Supply Chain Coordination with Guaranteed Auction Contracts

Abstract

This paper investigates the problem of contract coordination in a two-tier multi-unit auction supply chain consisting of a seller and an auction house. We theoretically show that the conventional commission-based mechanism distorts the transmission of demand information from the demand side to the supply side, thereby preventing effective supply chain coordination. In contrast, guaranteed auction contracts can achieve coordination under both cooperative and non-cooperative game frameworks. Under the cooperative game setting, profits are allocated according to a Nash bargaining solution, in which each party receives its disagreement payoff and a bargaining-power-weighted share of the surplus, with risks and returns being allocated symmetrically. Under the non-cooperative game setting, the supply chain leader can appropriate a larger share of the total profit while bearing relatively lower risk. These results indicate that, as the supply chain leader, the auction house can select different cooperation modes under guaranteed auction contracts according to its bargaining position, but profit allocation should be benchmarked against the cooperative game outcome in order to enhance the long-term competitiveness and stability of the supply chain.

1. Introduction

As third-party intermediaries, auction houses specialize in aggregating market information, managing auction processes, and facilitating the valuation and allocation of goods [1,2]. These capabilities enable auction houses to offer orderly, efficient, and convenient trading services for both buyers and sellers. Consequently, most sellers choose not to organize auctions independently but instead consign their goods to auction houses, particularly in markets involving large volumes of homogeneous goods [3]. Representative examples include the North American Fur Auctions in Toronto, the Aalsmeer Flower Auction in Amsterdam, the Colombo Tea Auction in Sri Lanka, and the Tsukiji tuna auction market in Tokyo. Across these markets, the annual value of consigned goods traded reaches tens of billions of dollars.

A typical multi-unit consignment auction is conducted under a commission-based mechanism. When a seller consigns a given quantity of goods to an auction house, a reserve price is specified in advance. The goods are sold when the auction price exceeds the reserve price, and the auction house charges a commission as a fixed proportion of the final price. If not, the goods remain unsold and are returned to the seller. Although the commission mechanism is simple to implement and widely adopted in practice, it has attracted substantial criticism. The auction literature shows that maximizing system-wide revenue requires the joint optimization of supply and selling decisions [4]. In practice, however, sellers’ choices of auction quantities and reserve prices, as well as auction houses’ commission rates, are often misaligned with the objective of optimizing overall supply chain performance [5]. Moreover, empirical evidence suggests that some sellers exhibit low willingness to cooperate with auction houses and may collude with buyers to bypass the auction process and engage in private transactions [6]. As a result, disputes arising from commission-based consignment auctions are frequently reported, undermining normal market order.

To address these issues and stabilize sellers’ supply, auction houses are increasingly adopting guaranteed auction mechanisms. These mechanisms promise sellers a minimum return. For example, the North American Fur Auctions and the Colombo Tea Auction cooperate with governments or industry associations to provide downside protection for sellers, whereby unsold lots are purchased at a predetermined minimum price by the auction house. In China, the Dounan Flower Electronic Trading Center (DFETC) in Kunming has recently begun experimenting with compensation schemes for unsold lots. In addition, large liquidation and bankruptcy service providers, such as Hilco Global and Gordon Brothers, frequently offer guaranteed minimum return arrangements. Such price guarantees require auction houses to commit their own capital. To compensate for this investment risk, these guarantees are typically accompanied by surplus-sharing mechanisms that allow auction houses to participate in upside gains. Nevertheless, compared with the risk-free commission-based mechanisms, a considerable number of auction houses remain cautious about adopting guaranteed auction arrangements.

Our study examines optimal supply and selling strategies in a consignment auction supply chain consisting of a seller and an auction house. We analyze why commission-based mechanisms are often resisted by sellers and how these mechanisms lead to coordination failures that undermine total supply chain profit. We further investigate whether guaranteed auction mechanisms constitute a superior contractual arrangement and, if so, how such contracts can be designed to benefit both parties.

To address these questions, we develop a game-theoretic model in which a seller consigns a given quantity of goods to an auction house for sale. Prices are discovered through auctions in an uncertain market, and auction revenues are shared between the seller and the auction house after the sale. We take the supply and selling strategies under centralized decision making as a benchmark, representing the best achievable performance of the consignment auction supply chain. We then examine the strategic decisions of supply chain members and the resulting performance under commission-based and guaranteed auction mechanisms, respectively. In particular, we focus on how profits and associated risks are allocated among members, as these factors are critical to the stability of cooperation within the supply chain.

Our analysis shows that the presence of auction commissions distorts the transmission of market demand information, thereby preventing the supply chain from achieving optimal outcomes. More importantly, the commission-based mechanism insulates the auction house from downside risk associated with the auction. Under this mechanism, the auction house merely extracts commissions from sellers and buyers, while the entire risk of unsold goods is borne by the seller. This asymmetry explains why sellers may obtain demand information from the auction house yet subsequently bypass the auction process in actual transactions. By doing so, they avoid commission payments and reduce their exposure to transaction failure risk. These findings highlight the need for an alternative mechanism that more closely aligns the incentives of the seller and the auction house, enabling them to jointly confront market uncertainty.

Unlike commission contracts, guaranteed auction mechanisms provide greater flexibility in profit allocation. Recognizing that auction houses do not always occupy a leading position in cooperative relationships and that some sellers may also be sufficiently influential to negotiate contract terms, we analyze supply and selling decisions under two decision regimes: non-cooperative and cooperative games. Our results show that both regimes can achieve supply chain coordination. However, outcomes under the cooperative game framework are characterized by a high-risk–high-return profile and yield a more balanced distribution of profits. In contrast, non-cooperative games tend to induce profit appropriation by the auction house at the expense of the seller, which may destabilize the collaborative supply chain relationship when the external competitive environment changes. Accordingly, we argue that guaranteed auction contracts are more appropriately negotiated within a cooperative game framework. Even when the auction house plays a leading role in the supply chain, profit allocation should be benchmarked against the cooperative game outcome in order to enhance supply chain competitiveness and long-term sustainability.

The remainder of the paper is organized as follows. Section 2 reviews the related literature. Section 3 introduces the basic multi-unit consignment auction model and derives the system-optimal solution under centralized decision making. Section 4 analyzes supply and selling decisions under the conventional commission contract and shows that such contracts fail to coordinate the supply chain. Section 5 develops coordination mechanisms based on guaranteed auction contracts under both cooperative and non-cooperative game frameworks and provides a comparative analysis of the two regimes. Section 6 presents numerical experiments to validate the theoretical results and to examine the risk–return trade-offs faced by supply chain members under different game structures. Section 7 concludes the paper by summarizing the main findings, discussing managerial implications, and outlining directions for future research. All proofs are provided in Appendix A.

2. Literature Review

This paper is related to three streams of literature: auction revenue allocation, optimization of multi-unit auction supply chains, and cooperative game-based approaches to supply chain coordination. We review the most relevant studies in these areas and compare them with our work to highlight our contributions.

2.1. Auction Revenue Allocation

The most straightforward revenue-sharing rule in auction practice is the consignor’s commission. Tamura [7] shows that when buyers’ and sellers’ value information is positively correlated, auction houses can earn higher commission revenues by adopting first-price auctions instead of second-price auctions. Tsuchihashi and Zennyo [8] further demonstrate that, from the auction house’s perspective, commissions charged to buyers and sellers contribute equally to expected profits and can therefore be viewed as substitutes. Using data from an online auction platform, Marra [9] finds that increasing seller commissions tends to exclude high-valuation sellers but improves the competitive environment faced by the remaining sellers, thereby mitigating their welfare losses. All of the above studies remain confined to the commission-based framework and do not examine operational performance from the perspective of the auction supply chain as a whole.

Guaranteed auctions, as an alternative mechanism for allocating auction revenues, have received relatively limited attention in the literature. Greenleaf et al. [10], together with the extension by Greenleaf et al. [11], provide the first, and, to date, the most authoritative, normative analysis of guarantee negotiations between an auction house and a consignor, showing that guaranteed auctions may adversely affect the auction house’s interests. Using auction data from Christie’s and Sotheby’s, Graddy and Hamilton [12] find that more expensive items are more likely to receive guarantees, but report insufficient evidence that guarantees themselves have a direct effect on hammer prices. Charlin and Cifuentes [13] propose an option-pricing-based framework to quantify the risks and returns faced by auction houses when offering guarantees. Existing studies on guaranteed auctions focus primarily on single-item auctions and do not consider multi-unit settings.

2.2. Optimization of Multi-Unit Auction Supply Chains

Scholars have long recognized the impact of supply and selling decisions on revenues in multi-unit auctions and have proposed various optimization approaches [4,14]. For example, Wang [5], in the context of online Vickrey auctions, analyzes conflicts of interest between sellers and auction houses and proposes managerial measures such as reducing listing fees, increasing commission rates, and shortening auction duration. Ning et al. [15] study how a digital product seller with unlimited supply determines optimal quantities and prices over an infinite horizon when consumers repeatedly participate in auctions. Yu et al. [16] investigate resource trading in cloud manufacturing environments and employ deep reinforcement learning to address the lot-sizing problem in sequential auctions. However, these studies typically impose the restrictive assumption that buyers demand only a single unit. Moreover, implementing optimal auction mechanisms often requires sellers to possess detailed knowledge of buyers’ valuation distributions, an assumption that is difficult to satisfy in practice [17]. These stringent assumptions limit the practical applicability of their approaches.

