Abstract
Mobile crowd sensing is a widely used sensing paradigm allowing applications on mobile smart devices to routinely obtain spatially distributed data on a range of user attributes: location, temperature, video and audio. Such data then typically forms the input to application specific machine learning tasks to achieve objectives such as improving user experience, targeting geo-localised query based searches to user interests and commercial aspects of targeted geo-localised advertising. We consider a scenario in which the sensing application purchases data from spatially distributed smartphone users. In many spatial monitoring applications, the crowdsourcer needs to incentivize users to contribute sensing data. This may help ensure collected data has good spatial coverage, which will enhance quality of service provided to the application user when used in machine learning tasks such as spatial regression. Privacy considerations should be addressed in such crowd sensing applications, and an incentive offered to “privacy-concerned” users to contribute data. A novel Stackelberg incentive mechanism is developed that allows workers to specify their location whilst satisfying their location privacy requirements. The Stackelberg and Nash equilibria are explored and an algorithm to demonstrate the approach is developed for a real data application.
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Appendices
Appendix 1. Proof of Lemma 2
A unique Nash equilibrium exists in the Followers game if for all \(i \in \mathcal {I}\) Rosen (1965): (i) the domain of the workers’ strategy set \(\{ {t}_{i} \}_{i \in \mathcal {I}}\) is convex and compact, and (ii) ui is continuous and strictly concave in ti. Indeed, the domain of the workers’ strategy set \(\{ {t}_{i} \}_{i \in \mathcal {I}}\) is convex and compact since ti is assumed to be bounded, and ui is continuous in ti, and strictly concave in ti as \(\frac {\partial ^{2} {u}_{i}}{\partial {{t}_{i}^{2}}} < 0\):
Therefore, a Nash equilibrium exists in the Followers game. □
Appendix 2. Proof of Theorem 3
By Lemma 2, there exists a unique strategy profile that maximizes the utility of each worker given the strategies of the other workers. Thus, if each worker i plays its best response strategy, it will achieve the unique Nash equilibrium point \({t}_{i}^{*}\). To prove Theorem 3, we derive \({t}_{i}^{*}\) by solving \(\frac {\partial {u}_{i}}{\partial {t}_{i}} = 0\) to obtain:
Next, we manipulate the expression in Eq. 23 to obtain:
where (i) we express \({t}_{i}^{*} {\rho }_{i}\) in terms of the other \({t}_{j}^{*} {\rho }_{j}\) values, (ii) we sum up the \({t}_{i}^{*} {\rho }_{i}\) values in Eq. 25 for all participating workers in region li, and (iii) we divide the previous expression by \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j}^{*} {\rho }_{j}\).
Finally, we substitute Eq. 26 into Eq. 25 to obtain the unique Nash equilibrium point for each worker i as required. □
Appendix 3. Proof of Lemma 4
To prove Lemma 4, we show that the set of participating workers \(\mathcal {Q}_{{l}}\) computed by Algorithm 1 always satisfies Eq. 7 for all regions \({l} \in {\mathscr{L}}\).
Consider the region l. Suppose the constraint in Eq. 7 is not satisfied, then the following constraints must be true:
From Eq. 27, it is obvious that if \( \frac {{c}_{k}}{{\rho }_{k}} \geq \frac {{c}_{i}}{{\rho }_{i}} \) and \( \frac {{c}_{i}}{{\rho }_{i}} \geq \frac { \sum \limits _{j \in \mathcal {Q}_{{l}}: j \neq i} \frac {{c}_{j}}{{\rho }_{j}} }{ |\mathcal {Q}_{{l}}| - 2 }\), then \( \frac {{c}_{k}}{{\rho }_{k}} \geq \frac { \sum \limits _{j \in \mathcal {Q}_{{l}}: j \neq i} \frac {{c}_{j}}{{\rho }_{j}} }{ |\mathcal {Q}_{{l}}| - 2 }\). Hence, the greedy computation of the set of participating workers \(\mathcal {Q}_{{l}}\) from the sorted \(\frac {{c}_{i}}{{\rho }_{i}}\) values in Algorithm 1 does not affect the set of participating workers \(\mathcal {Q}_{{l}}\) that achieves the unique Nash equilibrium solution of the Followers game. □
Appendix 4. Proof of Theorem 5
