A contract-based incentive mechanism for distributed meeting scheduling: Can agents who value privacy tell the truth?

Abstract

We consider a distributed meeting scheduling problem where agents negotiate with each other to reach a consensus over the starting time of the meeting. Each agent has a private preference over a set of time slots, and aims to select its own preferred slot while revealing as little information about its preference as possible. A key challenge in this canonical setting is whether it is possible to design a distributed mechanism where agents that value their privacy are motivated to tell the truth about their preferences. In this paper, we give a positive answer by proposing a novel incentive mechanism based on economic contract theory. A set of contracts are carefully designed for agents of different types, consisting of the required actions, corresponding rewards and the privacy leakage level. By selecting the contract that maximises its own utility, each agent will not deviate from the required actions and can avoid unnecessary privacy leakage. Other properties of the mechanism such as budget balance, no need for a central authority, and near-optimal social welfare are also theoretically proved. Our empirical evaluations show that our proposed mechanism reduces privacy leakage by 58% compared to a standard calendar-sharing scheme. The social welfare of the proposed mechanism reaches over 88% of the optimal centralized method, and is higher than the social welfare of the state-of-the-art schemes by between 16 and 82%. A better trade-off between the privacy leakage and the number of rounds for convergence is also achieved compared to a typical negotiation mechanism.

Appendices

Appendix 1: Proof of Proposition 1

For an agent n free at time slot t, rewards should satisfy (we omit the subscript n and t for simplicity)

$$\begin{aligned}&R_F - C(F,F) - \alpha \left| \theta - 0 \right| > R_O - C(F,O) - \alpha \left( \left| \theta - 0 \right| - \left| p_0 - 0 \right| \right) \end{aligned}$$

(50a)

$$\begin{aligned}&R_F - C(F,F) - \alpha \left| \theta - 0 \right| > 0 - 0 - \alpha \left( \left| \theta - 0 \right| - \left| 1 - 0 \right| \right) \end{aligned}$$

(50b)

such that this agent’s utility can be maximised only when it reports to free. Similarly, if this agent is OK with time slot t, rewards should satisfy

$$\begin{aligned}&R_O - C(O,O) - \alpha \left| \theta - p_0 \right| > R_F - C(O,F) - \alpha \left( \left| \theta - p_0 \right| - \left| 0 - p_0 \right| \right) \end{aligned}$$

(51a)

$$\begin{aligned}&R_O - C(O,O) - \alpha \left| \theta - p_0 \right| > 0 - 0 - \alpha \left( \left| \theta - p_0 \right| - \left| 1 - p_0 \right| \right) . \end{aligned}$$

(51b)

When the agent is busy at slot t, rewards should be designed in a way that

$$\begin{aligned}&R_B - C(B,B) - \alpha \left| \theta - 1 \right| > R_F - C(B,F) - \alpha \left( \left| \theta - 1 \right| - \left| 0 - 1 \right| \right) \end{aligned}$$

(52a)

$$\begin{aligned}&R_B - C(B,B) - \alpha \left| \theta - 1 \right| > R_O - C(B,O) - \alpha \left( \left| \theta - 1 \right| - \left| p_0 - 1 \right| \right) . \end{aligned}$$

(52b)

Given $R_B = 0$ and $C(B,B) = 0$, (51b) can rewritten by

$$\begin{aligned} R_O > C(O,O) + \alpha (1-p_0), \end{aligned}$$

(53)

which is exactly (21a). By substituting (21a) into (50a) and (50b), respectively, we find that these two constraints are equivalent, which can be written as (21b).

By substituting (21b) into (51a), we have

$$\begin{aligned} C(O,F) - C(O,O)> R_F - R_O + \alpha p_0 > C(F,F) - C(O,O) + 2\alpha p_0. \end{aligned}$$

(54)

Therefore, constraint (51a) is equivalent to (22a). Constraints (52a) and (52b) are shown in (21b) and (21c), respectively.

To make sure that (52b) does not conflict with (51b), the following inequality needs to be satisfied:

$$\begin{aligned} C(O,O) + \alpha (1-p_0) < C(B,O) - \alpha (1-p_0), \end{aligned}$$

(55)

which is equivalent to (22b). Similarly, to avoid the conflict between (21a) and (21b), the following inequality should hold:

$$\begin{aligned} R_O + C(F,F) - C(F,O) + \alpha p_0 < C(B,F) - \alpha . \end{aligned}$$

(56)

By substituting (21c) into the above inequality, we have (22c). That ends the proof.

