A Bayesian optimal social law synthesizing mechanism for strategical agents

Abstract

One of the effective and well studied approaches for coordinating multiagent systems is to synthesize social laws which restrict the behavior of individual agents. We show that when rational behavior of the agents and private information are considered, the optimal social law synthesizing problem naturally evolves into a setting which can be handled by the framework of algorithmic mechanism design. We focus on the Bayesian case in this paper, that is, the probability distribution of each agent’s private cost is known by the public. In this case, our problem closely relates to path/spanning-tree auctions and Myerson’s optimal auction mechanism, but the optimization objective is new, that is, we focus on profit maximization instead of payment maximization in a reverse auction, and as far as we know in this setting no existing mechanism can be directly applied. By studying this problem: We further extend the logic-based framework of social law optimization problem to the strategic case, and show that it becomes a new problem of algorithmic mechanism design; We find out a mechanism that is incentive compatible, individually rational and maximize the expected profit for all input cost profiles; However, we can show that this mechanism is computational intractable (FP$^{NP}$-complete); So, we try to specify Computation Tree Logic semantics as a set of linear-integer constraints, design a Integer-Linear Programming based algorithm for computing the proposed mechanism, enabling it to be handled by current ILP solvers, and finally find out a tractable 2-approximation mechanism.

Appendices

Appendix

Appendix A: Proof of Lemma 1

Proof

First of all, the following equivalence relations follow trivially from the definition of bnic and Eqs. (16) and (17):

The mechanism $\langle H,P\rangle$ is bnic

• iff $\forall i\in A, \xi _i, \gamma _i\in \Xi _i:\bar{u}_i(\xi _i,\xi _i)\ge \bar{u}_i(\xi _i,\gamma _i)$

• iff $\forall i\in A, \xi _i, \gamma _i\in \Xi _i: \hat{u}_i(\xi _i)\ge p_i(\gamma _i) - h_i(\gamma _i)\xi _i$

• iff $\forall i\in A, \xi _i, \gamma _i\in \Xi _i:\hat{u}_i(\xi _i)-\hat{u}_i(\gamma _i)\ge h_i(\gamma _i)(\gamma _i-\xi _i)$.

Now, we are ready to prove Lemma 1:

“$\Rightarrow$”: According to the above equivalence relation, for all i, $\xi _i$ and $\gamma _i$, we have both $\hat{u}_i(\xi _i)-\hat{u}_i(\gamma _i)\ge h_i(\gamma _i)(\gamma _i-\xi _i)$ and $\hat{u}_i(\gamma _i)-\hat{u}_i(\xi _i)\ge h_i(\xi _i)(\xi _i-\gamma _i)$. So we can further obtain

$$\begin{aligned} h_i(\xi _i)(\xi _i-\gamma _i)\le \hat{u}_i(\gamma _i)-\hat{u}_i(\xi _i)\le h_i(\gamma _i)(\xi _i-\gamma _i) \end{aligned}$$

(A1)

It follows that $h_i(\xi _i)\le h_i(\gamma _i)$ iff $\xi _i\ge \gamma _i$. Therefore, $h_i(\xi _i)$ is a monotone nonincreasing function.

To show the correctness of Eq. (21), we firstly divide the interval $[0,\xi _i]$ into L intervals of length $\delta = \frac{\xi _i}{L}$. Denote by $y^k = (k+1)\delta$ the rightmost end of the kth interval, and by $x^k = k\delta$ its leftmost end. Let $\xi _i = y^k$ and $\gamma _i = x^k$, then

$$\begin{aligned} \sum _{k=0}^{L-1} h_i(y^k)(y^k-x^k)\le \sum _{k=0}^{L-1} \hat{u}_i(x^k)-\hat{u}_i(y^k)\le \sum _{k=0}^{L-1} h_i(x^k)(y^k-x^k) \end{aligned}$$

(A2)

Noticing that, $y^k = x^{k+1}$ for all $0\le k\le L-1$, therefore

$$\begin{aligned} \sum _{k=0}^{L-1} \hat{u}_i(x^k)-\hat{u}_i(y^k) = \hat{u}_i(0) - \hat{u}_i(\xi _i) \end{aligned}$$

