is as follows:

Table 1 shows the determinate numbers, including 1, 2, 3, 4, 5, and the normalizing formula can be written as

$${\overline{w}}_{i,j}=q_{i,j}/Maxq$$

(7)

After the above processing, the normalized demand matrix can be obtained as follows:

$${\left[{\overline{w}}_{i,j}\right]}_{N\times J}=\left[\begin{array}{ccc}{\overline{w}}_{\mathrm{1,1}}& {\overline{w}}_{\mathrm{1,2}}& \begin{array}{cc}\dots & {\overline{w}}_{1,j}\end{array}\\ {\overline{w}}_{\mathrm{2,1}}& {\overline{w}}_{\mathrm{2,1}}& \begin{array}{cc}\dots & {\overline{w}}_{2,j}\end{array}\\ \begin{array}{c}\dots \\ {\overline{w}}_{N,1}\end{array}& \begin{array}{c}\dots \\ {\overline{w}}_{N,2}\end{array}& \begin{array}{cc}\begin{array}{c}\ddots \\ \dots \end{array}& \begin{array}{c}\dots \\ {\overline{w}}_{i,j}\end{array}\end{array}\end{array}\right]$$

(8)

In addition, sellers need to determine the weighting of each attribute on the basis of their expectations and the actual condition of the used car, which is a critical step in influencing the final bidding and decision [31]. When the attribute weights are set, most of the literature uses subjective weight to maximize their own interests, ignoring the fact that sellers often have a poor grasp of market conditions. Therefore, the platform should comprehensively consider the rationality of subjective demand and objective market situation, and then negotiate the minimum multi-attribute demand standards St.

At the end, the multi-attribute demand level is

$$r\left(\overline{w}\right)={{\displaystyle \sum }}_{k=1}^j{\overline{o}}_{k}r\left({\overline{w}}_k\right)$$

(9)

2.2. Multi-Attribute Combinatorial Auction Model

2.2.1. Multi-Attribute Preference of Bidders

In the combinatorial auction, bidders can clearly consider the complementarity and substitutability of used cars and express their preference for the relationship between the current auctioned vehicles on the basis of their assessment of the need for different attributes. Some assumptions are given below to establish the combinatorial auction model of bidders:

(1) $n$ bidders are risk neutral [33];

(2) Seller has $m$ indivisible heterogeneous objects (different used cars), and $M$ is the set of these objects. For the used cars, bidders have independent private values, and each bidder knows only its own valuation of different commodities [34];

(3) No cooperative relationship or collusion occurs among bidders [35].

The estimated value of item or item combination $S$ is $v_i\left(S\right)$, where $S\in M$ represents a combination. For any $S,C\subset M$, and if $C\subset S$, then $v_i\left(S\right)\ge v_i\left(C\right)$. That is, the bidder’s valuation increases as the number of combinations increases.

In reality, for different used cars, bidders determine preference on the basis of their multi-attribute demand, such as price attributes and non-price attributes (risk attribute, environment attribute, service attribute, and quality attribute). In other words, when the bidder joins the combinatorial auction for existing cars, he can express his preference between multiple items according to his own demand. Bidders give the relationship of complementarity and substitutability of each item combination [36], especially in non-price attributes, which is shown as follows:

$$v_i^{ab}=f(r\left(w_a\right),r\left(w_b\right))$$

(10)

$$v_i^{abc}=f(r\left(w_a\right),r\left(w_b\right),r\left(w_c\right))$$

(11)

If $a\in M$, $C\subset S\subseteq M$, and $a\notin C$, $a\notin S$, and bidder $i$ considers that complementarity exists between these items, then we can obtain

$$v_i\left(S\cup \left\{a\right\}\right)-v_i\left(S\right)\le v_i\left(C\cup \left\{a\right\}\right)-v_i\left(C\right)$$

(12)