Appendix 2: Proof of Lemma 1

As shown in equation (15), when an attendee selects the outer-layer contracts, it also considers which response $r_n(t)$ to make if a free or OK time slot t is proposed after an outer-layer contract is selected. Note that it only deviates from its real type ${\theta }$ either to obtain higher rewards at free and OK time slots or to suffer less privacy leakage. In other words, it has no motivation to select a type-$\widetilde{\theta }_n$ contract that it can only maximise its utility by claiming to be busy when it is actually free or OK. Therefore, the host only needs to guarantee that when an attendee selects a type-$\widetilde{\theta }_n$ contract, it does not have incentives to report to be free (or OK) when it is actually OK (or free). We do not consider the case of being busy because a busy attendee does not claim to be free or OK in any case. This can be achieved by setting the following constraints

$$\begin{aligned}&R_F(\widetilde{\theta }_n) - C(F,F| {\theta }_n) > R_O(\widetilde{\theta }_n) - C(O,O| {\theta }_n) + \alpha (\left| p_0 - 0 \right| ), \forall {\theta }_n, \end{aligned}$$

(57a)

$$\begin{aligned}&R_O(\widetilde{\theta }_n) - C(O,O| {\theta }_n)0 > R_F(\widetilde{\theta }_n) - C(O,F| {\theta }_n) + \alpha (\left| 0 - p_0 \right| ), \forall {\theta }_n, \end{aligned}$$

(57b)

which is equivalent to conditions (25). Thus, the following equation hold

$$\begin{aligned} \begin{aligned}&U_n\left( \widetilde{\theta }_n| {\theta }_n, s_n \right) \\&\quad = \max \limits _{r_n(t)\in \mathcal{{S}}} \left\{ R_{r_n(t)}\left( \widetilde{\theta }_n\right) - C\left( s_n(t), r_n(t)| {\theta }_n\right) - \alpha L_n\left( g_n\right) \right\} \\&\quad =\max \limits _{r_n(t)\in \mathcal{{S}}} \left\{ R_{r_n(t)}\left( \widetilde{\theta }_n\right) - C\left( s_n(t), r_n(t)| {\theta }_n\right) \right\} - \alpha L_n\left( g_n\right) \\&\quad =R_{s_n(t)}\left( \widetilde{\theta }_n\right) - C\left( s_n(t), s_n(t)| {\theta }_n\right) - \alpha L_n\left( g_n\right) . \end{aligned} \end{aligned}$$

(58)

By substituting (58) into (17), we have (26). That ends the proof.

Appendix 3: Proof of Eq. (27)

Since the host aims to minimize the rewards, according to (23b), the optimal $R_O^*(\theta )$ can be obtained at its minimum value,

$$\begin{aligned} R_O^*(\theta ) = C(O,O|\theta ) + \alpha (1-p_0) + \epsilon , \end{aligned}$$

(59)

which is exactly (27a). By substituting (27a) into (23a) and (25), respectively, we have

$$\begin{aligned}&R_F(\theta ) > C(O,O|\theta ) + C(F,F|\theta ) - C(F,O|\theta ) + \alpha + \epsilon , \end{aligned}$$

(60a)

$$\begin{aligned}&R_F(\theta ) > C(O,O|\theta ) + \max \limits _{\theta }\left[ C(F,F| \theta ) - C(F,O| \theta )\right] + \alpha . \end{aligned}$$

(60b)

Therefore, the optimal $R_F^*(\theta )$ is obtained at its minimum value, i.e.,

$$\begin{aligned} R_F^*(\theta ) = \max \limits _{\theta }\left[ C(F,F| \theta ) - C(F,O| \theta )\right] + C(O,O|\theta ) + \alpha + 2 \epsilon , \end{aligned}$$

(61)

which is exactly (27b). That ends the proof.

Appendix 4 Proof of Eq. (34)

We first explore the condition in which (29a) and (29b) hold at the same time. For convenience, we define

$$\begin{aligned} x(\theta ) \triangleq \frac{\kappa _1\alpha {\bar{l}}(\theta )}{1-\theta } = \frac{\kappa _1 \alpha (b_1\theta ^2 + b_2\theta + b_3)}{1-\theta }. \end{aligned}$$

(62)

Equation (29a) is then rewritten as

$$\begin{aligned} \frac{d W(\theta )}{d \theta } = \frac{1}{x(\theta )} \cdot \frac{d C\left( O,O; \theta \right) }{d \theta }, \end{aligned}$$

(63)

based on which (29a) can be rewritten by

$$\begin{aligned} \frac{d^2 C\left( O,O| \theta \right) }{d {\theta }^2} - \frac{x(\theta )}{\left[ x(\theta ) \right] ^2 }\cdot \left[ \frac{d^2 C\left( O,O| \theta \right) }{d {\theta }^2} \cdot x(\theta ) - \frac{d C\left( O,O| \theta \right) }{d {\theta }} \cdot \frac{d x(\theta )}{d\theta } \right] \le 0. \end{aligned}$$

(64)

Therefore, both (29a) and (29b) hold if the following inequality is satisfied:

$$\begin{aligned} \frac{d C\left( O,O| \theta \right) }{d {\theta }} \cdot \frac{d x(\theta )}{d\theta } \le 0. \end{aligned}$$

(65)

Given (5) and (62), (65) naturally stands since we have $\theta \in \left[ 0,1 \right]$ and $b_1+b_2+b_3 = 0$. In other words, the solution of (29a) also satisfies constraint (29b).