2.3. Cooperative Game-Based Approaches to Supply Chain Coordination

In the classical non-cooperative game framework, a supply chain leader induces followers to make decisions consistent with the system optimum by offering coordinating contracts such as buyback agreements and quantity discounts [18,19,20,21]. As market competition intensifies and information communication costs decline, partnership-based relationships have increasingly replaced traditional leader–follower structures. Accordingly, a growing body of literature has re-examined supply chain coordination from the perspective of cooperative game theory [22]. For example, He and Zhao [23] design a production cost subsidy contract and an advance order discount contract to coordinate a supply chain with output uncertainty and show that the Nash bargaining solutions for suppliers and retailers under these two coordination schemes are equivalent. In the context of a three-tier closed-loop supply chain consisting of a manufacturer, a distributor, and a retailer, Zheng et al. [24] develop three coordination mechanisms based on the Shapley value, the core, and the equal satisfaction principle, respectively. Li et al. [25] take the Nash bargaining solution as a fairness benchmark and investigate optimal decisions and coordination mechanisms in a low-carbon supply chain. Most existing studies treat cooperative game-based coordination as a distinct framework and pay little attention to the differences and connections between cooperative and non-cooperative coordination mechanisms.

Building on the foregoing discussion, our study examines the guaranteed auction contract as a mechanism for incentive alignment and makes four main contributions to the literature.

• Most existing studies on auction commissions focus either on local improvements within commission-based frameworks or on the effects of alternative auction formats. Our study, by contrast, adopts a systemic supply chain perspective. We show analytically that auction commissions do more than transfer revenue. They act as structural frictions that distort the transmission of genuine demand signals to the supply side and ultimately make global coordination failure unavoidable.
• Existing research on guaranteed auctions has focused on single-item settings. Multi-unit auctions, however, are fundamentally more than a simple aggregation of single-item sales. They generate endogenous dynamics in which changes in supply quantity reshape buyers’ valuations and ultimately determine the market-clearing price. Our model treats supply quantity as an indispensable endogenous coordination variable. It therefore recasts the coordination problem as a two-dimensional dynamic game over reserve price and supply quantity and broadens the analytical scope of guaranteed auction research to multi-unit settings.
• Although prior studies have proposed various coordination mechanisms for multi-unit auction supply chains, they usually rely on assumptions that are mathematically convenient but difficult to satisfy in practice. In particular, they assume that each buyer demands only one unit and that valuations are symmetric across buyers. Our model relaxes both restrictions by allowing for multi-unit demand and valuation asymmetry. As a result, the contract we develop is more robust and applicable to a broader range of real-world settings.
• A substantial body of supply chain research has examined coordinating contracts within cooperative game frameworks. Our study differs in that it analyzes the performance of the same contract under both cooperative and non-cooperative games. This comparison yields an important operational insight. Differences in coordination outcomes between the two mechanisms arise not from the contractual form itself, but from the underlying bargaining structure and the markedly different risk–return allocations it generates.

3. Model Settings and Benchmark

To investigate supply and selling cooperation between a seller and an auction house, Section 3.1 presents a normative description of a multi-unit consignment auction abstracted from industry practice. Section 3.2 analyzes the operation of a hypothetical centralized decision maker, whose resulting outcome serves as a benchmark against which the performance of conventional commission contracts and guaranteed auction contracts is evaluated.

3.1. Assumptions and Notations

We consider a two-tier auction supply chain consisting of a seller, denoted by $S$, and an auction house, denoted by $H$. The seller produces and supplies a monopolistic product with a constant unit production cost $c_0$. In each production cycle, the seller consigns $Q$ units of a homogeneous and indivisible product to the auction house, which auctions them to a large number of potential buyers. The organizational cost of conducting the auction is assumed to be zero, as incorporating such costs would not materially affect the main conclusions of this study.

The auction is conducted as a discriminatory price auction with a non-public reserve price $r$. Each buyer submits $Q$ sealed bids, with each bid referring to a single unit offered for sale, and the highest $Q$ bids are identified as the competitive bids. For unit $j\left(j=1,2,\dots ,Q\right)$, if the corresponding competitive bid $X_j\left(Q\right)\ge r$, the bid is deemed a winning bid. The buyer submitting this bid wins unit $j$ and pays $X_j\left(Q\right)$ to the supply chain, together with a premium charged at rate $c_B$. If $X_j\left(Q\right)<r$, unit $j$ remains unsold. In this case, the supply chain recovers a salvage value $v$ for the unit in the secondary market.

To avoid unnecessary complexity, we model market demand as a stationary stochastic process. All competitive bids are assumed to be independently and identically distributed according to a distribution function $F(x;Q)$ with support $x\in [x_l,x_h]$ and associated density function $f(x;Q)\equiv F^{\prime }(x;Q)$. We assume that when $Q_1<Q_2$, the distribution $F(x;Q_1)$ first-order stochastically dominates $F(x;Q_2)$, and that ${\mathrm{l}\mathrm{i}\mathrm{m}}_{Q\to \mathrm{\infty }}F(x_l;Q)=1$. These assumptions capture the commonly observed pattern that buyers’ bids decrease as the supplied quantity increases.

In the subsequent analysis, all supply chain participants make decisions to maximize their expected profits. They are assumed to have complete information and to be risk neutral. In addition, to ensure the internal consistency of the model, we impose the parameter ordering $x_l<v<c_0<x_h$.

3.2. Benchmark: Centralized Operation

As a benchmark for evaluating system performance, we first derive the optimal operation of the auction supply chain under centralized decision making, including the auction reserve price and the supply quantity, denoted by $\{r^o,Q^o\}$.

From the perspective of the supply chain as a whole, the profit generated from selling unit $j$ is

$$Y_j(r,Q)=\{\begin{array}{ll}{X}_j\left(Q\right)\left(1+c_B\right)-c_0,& r\le X_j\left(Q\right)<x_h;\\ v-c_0,& x_l\le X_j\left(Q\right)<r.\end{array}$$

The corresponding expected profit is

$$\mathbb{E}{Y}_j\left(r,Q\right)={\int }_r^{x_h}x\left(1+c_B\right)f\left(x;Q\right)\mathrm{d}x+vF\left(r;Q\right)-c_0,$$

where $\mathbb{E}(·)$ denotes the expectation operator. Accordingly, the expected total sales profit of the supply chain is given by

$$\begin{array}{cc}& \pi \left(r,Q\right)\hfill \\ =& \mathbb{E}{\displaystyle \sum _{j=1}^Q}{Y}_j(r,Q)\hfill \\ =& {\displaystyle \sum _{j=1}^Q}\mathbb{E}{Y}_j(r,Q)\hfill \\ =& {\int }_r^{x_h}x\left(1+c_B\right)Qf\left(x;Q\right)\mathrm{d}x+vQF\left(r;Q\right)-c_{0}Q.\hfill \end{array}$$

(1)

Under centralized decision making, the profits of individual supply chain members are not distinguished; instead, all auction profits are treated as a single aggregate. The optimal reserve price $r^o$ and supply quantity $Q^o$ are determined by maximizing the expected profit of the entire supply chain. This leads to Proposition 1. To maintain the flow of the paper, the proof is provided in Appendix A.

Proposition 1.

The optimal auction reserve price for the supply chain

$$r^o={\displaystyle \frac{v}{1+c_B}}.$$

(2)

Proposition 1 shows that the optimal auction reserve price $r^o$ is independent of both the supply quantity $Q$ and the distribution of competing bids $F\left(x;Q\right)$. This result simplifies reserve-price implementation for system decision makers, because the hidden reserve price need not be adjusted in response to demand fluctuations. The key criterion is whether the auction’s net revenue exceeds the item’s opportunity cost, defined as its buyer-premium-adjusted salvage value. Any bid satisfying this condition contributes positively to total supply chain profit and should therefore be accepted.

On the other hand, the presence of the abstract distribution function $F(x;Q)$ prevents us from deriving a closed-form expression for the optimal supply quantity $Q^o$. Nevertheless, the following proposition establishes the existence of such an optimal solution.

Proposition 2.

Given the reserve price $r^o$, there exists an optimal supply quantity $Q^o$ that maximizes the supply chain’s expected profit $\pi (r^o,Q)$.

Proposition 2 highlights a central operational trade-off arising from the scarcity effect in multi-unit auctions. A larger supply quantity $Q$ may raise sales volume, but it may also reduce buyers’ average valuations by weakening market scarcity. This result suggests that decision makers in the auction supply chain should avoid expanding supply excessively in an attempt to capture market share. Instead, supply should be managed carefully to balance the gains from higher sales volume against the erosion of scarcity-based premiums.

4. Supply Chain Performance Under Conventional Commission Contracts

In conventional multi-unit consignment auctions, auction revenues are typically allocated between the seller and the auction house according to a commission contract. The seller transfers a fraction $c_S$ of the transaction price to the auction house as a seller-side commission. In addition, the auction house may charge buyers a premium at rate $c_B$. All unsold units are returned to the seller, who can recover a salvage value $v$ per unit in the secondary market.

