Appendix
For notational compactness, we suppress the notations \(h_{t-1}^{\theta }\), \(h_{t-1}^{{\hat{\theta }}}\), and \(h^{a}_{t-1}\) from \(\alpha , \phi , \xi \), and \(\sigma \).
Proof of Proposition 1
The proof of the only if part directly follows from the optimality of truthful reporting and here we only provide the proof of the if part. Suppose, on the contrary, the truthful reporting strategy \(\sigma ^{*}\) satisfies (18) and (19) but not (16) and (17). Then there exists a reporting strategy \(\sigma '\) and a state \(\theta _{t}\), at period \(t\in {\mathbb {T}}\), such that \(V^{\alpha ,\phi ,\xi ,\rho }_{t}(\theta _{t};\sigma ')> V^{\alpha ,\phi ,\xi ,\rho }_{t}(\theta _{t};\sigma ^{*})\). Suppose that the optimal stopping rule with \(\sigma ^{*}\) calls for stopping and the agent decides to continue by using \(\sigma '\), i.e.,
$$\begin{aligned} J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(t,\theta _{t})< {\mathbb {E}}^{\varXi _{\alpha ;\sigma '}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ]. \end{aligned}$$
Hence, there exists some \(\varepsilon >0\) such that
$$\begin{aligned} {\mathbb {E}}^{\varXi _{\alpha ;\sigma '}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] \ge J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(t,\theta _{t}) + 2\varepsilon . \end{aligned}$$
(85)
Let \(\sigma ''\) be the reporting strategy such that if \(\sigma ''\) and \(\sigma '\) have the same reporting strategies from period t to \(t+k\), for some \(k\ge 0\), then
$$\begin{aligned} {\mathbb {E}}^{\varXi _{\alpha ;\sigma ''}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] \ge {\mathbb {E}}^{\varXi _{\alpha ;\sigma '}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] - \varepsilon . \end{aligned}$$
(86)
From (85) and (86), we have
$$\begin{aligned} {\mathbb {E}}^{\varXi _{\alpha ;\sigma ''}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] \ge J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(t,\theta _{t}) + \varepsilon . \end{aligned}$$
(87)
Here, (87) implies that any deviation(s) for the periods from t to \(t+k\) (reporting truthfully for all other periods) can improve the value \(V^{\alpha ,\phi ,\xi ,\rho }_{t}\).
Let \({\hat{\sigma }}^{s}\) denote the reporting strategy that differs only at period s from \(\sigma ^{*}\) and \({\hat{\sigma }}^{s}_{s} = \sigma ''_{s}\), for \(s\in [t,t+k]\). Then, we have
$$\begin{aligned} {\mathbb {E}}^{\varXi _{\alpha ;{\hat{\sigma }}^{t+k-1}}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] > J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(t,\theta _{t}). \end{aligned}$$
(88)
Now, we look at period \(t+k-1\). Because \(\sigma ^{*}\) satisfies (18) and (19), we have, for all \(\theta _{t+k-1}\in \varTheta _{t+k-1}\),
$$\begin{aligned} \begin{aligned}&{\mathbb {E}}^{\varXi _{\alpha ;{\hat{\sigma }}^{t+k-2}}[h^{\theta }_{t+k-1}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+k-1}(\theta _{t+k-1}) \big ] = V^{\alpha ,\phi ,\xi ,\rho }_{t+k-1}(\theta _{t+k-1})\\&\quad \ge \max \Big (J^{\alpha ,\phi ,\xi ,\rho }_{1,t+k-1}(t+k-1,\theta _{t+k-1}, {\hat{\sigma }}^{t+k-1}_{t+k-1}(\theta _{t+k-1})|h^{\theta }_{t+k-2}) , {\mathbb {E}}^{\varXi _{\alpha ;{\hat{\sigma }}^{t+k-1}}[h^{\theta }_{t+k-1}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+k}({\tilde{\theta }}_{t+k}) \big ] \Big )\\&\quad = V^{\alpha ,\phi ,\xi ,\rho }_{t+k-1}(\theta _{t+k-1};{\hat{\sigma }}^{t+k-1}_{t+k-1}). \end{aligned}\nonumber \\ \end{aligned}$$
(89)
Then,
$$\begin{aligned} \begin{aligned} {\mathbb {E}}^{\varXi _{\alpha ;{\hat{\sigma }}^{t+k-2}}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] \ge {\mathbb {E}}^{\varXi _{\alpha ;{\hat{\sigma }}^{t+k-1}}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ]. \end{aligned} \end{aligned}$$
(90)
From (88) and (90), we have
$$\begin{aligned} {\mathbb {E}}^{\varXi _{\alpha ;{\hat{\sigma }}^{t+k-2}}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] > J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(t,\theta _{t}). \end{aligned}$$
Backward induction yields
$$\begin{aligned} {\mathbb {E}}^{\varXi _{\alpha ;{\hat{\sigma }}^{t}}[h^{\theta }_{t}]}\big [V^{\alpha ,\phi ,\xi ,\rho }_{t+1}({\tilde{\theta }}_{t+1}) \big ] > J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(t,\theta _{t}), \end{aligned}$$
which contradicts the fact that \(\sigma ^{*}\) satisfies (18) and (19).