By substituting (5) and (31) into (29a), we have

$$\begin{aligned} \frac{d W(\theta )}{d \theta } = \frac{a_1 a_2 (1-\theta )}{{\kappa _1}{\alpha }(2-\theta )^{a_2+1}(b_1\theta ^2 + b_2\theta + b_3)}. \end{aligned}$$

(66)

Therefore, from (29), $W(\theta )$ can be obtained by

$$\begin{aligned} W(\theta ) = \int \frac{a_1 a_2}{\kappa _1 \alpha }\cdot \frac{(1-\theta )}{(2-\theta )^{a_2+1}(b_1\theta ^2+b_2\theta +b_3)}d \theta + w_{cons}, \end{aligned}$$

(67)

where $w_{cons}$ is a constant. To guarantee that the number of time slots to propose to each attendee is no smaller than 1, we have $\min {w_{cons}} = 1- \omega (0)$. That ends the proof.

Appendix 5 Proof of Remark 2

When a type-$\theta _n$ attendee n does not trust the host, it knows that the host does not explore more time slots even if it reports to be OK, and thus, it has to attend the meeting as if it is free. Its cost of telling the truth at an OK time slot turns to be $C(O,F| {\theta }_n)$. The utilities it obtains when reporting to be OK and busy, respectively, are

$$\begin{aligned} U_n^{O}&= R_O({\theta }_n) - C(O,F| {\theta }_n) - \alpha \left| {\theta }_n - p_0 \right| - H_n \end{aligned}$$

(68a)

$$\begin{aligned}&= C(O,O|\theta _n) - C(O,F|\theta _n) + \alpha \left( 1 - {\theta }_n \right) + \epsilon - H_n,\nonumber \\ U_n^B&= 0 - 0 - \alpha \left| {\theta }_n - p_0 \right| + \alpha \left| 1 - p_0 \right| - H_n. \end{aligned}$$

(68b)

Therefore, $U_n^{B} > U_n^{O}$ since $C(O,F| {\theta }_n) \gg C(O,O| {\theta }_n)$. Attendee n reports to be busy when an OK time slot is proposed in order to maximise its utility.

Appendix 6 Proof of Proposition 4

The first constraint is necessary because the IC guarantee only holds when $k < N$. If the host is required to guarantee all attendees’ social welfare, i.e., $k = N$, then each attendee has incentive to deviate from the T strategy. The free attendee may report to be OK so as to force the host to raise the reward to $R_C$. In contrast, if $k < N$, a free attendee would not report to be OK since it is possible that the host does not raise its reward, and thus, it may get a lower reward compared to the truth-telling case.

Now let us move to the second constraint. When the host deviates from strategy HO, the largest utility gain $G_{max}$ is

$$\begin{aligned} \begin{aligned}&\left[ \Gamma - \sum _{n\ne 0}R_O(\theta _n) - C(F,F| \theta _0)\right] - \left[ \Gamma - \sum _{i \in \mathcal {K}}R_C(\theta _i) - \sum _{n \notin \mathcal{{K}}\cup \{0\}}R_O(\theta _n)- C(O,F| \theta _0) \right] \\&\quad =\sum _{i \in \mathcal {K}} \left[ R_C(\theta _i)-R_O(\theta _i) \right] + C(O,F| \theta _0) - C(F,F| \theta _0). \end{aligned} \end{aligned}$$

(69)

We omit the privacy leakage here since it is much smaller than $R_C$ and C(O, F). After deviating from strategy HO, each attendee plays strategy NT for all future meetings. The minimum utility loss of the host in each meeting is

$$\begin{aligned} D_{min} = \sum _{n \in \mathcal{{K}}\cup \{0\}} \left[ R_C(\theta _n) - R_O(\theta _n) \right] - \left[ C(O,F| \theta _0) - C(F,F| \theta _0) \right] . \end{aligned}$$

(70)

To prevent the host deviate from strategy HO, it should hold that

$$\begin{aligned} \begin{aligned} G_{max} \le D_{min}\left( \delta + {\delta }^2 + \cdots \right) \Leftrightarrow \delta \ge \frac{G_{max}}{G_{max} + D_{min}}. \end{aligned} \end{aligned}$$

(71)

Substituting (69) and (70) into (71), $\delta$ can be approximated by k/N, and (38) can be obtained.