According to the definition of complementarity between items, when the bidder obtain more items, he will acquire more values from another item. Thus, the above formula is also equivalent to:

$$v_i\left(S\right)+v_i\left(C\right)\le v_i\left(C\cup S\right)+v_i\left(C\cap S\right)$$

(13)

If bidder $i$ considers that substitutability exists between demand items, then we can derive

$$v_i\left(S\right)+v_i\left(C\right)\ge v_i\left(C\cup S\right)+v_i\left(C\cap S\right).$$

(14)

We describe the complementarity and substitutability between two items and among three items by using $v_i^{ab}$ and $v_i^{abc}$, respectively

$$v_i^{ab}=\left\{\begin{array}{c}\epsilon \times \left|r\left(\overline{w_a}\right)-r\left(\overline{w_b}\right)\right|,if\left|r\left(\overline{w_a}\right)-r\left(\overline{w_b}\right)\right|\le threshold\\ \delta \times \left|r\left(\overline{w_a}\right)-r\left(\overline{w_b}\right)\right|,if\left|r\left(\overline{w_a}\right)-r\left(\overline{w_b}\right)\right|>threshold\end{array}\right..$$

(15)

where $\epsilon$ and $\delta$ represent the substitutability parameter and the complementarity parameter, respectively.

Set $\alpha =\max \left[\left|r\left(\overline{w_a}\right)-r\left(\overline{w_b}\right)\right|,\left|r\left(\overline{w_a}\right)-r\left(\overline{w_c}\right)\right|,\left|r\left(\overline{w_b}\right)-r\left(\overline{w_c}\right)\right|\right]$ and $\beta =\min \left[\left|r\left(\overline{w_a}\right)-r\left(\overline{w_b}\right)\right|,\left|r\left(\overline{w_a}\right)-r\left(\overline{w_c}\right)\right|,\left|r\left(\overline{w_b}\right)-r\left(\overline{w_c}\right)\right|\right]$.

$$v_i^{abc}=\left\{\begin{array}{c}\epsilon \times \left|\alpha -\beta \right|,if\left|\alpha -\beta \right|\le threshold\\ \delta \times \left|\alpha -\beta \right|,if\left|\alpha -\beta \right|>threshold\end{array}\right..$$

(16)

2.2.2. Social Welfare Maximization Combinatorial Auction Model

The type of bidders can be defined as a two-tuples array: ${\phi }_i\left(v_i(a_i,p_i),s_i\right)\in \left(\underset{\_}{v_i},\overline{v_i}\right)\times M={\Phi }_i$, and ${\Phi }_i$ is the set of all possible types of buyer i.

Additionally, $p_i$ denotes the price attribute, and $a_i$ denotes the non-price attribute demand rating, $a_i=r\left(q_{1i},q_{2i},q_{3i},q_{4i}\right)$, which includes risk attribute ($q_1$), pro-environment attribute ($q_2$), service attribute ($q_3$), and quality attribute ($q_4$). $v_i$ denotes the estimation value of multi-attribute demand, $s_i$ is the demand combination of bidder i, and $x_i=(x_i^1,x_i^2,\dots x_i^m)$ means the allocation to bidder i, $x_i^j(j=\mathrm{1,2}\dots m)$=0 or 1, which means whether the auction vehicle j is allocated to bidder i. $e_i$ is the payment of bidder i for the combination of winning items. If the item is obtained, then bidder i will pay the quoted price; if the item is not obtained, then no payment is made.

The benefit brought by any combination($s_i$) to bidder i can be shown as

$$V_{si}={{\displaystyle \sum }}_{j=1}^m\left(v_i^j{x}_i^j+{v_i^{ab}{x}_i^a{x}_i^b+v}_i^{abc}{x}_i^a{x}_i^b{x}_i^c\right),a\ne b\ne c,a,b,c=\mathrm{1,2}\dots ,m.$$

(17)

Therefore, the utility of bidder i can be expressed as

$$u_i\left({\phi }_i\right)=V_{si}\left({\phi }_i\right)x_i\left({\phi }_i\right)-e_i\left({\phi }_i\right).$$