Following the similar analysis, we can prove the cases when the optimal stopping rule with truthful \(\sigma ^{*}\) (1) calls for stopping and the agent decides to stop, (2) calls for continuing and the agent decides to continue, and (3) calls for continuing and the agent decides to stop. \(\square \)
Proof of Lemma 1
We prove (23) here. The proof of (24) can be done analogously. For any \(\tau \in {\mathbb {T}}\), the agent’s ex-ante expected payoff (2) can be written as
$$\begin{aligned} \begin{aligned} J^{\alpha ,\phi ,\xi ,\rho }_{1}(\tau )&= {\mathbb {E}}^{\varXi _{\alpha }}\Bigg [ \sum ^{\tau -1}_{t=1}\Big [ \delta ^{t+1}\big [ u_{1,t+1}({\tilde{\theta }}_{t+1}, \alpha _{t+1}( {\tilde{\theta }}_{t+1} )) +\xi _{t+1}({\tilde{\theta }}_{t+1})\big ]+\rho (t+1) \\&\quad + \delta ^{t}\big [ \phi _{t}({\tilde{\theta }}_{t}) -\xi _{t}({\tilde{\theta }}_{t}) \big ] - \rho (t)\Big ] + \delta \big [u_{1,1}({\tilde{\theta }}_{1},\alpha _{1}({\tilde{\theta }}_{1}))+\xi _{1}({\tilde{\theta }}_{1})] +\rho (1) \Bigg ]. \end{aligned} \end{aligned}$$
From law of total expectation, we have
$$\begin{aligned} \begin{aligned} J^{\alpha ,\phi ,\xi ,\rho }_{1}(\tau )&= {\mathbb {E}}^{\varXi _{\alpha }}\Bigg [ \sum ^{\tau -1}_{t=1}\Big [ {\mathbb {E}}^{\alpha ;\theta _{t}}\Big [ \delta ^{t+1}\big [ u_{1,t+1}({\tilde{\theta }}_{t+1}, \alpha _{t+1}( {\tilde{\theta }}_{t+1} )) +\xi _{t+1}({\tilde{\theta }}_{t+1})\big ]+\rho (t+1) \\&\qquad + \delta ^{t}\big [ \phi _{t}({\tilde{\theta }}_{t}) -\xi _{t}({\tilde{\theta }}_{t}) \big ]\Big ] - \rho (t) \Big ] + \delta \big [u_{1,1}({\tilde{\theta }}_{1},\alpha _{1}({\tilde{\theta }}_{1}))+\xi _{1}({\tilde{\theta }}_{1})] +\rho (1) \Bigg ]\\&\quad = {\mathbb {E}}^{\varXi _{\alpha }}\Big [ \sum _{s=1}^{\tau -1} L^{\alpha , \phi , \xi , \rho }_{s}({\tilde{\theta }}_{s}) -\rho (s)\Big ] + J^{\alpha , \phi , \xi , \rho }_{1}(1). \end{aligned} \end{aligned}$$
\(\square \)
Proof of Lemma 5
Let \({\hat{\sigma }}[t]\) be the one-shot deviation strategy that reports \({\hat{\theta }}_{t}\) for the true state \(\theta _{t}\) at t. Let \(\varOmega ^{*}[{\hat{\sigma }}[t]]\) be the optimal stopping time rule defined in (14) with the stopping region \(\varLambda ^{\alpha , \phi , \xi , \rho }_{1,t}(t;{\hat{\sigma }}[t])\) given in (13) (equivalently, (32)). Suppose that at period t the agent observes a state \(\theta _{t}\in \varLambda ^{\alpha , \phi , \xi , \rho }_{1,t}(t;{\hat{\sigma }}[t])\). Hence, the agent stops at t optimally. Then, we obtain, for every \(\theta '_{t}\le \theta _{t}\),
$$\begin{aligned} \rho (t) \ge {\bar{\mu }}^{\alpha , \phi , \xi , \rho }_{t}(\theta _{t}, {\hat{\theta }}_{t}) \ge {\bar{\mu }}^{\alpha , \phi , \xi , \rho }_{t}(\theta '_{t},{\hat{\theta }}_{t}), \end{aligned}$$
where the inequality is due to Lemma 4. Therefore, \(\theta '_{t}\in \varLambda ^{\alpha , \phi , \xi , \rho }_{1,t}(t;{\hat{\sigma }}[t])\) for every \(\theta '_{t}\le \theta _{t}\), which implies that \(\varLambda ^{\alpha , \phi , \xi , \rho }_{1,t}(t;{\hat{\sigma }}[t])\) is an interval left-bounded by \({\underline{\theta }}_{t}\). Since \(L^{\alpha , \phi , \xi , \rho }_{t}\) is continuous, \(\varLambda ^{\alpha , \phi , \xi , \rho }_{1,t}(t;{\hat{\sigma }}[t])\) is closed. Hence, according to Assumption 3, there exists some \(\eta (t)\in \varTheta _{t}\) such that \(\varLambda ^{\alpha , \phi , \xi , \rho }_{1,t}(t;{\hat{\sigma }}[t]) = [{\underline{\theta }}_{t}, \eta (t)]\). \(\square \)
Proof of Lemma 6
Let \(\varOmega [{\hat{\sigma }}[t]]|\eta \) and \(\varOmega [{\hat{\sigma }}[t]]|\eta '\) denote the optimal stopping rule with threshold functions \(\eta \) and \(\eta '\), respectively. Let \(\tau _{\eta }\) and \(\tau _{\eta '}\) denote the expected realized stopping time from \(\varOmega [{\hat{\sigma }}[t]]|\eta \) and \(\varOmega [{\hat{\sigma }}[t]]|\eta '\), respectively. Without loss of generality, suppose \(\eta (t)<\eta '(t)\) for some \(t\in {\mathbb {T}}\). Here, we obtain the probability of \(\tau _{\eta }=t\) as:
$$\begin{aligned} \begin{aligned} P_{r}(\tau _{\eta }=t) = P_{r}(\theta _{t}\le \eta (t), \tau _{\eta }>t-1) =&{\mathbb {E}}^{\varXi _{\alpha }}\Bigg [{\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [{\mathbf {1}}_{\{{\tilde{\theta }}_{t}\le \eta (t)\}} \Big ] {\mathbf {1}}_{\{\tau _{\eta }>t-1\}} \Bigg ]. \end{aligned} \end{aligned}$$