(A3)

Both the left part and right part of inequality (A2) are Rieman sums. By increasing L, $\delta$ gradually approaches 0, both of the left part and right part of inequality (A2) converge to $\int _{0}^{\xi _i} h_i(t_i) dt_i$. Therefore,

$$\begin{aligned} \hat{u}_i(0) - \hat{u}_i(\xi _i) = \int _{0}^{\xi _i} h_i(t_i)dt_i \end{aligned}$$

(A4)

Moreover, Eq. (17) follows

$$\begin{aligned} \hat{u}_i(0) = p_i(0) \end{aligned}$$

(A5)

Finally, the equation in item (2) of this lemma follows by combining the Eqs. (17), (A4) and (A5).

“$\Leftarrow$”: Equation (22) follows, for all $\gamma _i\in \Xi _i$

$$\begin{aligned} \hat{u}_i(\xi _i) - \hat{u}_i(\gamma _i) = \int _{0}^{\gamma _i}h_i(t_i)dt_i-\int _{0}^{\xi _i}h_i(t_i)dt_i= \int _{\xi _i}^{\gamma _i}h_i(t_i)dt_i \end{aligned}$$

(A6)

Since $h_i$ is monotone nonincreasing,

$$\begin{aligned} \int _{\xi _i}^{\gamma _i}h_i(t_i)dt_i\ge (\gamma _i-\xi _i)h_i(\gamma _i) \end{aligned}$$

(A7)

then we have $\hat{u}_i(\xi _i) - \hat{u}_i(\gamma _i) \ge (\gamma _i-\xi _i)h_i(\gamma _i)$.

Therefore, by the equivalence relation established in the beginning of this proof, $\langle H,P\rangle$ is bnic. $\square$

Appendix B: Proof of Lemma 13

Proof

• (1) $x^s_{p} = 1$ iff $K\dag \eta ,s \models p$ iff $p\in \pi (x)$;

• (2) $x^s_{\lnot \psi } = 1$ iff $K\dag \eta ,s \models \lnot \psi$ iff $K\dag \eta ,s \models \psi$ doesn’t hold iff $x^s_{\psi } = 0$;

• (3) $x^s_{\psi \vee \chi } = 1$ iff $K\dag \eta ,s \models \psi \vee \chi$ iff $K\dag \eta ,s \models \psi$ or $K\dag \eta ,s \models \chi$ iff $x^s_{\psi } = 1$ or $x^s_{\chi } = 1$;

• (4) $x^s_{\mathsf {E}\bigcirc \psi } = 1$ iff $K\dag \eta , s \models \mathsf {E}\bigcirc \psi$ iff $\exists s'\in \mathcal {N}(s): (s,s')\notin \eta$ and $K\dag \eta , s' \models \psi$ iff $\exists s'\in \mathcal {N}(s):$ $y^s_{s'} = 0$ and $x^{s'}_{\psi } = 1$;

• (5) $x^s_{\mathsf {E}(\psi \mathcal {U}\chi )} = 1$ iff $K\dag \eta , s \models \mathsf {E}(\psi \mathcal {U}\chi )$ iff $K\dag \eta , s \models \mathsf {E}(\psi \mathcal {U}\chi )$ iff $K\dag \eta , s \models \chi \vee (\psi \wedge \mathsf {E}\bigcirc \mathsf {E}(\psi \mathcal {U}\chi )))$ (by Proposition 12.1) iff $K\dag \eta , s \models \chi$ or $K\dag \eta , s \models \psi \wedge \mathsf {E}\bigcirc \mathsf {E}(\psi \mathcal {U}\chi )$ iff $K\dag \eta , s \models \chi$ or ($K\dag \eta , s \models \psi$ and $K\dag \eta , s \models \mathsf {E}\bigcirc \mathsf {E}(\psi \mathcal {U}\chi )$) iff $x^s_{\chi } = 1$ or ($x^s_{\psi } = 1$ and $x^{s}_{\mathsf {E}\bigcirc \mathsf {E}(\psi \mathcal {U}\chi )}=1$);