(18)

The above equations can be regarded as the utility of bidder i in the case that all bidders report the true type. When bidder i reports false type ${\phi }_i^{\prime }$ with others’ true type, the utility of bidder i is

$$u_i\left({\phi }_i^{\prime }\right)=V_{si}\left({\phi }_i^{\prime },{\phi }_{-i}\right)x_i\left({\phi }_i^{\prime },{\phi }_{-i}\right)-e_i\left({\phi }_i^{\prime },{\phi }_{-i}\right).$$

(19)

The seller’s goal is to maximize revenue. Thus, its utility is

$$U={{\displaystyle \sum }}_{i=1}^n{e}_i\left({\phi }_i\right).$$

(20)

Social welfare is maximized by requiring the utility of the whole system to be maximum, thus necessitating both buyer’s utility and seller’s utility to be considered in the auction. To sum up, when the auction mechanism is designed from the perspective of social welfare, its objective function is as follows:

$$W={{\displaystyle \sum }}_{i=1}^n{e}_i\left(p_i\right)+{{\displaystyle \sum }}_{i=1}^n{u}_i\left({\phi }_i\right)={{\displaystyle \sum }}_{i=1}^n{V}_{si}{x}_{si}\left({\phi }_i\right).$$

(21)

As can be seen from the above equation, only the bidder with a higher benefit valuation for non-price attributes wins, which is the optimal allocation pursued. Thus, if an assignment ($x$,$e$) satisfies the following conditions, then we think the result is feasible

$$x_i^j\in x_i,$$

(22)

$$x_i^j\in \arg \max {\sum }_{i=1}^n{V}_{si}\left(x_{si}\right).$$

(23)

Any mechanism is feasible only if it satisfies IR and IC. For IR, we explain it as follows: For any bidder, if he does not participate in the auction, then auctioning any used cars is not allowed and he does not have to pay. Thus, the expected income of not participating in the auction is 0. The expected income of participating in the auction should not be less than 0 to enable bidders to participate in the auction voluntarily.

$$u_i\left({\phi }_i\right)=V_{si}\left({\phi }_i\right)x_i\left({\phi }_i\right)-e_i\left({\phi }_i\right)\ge 0.$$

(24)

As for IC, if all bidders except bidder i submit their bid according to true valuation, then, when bidder i reports the true type, his utility function is

$$u_i\left({\phi }_i\right)=V_{si}\left({\phi }_i,{\phi }_{-i}\right)x_i\left({\phi }_i,{\phi }_{-i}\right)-e_i\left({\phi }_i,{\phi }_{-i}\right)$$

(25)

While he does not bid according to his own real valuation and other bidders offer real prices, his utility function is

$$u_i\left({\phi }_i^{\prime }\right)=V_{si}\left({\phi }_i^{\prime },{\phi }_{-i}\right)x_i\left({\phi }_i^{\prime },{\phi }_{-i}\right)-e_i\left({\phi }_i^{\prime },{\phi }_{-i}\right).$$

(26)

Then, in accordance with the IC condition, we can obtain

$$u_i\left({\phi }_i\right)\ge u_i\left({\phi }_i^{\prime }\right),$$

(27)

which is equivalent to

$$V_{si}\left({\phi }_i,{\phi }_{-i}\right)x_i\left({\phi }_i,{\phi }_{-i}\right)-e_i\left({\phi }_i,{\phi }_{-i}\right)\ge V_{si}\left({\phi }_i^{\prime },{\phi }_{-i}\right)x_i\left({\phi }_i^{\prime },{\phi }_{-i}\right)-e_i\left({\phi }_i^{\prime },{\phi }_{-i}\right).$$

(28)

Moreover, to ensure that the mechanism satisfies the IC, we need to describe the characteristics of the combination auction that satisfy the IC condition. According to the direct revelation principle, the feasibility conditions of the IC of the combined auction mechanism that we designed can be described by the following mathematical expressions:

In arbitrary allocation rules x, for bidder i, $V=V_{si}\left({\phi }_i,{\phi }_{-i}\right)$ and $V_{si}\left({\phi }_i^{\prime }\right)\ne V_{si}\left({\phi }_i\right)$ need to satisfy

$${{\displaystyle \int }}_{V_{si}\prime }^{V_{si}}\mathsf{\partial }{V}_{si}\left(x_{si}\left(z\right),z,V_{-i}\right)dz\le V_{si}\left(x_{si}\left(V\right),V\right)-V_{si}\left({\phi }_i^{\prime }\right)\left(x_{si}\left(V\right),V_{si}\prime ,V_{-i}\right)$$

(29)

The proof is given from two aspects.

① If the IC is established, then according to the envelope theorem, we can obtain

$$\begin{array}{c}\underset{V_{si}\prime }{\overset{V_{si}}{\int }}\mathsf{\partial }{V}_{si}\left(x_{si}\left(z\right),z,V_{-i}\right)dz=u_i\left(x_{si}\left(V\right),V_i\right)-u_i\left(x_{si}\prime \left(V_{si}\prime ,V_{-i}\right),V_{si}\prime \right)\hfill \\ =V_{si}\left(x_{si}\left(V\right),V\right)-e_i\left(V_{si}\right)-\left[V_{si}\left(x_{si}\prime \left(V_{si}\prime ,V_{-i}\right),V_{si}^{\prime },V_{-i}\right)-e_i\left(V_{si}^{\prime },V_{-i}\right)\right]\hfill \\ \le V_{si}\left(x_{si}\left(V\right),V\right)-e_i\left(V_{si}\right)-\left[V_{si}\left(x_{si}\left(V_{si}\right),V_{si}^{\prime },V_{-i}\right)-e_i\left(V_{si}\right)\right]\hfill \\ =V_{si}\left(x_{si}\left(V\right),V\right)-V_{si}\left(x_{si}\left(V_{si}\right),V_{si}^{\prime },V_{-i}\right)\hfill \end{array}$$

② If any allocation satisfies the feasibility condition and the envelope theorem holds, then we can obtain

$$\begin{array}{c}{u}_i\left(x_{si}\left(V\right),V_i\right)-u_i\left(x_{si}\prime \left(V_{si}\prime ,V_{-i}\right),V_{si}\prime \right)=\underset{V_{si}\prime }{\overset{V_{si}}{\int }}\mathsf{\partial }{V}_{si}\left(x_{si}\left(z\right),z,V_{-i}\right)dz\\ \le V_{si}\left(x_{si}\left(V\right),V\right)-V_{si}\left(x_{si}\left(V_{si}\right),V_{si}^{\prime },V_{-i}\right),\end{array}$$

which means IC holds.

Therefore, according to IC and IR, the combinatorial auction model based on social welfare maximization can be shown as follows:

$$\begin{array}{c}\underset{(x,e)}{\max }W\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{u}_i\left({\phi }_i\right)\ge u_i\left({\phi }_i^{\prime }\right)\\ u_i\left({\phi }_i\right)\ge 0\\ {\displaystyle \sum _{i=1}^n}{x}_i^j\left(v\right)\le 1,j=\mathrm{1,2},\dots m\\ {\displaystyle \sum _{j=1}^m}{x}_i^j\left(v\right)\le 3,i=\mathrm{1,2},\dots n\\ \max e_i\left(p_i\right)\ge r_{si}\\ x_i\in \left\{\mathrm{0,1}\right\},1\le i\le n\end{array}\right.\end{array}$$

The objective function of the model is from the perspective of social welfare maximization, and the utility maximization of buyer and seller is considered. The first constraint is IC, that is, to ensure that all individual buyers receive at least as much utility when they truly report their own type as they report other types. The second constraint condition is IR, that is, the utility of individual buyers participating in the auction is not negative. The third constraint is that the auction is for a single item only, and the number of transactions is 1. The fourth constraint is that the highest paid price (transaction price) must not be lower than the reserve price.

