## 4 Closed-Loop Supply-Chain Decision Making and Model Analysis
### Incentive Mechanism Design
#### 4.2.2 The Solution of the Information Screening Contract Model  
In the case of asymmetric information, the recyclers of different capabilities determine the optimal effort level of the second stage (see Table 1) according to the \"self-selection\" information screening contract to achieve the goal of maximizing their own interests. The best-effort level for the recycler with real recovery and dismantling abilities of \(r_{H}\) and \(r_{L}\) is:  
\[e_{HH}^{*}=\frac{r_{H}\left[f(v)\beta_{H}-p_{m}+p_{t}\right]}{k} \tag{14}\]  
\[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]  
The true recovery and dismantling ability is \(r_{H}\), the recycler selection contract is \((T_{L},\beta_{L})\), with ability type \(r_{L}\), real recovery and dismantling ability \(r_{L}\), and false reporting likelihood \(r_{H}\). When the recycler chooses the contract \((T_{H},\beta_{H})\), in order to maximize its own interests, the optimal level of effort made at that time is:  
\[e_{HL}^{*}=\frac{r_{H}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{16}\]  
\[e_{LH}^{*}=\frac{r_{L}\left[f(v)\beta_{H}-p_{m}+p_{t}\right]}{k} \tag{17}\]  
**Proposition 1**. : _Under the conditions of double information asymmetry, the optimal contract \((T_{L},\beta_{L})\), (\(T_{H},\beta_{H}\)) is:_  
\[\beta_{H}^{*}=\frac{r_{H}^{2}\left[f(v)\left(p_{n}-c_{r}\right)-p_{t}\right]}{ f(v)\left(r_{H}^{2}+k\rho\sigma^{2}\right)} \tag{18}\]  
\[\beta_{L}^{*}=\frac{r_{L}^{2}\left[(1-\delta)\left(f(v)\left(p_{n}-c_{r} \right)-p_{t}\right)+\delta\left(p_{t}-p_{m}\right)\right]-\delta r_{H}^{2} \left(p_{t}-p_{m}\right)}{f(v)\left(\delta r_{H}^{2}+(1-\delta)k\rho\sigma^{2 }+(1-2\delta)r_{L}^{2}\right)} \tag{19}\]  
\[T_{L}^{*}=-\frac{r_{L}^{2}\left[f(v)\beta_{L}^{*}-p_{m}+p_{t}\right]^{2}}{2k} +\frac{1}{2}\rho\sigma^{2}f(v)^{2}\beta_{L}^{2*}+\theta\left(-f(v)\beta_{L}^{ *}+p_{m}-p_{t}\right)+\pi \tag{20}\]  
\[T_{H}^{*}=\frac{f(v)\left(\beta_{H}^{*}-\beta_{L}^{*}\right)\left[f(v)\left( \beta_{H}^{*}+\beta_{L}^{*}\right)\left(k\rho\sigma^{2}-r_{H}^{2}\right)+2r_{H }^{2}p_{m}-2r_{H}^{2}p_{t}-2\theta k\right]+2 kT_{L}^{*}}{2k} \tag{21}\]  
Proof. : We solve the first-order partial derivative of Equations (10)-(13), which is 0, and obtain Equations (14)-(17). \[\frac{\partial E(\Pi_{T_{H}r_{H}})}{\partial e_{HH}^{*}}=r_{H}\left[f(v)\beta _{H}-p_{m}+p_{t}\right]-k\epsilon_{\rm HH}=0 \tag{22}\]  
\[\frac{\partial E(\Pi_{T_{t_{L}T_{L}}})}{\partial e_{LL}^{*}}=r_{L}\left[f(v) \beta_{L}-p_{m}+p_{t}\right]-k\epsilon_{\rm LL}=0 \tag{23}\]  
\[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]0  
\[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]1  
Substituting the Formulae (14)-(17) into Equation (5) and solving the available Proposition 1 with the second-level planning method, the proof is completed. **Proposition 2**. : _The high-capacity agent will obtain more profit. If the low recycling ability of the agent is greater than the retained utility, the high recycling ability of the agent must be greater than the retained utility._  
Proof. : Substituting Equations (14)-(17) into Equations (6) and (9) and solving it, we obtain:\[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]2 \[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]3 \[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]4 (27) \[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]5 Due to: \[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]6 (28) Therefore: \[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]7 (29) Which is: \[e_{LL}^{*}=\frac{r_{L}\left[f(v)\beta_{L}-p_{m}+p_{t}\right]}{k} \tag{15}\]8  
**Proposition 3**. : _Under the information screening contract model, the recycler will report the recycling and dismantling capability information truthfully.