• (6) $x^s_{\mathsf {E}\Box \varphi } = 1$ iff $K\dag \eta , s \models \mathsf {E}\Box \varphi$ iff $K\dag \eta , s \models \varphi \wedge \mathsf {E}\bigcirc \mathsf {E}\Box \varphi$ (by Proposition 12.2) iff $K\dag \eta , s \models \varphi$ and $K\dag \eta , s \models \mathsf {E}\bigcirc \mathsf {E}\Box \varphi$ iff $x^s_{\varphi } = 1$ and $x^{s}_{\mathsf {E}\bigcirc \mathsf {E}\Box \varphi } = 1$;

• (7) $x^s_{\mathsf {A}\bigcirc \psi } = 1$ iff $K\dag \eta , s \models \mathsf {A}\bigcirc \psi$ iff $\forall s'\in \mathcal {N}(s): (s,s')\in \eta$ or $K\dag \eta , s' \models \psi$ iff $\forall s'\in \mathcal {N}(s):$ $y^s_{s'} = 1$ or $x^{s'}_{\psi } = 1$;

• (8) $x^s_{\mathsf {A}(\psi \mathcal {U}\chi )} = 1$ iff $K\dag \eta , s \models \mathsf {A}(\psi \mathcal {U}\chi )$ iff $K\dag \eta , s \models \mathsf {A}(\psi \mathcal {U}\chi )$ iff $K\dag \eta , s \models \chi \vee (\psi \wedge \mathsf {A}\bigcirc \mathsf {A}(\psi \mathcal {U}\chi )))$ (by Proposition 12.3) iff $K\dag \eta , s \models \chi$ or $K\dag \eta , s \models \psi \wedge \mathsf {E}\bigcirc \mathsf {E}(\psi \mathcal {U}\chi )$ iff $K\dag \eta , s \models \chi$ or ($K\dag \eta , s \models \psi$ and $K\dag \eta , s \models \mathsf {A}\bigcirc \mathsf {A}(\psi \mathcal {U}\chi )$) iff $x^s_{\chi } = 1$ or ($x^s_{\psi } = 1$ and $x^{s}_{\mathsf {A}\bigcirc \mathsf {A}(\psi \mathcal {U}\chi )}=1$);

• (9) $x^s_{\mathsf {A}\Box \varphi } = 1$ iff $K\dag \eta , s \models \mathsf {A}\Box \varphi$ iff $K\dag \eta , s \models \varphi \wedge \mathsf {A}\bigcirc \mathsf {A}\Box \varphi$ (by Proposition 12.4) iff $K\dag \eta , s \models \varphi$ and $K\dag \eta , s \models \mathsf {E}\bigcirc \mathsf {A}\Box \varphi$ iff $x^s_{\varphi } = 1$ and $x^{s}_{\mathsf {A}\bigcirc \mathsf {A}\Box \varphi } = 1$;

• (10) $x^s_{\mathsf {E}\Diamond \varphi } = 1$ iff $K\dag \eta , s \models \mathsf {E}\Diamond \varphi$ iff $K\dag \eta , s \models \varphi \vee \mathsf {E}\bigcirc \mathsf {E}\Diamond \varphi$ (by Proposition 12.5) iff $K\dag \eta , s \models \varphi$ or $K\dag \eta , s \models \mathsf {E}\bigcirc \mathsf {E}\Diamond \varphi$ iff $x^s_{\varphi } = 1$ or $x^{s}_{\mathsf {E}\bigcirc \mathsf {E}\Diamond \varphi } = 1$;

• (11) $x^s_{\mathsf {A}\Diamond \varphi } = 1$ iff $K\dag \eta , s \models \mathsf {A}\Diamond \varphi$ iff $K\dag \eta , s \models \varphi \vee \mathsf {A}\bigcirc \mathsf {A}\Diamond \varphi$ (by Proposition 12.6) iff $K\dag \eta , s \models \varphi$ or $K\dag \eta , s \models \mathsf {A}\bigcirc \mathsf {A}\Diamond \varphi$ iff $x^s_{\varphi } = 1$ or $x^{s}_{\mathsf {A}\bigcirc \mathsf {A}\Diamond \varphi } = 1$. $\square$