On the basis of the above combined auction model, the auctioneer needs to determine an optimal set of allocation schemes and payment rules $\left(x^{*},\tau \right)$. The allocation of the bidders who do not win is $x_i^j=0$. Under this allocation scheme, the sum of the valuations of all the winning bidder on the used car combination is the highest, that is,

$$x_i^j\in \arg \max {{\displaystyle \sum }}_{i=1}^n{V}_{si}\left(x_{si}\right).$$

Under this optimal allocation, if the bidder i is one of the winners in this combination auction, then his social surplus (payment) is

$$e_i={{\displaystyle \sum }}_{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{V}_j\left(x_{-i}\right)-{{\displaystyle \sum }}_{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{V}_j\left(x_j\right)$$

where ${\displaystyle \sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{V}_j\left(x_{-i}\right)$ denotes the social surplus from the optimal allocation ($x_{-i}$) when bidder i does not participate in the auction; ${\displaystyle \sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{V}_j\left(x_j\right)$ represents the total social surplus created by other bidders except bidder i under the optimal allocation when bidder i participates in the auction. Therefore, the price that bidder i needs to pay is equal to the difference between the social surplus created by other bidders when he participates in the auction and when he does not.

3. Numerical Application and Results

The assumption was that there were three used cars for auction, and seven bidders estimated any combination of items (single item, two items, and three items), anyone of whom can obtain up to three items in the auction. Each bidder’s estimated value on a single car is denoted by ${\mathrm{v}}_{\mathrm{i}}^1$, ${\mathrm{v}}_{\mathrm{i}}^2$, and ${\mathrm{v}}_{\mathrm{i}}^3$, all of which obey the same distribution $\mathrm{U}\left(\mathrm{5,10}\right)$. Any combination of cars considering complementarity and substitutability are denoted by ${\mathrm{v}}_{\mathrm{i}}^{12}$, ${\mathrm{v}}_{\mathrm{i}}^{13}$, ${\mathrm{v}}_{\mathrm{i}}^{23}$, and ${\mathrm{v}}_{\mathrm{i}}^{123}$; substitutability parameter ($\mathsf{\epsilon }$) and complementarity parameter($\mathsf{\delta }$): $\mathsf{\epsilon }=-2$, $\mathsf{\delta }=1.5$; and $\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}=0.06$. The reserved prices of different cars combination are shown in

Table 3. Reserved price of combinational auction.

Combination	1	2	3	1 + 2	1 + 3	2 + 3	1 + 2 + 3
Reserved price	5.5	6.5	8.5	11	13	14	19

.

3.1. Initial Value

For the convenience of analysis, we assumed that seven bidders participated in the multi-attribute auction of a used car, whose evaluations of each used car are shown in

Table 4. Bidders’ evaluations of each used car.

