# Contract-Based Incentive Mechanism for Mobile Crowdsourcing Networks
## 4 Analysis and Discussion  
In this section, the impact of the crowdsourcing participants' risk preferences on the incentive mechanism is illustrated.  
First, the optimal incentive mechanism of the risk-averse MUs is considered with \(\eta_{M}\
eq 0\). From Equation (23), we have  
\[\frac{\partial\beta_{i}^{*}}{\partial\eta_{M}} = \frac{-\sigma^{2}c_{i}(\theta_{i}^{2}+\eta_{S}\sigma^{2}c_{i})}{ \left(\theta_{i}^{2}+\eta_{M}\sigma^{2}c_{i}+\eta_{S}\sigma^{2}c_{i}\right)^{2 }}<0 \tag{24}\] \[\frac{\partial\beta_{i}^{*}}{\partial\eta_{S}} = \frac{\eta_{M}\sigma^{4}c_{i}^{2}}{\left(\theta_{i}^{2}+\eta_{M} \sigma^{2}c_{i}+\eta_{S}\sigma^{2}c_{i}\right)^{2}}>0 \tag{25}\]  
and  
\[\frac{\partial\beta_{i}^{*}}{\partial c_{i}} = \frac{-\eta_{M}\sigma^{2}\theta_{i}^{2}}{\left(\theta_{i}^{2}+ \eta_{M}\sigma^{2}c_{i}+\eta_{S}\sigma^{2}c_{i}\right)^{2}}<0 \tag{26}\]  
Thus, from the above formulas, we have that the \(i\)th MU's optimal bonus coefficient \(\beta_{i}^{*}\) is decreasing in its absolute risk-averse coefficient \(\eta_{M}\) and in its crowdsourcing cost, andis increasing  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Settings** \\ \hline \(\beta_{i}^{*}\) & \(\frac{\theta_{i}^{2}+\eta_{S}\sigma^{2}c_{i}}{\theta_{i}^{2}+\eta_{M}\sigma^{2 }c_{i}+\eta_{S}\sigma^{2}c_{i}}\) \\ \hline \(c_{i}^{*}\) & \(\frac{\hat{\beta}_{i}^{*}\theta_{i}}{c_{i}}\) \\ \hline \(\alpha_{i}^{*}\) & \(\Omega-\beta_{i}^{*}\theta_{i}\epsilon_{i}^{*}+\frac{\zeta_{i}}{2}(\epsilon_{i} ^{*})^{2}+\frac{\eta_{M}}{2}(\beta_{i}^{*})^{2}\sigma^{2}\) \\ \hline \(E\left[u\left(W_{M_{i}}\right)\right]^{*}\) & \(-e^{-\eta_{M}\Omega}\) \\ \hline \(f_{S}^{*}\) & \(\sum\limits_{i=1}^{N}\left[\left(1-\beta_{i}^{*}\right)\theta_{i}\epsilon_{i}^ {*}-a_{i}^{*}-\frac{\eta_{S}}{2}\sigma^{2}(1-\beta_{i}^{*})^{2}\right]\) \\ \hline \(E\left[W_{S}\right]^{*}\) & \(-e^{-\eta_{S}f_{S}^{*}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimal contract design parameters and settings.  
in the SP's absolute risk-averse coefficient \(\eta_{S}\). Given the SP's absolute risk-averse coefficient \(\eta_{S}\), an increasing \(\eta_{M}\) can reduce the MU's optimal bonus coefficient. The greater the SP's absolute risk-averse coefficient \(\eta_{S}\), the greater the risk transferred to the MUs, and the greater the MU's optimal bonus coefficient \(\beta_{i}^{*}\). Therefore, the MUs need to take a greater risk to obtain more utility.  
Then, because \(e_{i}^{*}=\frac{\beta_{i}^{*}\theta_{i}}{c_{i}}\), from the above illustrations, we can also have that the \(i\)th MU's optimal effort \(e_{i}^{*}\) is decreasing in its absolute risk-averse coefficient \(\eta_{M}\) and increasing in the SP's absolute risk-averse coefficient \(\eta_{S}\). The greater the MU's absolute risk-averse coefficient \(\eta_{M}\), the lesser the value its crowdsourcing risk will take, and the lower the expected level of its effort.  
In particular, the optimal incentive mechanism of the risk-neutral MUs is considered with \(\eta_{M}=0\). From Equation (23), we have \(\beta_{i}^{*}=1\), \(e_{i}^{*}=\frac{\theta_{i}}{c_{i}}\) and \(f_{S}^{*}=\sum\limits_{i=1}^{N}\left[\frac{\theta_{i}^{2}}{2c_{i}}-\bar{U}\right]\). We notice that the optimal expected utility of the SP has no relation to the MUs' crowdsourcing effort. Furthermore, the greater the MU's crowdsourcing cost, the lower the level of its effort.