We can obtain \(P_{r}(\tau _{\eta '}=t)\) in a similar way. Then,
$$\begin{aligned} \begin{aligned} P_{r}(\tau _{\eta '}=t)-P_{r}(\tau _{\eta }=t)&= {\mathbb {E}}^{\varXi _{\alpha }}\Bigg [{\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [{\mathbf {1}}_{\{{\tilde{\theta }}_{t}\le \eta '(t)\}} \Big ] {\mathbf {1}}_{\{\tau _{\eta '}>t-1\}} \Bigg ]\\&\quad -{\mathbb {E}}^{\varXi _{\alpha }}\Bigg [{\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [{\mathbf {1}}_{\{{\tilde{\theta }}_{t}\le \eta (t)\}} \Big ] {\mathbf {1}}_{\{\tau _{\eta }>t-1\}} \Bigg ]\\&= {\mathbb {E}}^{\varXi _{\alpha }}\Bigg [ {\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [{\mathbf {1}}_{\{\eta (t)\le {\tilde{\theta }}_{t} \le \eta '(t) \} } \Big ] {\mathbf {1}}_{\tau _{\eta }>t-1} \Bigg ]. \end{aligned} \end{aligned}$$
(91)
Since \(\tau _{\eta }=\tau _{\eta '}\), the probabilities \(P_{r}(\tau _{\eta '}=t)\) and \(P_{r}(\tau _{\eta }=t)\) are equal, i.e., (91) equals 0. However, from Assumption 3, we know \({\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [{\mathbf {1}}_{\{\eta (t)\le {\tilde{\theta }}_{t} \le \eta '(t) \} } \Big ]>0\) and \(P_{r}(\tau _{\eta }>t-1)>0\), which implies that
$$\begin{aligned} {\mathbb {E}}^{\varXi _{\alpha }}\Bigg [ {\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [{\mathbf {1}}_{\{\eta (t)\le {\tilde{\theta }}_{t} \le \eta '(t) \} } \Big ] {\mathbf {1}}_{\tau _{\eta }>t-1} \Bigg ]>0. \end{aligned}$$
This contradiction implies that \(\eta \) is unique. \(\square \)
Proof of Proposition 2
From the construction of \(\xi \) in (39), we have
$$\begin{aligned} \begin{aligned} \xi _{t}({\hat{\theta }}_{t}) - \xi _{t}(\theta _{t})&= \delta ^{-t}\beta ^{\alpha }_{S,t}({\hat{\theta }}_{t}) -\delta ^{-t}\beta ^{\alpha }_{S,t}(\theta _{t}) + u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) - u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}))\\&= \delta ^{-t}\beta ^{\alpha }_{S,t}({\hat{\theta }}_{t}) -\delta ^{-t}\beta ^{\alpha }_{S,t}(\theta _{t}) -(u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}))- u_{1,t}(\theta _{t}, \alpha _{t}({\hat{\theta }}_{t})))\\&\quad + u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) - u_{1,t}(\theta _{t}, \alpha _{t}({\hat{\theta }}_{t}))). \end{aligned} \end{aligned}$$
(92)
From the definition of \(\ell ^{\alpha }_{S,t}\) in (36) and the condition (41),
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (92) }&=\delta ^{-t}\beta ^{\alpha }_{S,t}({\hat{\theta }}_{t}) -\delta ^{-t}\beta ^{\alpha }_{S,t}(\theta _{t}) + u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) -\ell ^{\alpha }_{S,t}({\hat{\theta }}_{t}, \theta _{t}) \\&\quad + u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) - u_{1,t}(\theta _{t}, \alpha _{t}({\hat{\theta }}_{t})))\\&\le u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) - u_{1,t}(\theta _{t}, \alpha _{t}({\hat{\theta }}_{t}))), \end{aligned} \end{aligned}$$
which implies
$$\begin{aligned} u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) +\xi _{t}(\theta _{t}) \ge u_{1,t}(\theta _{t}, \alpha _{t}({\hat{\theta }}_{t}))) + \xi _{t}({\hat{\theta }}_{t}), \end{aligned}$$
(93)
i.e.,
$$\begin{aligned} U^{\alpha ,\phi ,\xi ,\rho }_{S, t}(\theta _{t}|h^{\theta }_{t-1})\ge U^{\alpha ,\phi ,\xi ,\rho }_{S, t}(\theta _{t},{\hat{\theta }}_{t}|h^{\theta }_{t-1}). \end{aligned}$$
From the construction of \(\phi \) in (38), we have, for any \(\tau \in {\mathbb {T}}_{t+1}\),
$$\begin{aligned} \begin{aligned} \phi _{t}({\hat{\theta }}_{t}) - \phi _{t}(\theta _{t})&= \beta ^{\alpha }_{{\bar{S}},t}({\hat{\theta }}_{t}) - {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [\beta _{{\bar{S}},t+1}({\tilde{\theta }}_{t+1})\Big ] - u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}))\\&\quad -\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t}) + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\beta _{{\bar{S}},t+1}({\tilde{\theta }}_{t+1})\Big ] + u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t}))\\&= \beta ^{\alpha }_{{\bar{S}},t}({\hat{\theta }}_{t}) - \beta ^{\alpha }_{{\bar{S}},t}(\theta _{t}) + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t}^{T}\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{T-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{T}({\tilde{\theta }}_{T})\Big ]\\&\quad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau -1}\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau }({\tilde{\theta }}_{\tau }) \Big ]. \end{aligned}\nonumber \\ \end{aligned}$$