Bidder	Used Car	$\mathbf{Price} \mathbf{Attribute} \mathbf{\left(}\mathit{p}\mathbf{\right)}$	Non-Price Attributes
			$\mathbf{\left(}{\mathit{p}}_1\mathbf{\right)}$	$\mathbf{\left(}{\mathit{p}}_2\mathbf{\right)}$	$\mathbf{\left(}{\mathit{p}}_3\mathbf{\right)}$	$\mathbf{\left(}{\mathit{p}}_4\mathbf{\right)}$
1	1	5.71	(5.03 10.48 12.84)	medium	3	[3, 5]
	2	7.15	(5.49 10.73 13.77)	superior	4	[4, 5]
	3	8.73	(5.36 7.00 12.56)	superior	4	[3, 4]
2	1	5.82	(5.03 10.48 12.56)	medium	3	[2, 4]
	2	6.86	(5.67 7.01 12.56)	superior	4	[4, 5]
	3	8.64	(5.12 9.58 12.84)	superior	4	[3, 5]
3	1	6.21	(4.03 10.36 11.53)	low	2	[3, 5]
	2	6.92	(5.58 10.36 12.57)	superior	5	[2, 5]
	3	8.78	(5.57 10.38 12.55)	superior	5	[3, 5]
4	1	5.55	(4.88 9.68 11.34)	high	3	[3, 4]
	2	7.22	(5.58 10.47 12.31)	high	4	[5, 1]
	3	8.81	(5.26 10.55 12.72)	inferior	4	[4, 5]
5	1	5.72	(5.03 10.36 12.56)	inferior	3	[2, 5]
	2	6.63	(6.13 9.11 12.13)	superior	4	[3, 5]
	3	8.90	(4.89 8.22 12.56)	high	4	[3, 4]
6	1	5.97	(5.88 9.73 12.77)	superior	2	[4, 5]
	2	7.36	(4.58 9.36 13.56)	high	5	[4, 5]
	3	8.63	(5.12 9.58 12.84)	superior	4	[3, 5]
7	1	5.82	(5.14 10.53 11.37)	medium	3	[3, 5]
	2	6.87	(4.36 8.02 12.78)	superior	4	[4, 5]
	3	8.74	(4.66 9.87 13.45)	inferior	4	[3, 5]

. The table contains price attribute $\left(p\right)$, risk attribute ($q_1$), pro-environment attribute ($q_2$), service attribute ($q_3$), and quality attribute ($q_4$).

According to the agreement between the platform and the seller, the weights of non-price attributes were 0.1, 0.2, 0.2, and 0.5, and the minimum standard negotiated was St = 0.63. If the pro-environment attribute was not considered, then the weights of the non-price attributes were 0.2, 0.3, and 0.5. According to the conditions of the used cars and the current vehicle emission standards required by the state, the environmental attribute standard was set as 0.55.

3.2. Multi-Attribute Demand of Bidders

The auction of three used cars had seven bidders. For the sake of analysis, bidder 1 was taken as an example to show the multi-attribute demand. (Related information is shown in Table 4.)

According to Equations (1)–(8), the demand matrix of bidder 1 is

$${\left[{\overline{w}}_{i,j}\right]}_{3\times 4}=\left[\begin{array}{c}0.47 0.53 0.6 0.69\\ 0.48 0.56 0.8 0.70\\ 0.56 0.56 0.8 0.70\end{array}\right]$$

Therefore, the demand non-price attributes of bidder 1 are

$$r\left(\overline{w}\right)=\left[\begin{array}{c}0.47 0.53 0.6 0.69\\ 0.48 0.56 0.8 0.70\\ 0.56 0.56 0.8 0.70\end{array}\right]\times \left[\begin{array}{c}\begin{array}{c}0.1\\ 0.2\\ 0.2\end{array}\\ 0.5\end{array}\right]={\left(0.62 0.67 0.68\right)}^T$$

The above is a multi-attribute demand for one bidder whose calculation process can be adapted to other bidders. The non-price multi-attribute demand level $r\left(w\right)$ of seven bidders is summarized in

Table 5. Bidders’ non-price attribute demand.

Bidders	Used Car 1	Used Car 2	Used Car 3
1	0.62	0.67	0.68
2	0.61	0.68	0.66
3	0.51	0.69	0.70
4	0.64	0.72	0.65
5	0.58	0.66	0.68
6	0.59	0.73	0.66
7	0.62	0.68	0.64

.