Under this revenue allocation scheme, the profit that the seller obtains from the sale of unit $j$ is

$$Y_j^{SC}\left(r,Q\right)=\{\begin{array}{ll}{X}_j\left(Q\right)\left(1-c_S\right)-c_0,& r\le X_j\left(Q\right)<x_h;\\ v-c_0,& x_l\le X_j\left(Q\right)<r.\end{array}$$

(3)

The corresponding expected profit is

$$\mathbb{E}{Y}_j^{SC}\left(r,Q\right)={\int }_r^{x_h}x\left(1-c_S\right)f\left(x;Q\right)\mathrm{d}x+vF\left(r;Q\right)-c_0.$$

Accordingly, the seller’s expected profit from selling all units is given by

$$\begin{array}{cc}& {\pi }^{SC}\left(r,Q\right)\hfill \\ =& \mathbb{E}{\displaystyle \sum _{j=1}^Q}{Y}_j^{SC}(r,Q)\hfill \\ =& {\displaystyle \sum _{j=1}^Q}\mathbb{E}{Y}_j^{SC}(r,Q)\hfill \\ =& {\int }_r^{x_h}x\left(1-c_S\right)Qf\left(x;Q\right)\mathrm{d}x+vQF\left(r;Q\right)-c_{0}Q.\hfill \end{array}$$

(4)

On the other hand, the profit earned by the auction house from the sale of unit $j$ is

$$Y_j^{HC}\left(r,Q\right)=\{\begin{array}{ll}{X}_j\left(Q\right)\left(c_S+c_B\right),& r\le X_j\left(Q\right)<x_h;\\ 0,& x_l\le X_j\left(Q\right)<r.\end{array}$$

The corresponding expected profit is

$$\mathbb{E}{Y}_j^{HC}\left(r,Q\right)={\int }_r^{x_h}x\left(c_S+c_B\right)f\left(x;Q\right)\mathrm{d}x.$$

Accordingly, the auction house’s expected profit from accepting the auction consignment is given by

$$\begin{array}{cc}& {\pi }^{HC}\left(r,Q\right)\hfill \\ =& \mathbb{E}{\displaystyle \sum _{j=1}^Q}{Y}_j^{HC}(r,Q)\hfill \\ =& {\displaystyle \sum _{j=1}^Q}\mathbb{E}{Y}_j^{HC}(r,Q)\hfill \\ =& {\int }_r^{x_h}x\left(c_S+c_B\right)Qf\left(x;Q\right)\mathrm{d}x.\hfill \end{array}$$

(5)

Under the commission auction contract, both the supply quantity $Q$ and the auction reserve price $r$ are determined by the seller. Acting in its own interest, the seller chooses the optimal consignment strategy $\left\{r_{SC}^{*}\right|Q_{SC}^{*}\}$ at the beginning of each production cycle to maximize its expected profit function ${\pi }^{SC}(r,Q)$.

Proposition 3.

Under the commission auction contract, the seller’s optimal auction reserve price

$$r_{SC}^{*}={\displaystyle \frac{v}{1-c_S}}.$$

(6)

Comparing Equations (2) and (6), we observe that $r_{SC}^{*}>r^o$. This result arises because the seller must transfer a portion of buyers’ bids to the auction house in the form of commissions. Consequently, the seller sets a higher reserve price to ensure that its net auction revenue is no lower than the salvage value $v$. However, doing so also causes some units that could have generated positive surplus for the supply chain as a whole to remain unsold. This leads to the following result.

Proposition 4.

The commission auction contract fails to coordinate the supply chain.

The fundamental reason for the failure of the conventional commission mechanism to coordinate the supply chain is that the seller’s auction revenue is net of the seller’s commission and the buyer’s premium. As a result, the revenue benchmark used by the seller when setting the reserve price does not reflect buyers’ true willingness to pay. The seller’s optimal reserve price $r_{SC}^{*}$ therefore inevitably deviates from the system-optimal reserve price $r^o$. In other words, auction commissions distort market demand information as it is transmitted from the demand side to the supply side.

5. Supply Chain Coordination Under Guaranteed Auction Contracts

Section 4 has shown that conventional commission contracts fail to coordinate the supply chain. In this section, we examine whether guaranteed auction contracts can deliver superior supply chain performance. Under a guaranteed auction arrangement, the auction house offers a guaranteed price $g$ for each unit after accepting the seller’s consignment. If a unit remains unsold, the auction house is obligated to purchase the unit from the seller at price $g$, thereby ensuring a minimum auction revenue for the seller. In return for providing the guarantee and forgoing commission revenues, the auction house receives a fraction $1-\psi$ of the surplus whenever the auction price exceeds the guaranteed price $g$.

Under the guaranteed auction contract, the seller’s auction revenue does not depend on whether a unit remains unsold, but only on whether the auction price exceeds the guaranteed price $g$. If the competitive bid for unit $j$ satisfies $X_j\left(Q\right)\ge g$, i.e., the auction price exceeds the guaranteed price, the seller receives the guaranteed price $g$ plus a fraction $\psi$ of the surplus. If $X_j\left(Q\right)<g$, then regardless of whether the unit is sold through the auction, the seller receives a payment of $g$ from the auction house. Subtracting the production cost yields the seller’s profit from consigning unit $j$:

$$Y_j^{SG}\left(Q,g,\psi \right)=\{\begin{array}{ll}g+\left(X_j\left(Q\right)-g\right)\psi -c_0,& g\le X_j\left(Q\right)\le x_h;\\ g-c_0,& x_l\le X_j\left(Q\right)<g.\end{array}$$

(7)

The corresponding expected profit is

$$\mathbb{E}{Y}_j^{SG}\left(Q,g,\psi \right)={\int }_g^{x_h}\left(x-g\right)\psi f\left(x;Q\right)\mathrm{d}x+g-c_0.$$

Accordingly, the seller’s expected profit under the guaranteed auction contract is given by

$$\begin{array}{cc}& {\pi }^{SG}\left(Q,g,\psi \right)\hfill \\ =& \mathbb{E}{\displaystyle \sum _{j=1}^Q}{Y}_j^{SG}(Q,g,\psi )\hfill \\ =& {\displaystyle \sum _{j=1}^Q}\mathbb{E}{Y}_j^{SG}(Q,g,\psi )\hfill \\ =& {\int }_g^{x_h}\left(x-g\right)\psi Qf\left(x;Q\right)\mathrm{d}x+gQ-c_{0}Q.\hfill \end{array}$$

(8)

On the other hand, the auction house’s profit from auctioning unit $j$ depends on the corresponding competitive bid $X_j\left(Q\right)$ and can be classified into three cases. First, if $X_j\left(Q\right)\ge g$, i.e., the auction price exceeds the guaranteed price, the auction house receives the bid payment $X_j\left(Q\right)$ and the buyer’s premium $c_B{X}_j\left(Q\right)$, and then transfers the guaranteed price $g$ as well as a fraction $\psi$ of the surplus to the seller. Second, if $r\le X_j\left(Q\right)<g$, unit $j$ is successfully sold but at a price below the guaranteed price $g$. In this case, the auction house must transfer the entire auction price $X_j\left(Q\right)$ to the seller and can rely only on the buyer’s premium to cover the gap between the auction price and the guaranteed price. If the buyer’s premium is insufficient, the auction house incurs a loss. Finally, if $X_j\left(Q\right)<r$, unit $j$ remains unsold, but under the terms of the contract, the auction house must purchase the unit from the seller at the guaranteed price $g$. This outcome is clearly unfavorable for the auction house, which can resell the unit in the secondary market at price $v$ to partially recover its loss. Taken together, the auction house’s profit from auctioning unit $j$ is given by

$$Y_j^{HG}\left(r,Q,g,\psi \right)=\{\begin{array}{ll}{X}_j\left(Q\right)\left(1+c_B\right)-\left[g+\left(X_j\left(Q\right)-g\right)\psi \right],& g\le X_j\left(Q\right)\le x_h;\\ X_j\left(Q\right)\left(1+c_B\right)-g,& r\le X_j\left(Q\right)<g;\\ v-g,& x_l\le X_j\left(Q\right)<r.\end{array}$$

The corresponding expected profit is

$$\mathbb{E}{Y}_j^{HG}\left(r,Q,g,\psi \right)={\int }_r^{x_h}x\left(1+c_B\right)f\left(x;Q\right)\mathrm{d}x-{\int }_g^{x_h}\left(x-g\right)\psi f\left(x;Q\right)\mathrm{d}x+vF\left(r;Q\right)-g.$$

Accordingly, the auction house’s expected profit under the guaranteed auction contract is

$$\begin{array}{cc}& {\pi }^{HG}\left(r,Q,g,\psi \right)\hfill \\ =& \mathbb{E}{\displaystyle \sum _{j=1}^Q}{Y}_j^{HG}\left(r,Q,g,\psi \right)\hfill \\ =& {\displaystyle \sum _{j=1}^Q}\mathbb{E}{Y}_j^{HG}\left(r,Q,g,\psi \right)\hfill \\ =& {\int }_r^{x_h}x\left(1+c_B\right)Qf\left(x;Q\right)\mathrm{d}x-{\int }_g^{x_h}\left(x-g\right)\psi Qf\left(x;Q\right)\mathrm{d}x+vQF\left(r;Q\right)-gQ.\hfill \end{array}$$

(9)

In fact, guaranteed auction contracts can facilitate the coordination of supply chain members’ decisions under both cooperative and non-cooperative game frameworks. In Section 5.1 and Section 5.2, we demonstrate how supply chain coordination mechanisms are constructed under these two frameworks, respectively. Section 5.3 then compares the similarities and differences in the two coordination mechanisms.

5.1. Supply Chain Coordination Under a Cooperative Game

Under a cooperative game framework, the seller and the auction house bargain over the rights and obligations embedded in the guaranteed auction contract and jointly determine the guaranteed price $g$ and the surplus-sharing ratio $\psi$. If the parties fail to reach a mutually acceptable pair of contract parameters $\{g,\psi \}$, i.e., the guaranteed auction bargaining breaks down, auction revenues revert to being allocated under the conventional commission-based mechanism.

For convenience of exposition, we formalize the bargaining process as follows. Let ${\mathit{\pi }}^G=\left({\pi }^{SG}\left(Q,g,\psi \right),{\pi }^{HG}\left(r,Q,g,\psi \right)\right)$ denote a profit allocation of the supply chain under the guaranteed auction contract, where ${\pi }^{SG}$ and ${\pi }^{HG}$ represent the profits of the seller and the auction house, respectively. The feasible set of supply chain profit allocations is given by

$$U=\left\{{\mathit{\pi }}^G\in {\mathbb{R}}^2\right|\begin{array}{l}{\pi }^{SG}\left(Q,g,\psi \right)\ge 0\\ {\pi }^{HG}\left(r,Q,g,\psi \right)\ge 0\\ {\pi }^{SG}\left(Q,g,\psi \right)+{\pi }^{HG}\left(r,Q,g,\psi \right)\le \pi (r^o,Q^o)\end{array}\}.$$

Let ${\mathit{\pi }}^d=\left({\pi }^{SC}\left(r_{SC}^{*},Q_{SC}^{*}\right), {\pi }^{HC}(r_{SC}^{*},Q_{SC}^{*})\right)$ denote the disagreement point of the bargaining, corresponding to the profit allocation under the conventional commission contract. Then, the pair $(U,{\mathit{\pi }}^d)$ constitutes the bargaining game between the seller and the auction house over the guaranteed auction contract.