(94)
From the condition (42), we have
$$\begin{aligned} \begin{aligned} \text {R.H.S. of 94}&\le \inf _{\tau \in {\mathbb {T}}_{t}}\Big \{ {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}} \Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ]\\&\quad -{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ] \Big \}-\sup _{\tau \in {\mathbb {T}}_{t}}\rho (\tau )\\&\quad + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t}^{T}\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{T-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{T}\xi _{T}({\tilde{\theta }}_{T})\Big ]\\&\quad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau -1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau }({\tilde{\theta }}_{\tau }) \Big ]. \end{aligned}\nonumber \\ \end{aligned}$$
(95)
From the condition (43), \(\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t})\ge \beta ^{\alpha }_{S,t}(\theta _{t})\), for all \(\theta _{t}\in \varTheta _{t}\), \(t\in {\mathbb {T}}\). Hence, \(\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t})\ge \xi _{t}(\theta _{t})\) \(+ u_{1,t}(\theta _{t},\alpha _{t}(\theta _{t}))\), for all \(\theta _{t}\in {\mathbb {T}}\), \(t\in {\mathbb {T}}\). Then,
$$\begin{aligned} \begin{aligned}&\inf _{\tau \in {\mathbb {T}}_{t}}\Big \{ {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}} \Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ]\\&\qquad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau -1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau }({\tilde{\theta }}_{\tau }) \Big ]\Big \}\\&\quad \le \inf _{\tau \in {\mathbb {T}}_{t}}\Big \{ {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}} \Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ]\\&\qquad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ]\Big \}\\&\quad = 0. \end{aligned} \end{aligned}$$
(96)
Hence, from (96), we have
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (95) }&\le {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{T}\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{T-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{T}\xi _{T}({\tilde{\theta }}_{T})\Big ]\\&\quad +\inf _{\tau \in {\mathbb {T}}_{t}}\Big \{-{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ] \Big \} \\&\quad -\sup _{\tau \in {\mathbb {T}}_{t}}\rho (\tau ). \end{aligned}\nonumber \\ \end{aligned}$$
(97)
From the construction of \(\rho \) in (40) and Lemma 2, we have, for some \(\tau '\in {\mathbb {T}}_{t}\)
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (97)}&\le {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau '}\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau '-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau '}\xi _{\tau '}({\tilde{\theta }}_{\tau '}) +\rho (\tau ')\Big ]\\&\quad +\inf _{\tau \in {\mathbb {T}}_{t}}\Big \{-{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ] \Big \}\\&\quad +\inf _{\tau \in {\mathbb {T}}_{t}}\Big \{-\rho (\tau )\Big \}, \end{aligned}\nonumber \\ \end{aligned}$$
(98)
which can be further bounded as
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (98)}&\le \sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ]\Big \}\\&\quad +\inf _{\tau \in {\mathbb {T}}_{t}}\Big \{-{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ] \Big \}\\&= \sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ]\Big \}\\&\quad -\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ] \Big \}. \end{aligned} \end{aligned}$$
Hence, from the definition of \(U^{\alpha ,\phi ,\xi ,\rho }_{{\bar{S}},t}\), we have
$$\begin{aligned} U^{\alpha ,\phi ,\xi ,\rho }_{{\bar{S}},t}(\theta _{t}|h^{\theta }_{t-1})\ge U^{\alpha ,\phi ,\xi ,\rho }_{{\bar{S}},t}(\theta _{t}, {\hat{\theta }}_{t}|h^{\theta }_{t-1}). \end{aligned}$$
Therefore, we can conclude that the mechanism is DIC. \(\square \)
Proof of Lemma 7
Let \({\tilde{m}}_{t}\) be uniformly distributed over (0, 1). Given the kernel \(K_{t}\), define the inverse of \(F_{t}(\cdot |\theta _{t-1}, a_{t-1})\) as follows:
$$\begin{aligned} \begin{aligned} F^{-1}_{t}(m_{t}|\theta _{t-1}, a_{t-1})= \inf \{\theta _{t}\in \varTheta _{t}: F_{t}(\theta _{t}|\theta _{t-1}, a_{t-1})\ge m_{t}\}. \end{aligned} \end{aligned}$$
