On the Profit-Maximizing for Transaction Platforms in Crowd Sensing

Abstract

Crowd sensing is a novel sensing paradigm, in which a challenging task is to balance benefits of various participants, i.e., data requesters, data providers and transaction platforms for attracting sufficient participants. Little attention in literature has been paid to the transaction platform’s profit which is one of the major issues for maintaining a crowd sensing system consistently. In this paper, we aim to propose a mechanism design for optimizing the platform’s profit. For that, we first model the interactions in crowd sensing by leveraging tools of game theory, and then we prove the best strategy for maximizing the benefit of transaction platforms with satisfying individual rationality constraint and incentive compatibility constraint. Finally, we propose two practical algorithms based on the best strategy. Our simulations show that the algorithms are effective in terms of keeping the platform’s profit and time efficiency.

Appendix: Proof of Lemma 1

Proof. The original problem that we formulated is as follows,

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We first have $\mathbb {E}_V\{\sum _{j=1}^{m}p^R_j(\mathbf v )\} = \sum _{j=1}^{m} \int _{0}^{\bar{v_j}} \mathbb {E}_{V_{R_j}} \{p^R_j(u,\mathbf v _{j}\} f^R_j(u)du$.

For data requester j, we discover $\mathbb {E}_{V_{R_j}} \{p^R_j(u,\mathbf v _{j}\} = \mathbb {E}_{V_{R_j}}\{u(1-x^R_j(u,\mathbf v _{j})\} - U^R_j(u)$ according to its utility function. Recalling in incentive compatibility constraint, we manage to transform utility function in to following integration form, $U^R_j(u) = U^R_j(0)+\int _{0}^{u}\mathbb {E}_{V_{R_j}}\{ (1-x^R_j(u,\mathbf v _{j})\}du$

We then substitute the term in the integration with the above two equations.

$$\begin{aligned} \begin{aligned} \mathbb {E}_V\{\sum _{j=1}^{m}p^R_j(\mathbf v )\}&= \sum _{j=1}^{m} \int _{0}^{\bar{v_j}} \mathbb {E}_{V_{R_j}}\{u(1-x^R_j(u,\mathbf v _{j}) - U^R_j(u)\}f^R_j(u)du \\&= \sum _{j=1}^{m} \mathbb {E}_V\{u(1-x^R_j(u,\mathbf v _{j})- U^R_j(0)\} \\&-\sum _{j=1}^{m}\int _{0}^{\bar{v_j}}\int _{0}^{u}\mathbb {E}_{V_{R_j}}\{ (1-x^R_j(x,\mathbf v _{j})\}dx f^R_j(u)du \end{aligned} \end{aligned}$$

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We can apply integration by parts ($f'= f^R_j(u), \quad g = \int _{0}^{u}\mathbb {E}_{V_{R_j}}\{ (1-x^R_j(x,\mathbf v _{j})\}dx$) to the second term and rewrite the above equation, $\sum _{j=1}^{m} \int _{0}^{\bar{v_j}} \int _{0}^{u} \mathbb {E}_{V_{R_j}} \{ (1-x^R_j(x,\mathbf v _{j}) \} dx f^R_j(u)du = \sum _{j=1}^{m} \mathbb {E}_V\{\frac{1-F^R_j(u)}{f^R_j(u)}(1-x^R_j(u,\mathbf v _{j})\}$.

So, we have rewrite the payment of data requesters with definition of $Q^R_j(v_j) = v_j - \frac{1-F^R_j(v_j)}{f^R_j(v_j)}$: $\mathbb {E}_V\{\sum _{j=1}^{m}p^R_j(\mathbf v )\} = \mathbb {E}_V\{\sum _{j=1}^{m} \{Q^R_j(v_j)-x^R_j(\mathbf v )Q^R_j(v_j)-U^R_j(0)\}$.

The deduction of data providers’ payment is almost the same. We first have: $\mathbb {E}_V\{\sum _{i=1}^{n}p^W_i(\mathbf v )\} = \sum _{i=1}^{n} \int _{V_i} \mathbb {E}_{V_{W_i}} \{p^W_i(\mathbf v )\} f^W_i(\mathbf u )d\mathbf u$. Apply again integration form of utility function in individual rationality constraint to substitute the term in the integration.

$$\begin{aligned} \begin{aligned} \mathbb {E}_V\{\sum _{i=1}^{n}p^W_i(\mathbf v )\}&= \sum _{i=1}^{n} \int _{V_i} \{U^W_i(\mathbf u )+ \mathbb {E}_{V_{W_i}} \{ \sum _{j=1}^{m} x^W_{i,j}c_{i,j} \}\} f^W_i(\mathbf u )d\mathbf u \\&= \sum _{i=1}^{n} \mathbb {E}_V\{\sum _{j=1}^{m} x^W_{i,j}c_{i,j}+U^W_i(\bar{\mathbf{c _\mathbf i }})\}\\&-\sum _{i=1}^{n}\int _{V_i}\sum _{j=1}^{m} \int _{u_{i,j}}^{\bar{c_{i,j}}} \mathbb {E}_{V_{W_i}} \{x^W_{i,j}(\mathbf x ,\mathbf v )\}d\mathbf x \}f^W_i(\mathbf u )d\mathbf u \end{aligned} \end{aligned}$$

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The second term needs to simplify further. Replace $V_i$ and $f^W_i$ by their definition and apply integration by parts for every components in summation and $f^W_{i,j}$, and the results is as follows: $\mathbb {E}_V\{\sum _{i=1}^{n}p^W_i(\mathbf v )\} = \sum _{i=1}^{n}\mathbb {E}_V\{\sum _{j=1}^{m} \frac{F^W_{i,j}(c_j)}{f^W_{i,j}(c_j)}x^W_{i,j} +\sum _{j=1}^{m}x^W_{i,j}c_{i,j}+U^W_i(\bar{\mathbf{c }}_\mathbf i )\}$. Let $Q^W_{i,j}(c_{i,j})$ represent $\frac{F^W_{i,j}(c_{i,j})}{f^W_{i,j}(c_{i,j})} + c_{i,j}$, we have

$\mathbb {E}_V\{\sum _{i=1}^{n}p^W_i(\mathbf v )\} = \sum _{i=1}^{n}\mathbb {E}_V\{\sum _{j=1}^{m} Q^W_{i,j} (c_{i,j})x^W_{i,j} +U^W_i(\bar{\mathbf{c }}_\mathbf i )\}$. In order to satisfy the individual rationality constraint and maximizing the profit of platform, our best choice is set all $U^W_i(\mathbf c _\mathbf i )$ and $U^R_j(0)$ equal to zero according to Eqs. (14) and (15).

The final problem of the platform’s profit is as follows.

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