To show that the obtained solution from Algorithm 1 is the unique Nash equilibrium (NE) solution of the workers, we proof that the \(\frac {\partial {u}_{i}}{\partial {t}_{i}} = 0\) (stationary point) condition given in Eq. 24 is satisfied by the set of participating workers \(i \in \mathcal {Q}\). By Lemma 2, there exists a unique NE in the followers game. Hence, any stationary point is the unique NE point for the workers. From Algorithm 1, we have \( {t}_{i}^{*} = \frac {|\mathcal {Q}_{{l}_{i}}| -1 }{ \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} \frac {{c}_{j}}{{\rho }_{j}} } \left (1 - \frac {(|\mathcal {Q}_{{l}_{i}}| -1) \frac {{c}_{i}}{{\rho }_{i}} } {\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} \frac {{c}_{j}}{{\rho }_{j}} } \right ) \frac {{R}_{{l}_{i}} }{ {\rho }_{i} } \) if \(i \in \mathcal {Q}_{{l}_{i}},\) and \( {t}_{i}^{*} = 0\) otherwise. In addition, we have the expression \( \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j}^{*} {\rho }_{j} = \frac {|\mathcal {Q}_{{l}_{i}}| -1 }{ \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} \frac {{c}_{j}}{{\rho }_{j}} } {R}_{{l}_{i}} \) from Eq. 26. We substitute the expressions for \({t}_{i}^{*}\) and \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j}^{*} {\rho }_{j}\) into the \(\frac {\partial {u}_{i}}{\partial t_{i}}\) expression in Eq. 24 to obtain the following equality:
Since Eq. 28 satisfies the expression for \(\frac {\partial {u}_{i}}{\partial {t}_{i}} = 0\), we conclude that Algorithm 1 correctly outputs the unique Nash equilibrium solution of the workers in \(\mathcal {Q}_{{l}}\). Also, by Lemma 4, Algorithm 1 also correctly computes the set of participating workers \(\mathcal {Q}_{{l}}\) as used in Eq. 28. □
Appendix 5. Proof of Lemma 8
From Eq. 11, we have the following expression for all workers \(i \in \mathcal {Q}_{{l}}\):
From Eq. 29, we set \(\frac {\partial {u}_{i}}{\partial {c}^{\prime }_{i}}=0\) to obtain the critical point:
We substitute the expression of the critical point derived in Eq. 30 into the \(\frac {\partial ^{2} {u}_{i}}{\partial ({c}^{\prime }_{i})^{2}}\) expression in Eq. 29 to obtain:
The denominator term \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} \frac {{c}^{\prime }_{j}}{{\rho }_{j}}\) in the \(\frac {\partial ^{2} {u}_{i}}{\partial ({c}^{\prime }_{i})^{2}}\) expression from Eq. 29 is positive due to both ci and ρi being positive. Since we assume that \(|\mathcal {Q}_{{l}_{i}}| > 2\), then we have \(\frac {\partial ^{2} {u}_{i}}{\partial ({c}^{\prime }_{i})^{2}} < 0\). This leads us to conclude that the critical point is a maximum point. Hence, the proof is complete. __
Appendix 6. Proof of Theorem 16
To prove Theorem 16, we make the following two claims.
Claim
Suppose that the αi, ρi, and ci values are constant for all workers \(i \in \mathcal {I}\), the Stackelberg equilibrium of the proposed Stackelberg incentive model is Pareto efficient.
Proof
To prove the claim, we use the following key observations:
Consider a strategy profile (R,t)≠(RSE,tSE). If \({U_{CS}}(\mathbf {{R}}; \mathbf {{t}}) > {U_{CS}}(\mathbf {{R}}^{SE}; \mathbf {{t}}^{SE})\), then ∃i where \({{u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) < {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}) }\). It can be shown that the previous statement is true since \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j} > \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j}^{SE}\) is a necessary condition for \({U_{CS}}(\mathbf {{R}}; \mathbf {{t}}) > {U_{CS}}(\mathbf {{R}}^{SE}; \mathbf {{t}}^{SE})\) when the αi, ρi, and ci values are constant for all workers \(i \in \mathcal {I}\). From Eq. 32, we know that \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {u}_{j}({t}_{j}; \mathbf {{t}_{-j}}, {R}_{{l}_{j}})\) is inversely proportional to \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {c} {t}_{j}\). This means that if \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j} > \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j}^{SE}\) is true, then we have \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {u}_{j}({t}_{j}; \mathbf {{t}_{-j}}, {R}_{{l}_{j}}) < \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {u}_{j}({t}_{j}^{SE}; \mathbf {{t}_{-j}}^{SE}, {R}_{{l}_{j}}^{SE})\). Hence, we conclude that ∃i where \({u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) < {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}) \).
