# Contract-Based Incentive Mechanism for Mobile Crowdsourcing Networks
## 2 System Model for Crowdsourcing Incentive Mechanism
### Utility of Mobile Users  
Suppose that the \(i\)th MU offers its crowdsourcing effort \(e_{i}\) to obtain the reward from the SP. The SP can achieve profit \(\pi_{i}\) with the help of the \(i\)th MU. As a result of some measurement errors, the SP's achieved profit may be slightly different from the actual effort exerted by the MU. Therefore, we assume that the SP's actual achieved profit \(\pi_{i}\) is a noisy signal, which is given as  
\[\pi_{i}=\theta_{i}e_{i}+\delta \tag{1}\]  
where \(\theta_{i}\) is the profit per unit crowdsourcing effort, and \(\delta\) is a normally distributed random variable with \(\delta\sim\mathcal{N}(0,\sigma^{2})\).  
The more the crowdsourcing resources the MUs consume, the greater the crowdsourcing cost the MUs pay. Moreover, we assume that \(C_{i}(e_{i})\) grows more rapidly in the large crowdsourcing effort than it does in the small crowdsourcing effort. Therefore, \(C_{i}^{\prime}(e_{i})>0\) and \(C_{i}^{\prime\prime}(e_{i})>0\). For simplicity, the crowdsourcing cost \(C_{i}(e_{i})\) of the \(i\)th MU is assumed to be quadratic:  
\[C_{i}(e_{i})=c_{i}e_{i}^{2}/2 \tag{2}\]  
Figure 1: Mobile crowdsourcing network (MCN).  
where \(c_{i}\) is the \(i\)th MU's crowdsourcing cost coefficient, which can describe the \(i\)th MU's crowdsourcing cost information (i.e., battery, memory, and computing power). We note that different crowdsourcing scenarios may have a different crowdsourcing cost.  
We assume that the SP offers the payment \(W_{M_{i}}\) to the \(i\)th MU in the linear form [25] defined as  
\[W_{M_{i}}=\alpha_{i}+\beta_{i}\pi_{i}-\frac{c_{i}}{2}e_{i}^{2} \tag{3}\]  
where \(\alpha_{i}\) is the \(i\)th MU's basic wage, and \(\beta_{i}\in[0,1]\) is the bonus coefficient related to the crowdsourcing performance. Considering the different crowdsourcing actions, MUs may obtain different bonuses.  
The payment \(W_{M_{i}}\) is approximately normally distributed with means  
\[E\left[W_{M_{i}}\right]=\alpha_{i}+\beta_{i}\theta_{i}e_{i}-\frac{c_{i}}{2}e_ {i}^{2} \tag{4}\]  
and variances  
\[V_{ar}\left[W_{M_{i}}\right]=\beta_{i}^{2}\sigma^{2} \tag{5}\]  
In this section, we assume that each MU has a constant absolute risk-averse (CARA) preference; then, the \(i\)th MU's negative exponential utility is defined as  
\[u\left(W_{M_{i}}\right)=-e^{-\eta_{M}W_{M_{i}}} \tag{6}\]  
where \(\eta_{M}>0\) is the \(i\)th MU's coefficient of absolute risk aversion (\(\eta_{M}=-u^{\prime\prime}\left(W_{M_{i}}\right)/u^{\prime}\left(W_{M_{i}}\right)\)). A larger value of \(\eta_{M}>0\) means that the MU has less incentive to participate in crowdsourcing tasks; \(\eta_{M}=0\) means that the MU is risk-neutral. A risk-neutral MU's decision is not affected by the degree of crowdsourcing uncertainty.  
Then, the \(i\)th MU's expected utility \(u\left(W_{M_{i}}\right)\) can be written as  
\[E\left[u\left(W_{M_{i}}\right)\right] =E\left[-e^{-\eta_{M}W_{M_{i}}}\right] \tag{7}\] \[=\frac{-1}{\sqrt{2\pi V_{ar}\left[W_{M_{i}}\right]}}\int_{-\infty }^{\infty}e^{-\frac{W_{M_{i}}^{2}-2E\left[W_{M_{i}}\right]|W_{M_{i}}+(E\left[ W_{M_{i}}\right])^{2}+2V_{ar}\left[W_{M_{i}}\right]_{M}W_{M_{i}}}{2V_{ar}\left[W_{M_{i}} \right]}}dW_{M_{i}}\] \[=\frac{-1}{\sqrt{2\pi V_{ar}\left[W_{M_{i}}\right]}}\cdot e^{2} V_{ar}\left[W_{M_{i}}\right]_{M_{i}}^{2}-E\left[W_{M_{i}}\right]_{M_{i}}\cdot \int_{-\infty}^{\infty}e^{-\frac{\left(W_{M_{i}}-E\left[W_{M_{i}}\right]+V_{ ar}\left[W_{M_{i}}\right]_{M}\right)^{2}}{2V_{ar}\left[W_{M_{i}}\right]}}dW_{M_{i}}\] \[=-e^{-\eta_{M}\left[E\left[W_{M_{i}}\right]-\frac{1}{2}V_{ar} \left[W_{M_{i}}\right]_{M_{i}}\right]},\] \[-e^{-\eta_{M}\left[\alpha_{i}+\beta_{i}\theta_{i}e_{i}-\frac{c_{i }}{2}\sigma_{i}^{2}-\frac{\eta_{M}}{2}\beta_{i}^{2}\sigma^{2}\right]}\]