For the complementarity and substitutability,

$${\mathrm{v}}_{\mathrm{i}}^{\mathrm{a}\mathrm{b}}=\left\{\begin{array}{c}-2\times \left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)\right|,if\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)\right|\le 0.06\\ 1.5\times \left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)\right|,if\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)\right|>0.06\end{array}\right..$$

$$\begin{array}{c}\mathrm{Set} \mathsf{\alpha }=\max \left[\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)\right|,\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{c}}}\right)\right|,\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{c}}}\right)\right|\right] \mathrm{a}\mathrm{n}\mathrm{d}\\ \mathsf{\beta }=\min \left[\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)\right|,\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{a}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{c}}}\right)\right|,\left|\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{b}}}\right)-\mathrm{r}\left(\overline{{\mathrm{w}}_{\mathrm{c}}}\right)\right|\right].\\ {\mathrm{v}}_{\mathrm{i}}^{\mathrm{a}\mathrm{b}\mathrm{c}}=\left\{\begin{array}{c}-2\times \left|\mathsf{\alpha }-\mathsf{\beta }\right|,\mathrm{i}\mathrm{f}\left|\mathsf{\alpha }-\mathsf{\beta }\right|\le 0.06\\ 1.5\times \left|\mathsf{\alpha }-\mathsf{\beta }\right|,\mathrm{i}\mathrm{f}\left|\mathsf{\alpha }-\mathsf{\beta }\right|>0.06\end{array}\right..\end{array}$$

In accordance with the above analysis, we can obtain the complementary or substitutability relationship of bidders among different combinations of used cars, as shown in

Table 6. Bidders’ valuations of different used cars and their combination relationships.

	Combinations	Price Attribute			Non-Price Attributes
Bidders		1	2	3	1 + 2	1 + 3	2 + 3	1 + 2 + 3
1		5.71	7.15	8.73	−0.1	−0.12	−0.02	−0.1
2		5.82	6.86	8.64	0.105	−0.1	−0.04	−0.1
3		5.91	6.92	8.51	0.27	0.285	0.015	0.27
4		5.55	7.22	8.81	0.12	−0.02	0.105	0.105
5		5.72	6.63	8.90	0.12	0.15	−0.04	0.12
6		5.97	7.36	8.63	0.21	0.105	0.105	0.105
7		5.82	6.87	8.74	−0.12	−0.04	−0.08	−0.08

.

3.3. Allocation Results

From the above table, the valuation of seven bidders on the used car combination based on the demand of buyers is summarized in

Table 7. Bidders’ demand for different used cars and valuation of the combination relationship.

Bidders	Car 1	Car 2	Car 3	Price Estimation	Multi-Attribute Estimation
1	0	0	1	8.73	8.73
2	1	0	0	5.82	5.82
3	1	0	1	14.42	14.705
4	0	1	0	7.22	7.22
5	1	1	0	12.35	12.47
6	0	1	1	15.99	16.095
7	1	1	1	21.43	21.35

.

We list the allocation results as follows (

Table 8. Valuation and quotation of bidder under different allocation.

Allocation Bidder	1,5	1,2,4	2,6	3,4	7
Allocation result	(5,5,1)	(2,4,1)	(2,6,6)	(3,4,3)	(7,7,7)
Social welfare	21.2	21.77	21.915	21.925	21.43
price	21.08	21.77	21.81	21.64	21.43

):

(1) Allocation result considering multiple attributes.

According to the goal of maximizing social welfare, the allocation is selected as (3,4,3), and all bidders need to pay 21.905, including the payment of bidder 3 (21.915 − 7.22 = 14.695) and the payment of bidder 4 (21.915 − 14.705 = 7.21), Thus, the seller’s income is 21.905. Bidder 3’s utility is 0.01, and bidder 4’s utility is 0.01. Therefore, all bidders’ utility is 0.02. The corresponding social welfare is 21.925.

(2) Allocation result without considering multiple attributes.

If the preference of non-price attributes (especially pro-environment attribute) is not considered, then the combinatorial auction is analyzed only from the price dimension. From Table 6, the highest price from all bidders was 21.81, which was from bidder 2 and bidder 6. Bidder 2 needs to pay 5.82, and bidder 6 needs to pay 15.99, with social welfare 21.915 being lower than the allocation result considering multiple attributes.

