A matching method for second-hand goods exchange considering loss aversion of buyer and seller in e-brokerage

Abstract

The second-hand goods exchange matching problem with intelligent exchange platform is a valuable research topic, while research in the related field is relatively lacking. Thus, the goal of this paper is to develop a two-sided matching method for the second-hand goods exchange matching. In the method, not only the characteristic of loss aversion of buyers/sellers and the benefit of e-brokerage are considered, but also the suggested exchange prices that meet the requirements of buyers and sellers are determined. Specifically, the description of second-hand goods exchange matching problem is firstly presented, in which the 2-tuple including the ideal level and the lowest acceptable level is used to quantify the expected demands of buyers and sellers for attributes. Then, several models maximizing the perceived values of buyers and sellers are derived to determine the suggested exchange price of goods. Next, according to the expectation information of buyers and sellers, and the real information of goods, the perceived value matrix for each attribute of buyers and sellers can be obtained by using the prospect theory. Furthermore, a model can be constructed and solved to determine the maximum number of exchange of buyers and sellers in e-brokerage, and an optimization model with the constraint of the exchange number threshold is constructed, and the optimal exchange matching results can be obtained by solving the optimization model. Finally, an example is given to illustrate the practicality of the proposed method. The method can be used to improve the operational performance of intelligent exchange platforms.

Appendix

Proof of Theorem 1

Let $f_{1} (p_{ij}^{{}} ) = (e_{i1} - p_{ij} )^{\alpha } + (p_{ij} - \overline{e}_{j} )^{\alpha }$ be the function about the suggested exchange price $p_{ij}$, $\overline{e}_{j} \le p_{ij} \le e_{i1}$, the first derivative function $f_{1} ^{\prime}$ can be shown as

$$ f_{1}^{\prime } \left( {p_{ij} } \right) = - \alpha \left( {e_{i1} - p_{ij} } \right)^{\alpha - 1} + \alpha \left( {p_{ij} - \overline{e}_{j} } \right)^{\alpha - 1} $$

Let $f_{1} ^{\prime}(p_{ij} ) = 0$, we have $p_{ij} = \frac{{e_{i1} + \overline{e}_{j} }}{2}$. The second derivative function $f_{1} ^{\prime\prime}$ satisfies

$$ f_{1}^{\prime \prime } \left( {p_{ij} } \right) = \alpha (\alpha - 1)\left( {e_{i1} - p_{ij} } \right)^{\alpha - 2} + \alpha (\alpha - 1)\left( {p_{ij} - \overline{e}_{j} } \right)^{\alpha - 2} < 0 $$

Thus, if $\overline{e}_{j} \le p_{ij} \le e_{i1}$, the suggested exchange price $p_{ij} = \frac{{e_{i1} + \overline{e}_{j} }}{2}$ maximizes the agents' PVs.

Proof of Theorem 2

Let $f_{2} (p_{ij}^{{}} ) = (e_{i1} - p_{ij} )^{\alpha } - \lambda (\overline{e}_{j} - p_{ij} )^{\beta }$ be the function about the suggested exchange price $p_{ij}$, $p_{ij} < e_{i1}$. The first derivation function $f_{2} ^{\prime}$ satisfies.

$f_{2}^{\prime } \left( {p_{ij} } \right) = - \alpha \left( {e_{i1} - p_{ij} } \right)^{\alpha - 1} + \lambda \beta \left( {\overline{e}_{j} - p_{ij} } \right)^{\beta - 1} > 0.$

Thus, the maximum PV $Z_{ij}^{(1)}$ can be obtained if $p_{ij} = e_{i1}$.

Proof of Theorem 3

Let $f_{3} (p_{ij}^{{}} ) = - \lambda (p_{ij} - e_{i1} )^{\beta } - \lambda (\overline{e}_{j} - p_{ij} )^{\beta }$ be the function about the suggested exchange price $p_{ij}$, $e_{i1} < p_{ij} < \overline{e}_{j}$, $\overline{a}_{j} \le p_{ij} \le a_{i1}$. The first derivative function $f_{3} ^{\prime}$ is:

$$ f_{3}^{\prime } \left( {p_{ij} } \right) = - \lambda \beta \left( {p_{ij} - e_{i1} } \right)^{\beta - 1} + \lambda \beta \left( {\overline{e}_{j} - p_{ij} } \right)^{\beta - 1} $$

Let $f_{3} ^{\prime}(p_{ij} ) = 0$, we have $p_{ij} = \frac{{e_{i1} + \overline{e}_{j} }}{2}$. The second derivative function $f_{3} ^{\prime\prime}$ satisfies

$$ f_{3}^{\prime \prime } \left( {p_{ij} } \right) = - \lambda \beta (\beta - 1)\left( {p_{ij} - e_{i1} } \right)^{\beta - 2} - \lambda \beta (\beta - 1)\left( {\overline{e}_{j} - p_{ij} } \right)^{\beta - 2} < 0 $$

Thus, if $e_{i1} < p_{ij} < \overline{e}_{j}$, the price $p_{ij} = \frac{{e_{i1} + \overline{e}_{j} }}{2}$ maximizes the agents' PVs.

Proof of Theorem 4

Let $f_{4} (p_{ij}^{{}} ) = - \lambda (p_{ij} - e_{i1} )^{\beta } + (p_{ij} - \overline{e}_{j} )^{\alpha }$ be the function about the suggested exchange price $p_{ij}^{{}}$, $p_{ij} \ge \overline{e}_{j}$. The first derivation $f_{4} ^{\prime}$ satisfies

$$ f_{4}^{\prime } \left( {p_{ij} } \right) = - \lambda \beta \left( {p_{ij} - e_{i1} } \right)^{\beta - 1} + \alpha \left( {p_{ij} - \overline{e}_{j} } \right)^{\alpha - 1} < 0 $$

Thus, the maximum PV $Z_{ij}^{(3)}$ can be obtained if $p_{ij} = \overline{e}_{j}$.