Let \(\theta _{t}\in \varTheta _{t}\) and \(\theta _{t+1}\in \varTheta _{t+1}\) be any two realized states at two adjacent periods, for any \(t\in {\mathbb {T}}\backslash \{T\}\). Then, we have
$$\begin{aligned} \begin{aligned} \frac{\partial \theta _{t+1}}{\partial r}\Big |_{r= \theta _{t}} =&\frac{\partial F^{-1}_{t+1}(m_{t+1}|r, a_{t}) }{\partial r }\Big |_{r=\theta _{t}}= \frac{-\partial F_{t+1}(\theta _{t+1}| r, a_{t})}{f_{t+1}(\theta _{t+1}|\theta _{t}, a_{t}) \partial r}\Big |_{r=\theta _{t}}. \end{aligned} \end{aligned}$$
Then, for any sequence of realized states \(\{\theta _{t},\theta _{t+1},\dots , \theta _{t+k}\}\), for some \(k>1\), we have
$$\begin{aligned} \begin{aligned} \frac{\partial \theta _{t+k}}{\partial r}\Big |_{r= \theta _{t}} =&\prod _{s=t+1}^{t+k}\frac{\partial F^{-1}_{s}(m_{s}|r, a_{s-1}) }{\partial r }\Big |_{r=\theta _{s-1}}= \prod _{s=t+1}^{t+k}\Big [\frac{-\partial F_{s}(\theta _{s}| r, a_{s-1})}{f_{s}(\theta _{s}|\theta _{s-1}, a_{s-1}) \partial r}\Big |_{r=\theta _{s-1}}\Big ]. \end{aligned} \end{aligned}$$
In any DIC mechanism, truthful reporting strategy is optimal. Then, the envelope theorem yields the following:
$$\begin{aligned} \begin{aligned} \frac{\partial U^{\alpha ,\phi ,\xi ,\rho }_{t}(\tau , r|h^{\theta }_{t-1})}{\partial r}\Big |_{r= \theta _{t}} =&{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s}))}{\partial r }\Big |_{r={\tilde{\theta }}_{s}} \cdot \frac{\partial {\tilde{\theta }}_{s}}{ \partial l }\Big |_{l=\theta _{t}}\Big ]\\ =&{\mathbb {E}}^{\alpha |\theta _{t}}\Bigg [\sum _{s=t}^{\tau }\frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s}))}{\partial r }\Big |_{r={\tilde{\theta }}_{s}} \cdot \prod _{k=t+1}^{s}\Big [\frac{-\partial F_{k}(\theta _{k}| r, a_{k-1})}{f_{k}(\theta _{k}|\theta _{k-1}, a_{k-1}) \partial r}\Big |_{r=\theta _{k-1}}\Big ]\Bigg ]\\ =&{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s}))}{\partial r }\Big |_{r={\tilde{\theta }}_{s}} \cdot G_{t,s}(h^{\theta }_{t,s})\Big ]. \end{aligned} \end{aligned}$$
\(\square \)
Proof of Proposition 3
Since \(u_{1,t}(\theta _{t}, a_{t})\) is a non-decreasing function of \(\theta _{t}\), then \( \frac{\partial u_{1,t}(r, a_{t})}{\partial r}\Big |_{r = \theta _{t}} \ge 0 \) , for all \(t\in {\mathbb {T}}\). From Assumption 4, we have \(\frac{\partial F_{t+1}(\theta _{t+1}|r, a^{t})}{\partial r}\Big |_{r=\theta _{t}} \le 0\). Therefore, from Lemma 46, the term \(\gamma ^{\alpha }_{t}(\tau , \theta _{t}|h^{\theta }_{t-1})\) is nonnegative.
From the definition of \(\chi ^{\alpha , \phi , \xi }_{1,t}(\theta _{t})\) in (7), we have
$$\begin{aligned} \begin{aligned} \chi ^{\alpha , \phi , \xi }_{1,t}(\theta _{t})&= Z^{\alpha , \phi , \xi }_{1,t}(t+1, \theta _{t}|h^{\theta }_{t-1}) - Z^{\alpha , \phi , \xi }_{1,t}(t, \theta _{t}|h^{\theta }_{t-1})\\&= {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{t+1}\delta ^{s} u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s})) + \delta ^{t+1}\xi _{t+1}({\tilde{\theta }}_{t+1}) + \delta ^{t}\phi _{t}(\theta _{t})\Big ] - [\delta ^{t}u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t}))+ \delta ^{t}\xi _{t}(\theta _{t})]. \end{aligned} \end{aligned}$$
Substituting the constructions of \(\phi \) and \(\xi \) given by (38) and (39), respectively, yields
$$\begin{aligned} \begin{aligned} \chi ^{\alpha , \phi , \xi }_{1,t}(\theta _{t}) =&{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\beta _{{\bar{S}},t}(\theta _{t}) -\beta _{{\bar{S}},t+1}({\tilde{\theta }}_{t+1}) \Big ] + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\beta _{S,t+1}({\tilde{\theta }}_{t+1}) - \beta _{S,t}(\theta _{t}) \Big ]. \end{aligned} \end{aligned}$$
(99)
Given the formulations of \(\beta _{S,t}\) and \(\beta _{{\bar{S}},t}\) in (47) and (48), respectively, we have
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (99) }&= {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{\int ^{\theta _{t}}_{\theta _{\epsilon ,t}} \gamma ^{\alpha }_{t}(\tau , r|h^{\theta }_{t-1})dr\Big \} -\sup _{\tau \in {\mathbb {T}}_{t+1}}\Big \{\int ^{{\tilde{\theta }}_{t+1} }_{\theta _{\epsilon ,t+1}} \gamma ^{\alpha }_{t+1}(\tau , r|h^{\theta }_{t})dr\Big \} \Big ] \\&\quad + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\int ^{{\tilde{\theta }}_{t+1}}_{\theta _{\epsilon ,t+1}} \gamma ^{\alpha }_{t+1}(t+1, r|h^{\theta }_{t})dr - \int ^{\theta _{t}}_{\theta _{\epsilon ,t}} \gamma ^{\alpha }_{t}(t, r|h^{\theta }_{t-1})dr\Big ]. \end{aligned} \end{aligned}$$