A comparison of both allocation situations showed that multi-attribute demand from bidders is vital to combinatorial auction, especially the pro-environment attribute, under the goal of dual carbon. When multiple attributes are not considered, social welfare may be low, which does not benefit the trading and circulation of vehicles with low emission standards and does not provide any assistance to carbon emission. Therefore, the benefits brought by non-price attributes is the key to pursuing the goal of maximizing social welfare. In other words, strengthening the emphasis on non-price attributes and improving bidders’ perception of benefits can better improve social welfare.

4. Discussion

This paper proposed a multi-attribute auction mechanism for the bidders of automobile enterprises on the online used car platform, in which price, risk, pro-environment, quality, and service are considered based on an analysis of complementarity and substitution. We proved that the multi-attribute auction mechanism satisfied IR and IC, and we used numerical application to compare the allocation results of considering multiple attributes with those of the method that did not consider multiple attributes. The results showed that the multi-attribute combinatorial auction mechanism was fairer and more effective for bidders from the perspective of social welfare. The differences between this paper and previous studies are as follows:

(1)

For combinatorial auction, most studies considered only the optimal combination of bundle sale, ignoring the complementarity and substitution of some heterogeneous items. Erbil and Ali [37] proposed an energy-aware virtual-machine-scheduling model based on the multi-unit combinatorial auction. When the selling object is nondiscriminatory, the complementarity and substitution become trivial. David [38] analyzed complementarities to design a combinatorial auction to sell TV broadcasting rights, which is unsuitable for used cars. Therefore, the preference relationship (complementarity and substitutability) of bidders needs to be incorporated to explore the optimal revenue in the used car combinatorial auction.

(2)

Unlike in previous studies, a multi-attribute auction mechanism with IR and IC was designed in this paper, which considers price attribute and non-price attribute (risk, pro-environment, quality, service) from the perspective of social welfare. Albert et al. [39] considered different types of attributes in a multi-attribute auction but only roughly divided them into three categories. For the used car, its pro-environment attribute is more vital to circular economy development and the achievement of the dual carbon goal. Therefore, the combinatorial auction mechanism considering multi-attribute demand is significant.

(3)

In the used car market, each car is unique and heterogeneous. As a result, consumer demand is more difficult to determine [40]. From the real needs of consumers, this paper used fuzzy number to describe the different evaluation and perception of multiple attributes of used cars, which is closer to reality. In case of different types of data and actual situations, a variety of fuzzy number descriptions were given. In this way, the mechanism can support the accurate evaluation of multiple attributes and be closer to actual needs.

To sum up, our contributions are as follows: (1) The preference relationship (complementarity and substitutability) of bidders were incorporated to explore the optimal revenue in a used car combinatorial auction. (2) The combinatorial auction mechanism considering multi-attribute demand was designed, which was proved to satisfy IR and IC, and it can achieve higher social welfare than when only the price is considered. (3) A variety of fuzzy number descriptions were given in case of different type of data and actual situations.

5. Conclusions

In this paper, the combinatorial auction mechanism was designed for the bidders of automobile enterprises on an online used-car platform according to their multi-attribute demand. We proved the validity of the mechanism and adopted numerical application to show the importance of considering multiple attributes. Specifically, the combinatorial auction can provide a reference for a more fair and effective allocation of used cars to bidders and offer assistance to better promote both buyer’s utility and seller’s income. From the perspective of social welfare, the allocation results can be fairer and more effective for bidders, which can improve the efficiency of resource allocation, ensure that the transaction price is more likely to meet the satisfaction of both parties, and better promote the improvement of both buyer’s utility and seller’s income.

However, our paper had some limitations. First, buyers have differences not only in vehicle demand but also in their psychological perception of multi-attribute benefits. With psychological factors taken into consideration in future research, the heterogeneity can be better revealed and be more realistic. Theories such as prospect theory and regret theory can ensure that the valuation is closer to the actual situation. Second, the mathematical modeling used in this paper was theoretically feasible, but it lacked practical case support, which was reflected in the numerical application. Our future work will focus on a specific platform and will involve collecting a large amount of transaction data.