Based on the Nash axioms [26,27], the equilibrium outcome of this bargaining game exhibits the following properties.

Proposition 5.

Under the Nash bargaining solution, the seller and the auction house jointly choose the system-optimal supply and selling strategy $\{r^o,Q^o\}$ and equally share the additional surplus generated by adopting the guaranteed auction contract instead of the commission contract.

Proposition 5 indicates that the bargaining process induces the seller and the auction house to adopt the system-optimal supply and selling strategy $\{r^o,Q^o\}$. This result is both reasonable and intuitive, as the potential gains available to each participant are maximized only when the supply chain as a whole achieves its maximum profit. The second part of Proposition 5 characterizes how this system profit is allocated between the seller and the auction house, showing that the two parties equally share the additional surplus generated by operating at the system optimum. This allocation outcome reflects the symmetry in bargaining power between the seller and the auction house. To induce the adoption of the system-optimal strategy and thereby realize system-wide profit maximization, both parties must accept the revised revenue-sharing arrangement, as rejection by either party would lead to a breakdown of cooperation. Consequently, the seller and the auction house make equal marginal contributions to achieving the system optimum and therefore split the gains from cooperation equally.

Proposition 6.

Under the cooperative game framework, the guaranteed auction contract coordinates the supply chain. The coordinating contract parameters $\{g^o,{\psi }^o\}$ satisfy

$${\psi }^o={\displaystyle \frac{{\pi }^{SC}\left(r_{SC}^{\mathrm{*}},Q_{SC}^{\mathrm{*}}\right)-{\pi }^{HC}\left(r_{SC}^{\mathrm{*}},Q_{SC}^{\mathrm{*}}\right)+\pi \left(r^o,Q^o\right)+2\left(c_0-g^o\right)Q^o}{2{\int }_{g^o}^{x_h}\left(x-g^o\right)Q^{o}f\left(x;Q^o\right)\mathrm{d}x}}.$$

(10)

Proposition 6 yields several important managerial insights, which are discussed below.

• There generally exists more than one combination of $g^o$ and ${\psi }^o$ that can coordinate the supply chain, and all such combinations form a nonlinear curve. Note that

  $${\displaystyle \frac{\mathrm{d}{g}^o}{\mathrm{d}{\psi }^o}}={\displaystyle \frac{{\int }_{g^o}^{x_h}\left(x-g^o\right)Q^{o}f\left(x;Q^o\right)\mathrm{d}x}{-2Q^o+{\psi }^o{Q}^o\left(1-F\left(g^o;Q^o\right)\right)}}<0$$

  and

  $${\displaystyle \frac{{\mathrm{d}}^2{g}^o}{\mathrm{d}{{\psi }^o}^2}}={\displaystyle \frac{-Q^o\left(1-F\left(g^o;Q^o\right)\right){\int }_{g^o}^{x_h}\left(x-g^o\right)Q^{o}f\left(x;Q^o\right)\mathrm{d}x}{{\left[-2Q^o+{\psi }^o{Q}^o\left(1-F\left(g^o;Q^o\right)\right)\right]}^2}}<0.$$

  These properties imply that the seller’s share of the surplus, ${\psi }^o$, decreases as the corresponding guaranteed price $g^o$ increases, and that the rate of decrease becomes progressively steeper. This trade-off reflects a mutually acceptable outcome for both the seller and the auction house when negotiating the guaranteed auction contract.
• Compared with the conventional commission-based mechanism, guaranteed auctions reduce the variance of the seller’s profit. On the one hand, guaranteed auctions shield the seller from low buyer bids. As shown in Equation (7), bids below the guaranteed price $g^o$ have no impact on the seller’s profit, and the lower bound of the seller’s profit is effectively locked in at $g^o-c_0$. On the other hand, when buyer bids exceed the guaranteed price $g^o$, the auction house captures a fraction $1-{\psi }^o$ of the surplus, which limits how much the seller’s profit can rise above $g^o$. As a result, relative to the commission-based mechanism, both the lower and upper bounds of the seller’s profit are compressed under the guaranteed auction contract, while the probability of receiving the guaranteed price $g^o$ increases substantially. This contraction of the profit range leads to a lower variance of the seller’s profit and thereby enhances income stability.
• According to Equation (10), determining the values of $g^o$ and ${\psi }^o$ requires knowledge of the distribution of competitive bids $F(x;Q)$, which may be difficult to obtain in practice. However, the proof of Proposition 6 in Appendix A reveals that the key to supply chain coordination lies in the equal sharing of the additional surplus generated by the guaranteed auction. Equation (10) characterizes the necessary condition for achieving such an allocation. Therefore, under symmetric information, even in the absence of explicit knowledge of the bid distribution $F(x;Q)$, the seller and the auction house can iteratively adjust the parameters of the guaranteed auction contract in practice. By ensuring that each adjustment yields equal incremental gains for both parties, the supply chain can gradually approach the system-optimal profit level and ultimately achieve coordination.

Proposition 7.

Under the cooperative game framework, when the supply chain is coordinated through a guaranteed auction contract, the proportion of the seller’s expected profit in the total supply chain profit is

$${\lambda }_S^o={\displaystyle \frac{1}{2}}+{\displaystyle \frac{{\pi }^{SC}\left(r_{SC}^{*},Q_{SC}^{*}\right)-{\pi }^{HC}\left(r_{SC}^{*},Q_{SC}^{*}\right)}{2\pi \left(r^o,Q^o\right)}},$$

(11)

while the proportion of the auction house’s expected profit in the total supply chain profit is

$${\lambda }_H^o={\displaystyle \frac{1}{2}}-{\displaystyle \frac{{\pi }^{SC}\left(r_{SC}^{*},Q_{SC}^{*}\right)-{\pi }^{HC}\left(r_{SC}^{*},Q_{SC}^{*}\right)}{2\pi \left(r^o,Q^o\right)}}.$$

Proposition 6 characterizes cooperative operations under the guaranteed auction contract, whereas Proposition 7 clarifies the resulting profit allocation under such cooperation. The shares of expected profits allocated to supply chain members depend on their respective disagreement payoffs, which correspond to their expected profits under the conventional commission-based mechanism. These disagreement payoffs constitute each party’s effective threat point in the bargaining process. A higher threat point implies a stronger bargaining position and, consequently, a larger share of the total profit secured in equilibrium.

5.2. Supply Chain Coordination Under a Non-Cooperative Game

Under a non-cooperative game framework, the seller and the auction house make decisions independently so as to maximize their own expected profit functions. Treating the auction house as the leader of the supply chain, the sequence of decisions unfolds as follows. First, the auction house announces the parameters of the guaranteed auction contract, $\{g,\psi \}$. Next, given the guaranteed price $g$ and the surplus-sharing ratio $\psi$, the seller chooses the optimal supply quantity $Q_{SG}^{*}$. Finally, after receiving the consigned goods, the auction house selects the optimal auction reserve price $r_{HG}^{*}$ based on the distribution of buyers’ bids $F(x;Q_{SG}^{*})$. This decision process can be modeled as a Stackelberg game. Applying backward induction, we begin by solving for the auction house’s optimal strategy.

Proposition 8.

Under the guaranteed auction contract, the auction house’s optimal auction reserve price

$$r_{HG}^{*}={\displaystyle \frac{v}{1+c_B}}.$$

(12)

Proposition 8 highlights the advantage of transferring the reserve price decision from the seller to the auction house in consignment auctions. Unlike the seller, who captures only a fraction of the auction revenue, the auction house directly receives the full payment from winning buyers, including both the transaction price and the buyer’s premium. Leveraging this advantage, the auction house can fully extract buyers’ willingness to pay. Notably, the auction house’s optimal reserve price $r_{HG}^{*}$ coincides with the system-optimal reserve price $r^o$ under centralized decision making. This equivalence creates the possibility for guaranteed auction contracts to coordinate the supply chain even within the non-cooperative game framework.

Proposition 9.

Under the non-cooperative game framework, the guaranteed auction contract coordinates the supply chain. If the seller’s expected profit accounts for a proportion ${\lambda }_S$ of the total supply chain profit, then the coordinating contract parameters $\{g^{*},{\psi }^{*}\}$ satisfy

$${\displaystyle \frac{c_0-g^{*}}{{\psi }^{*}}}\ge {\int }_{g^{*}}^{x_h}\left(x-g^{*}\right)\left(Q^o+1\right)f\left(x;Q^o+1\right)dx-{\int }_{g^{*}}^{x_h}\left(x-g^{*}\right)Q^{o}f\left(x;Q^o\right)dx,$$

(13)

$${\displaystyle \frac{c_0-g^{*}}{{\psi }^{*}}}\le {\int }_{g^{*}}^{x_h}\left(x-g^{*}\right)Q^{o}f\left(x;Q^o\right)dx-{\int }_{g^{*}}^{x_h}\left(x-g^{*}\right)\left(Q^o-1\right)f\left(x;Q^o-1\right)dx,$$

(14)

and

$${\psi }^{*}={\displaystyle \frac{{\lambda }_S\pi \left(r^o,Q^o\right)+\left(c_0-g^{*}\right)Q^o}{{\int }_{g^{*}}^{x_h}\left(x-g^{*}\right)Q^{o}f\left(x;Q^o\right)dx}}.$$

(15)

Equations (13) and (14) in Proposition 9 restrict the feasible ranges of the coordinating contract parameters $g^{\mathrm{*}}$ and ${\psi }^{\mathrm{*}}$, while Equation (15) characterizes the functional relationship between them. For any given profit-sharing ratio ${\lambda }_S:{\lambda }_H$ of the supply chain, Equation (15) identifies a unique relationship between $g^{\mathrm{*}}$ and ${\psi }^{\mathrm{*}}$; however, the parameter pair $\{g^{*},{\psi }^{*}\}$ itself is not uniquely determined. This non-uniqueness arises from the indivisibility of the auctioned units, which requires the seller’s supply decision to take integer values. As a result, there remains flexibility in adjusting the continuously valued contract parameters $\{g^{*},{\psi }^{*}\}$ while preserving supply chain coordination.