Similarly, it can be shown that if ∃i where \({{u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) > {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}) }\) and \({{u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) \geq {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}), \forall i \in \mathcal {I} }\), then \({U_{CS}}(\mathbf {{R}}; \mathbf {{t}}) < {U_{CS}}(\mathbf {{R}}^{SE}; \mathbf {{t}}^{SE})\). This is because if ∃i where \({{u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) > {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}) }\) and \({u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}})\) \( \geq {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}), \forall i \in \mathcal {I} \), then we have \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {u}_{j}({t}_{j}; \mathbf {{t}_{-j}}, {R}_{{l}_{j}}) > \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {u}_{j}\) \(({t}_{j}^{SE}; \mathbf {{t}_{-j}}^{SE}, {R}_{{l}_{j}}^{SE})\). This is only possible if \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {c} {t}_{j} < \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {c} {t}_{j}^{SE}\), which means \(\sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j} < \sum \limits _{j \in \mathcal {Q}_{{l}_{i}}} {t}_{j}^{SE}\). Therefore, from Eq. 33, we conclude that \({U_{CS}}(\mathbf {{R}}; \mathbf {{t}}) < {U_{CS}}(\mathbf {{R}}^{SE}; \mathbf {{t}}^{SE})\).
Hence, the proof is complete. □
Claim
The proposed Stackelberg incentive model may not have a Pareto efficient Stackelberg equilibrium. In other words, suppose that (R,t)≠(RSE,tSE) and \({U_{CS}}(\mathbf {{R}}; \mathbf {{t}}) > {U_{CS}}(\mathbf {{R}}^{SE}; \mathbf {{t}}^{SE})\), we have \({{u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) \geq {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}), \forall i \in \mathcal {I} }\).
Proof
Consider the scenario where there are 2 regions l1 and l3 with 2 workers each. Let ρi = 1,ci = 1 for the workers in the 2 regions. Let l1 = l2 and l3 = l4, α1 = α2,α3 = α4, and α1 < α3. Intuitively, given that the worker costs are the same but α1 < α3, then \({R}_{{l}_{1}}^{SE} < {R}_{{l}_{3}}^{SE}\).
Suppose we have \({R}_{{l}_{3}} > {R}_{{l}_{3}}^{SE}\) and \({R}_{{l}_{1}} < {R}_{{l}_{1}}^{SE}\) (recall that the total rewards is bounded, so \({R}_{{l}_{1}}\) must decrease if \({R}_{{l}_{3}}\) increases). Let \({R}_{{l}_{3}} = {R}_{{l}_{3}}^{SE} + {\Delta }\) and \({R}_{{l}_{1}} = {R}_{{l}_{1}}^{SE} - {\Delta }\) where Δ > 0.
First, we obtain the closed-form expression for ti at the Stackelberg equilibrium. We substitute the ρi and ci values into Eq. 6 and obtain:
Next, we substitute the ti expression in Eq. 34 into Eq. 3 to obtain
Suppose \({u}_{1}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) \geq {u}_{1}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE})\), from Eq. 3, we have:
where \({\Delta }^{\prime } \geq 0\).
Similarly, if \({u}_{3}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) \geq {u}_{3}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE})\), from Eq. 3, we have:
where \({\Delta }^{\prime } \geq 0\).
If \({U_{CS}}(\mathbf {{R}}; \mathbf {{t}}) > {U_{CS}}(\mathbf {{R}}^{SE}; \mathbf {{t}}^{SE}) \), and \({u}_{i}({t}_{i}; \mathbf {{t}_{-i}}, {R}_{{l}_{i}}) \geq {u}_{i}({t}_{i}^{SE}; \mathbf {{t}_{-i}}^{SE}, {R}_{{l}_{i}}^{SE}), \forall i \in \mathcal {I}\) is true, then the following condition on UCS must be true. From Eq. 8, we have:
□
Therefore, Claim 2 shows a scenario where the proposed Stackelberg equilibrium may not be Pareto efficient for the crowdsourcer.
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Koh, J.Y., Peters, G.W., Nevat, I. et al. Privacy Considerations in Participatory Data Collection via Spatial Stackelberg Incentive Mechanisms. Methodol Comput Appl Probab 23, 1097–1128 (2021). https://doi.org/10.1007/s11009-020-09798-7
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DOI: https://doi.org/10.1007/s11009-020-09798-7