(100)
Since \(\gamma ^{\alpha }_{t}\) is nonnegative for all \(t\in {\mathbb {T}}\), then
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (100) }&= {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\int ^{\theta _{t}}_{\theta _{\epsilon ,t}} \gamma ^{\alpha }_{t}(T, r|h^{\theta }_{t-1})dr -\int ^{{\tilde{\theta }}_{t+1} }_{\theta _{\epsilon ,t+1}} \gamma ^{\alpha }_{t+1}(T, r|h^{\theta }_{t})dr \Big ]\\&\quad +{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\int ^{{\tilde{\theta }}_{t+1}}_{\theta _{\epsilon ,t+1}} \gamma ^{\alpha }_{t+1}(t+1, r|h^{\theta }_{t})dr - \int ^{\theta _{t}}_{\theta _{\epsilon ,t}} \gamma ^{\alpha }_{t}(t, r|h^{\theta }_{t-1})dr\Big ]. \end{aligned} \end{aligned}$$
(101)
Taking partial derivative of \(\chi ^{\alpha , \phi ,\zeta }_{1,t}\) given in (101) with respect to \(\theta _{t}\) gives
$$\begin{aligned} \begin{aligned} \frac{\partial \chi ^{\alpha , \phi , \xi }_{1,t}(r) }{\partial r}\Big |_{r=\theta _{t}}&= {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \gamma ^{\alpha }_{t}(T,\theta _{t}|h^{\theta }_{t-1}) - \gamma ^{\alpha }_{t+1}(T, {\tilde{\theta }}_{t+1}|h^{\theta }_{t})G_{t,t+1}({\tilde{\theta }}_{t+1}) \Big ] \\&\quad + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\gamma ^{\alpha }_{t+1}(t+1, {\tilde{\theta }}_{t+1}|h^{\theta }_{t})G_{t,t+1}({\tilde{\theta }}_{t+1}) - \gamma ^{\alpha }_{t}(t, \theta _{t}|h^{\theta }_{t-1}) \Big ]. \end{aligned} \end{aligned}$$
From Lemma 7, we have
$$\begin{aligned} \begin{aligned}&{\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \gamma ^{\alpha }_{t}(T,\theta _{t}|h^{\theta }_{t-1})drx - \gamma ^{\alpha }_{t}(t, \theta _{t}|h^{\theta }_{t-1}) \Big ]\\&\quad = \max \Big \{ {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t+1}^{T} \delta ^{s}\frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s})) }{\partial r }\Big |_{r={\tilde{\theta }}_{s}} G_{t,s}(h^{{\tilde{\theta }}}_{t,s}) \Big ], \;\; 0 \Big \}\\&\quad = {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t+1}^{T} \delta ^{s}\frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s})) }{\partial r }\Big |_{r={\tilde{\theta }}_{s}} G_{t,s}(h^{{\tilde{\theta }}}_{t,s}) \Big ], \end{aligned} \end{aligned}$$
where the second equality is from the fact that \(\gamma ^{\alpha }_{t}\) is nonnegative; and
$$\begin{aligned} \begin{aligned} {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \gamma ^{\alpha }_{t+1}(T, {\tilde{\theta }}_{t+1}|h^{\theta }_{t}) G_{t,s}(h^{{\tilde{\theta }}}_{t,s}) \Big ]&={\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t+1}^{T} \delta ^{s}\frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s})) }{\partial r }\Big |_{r={\tilde{\theta }}_{s}} G_{t+1,s}(h^{{\tilde{\theta }}}_{t+1,s})G_{t,s}(h^{{\tilde{\theta }}}_{t,s}) \Big ]\\&= {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t+1}^{T} \delta ^{s}\frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s})) }{\partial r }\Big |_{r={\tilde{\theta }}_{s}} G_{t,s}(h^{{\tilde{\theta }}}_{t,s}) \Big ]. \end{aligned} \end{aligned}$$
Also, Assumption 4 implies that \(G_{t,t+1}(\theta _{t+1})\ge 0\) for all \(\theta _{t+1}\in \varTheta _{t+1}\). Hence, we have
$$\begin{aligned} \begin{aligned} \frac{\partial \chi ^{\alpha , \phi , \xi }_{1,t}(r) }{\partial r}\Big |_{r=\theta _{t}} =&{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\gamma ^{\alpha }_{t+1}(t+1, {\tilde{\theta }}_{t+1}|h^{\theta }_{t})G_{t,t+1}({\tilde{\theta }}_{t+1}) \Big ] \ge 0. \end{aligned} \end{aligned}$$
Therefore, the constructions of potential functions given in (47) and (48) satisfy the monotonicity condition specified by Assumption 2, i.e., the statement (iii) in Proposition 2 is satisfied. \(\square \)
Proof of Proposition 5
Fix an arbitrary \({\hat{\theta }}_{\epsilon ,t}\in \varTheta _{\epsilon }\). We discuss the following two cases:
1.1 1. \(\varvec{\theta _{t}\in \varLambda _{t}(t)}:\)
Let \({\hat{\theta }}_{t} \in \varLambda _{t}(t)\). Without loss of generality, suppose \({\hat{\theta }}_{t}\le \theta _{t}\). Let \(\theta \), \(\theta ^{1}\), \(\theta ^{2}\in \bar{\varTheta }_{t} \equiv [\hat{\theta }_{t}, \theta _{t}]\). Since the mechanism is DIC, there exists \(\xi \) such that
$$\begin{aligned} \delta ^{t}\big [ u_{1,t}(\theta _{t }, \alpha _{t}(\theta _{t})) + \xi _{t}(\theta _{t})\big ] +\rho (t) \ge \delta ^{t}\big [ u_{1,t}(\theta _{t }, \alpha _{t}(\hat{\theta }_{t})) + \xi _{t}(\hat{\theta }_{t})\big ]+\rho (t). \end{aligned}$$