It is worth noting that the guaranteed auction contract is a two-parameter contract. Compared with the single-parameter commission contract, guaranteed auctions offer substantially greater flexibility in allocating auction revenues. This flexibility not only enables supply chain coordination under a non-cooperative game framework, but also allows the profit-sharing ratio between the seller and the auction house to be adjusted within a certain range.

5.3. Comparing Coordination Mechanisms Under Cooperative and Non-Cooperative Games

Both the cooperative game mechanism developed in Section 5.1 and the non-cooperative game mechanism developed in Section 5.2 are capable of guiding the multi-unit consignment auction supply chain toward coordination. They represent two alternative approaches to addressing the same contract coordination problem, sharing certain common features while also exhibiting important differences.

5.3.1. Mapping Equivalence and Orientation Difference

Regardless of whether coordination is achieved under the cooperative or the non-cooperative game framework, once the supply chain attains channel coordination through the guaranteed auction contract, the functional relationship between the coordinating parameters $g$ and $\psi$ is identical. In fact, Equations (10) and (11) together yield

$$\begin{array}{cc}& {\psi }^o\hfill \\ =& {\displaystyle \frac{{\pi }^{SC}\left(r_{SC}^{\mathrm{*}},Q_{SC}^{\mathrm{*}}\right)-{\pi }^{HC}\left(r_{SC}^{\mathrm{*}},Q_{SC}^{\mathrm{*}}\right)+\pi \left(r^o,Q^o\right)+2\left(c_0-g^o\right)Q^o}{2{\int }_{g^o}^{x_h}\left(x-g^o\right)Q^{o}f\left(x;Q^o\right)\mathrm{d}x}}\hfill \\ =& {\displaystyle \frac{\left(\frac{1}{2}+\frac{{\pi }^{SC}\left(r_{SC}^{\mathrm{*}},Q_{SC}^{\mathrm{*}}\right)-{\pi }^{HC}\left(r_{SC}^{\mathrm{*}},Q_{SC}^{\mathrm{*}}\right)}{2\pi \left(r^o,Q^o\right)}\right)\pi \left(r^o,Q^o\right)+\left(c_0-g^o\right)Q^o}{{\int }_{g^o}^{x_h}\left(x-g^o\right)Q^{o}f\left(x;Q^o\right)\mathrm{d}x}}\hfill \\ =& {\displaystyle \frac{{\lambda }_S^o\pi \left(r^o,Q^o\right)+\left(c_0-g^o\right)Q^o}{{\int }_{g^o}^{x_h}\left(x-g^o\right)Q^{o}f\left(x;Q^o\right)\mathrm{d}x}}.\hfill \end{array}$$

(16)

Comparing Equations (15) and (16), we observe that when the profit-sharing ratios ${\lambda }_S:{\lambda }_H$ under the cooperative and non-cooperative coordination mechanisms are identical, the corresponding guaranteed auction strategies $\{g,\psi \}$ are also identical. Because a change in the game framework does not alter the revenue allocation rule embedded in the guaranteed auction contract, there exists a one-to-one correspondence between the supply chain’s guaranteed auction strategy and its profit-sharing outcome.

The key difference between the two coordination mechanisms lies in their orientation. The cooperative game is outcome oriented in that the seller and the auction house focus on the total profit and its allocation. The cooperative guaranteed auction strategy $\{g^o,{\psi }^o\}$ serves primarily as an action guideline for achieving that outcome. In contrast, the non-cooperative game is strategy oriented. The seller and the auction house focus on the set of feasible strategies available to them and on identifying those that maximize their individual payoffs, and the resulting profit allocation ${\lambda }_S:{\lambda }_H$ reflects the system outcome induced by the equilibrium of their strategic interactions.

5.3.2. Implementation Difficulty and Parameter Adjustability

Note that the cooperative coordination mechanism requires the seller and the auction house to jointly negotiate and determine the system-optimal supply quantity $Q^o$ and reserve price $r^o$. By contrast, under the non-cooperative coordination mechanism, it suffices for the auction house to announce the guaranteed auction contract parameters $\{g^{*},{\psi }^{*}\}$. Driven by profit maximization, the seller and the auction house then independently choose the system-optimal operating decisions $\{r^o,Q^o\}$. From this perspective, the non-cooperative coordination mechanism is easier to implement than its cooperative counterpart. Moreover, as the supply chain leader, the auction house can further adjust the guaranteed auction parameters $\{g^{*},{\psi }^{*}\}$ to exert partial control over the profit-sharing ratio ${\lambda }_S:{\lambda }_H$ within the supply chain.

However, compared with the cooperative coordination mechanism, the feasible set of guaranteed auction contract parameters $\{g^{*},{\psi }^{*}\}$ under the non-cooperative coordination mechanism is much narrower. This difference stems from the distinct equilibrium formation processes underlying the two game frameworks. Under the cooperative mechanism, coordination is achieved by first reaching a mutual agreement on a desired profit-sharing outcome and then selecting appropriate contract parameters $\{g^o,{\psi }^o\}$ to implement that outcome. As long as the agreed profit allocation remains unchanged, the corresponding parameter pair can vary within a relatively broad range. In contrast, the non-cooperative coordination mechanism relies entirely on the careful design of the guaranteed auction rules, exploiting supply chain participants’ profit-seeking behavior to induce autonomous adjustments in the supply and selling decisions $\{r,Q\}$ and thereby achieve coordination. This approach imposes much stricter requirements on the coordinating contract parameters $\{g^{*},{\psi }^{*}\}$. In the mathematical formulation, this restriction is reflected in the fact that their feasible ranges are constrained not only by Equations (13) and (14), but also by the requirement that the guaranteed price $g^{*}$ must not exceed the unit production cost $c_0$. Otherwise, driven by the seller’s profit-maximizing incentives, the optimal supply quantity would become unbounded.

5.3.3. Stability and Suitability Under Intensified Competition

Although early studies on supply chain coordination were largely grounded in non-cooperative game theory, this framework fails to address a critical issue: how supply chain profits should be allocated. Coordination mechanisms based on non-cooperative games often grant excessive control over profit allocation to the supply chain leader. While such control reflects the strong decision-making influence historically exercised by firms with superior capital or technological capabilities over followers in distribution channels, excessive control may also lead to the exploitation of weaker parties [28,29]. As a result, coordination mechanisms based on non-cooperative games tend to be unstable. When the internal or external decision environment of the supply chain changes, followers are incentivized to deviate from the prevailing equilibrium in pursuit of higher profits, thereby undermining supply chain coordination [30].

To place the foregoing discussion in a broader context, it is instructive to briefly revisit several well-known industry episodes. In 2012, retail giant Carrefour adopted a tough stance in its annual procurement contracts regarding pricing and profit rebates, which led to conflicts with Master Kong and COFCO, bringing their retail–supplier relationships close to breakdown. In 2016, leveraging its terminal channel power, Gome demanded lower wholesale prices from Gree and required Gree to bear promotional expenses, severely disrupting Gree’s established pricing system and eventually prompting the suspension of supplies to Gome. More recently, in 2023, Xiaomi compressed the profit margins offered to offline retailers to only 8–10%, triggering collective resistance from distributors across several southern Indian states, who demanded higher rebate rates and threatened to halt purchases otherwise.

Admittedly, these real-world episodes involve multi-product portfolios, information asymmetries, and institutional complexities beyond the scope of our stylized model. Even so, they still reveal a clear pattern. In non-cooperative coordination settings, aggressive profit extraction by channel leaders tends to provoke follower resistance and is becoming increasingly unsustainable. Two broad forces underlie this trend. First, the development of the commodity economy has intensified market competition. Competition has become increasingly fierce among both upstream firms and downstream channel participants, making it difficult for any single firm to exert sufficient influence over the entire market [31,32]. As formerly monopolistic positions erode, leading firms have come to recognize that only through cooperation and mutual benefit can they attract and retain reliable partners. Second, advances in communication technologies have substantially reduced information and coordination costs. Whereas business negotiations in the past often required time-consuming travel, modern online communication tools, such as video conferencing, have become ubiquitous, significantly improving the timeliness and effectiveness of information exchange [33]. This reduction in coordination costs has strengthened the willingness of upstream and downstream partners to engage in negotiations aimed at establishing long-term cooperative relationships [34].

Against this backdrop of social and economic transformation, supply chain coordination mechanisms grounded in cooperative game theory demonstrate clear advantages. Unlike non-cooperative mechanisms, which often entail an implicit element of coercion, cooperative approaches emphasize joint value creation and mutual benefit. By allocating supply chain profits according to principles of fairness, cooperative mechanisms yield outcomes that all participating members can accept and endorse. As a result, they exhibit greater stability and are more effective in strengthening supply chain cohesion and enhancing overall competitiveness.

6. Numerical Experiments

This section draws on survey data from an actual flower auction market to validate and illustrate the analytical results derived above. Section 6.1 introduces the empirical background of the numerical experiments and explains how the parameter settings are calibrated from the field data. Section 6.2 presents the design of coordinating contracts under the guaranteed auction mechanism, and Section 6.3 further compares the risk–return profiles of individual supply chain members across alternative contractual arrangements.

6.1. Empirical Background and Parameter Settings

The data used in the subsequent experiments were collected from DFETC, the largest flower auction market in Asia, with an average daily trading volume of 15.68 million stems. In the early hours of each day, flower growers consign batches of graded, homogeneous fresh-cut flowers to DFETC, which subsequently auctions them to a large pool of buyers, mainly brokers and wholesalers. This two-tier consignment structure closely corresponds to the seller–auction house supply chain examined in our model, and the Dutch clock auction adopted by DFETC is strategically equivalent to the discriminatory price auction analyzed in this paper.

Specifically, we consider Grade-B roses of a particular brand traded in DFETC. Owing to the strong reputation of this brand, Grade-B flowers remain highly attractive to buyers despite being slightly inferior to Grade-A flowers of the same brand in stem length and bloom size. It is therefore reasonable to assume that the supply of this product is largely unaffected by competition from other flowers.

DFETC (hereafter, the auction house) charges the supplier (hereafter, the seller) a commission rate of $c_S=0.1$ and charges buyers a premium rate of $c_B=0.05$. The seller’s unit cost, including seedlings, agricultural inputs, logistics and packaging, and labor, is $c_0=0.5$ yuan/stem. If the product remains unsold, its highly perishable nature implies that it can only be disposed of at a low price in the free-trading area outside the auction market. This salvage value typically ranges from a few fen to about 0.2 yuan/stem, and we set its average value at $v=0.1$ yuan/stem. Because buyers’ bids are not fully disclosed, the competitive bid density and distribution functions reported below are calibrated from historical average transaction prices:

$$f\left(x;Q\right)={\displaystyle \frac{5Q}{4}}{\left({\displaystyle \frac{x}{2}}\right)}^{{\displaystyle \frac{3}{2}}}{\left[1-{\left({\displaystyle \frac{x}{2}}\right)}^{{\displaystyle \frac{5}{2}}}\right]}^{Q-1},F\left(x;Q\right)=1-{\left[1-{\left({\displaystyle \frac{x}{2}}\right)}^{{\displaystyle \frac{5}{2}}}\right]}^Q,x\in \left[0,2\right].$$

Based on the density functions above, Figure 1 depicts the relationship between competitive bids and supply quantity. As supply increases, buyers’ bids gradually decline and converge to zero, which accords with standard economic intuition. It should also be noted that the main conclusions of this paper do not depend on the specific functional form assumed for the bid distribution.

The minimum trading unit of the flower auction is one bucket, and each bucket of the product discussed here contains 120 stems. Because the auction house quotes prices on a per-stem basis, substituting the parameter values above into the analytical expressions developed in this paper yields profit figures measured in yuan·bucket/stem. The actual settlement amount can therefore be obtained by multiplying these figures by 120. For ease of interpretation and direct comparison with the theoretical results in Section 3, Section 4 and Section 5, all profit figures reported below are expressed in yuan·bucket/stem, without further conversion.

6.2. Design of Coordinating Contracts Under Guaranteed Auctions

First, we verify that the conventional commission contract fails to coordinate the supply chain. According to Proposition 1, together with standard optimization techniques, the system-optimal supply and selling strategy is given by $r^o=0.0952$ and $Q^o=8$. Based on Equation (1), the corresponding expected profit of the supply chain is $\pi (0.0952,8)=2.2739$. When the supply chain operates under the commission auction contract, Proposition 3 implies that the seller’s optimal strategy is $r_{SC}^{*}=0.1111$ and $Q_{SC}^{*}=5$. According to Equation (4), the seller’s expected profit is ${\pi }^{SC}(0.1111,5)=1.4794$, while Equation (5) yields the auction house’s expected profit ${\pi }^{HC}(0.1111,5)=0.6629$. Consequently, the expected profit of the entire supply chain is $\pi (0.1111,5)=2.1423$. Since $\pi (0.1111,5)<\pi (0.0952,8)$, the commission contract fails to coordinate the supply chain, thereby validating Proposition 4.

We next present the design of coordinating contracts based on guaranteed auctions under the cooperative and non-cooperative game frameworks, respectively.

6.2.1. Coordinating Contract Design Under a Cooperative Game

Based on Proposition 6, under the cooperative game framework, all pairs of guaranteed auction contract parameters $\{g^o,{\psi }^o\}$ that induce supply chain coordination satisfy the following relationship:

$${\psi }^o={\displaystyle \frac{0.6931-g^o}{10{\int }_{g^o}^2\left(x-g^o\right){\left(\frac{x}{2}\right)}^{\frac{3}{2}}{\left[1-{\left(\frac{x}{2}\right)}^{\frac{5}{2}}\right]}^7\mathrm{d}x}}.$$

(17)

Note that guaranteed auctions typically require the guaranteed price $g^o$ to be no lower than the salvage value $v$, and the surplus-sharing ratio ${\psi }^o$ to be non-negative. As a result, the feasible range of $g^o$ is restricted to the interval $[0.10,0.69]$. Based on Equation (17), the functional relationship between $g^o$ and ${\psi }^o$ is depicted in Figure 2. The figure clearly shows that the surplus-sharing ratio ${\psi }^o$ decreases as the guaranteed price $g^o$ increases, and that the rate of decrease becomes progressively steeper. This pattern is fully consistent with the analytical insights derived earlier.

Figure 2 indicates that there exist infinitely many combinations of $g^o$ and ${\psi }^o$ that can coordinate the supply chain. We select one such combination, $\{0.45,0.7591\}$, for further analysis. Substituting this pair into Equations (8) and (9) yields the profit allocation reported in

Table 1. Profit allocation among supply chain members under different auction contracts.

Auction Contract	$\mathit{r}$	$\mathit{Q}$	Seller’s Expected Profit	Auction House’s Expected Profit	Total Expected Supply Chain Profit
Conventional Commission Contract	$0.1111$	5	$1.4794$	$0.6629$	$2.1423$
Guaranteed Auction Contract	$0.0952$	8	$1.5452$	$0.7287$	$2.2739$

. For ease of comparison, the table also presents the expected profit distribution under the commission contract.

The above numerical results demonstrate that the guaranteed auction indeed generates a Pareto improvement for both the seller and the auction house, successfully steering the auction supply chain toward a coordinated outcome. Note that

$${\pi }^{SG}(8,0.45,0.7591)-{\pi }^{SC}(0.1111,5)={\pi }^{HG}(0.0952,8,0.45,0.7591)-{\pi }^{HC}(0.1111,5).$$

This equality implies that, after switching from the commission contract to the guaranteed auction contract, the increases in expected profits for the seller and the auction house are identical, reflecting the symmetry in their bargaining power. Moreover, since ${\pi }^{SC}(0.1111,5)>{\pi }^{HC}(0.1111,5)$, the seller’s disagreement payoff exceeds that of the auction house. Consequently, despite equal bargaining power, the final profit allocation favors the seller, with shares of $0.6795$ and $0.3205$ for the seller and the auction house, respectively.

6.2.2. Coordinating Contract Design Under a Non-Cooperative Game

According to Equation (15), under the non-cooperative game framework, the guaranteed auction contract parameters $\{g^{*},{\psi }^{*}\}$ that can coordinate the supply chain satisfy the following relationship:

$${\psi }^{*}={\displaystyle \frac{{\lambda }_S\times 2.2739+\left(0.5-g^{*}\right)8}{{\int }_{g^{*}}^2\left(x-g^{*}\right)8f\left(x;8\right)\mathrm{d}x}}.$$

If we set ${\lambda }_S={\lambda }_S^o=0.6795$, that is, if the profit allocation outcome under the non-cooperative game coincides with that under the cooperative game, the above relationship reduces to

$${\psi }^{*}={\displaystyle \frac{0.6931-g^{*}}{10{\int }_{g^{*}}^2\left(x-g^{*}\right){\left(\frac{x}{2}\right)}^{\frac{3}{2}}{\left[1-{\left(\frac{x}{2}\right)}^{\frac{5}{2}}\right]}^7\mathrm{d}x}}.$$

Comparing this expression with Equation (17), we find that the coordinating contract parameters under the non-cooperative game framework exhibit the same functional structure as those under the cooperative framework, which is consistent with our earlier analytical predictions. It is worth noting that although the relationship between $g^{*}$ and ${\psi }^{*}$ mirrors that in the cooperative case, the feasible range of the guaranteed price $g^{*}$ is further restricted by Equations (13) and (14) to the interval $[0.4236,0.4612]$. This additional restriction on the coordinating contract parameters can be interpreted as the necessary compromise arising from allowing the seller and the auction house to make decisions independently.

Under the non-cooperative game framework, the auction house, as the supply chain leader, can influence the profit-sharing ratio ${\lambda }_S:{\lambda }_H$ among participants, as illustrated more clearly by the numerical example in Section 6.3.3.

6.3. Risk–Return Profiles Under Alternative Contract Mechanisms

In multi-unit consignment auctions, uncertainty in buyers’ bids exposes the supply chain to the risk of asset losses while simultaneously creating opportunities to earn returns beyond expectations. This dual effect naturally raises the question of how risks and returns are allocated among supply chain members under commission-based and guaranteed auction mechanisms, and whether these mechanisms are capable of supporting stable long-term cooperation.

We measure the risk borne by each supply chain member using Conditional Value-at-Risk (CVaR) [35]. Let $L$ denote the loss incurred by a supply chain member in a given auction. The Value-at-Risk (VaR) at level $\alpha$ $(0<\alpha <1)$ is defined as

$$\mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }\left(L\right)=\min \left\{l\right|Prob\left\{L\le l\right\}\ge 1-\alpha \},$$

which represents the maximum loss that the member may incur with probability $1-\alpha$. The corresponding CVaR is given by

$$\mathrm{C}\mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }\left(L\right)=\mathbb{E}\left(L\right|L\ge \mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }\left(L\right)),$$

(18)

which captures the expected loss conditional on the loss exceeding the VaR threshold. This risk measure can be viewed as an extension of the approach adopted by Charlin and Cifuentes [13]. Unlike their measures that rely solely on expected losses, CVaR evaluates tail losses at a given confidence level, thereby providing a more flexible and accurate representation of the heterogeneous risk tolerance across supply chain members.

Based on Equation (18), if we replace the loss variable $L$ with a supply chain member’s profit $\Pi$, the corresponding CVaR can be expressed as

$$\mathrm{C}\mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }\left(\Pi \right)=\mathbb{E}\left(\Pi \right|\Pi \ge \mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }\left(\Pi \right)).$$

(19)

This measure captures the average profit conditional on the profit exceeding $\mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }\left(\Pi \right)$, that is, the expected profit under extremely favorable outcomes. To distinguish it from the conventional CVaR used to assess downside risk, we refer to this metric as Conditional Value-at-Return, denoted by CVaR⁺.

We next apply Monte Carlo simulation techniques [36,37] to simulate auction outcomes under the numerical settings described above. Using CVaR and CVaR⁺, we evaluate and compare the risks and returns borne by the seller and the auction house before and after the auction contract is modified.

6.3.1. Risk–Return Profiles Under the Commission Contract

To simulate the outcomes of consignment auctions, we independently draw $n\times Q_{SC}^{\mathrm{*}}$ samples of the competitive bid $X\left(Q_{SC}^{\mathrm{*}}\right)$, thereby generating $n$ bid vectors of dimension $Q_{SC}^{\mathrm{*}}$, denoted by ${\mathit{x}}_i=\left(x_{i1},x_{i2},\dots ,x_{i{Q}_{SC}^{*}}\right)$ ($i=1,2,\dots ,n$). Each bid vector corresponds to one simulated auction outcome. Let $y_{ij}^{SC}\left(x_{ij}\right)$ ($j=1,2,\dots ,Q_{SC}^{*}$) denote the profit obtained by the seller from bid $x_{ij}$. According to Equation (3), we have

$$y_{ij}^{SC}(x_{ij})=\{\begin{array}{ll}{x}_{ij}\left(1-c_S\right)-c_0,& r_{SC}^{*}\le x_{ij}\le x_h;\\ v-c_0,& x_l\le x_{ij}<r_{SC}^{*}.\end{array}$$

Thus ${\pi }_i^{SC}({\mathit{x}}_i)={\sum }_{j=1}^{Q_{SC}^{*}}{y}_{ij}^{SC}\left(x_{ij}\right)$ denotes the seller’s total profit corresponding to bid vector ${\mathit{x}}_i$. We then reorder the realizations $\left\{{\pi }_i^{SC}\right({\mathit{x}}_i){\}}_{i=1}^n$ in ascending order, and denote the $i$-th order statistic by ${\pi }_{\left(i\right)}^{SC}({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_n)$. Accordingly, based on Equation (18), the seller’s $\alpha$-level CVaR for multi-unit consignment auctions can be expressed as

$$\mathrm{C}\mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }\left(L^{SC}\right)=-{\displaystyle \frac{1}{\lfloor \alpha n\rfloor }}\sum _{i=1}^{\lfloor \alpha n\rfloor }{\pi }_{\left(i\right)}^{SC}\left({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_n\right),$$

where $\lfloor ·\rfloor$ denotes the floor function. According to Equation (19), the seller’s $\alpha$-level CVaR⁺ is given by

$$\mathrm{C}\mathrm{V}\mathrm{a}{\mathrm{R}}_{\alpha }^{+}\left({\Pi }^{SC}\right)={\displaystyle \frac{1}{\lfloor \alpha n\rfloor }}\sum _{i=n+1-\lfloor \alpha n\rfloor }^n{\pi }_{\left(i\right)}^{SC}({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_n).$$

The auction house’s CVaR and CVaR⁺ can be derived in an analogous manner.

For the above experiment, we set $n=1000$, corresponding to $1000$ simulated realizations of multi-unit consignment auctions. Based on these simulations, we estimate the CVaR and CVaR⁺ for both the seller and the auction house under the commission contract at three confidence levels, namely $0.90$, $0.95$, and $0.99$ ($\alpha =0.1$, $0.05$ and $0.01$). The resulting estimates are reported in

Table 2. CVaR and CVaR⁺ of supply chain participants under the commission contract.

$\mathit{\alpha }$	Seller’s CVaR	Seller’s CVaR+	Auction House’s CVaR	Auction House’s CVaR+
$0.10$	$-0.3094$	$2.7765$	$-0.4675$	$0.8794$
$0.05$	$-0.0959$	$2.9744$	$-0.4314$	$0.9124$
$0.01$	$0.2826$	$3.3429$	$-0.3679$	$0.9738$

.

As shown in Table 2, as $\alpha$ decreases from $0.10$ to $0.01$, both CVaR and CVaR⁺ increase for the seller and the auction house. This pattern arises because lower values of $\alpha$ capture more extreme loss and gain outcomes. For the seller, CVaR remains negative at $\alpha =0.10$ and $\alpha =0.05$, indicating that the seller is profitable in most cases and thereby supporting the practical viability of the commission contract. However, as $\alpha$ declines further, the seller becomes exposed to the risk of loss. By contrast, the auction house’s CVaR remains negative throughout, and this pattern persists even at lower values of $\alpha$. This contrast confirms our earlier conjecture that, under the conventional commission contract, market uncertainty is borne entirely by the seller. The auction house merely serves as an intermediary, earning revenue from the seller’s commission and the buyer’s premium, and thus operates in a risk-free manner.

6.3.2. Risk–Return Profiles Under Guaranteed Auctions with Cooperative Coordination

Since guaranteed auctions coordinate the supply chain, both the supply quantity and the reserve price differ from those under the commission-based mechanism. It is therefore necessary to re-simulate the auction outcomes. Specifically, we independently draw $n\times Q^o$ samples of the competitive bid $X\left(Q^o\right)$, thereby generating $n$ bid vectors of dimension $Q^o$, denoted by ${\mathit{x}}_i=\left(x_{i1},x_{i2},\dots ,x_{i{Q}^o}\right)$ ($i=1,2,\dots ,n$). The procedures for estimating the CVaR and CVaR⁺ for both the seller and the auction house are analogous to those described in the previous subsection and are therefore omitted for brevity.

Similarly, we analyze $1000$ randomly generated sets of buyers’ bid data. Note that there exist infinitely many combinations of coordinating contract parameters $\{g^o,{\psi }^o\}$. We select five representative cases and compute the CVaR and CVaR⁺ at the $0.10$, $0.05$, and $0.01$ levels for both the seller and the auction house under the corresponding guaranteed auction arrangements. The results are reported in

Table 3. CVaR and CVaR⁺ of supply chain participants under guaranteed auctions with different coordinating parameters $\{g^o,{\psi }^o\}$.

${\mathit{g}}^{\mathit{o}}$	${\mathit{\psi }}^{\mathit{o}}$	$\mathit{\alpha }$	Seller’s CVaR	Seller’s CVaR+	Auction House’s CVaR	Auction House’s CVaR+
$0.35$	$0.8436$	$0.10$	$-0.4249$	$2.7762$	$-0.3011$	$1.1014$
$0.35$	$0.8436$	$0.05$	$-0.2308$	$2.9761$	$-0.2087$	$1.1527$
$0.35$	$0.8436$	$0.01$	$0.1229$	$3.3588$	$-0.0242$	$1.2524$
$0.40$	$0.8086$	$0.10$	$-0.5079$	$2.7000$	$-0.2055$	$1.1829$
$0.40$	$0.8086$	$0.05$	$-0.3231$	$2.8893$	$-0.0961$	$1.2446$
$0.40$	$0.8086$	$0.01$	$-0.0102$	$3.2624$	$0.1211$	$1.3605$
$0.45$	$0.7591$	$0.10$	$-0.6102$	$2.6001$	$-0.0914$	$1.2883$
$0.45$	$0.7591$	$0.05$	$-0.4445$	$2.7764$	$0.0367$	$1.3656$
$0.45$	$0.7591$	$0.01$	$-0.1827$	$3.1317$	$0.2950$	$1.5011$
$0.50$	$0.6890$	$0.10$	$-0.7386$	$2.4707$	$0.0384$	$1.4216$
$0.50$	$0.6890$	$0.05$	$-0.5983$	$2.6320$	$0.1886$	$1.5179$
$0.50$	$0.6890$	$0.01$	$-0.3870$	$2.9553$	$0.4915$	$1.6862$
$0.55$	$0.5895$	$0.10$	$-0.8986$	$2.3040$	$0.1895$	$1.5891$
$0.55$	$0.5895$	$0.05$	$-0.7934$	$2.4433$	$0.3672$	$1.7112$
$0.55$	$0.5895$	$0.01$	$-0.6350$	$2.7220$	$0.7114$	$1.9304$

.

As shown in Table 3, focusing on the rows corresponding to $\alpha =0.10$ reveals that as the guaranteed price $g^o$ increases, the seller’s CVaR decreases, whereas the auction house’s CVaR increases. A similar pattern can be observed at $\alpha =0.05$ and $\alpha =0.01$. This pattern indicates that the guaranteed auction contract reallocates the downside risk associated with unsold units between the seller and the auction house through the guaranteed price $g^o$. At the same time, the surplus-sharing ratio ${\psi }^o$ decreases as $g^o$ increases, ensuring that when the auction house bears a larger share of downside risk, it is compensated by a greater share of the upside gains generated by high buyer bids. Consequently, as reported in Table 3, for both the seller and the auction house, CVaR⁺ always moves in the same direction as the corresponding CVaR. These results indicate that, under the cooperative game framework, guaranteed auctions constitute a high-risk–high-return profit-sharing mechanism that is consistent with commonly accepted notions of fairness.

6.3.3. Risk–Return Profiles Under Guaranteed Auctions with Non-Cooperative Coordination

Under the non-cooperative game framework, the profit allocation rule embedded in the guaranteed auction contract remains unchanged; what differs are the specific values of the contract parameters $\{g^{*},{\psi }^{*}\}$. Accordingly, we continue to employ the Monte Carlo simulation procedure and the simulated data described in Section 6.3.2 to evaluate the CVaR and CVaR⁺ for both the seller and the auction house under the non-cooperative setting.

As discussed earlier, under the non-cooperative game framework, the auction house can influence the profit shares allocated between supply chain members. To illustrate this effect, we consider five alternative values of ${\lambda }_S$. In all cases, the guaranteed price is fixed at $g^{*}=0.45$, and we compute the CVaR and CVaR⁺ at the $0.10$, $0.05$, and $0.01$ levels for both the seller and the auction house. The resulting estimates are reported in

Table 4. CVaR and CVaR⁺ of supply chain participants under guaranteed auctions with different profit-sharing ratios.

${\mathit{\lambda }}_{\mathit{S}}$	${\mathit{\psi }}^{\mathbf{*}}$	$\mathit{\alpha }$	Seller’s CVaR	Seller’s CVaR+	Auction House’s CVaR	Auction House’s CVaR+
$0.6506$	$0.7334$	$0.10$	$-0.5760$	$2.4986$	$-0.1324$	$1.3883$
$0.6506$	$0.7334$	$0.05$	$-0.4160$	$2.6689$	$-0.0011$	$1.4714$
$0.6506$	$0.7334$	$0.01$	$-0.1630$	$3.0123$	$0.2634$	$1.6194$
$0.6795$	$0.7591$	$0.10$	$-0.6102$	$2.6001$	$-0.0914$	$1.2883$
$0.6795$	$0.7591$	$0.05$	$-0.4445$	$2.7764$	$0.0367$	$1.3656$
$0.6795$	$0.7591$	$0.01$	$-0.1827$	$3.1317$	$0.2950$	$1.5011$
$0.7085$	$0.7848$	$0.10$	$-0.6443$	$2.7015$	$-0.0501$	$1.1884$
$0.7085$	$0.7848$	$0.05$	$-0.4731$	$2.8838$	$0.0756$	$1.2601$
$0.7085$	$0.7848$	$0.01$	$-0.2024$	$3.2512$	$0.3272$	$1.3830$
$0.7374$	$0.8105$	$0.10$	$-0.6785$	$2.8030$	$-0.0085$	$1.0889$
$0.7374$	$0.8105$	$0.05$	$-0.5017$	$2.9913$	$0.1148$	$1.1548$
$0.7374$	$0.8105$	$0.01$	$-0.2221$	$3.3707$	$0.3594$	$1.2662$
$0.7663$	$0.8361$	$0.10$	$-0.7127$	$2.9045$	$0.0331$	$0.9897$
$0.7663$	$0.8361$	$0.05$	$-0.5302$	$3.0987$	$0.1547$	$1.0495$
$0.7663$	$0.8361$	$0.01$	$-0.2418$	$3.4901$	$0.3935$	$1.1494$

.

Table 4 contains two benchmark groups. The first is represented by the first three rows, where ${\lambda }_S=0.6506$, implying that the seller maintains the same expected profit as under the conventional commission contract. The second is represented by rows 4–6, where ${\lambda }_S=0.6795$, for which both the profit allocation and the associated risk–return profile coincide with those under the cooperative game framework when $g^o=0.45$. From the perspective of the auction house as the supply chain leader, however, the latter allocation is unlikely to be chosen. Instead, the auction house has a stronger incentive to adopt ${\lambda }_S=0.6506$, thereby appropriating the additional gains generated by shifting from the commission contract to the coordinating guaranteed auction contract. If feasible, it may even prefer a lower value of ${\lambda }_S$, capturing a larger share of profit at the seller’s expense and weakening the seller’s financial buffer against risk.

Comparing the rows associated with a given value of $\alpha$ clarifies the logic behind the auction house’s preference. As the seller’s profit share ${\lambda }_S$ decreases, the seller’s CVaR increases, whereas its CVaR⁺ gradually decreases. By contrast, the auction house’s CVaR decreases, while its CVaR⁺ steadily increases. This risk–return pattern indicates that, under the non-cooperative game framework, a lower seller profit share allows the auction house to secure not only a larger share of total profit but also a more favorable risk exposure, characterized by lower downside risk and greater upside potential.

Yet the very asymmetry that favors the auction house also makes the supply chain more fragile. Under such an extremely asymmetric arrangement, even a small and unanticipated decline in demand, or a shift in the external competitive environment, may undermine the fragile balance of supply chain cooperation and trigger severe supply disruptions. The numerical results therefore suggest that it is myopic for the supply chain leader to rely solely on the non-cooperative mechanism to appropriate the entire coordination surplus. Because tail risks are distributed so unevenly, even a dominant platform should move toward the cooperative benchmark, which restores a more balanced allocation of risk and thereby strengthens the supply chain’s ability to withstand external shocks and sustain long-term operations.

7. Concluding Remarks

7.1. Findings and Implications

To examine how conventional commission contracts and guaranteed auction contracts affect the performance of auction supply chains, this paper investigates supply and selling cooperation in a two-tier multi-unit consignment auction supply chain consisting of a seller and an auction house. We focus on the strategic behaviors, profit allocation, and risk sharing between the two parties. The main findings are summarized as follows.

• Conventional commission contracts fail to coordinate the supply chain. Auction commissions distort the transmission of market demand information from the distribution side to the supply side, thereby inducing the seller to adopt overly conservative supply and selling strategies. At the same time, the commission-based mechanism effectively insulates the auction house from the risk of unsold goods and shifts this risk entirely onto the seller. Such a loosely coupled principal–agent relationship is detrimental to the optimization of overall supply chain performance.
• Guaranteed auction contracts can coordinate the supply chain under both cooperative and non-cooperative game frameworks, and the functional relationship between the coordinating parameters $g$ and $\psi$ is identical across the two settings. Under the cooperative framework, profit allocation is jointly determined by participants’ disagreement payoffs and their bargaining power. Under the non-cooperative framework, by contrast, the profit-sharing ratio ${\lambda }_S:{\lambda }_H$ is controlled by the supply chain leader and can be adjusted within a limited range.
• Cooperative and non-cooperative coordination mechanisms differ fundamentally in terms of risk sharing. Under the cooperative coordination mechanism, the CVaR⁺ of both the seller and the auction house moves in the same direction as their respective CVaR. By adjusting the guaranteed level for unsold goods, the supply chain can flexibly allocate risk and return. In contrast, under the non-cooperative mechanism, participants’ CVaR⁺ moves in the opposite direction to their CVaR. As a result, the auction house tends to extract profit at the seller’s expense, thereby undermining the stability of supply chain cooperation.

Based on the above findings, the following managerial implications may be of practical relevance to supply chain managers.

• In intermediary supply chains, such as consignment platforms and e-commerce agency channels, contract design should facilitate effective information transmission. In our model, under the conventional commission contract, the auction house captures part of buyers’ willingness to pay, preventing the seller from making system-optimal decisions. By contrast, the guaranteed auction contract separates and reallocates decision rights, allowing the auction house to determine the optimal reserve price $r$ while using the guaranteed payment $g$ to induce the seller to provide the system-optimal supply quantity $Q$. Coordination is therefore achieved.
• The long-term viability of supply chain coordination depends not only on the existence of a coordinating contract, but more fundamentally on whether the bargaining structure promotes symmetry between risk and return. In our model, both the cooperative and non-cooperative frameworks can achieve coordination in a mathematical sense. However, coordination under the non-cooperative leader–follower structure often relies on implicit pressure on the weaker party, thereby decoupling risk from return. By contrast, a cooperative bargaining structure, based on a fair allocation of coordination gains, aligns risk with return more naturally and is therefore better able to sustain long-term cooperation.
• Auction houses and similar powerful intermediaries should move beyond the conventional view that their sole objective is to maximize their own profit. Under a non-cooperative framework, exploiting a dominant bargaining position can seriously undermine the long-term sustainability of established cooperative relationships. Profit allocation should therefore be benchmarked against the cooperative outcome. In this sense, a fair allocation is not a sacrifice of profitability, but a strategic investment in supply chain resilience and long-term competitiveness.

7.2. Limitations and Future Research

Although this study provides a comprehensive analysis of contract design and coordination mechanisms in consignment auction supply chains, several limitations should be acknowledged. These limitations also point to several promising directions for future research.

• The optimal strategies derived from our game-theoretic model rest on the assumption that both the seller and the auction house are risk neutral. Although our numerical analysis evaluates the risk–return profile ex post using CVaR and CVaR⁺, incorporating risk preferences, such as risk aversion, directly into the participants’ objective functions remains an important direction for future research. Given that many consignors are small and medium-sized enterprises with limited risk-bearing capacity, examining how different degrees of risk aversion reshape the optimal guaranteed price and surplus-sharing ratio would substantially enrich the literature on contract design for multi-unit auctions.
• This paper restricts attention to guaranteed auctions in which the auction house itself provides the guarantee. In practice, however, an auction house typically serves multiple sellers simultaneously. When guarantees across multiple consignment auctions are aggregated, the associated capital requirements and exposure to the risk of unsold goods can become substantial. In response, some auction houses have increasingly relied on third-party capital providers to underwrite such guarantees. The introduction of additional strategic players further complicates operational decision making as well as profit allocation in guaranteed auctions. Nevertheless, this setting represents a promising direction for future research.
• Our structural model and Monte Carlo simulations demonstrate the validity and effectiveness of guaranteed auction contracts. However, both approaches assume fully rational decision makers. In practice, supply chain participants may exhibit bounded rationality, and contract negotiations may be shaped by behavioral factors such as cognitive biases and reputation concerns. Future research should therefore examine guaranteed auction contracts through empirical studies or behavioral field experiments in real-world multi-unit auction markets, such as flower, agricultural product, and aquatic product auction centers. Such evidence would help clarify how these mechanisms perform in actual market settings and provide a firmer basis for their practical implementation.