(102)
Define
$$\begin{aligned} B_{t}(\theta ) \equiv \max _{x\in \bar{\varTheta }_{t} }\delta ^{t}\Big [ u_{1,t}(\theta , \alpha _{t}(x)) +\xi _{t}(x) \Big ]. \end{aligned}$$
(103)
DIC implies that
$$\begin{aligned} \theta \in \mathop {\mathrm{{argmax}}}\limits _{x\in \bar{\varTheta }_{t}}\delta ^{t}\Big [ u_{1,t}(\theta , \alpha _{t}(x)) +\xi _{t}(x) \Big ]. \end{aligned}$$
Then, we obtain
$$\begin{aligned} \begin{aligned} |B_{t}(\theta ^{2})- B_{t}(\theta ^{1}) |&\le \max _{x\in \bar{\varTheta }_{t}} \delta ^{t}\big | u_{1,t}( \theta ^{2}, \alpha _{t}(x)) - u_{1,t}( \theta ^{1}, \alpha _{t}(x)) \big |\\&=\max _{x\in \bar{\varTheta }_{t}}\delta ^{t}\Big |\int _{\theta ^{1}}^{\theta ^{2}} \frac{\partial u_{1,t}(y, \alpha _{t}(x)) }{ \partial y } \big |_{y = \theta } d \theta \Big |\\&= \max _{x\in \bar{\varTheta }_{t}}\delta ^{t}\Big |\beta ^{\alpha }_{S,t}(\theta ^{2}) - \beta ^{\alpha }_{S,t}(\theta ^{1}) \Big |. \end{aligned} \end{aligned}$$
By Assumption 1, we have that \(B_{t}\) is Lipschitz continuous. Thus, \(B_{t}\) is differentiable almost everywhere. Therefore, we have
$$\begin{aligned} B_{t}(\theta _{t}) - B_{t}(\hat{\theta }_{t}) = \int ^{\theta _{t}}_{{\hat{\theta }}_{t}} \frac{d B_{t}(y)}{d y} \big |_{y = \theta } d\theta . \end{aligned}$$
Applying envelope theorem to \(B_{t}\) yields
$$\begin{aligned} \begin{aligned} \frac{d B_{t}(y)}{d y} \big |_{y = \theta } =&\frac{\partial }{\partial x }\big [ \delta ^{t} u_{1,t}(x, \alpha _{t}(\theta )) + \xi _{t}(\theta ) \big ] \Big |_{x= \theta }\\ =&\frac{\partial }{\partial x } \delta ^{t} u_{1,t}(x, \alpha _{t}(\theta ) )\Big |_{x= \theta }\\ =&\gamma ^{\alpha }_{t} (t, \theta |h^{\theta }_{t-1}). \end{aligned} \end{aligned}$$
Therefore, we have
$$\begin{aligned} \begin{aligned} \beta ^{\alpha }_{S,t}(\theta _{t}) - \beta ^{\alpha }_{S,t}({\hat{\theta }}_{t}) =&B_{t}(\theta _{t}) - B_{t}(\hat{\theta }_{t}) \\ =&\delta ^{t}\big [ u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) + \xi _{t}(\theta _{t})\big ] - \delta ^{t}\big [u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t})) + \xi _{t}({\hat{\theta }}_{t})\big ] \end{aligned} \end{aligned}$$
From the definition of \(\ell ^{\alpha }_{S,t}(\theta _{t},{\hat{\theta }}_{t})\), we have
$$\begin{aligned} \begin{aligned} \ell ^{\alpha }_{S,t}(\theta _{t}, \hat{\theta }_{t}) =&\delta ^{t}u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) - \delta ^{t} u_{1,t}(\hat{\theta }_{t}, \alpha _{t}(\theta _{t}))\\ =&\delta ^{t}u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) - \delta ^{t} u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}) ) +\delta ^{t} u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}) ) - \delta ^{t} u_{1,t}(\hat{\theta }_{t}, \alpha _{t}(\theta _{t})) \\ \ge&\delta ^{t}\big [u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t})) - u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}) ) + \xi _{t}(\theta _{t}) - \xi _{t}(\hat{\theta }_{t})\big ]\\ =&\beta ^{\alpha }_{S,t}(\theta _{t}) - \beta ^{\alpha }_{S,t}({\hat{\theta }}_{t}). \end{aligned} \end{aligned}$$
1.2 2. \(\theta _{t}\not \in \varLambda _{t}(t):\)
Similar to the case when \(\theta _{t}\in \varLambda _{t}(t)\), DIC implies the existence of \(\phi \) and \(\xi _{s}\) such that
$$\begin{aligned} \begin{aligned}&\delta ^{t}\big [ u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t}))+ \xi _{t}(\theta _{t}) \big ]+{\bar{\mu }}^{\alpha ,\phi , \xi ,\rho }_{t}(\theta _{t}) \\&\quad \ge \delta ^{t}\big [ u_{1,t}(\theta _{t}, \alpha _{t}({\hat{\theta }}_{t}))+ \xi _{t}({\hat{\theta }}_{t}) \big ]+{\bar{\mu }}^{\alpha ,\phi , \xi ,\rho }_{t}(\theta _{t}, {\hat{\theta }}_{t}). \end{aligned} \end{aligned}$$
Define
$$\begin{aligned} B'_{t}(\theta ) \equiv \max _{x\in \bar{\varTheta }_{t} }\Big [ \delta ^{t}\big [ u_{1,t}(\theta , \alpha _{t}(x)) + \xi _{t}(x)\big ]+{\bar{\mu }}^{\alpha ,\phi , \xi ,\rho }_{t}(\theta , x) \Big ]. \end{aligned}$$
(104)
Since the mechanism is DIC, we have
$$\begin{aligned} \theta \in \mathop {{\mathrm{argmax}}}\limits _{x\in \bar{\varTheta }_{t}} \Big [ \delta ^{t}\big [ u_{1,t}(\theta , \alpha _{t}(x)) + \xi _{t}(x)\big ]+{\bar{\mu }}^{\alpha ,\phi , \xi ,\rho }_{t}(\theta , x) \Big ]. \end{aligned}$$
Envelope theorem yields the following:
$$\begin{aligned} \begin{aligned} \beta ^{\alpha }_{{\bar{S}},t}(\hat{\theta }_{t})-\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t})&= B'_{t}(\hat{\theta }_{t})-B'_{t}(\theta _{t})\\&= \delta ^{t}\big [ u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}))+\xi _{t}({\hat{\theta }}_{t}) \big ]+{\bar{\mu }}^{\alpha ,\phi , \xi ,\rho }_{t}({\hat{\theta }}_{t})\\&\quad - \delta ^{t}\big [ u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t}))+ \xi _{t}(\theta _{t}) \big ]-{\bar{\mu }}^{\alpha ,\phi , \xi ,\rho }_{t}(\theta _{t}). \end{aligned} \end{aligned}$$
(105)
From the definition of \({\bar{\mu }}^{\alpha ,\phi , \xi ,\rho }_{t}\), (105) can be extended as follows:
$$\begin{aligned} \begin{aligned}&\beta ^{\alpha }_{{\bar{S}},t}(\hat{\theta }_{t})-\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t}) \\&\quad = \sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s})) +\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \}\\&\qquad - \sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))+\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \} \\&\qquad + \phi _{t}({\hat{\theta }}_{t}) -\phi _{t}(\theta _{t}). \end{aligned} \end{aligned}$$
(106)
Since the mechanism is DIC, we have
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (106) }&\le \sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s})) +\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \}\\&\quad - \sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))+\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \} \\&\quad + \sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))+\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \} \\&\quad -\sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))+\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \}, \end{aligned} \end{aligned}$$
which is equal to
$$\begin{aligned} \begin{aligned}&\sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s})) +\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \}\\&\qquad -\sup _{\tau \in {\mathbb {T}}_{t+1}}\Bigg \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))+\sum _{s=t+1}^{\tau -1} \phi _{s}({\tilde{\theta }}_{s}) +\xi _{\tau }({\tilde{\theta }}_{\tau }) \big ] +\rho (\tau )\Big ] \Bigg \} \\&\quad = \sup _{\tau \in {\mathbb {T}}_{t+1}}\Big \{\pi ^{\alpha }_{t}({\hat{\theta }}_{t}) \Big \} - \sup _{\tau \in {\mathbb {T}}_{t+1}}\Big \{\pi ^{\alpha }_{t}(\theta _{t},{\hat{\theta }}_{t}) \Big \}. \end{aligned} \end{aligned}$$
Hence, the condition (55) is satisfied. \(\square \)
Proof of Proposition 4
Let \(\theta ^{a}_{\epsilon ,t}\), \(\theta ^{b}_{\epsilon ,t}\in \varTheta \) associate with \(\beta ^{\alpha ;a}_{{\bar{S}}, t}\) and \(\beta ^{\alpha ;b}_{{\bar{S}}, t}\), respectively. For any period \(t\in {\mathbb {T}}\backslash \{1\}\), time horizon \(\tau \in {\mathbb {T}}\),
$$\begin{aligned} \begin{aligned}&J{}^{\alpha ,\phi ^{a},\xi ,\rho }_{1,t}(\tau ,\theta _{t}|h^{\theta }_{t-1}) - J^{\alpha ,\phi ^{a},\xi ,\rho }_{1,t-1}(\tau ,\theta _{t-1}|h^{\theta }_{t-2}) \\&\quad = J^{\alpha ,\phi ^{a},\xi ,\rho }_{1,t}(\tau ,\theta _{t}|h^{\theta }_{t-1}) - {\mathbb {E}}^{F_{t}(\theta _{t-1},a_{t-1})}\Big [J^{\alpha ,\phi ^{a},\xi ,\rho }_{1,t-1} (\tau , {\tilde{\theta }}_{t}|h^{\theta }_{t-2}) \Big ]\\&\quad = {\mathbb {E}}^{F_{t}(\theta _{t-1},a_{t-1})}\Big [ J^{\alpha ,\phi ^{a},\xi ,\rho }_{1,t}(\tau ,\theta _{t}|h^{\theta }_{t-1}) - J^{\alpha ,\phi ^{a},\xi ,\rho }_{1,t-1} (\tau , {\tilde{\theta }}_{t}|h^{\theta }_{t-1}) \Big ]\\&\quad = {\mathbb {E}}^{F_{t}(\theta _{t-1},a_{t-1})}\Big [ \int ^{\theta _{t}}_{\theta ^{a}_{\epsilon ,t}} \gamma ^{\alpha }_{t-1}(\tau ,r|h^{\theta }_{t-2})dr - \int ^{{\tilde{\theta }}_{t}}_{\theta ^{a}_{\epsilon ,t}} \gamma ^{\alpha }_{t-1}(\tau ,r|h^{\theta }_{t-2})dr \Big ]\\&\quad = {\mathbb {E}}^{F_{t}(\theta _{t-1},a_{t-1})}\Big [ \int ^{\theta _{t}}_{{\tilde{\theta }}_{t}} \gamma ^{\alpha }_{t-1}(\tau ,r|h^{\theta }_{t-2})dr \Big ]\\&\quad = {\mathbb {E}}^{F_{t}(\theta _{t-1},a_{t-1})}\Big [ \int ^{\theta _{t}}_{\theta ^{b}_{\epsilon ,t}} \gamma ^{\alpha }_{t-1}(\tau ,r|h^{\theta }_{t-2})dr - \int ^{{\tilde{\theta }}_{t}}_{\theta ^{b}_{\epsilon ,t}} \gamma ^{\alpha }_{t-1}(\tau ,r|h^{\theta }_{t-2})dr \Big ]\\&\quad = J^{\alpha ,\phi ^{b},\xi ,\rho }_{1,t}(\tau ,\theta _{t}|h^{\theta }_{t-1}) - J^{\alpha ,\phi ^{b},\xi ,\rho }_{1,t-1}(\tau ,\theta _{t-1}|h^{\theta }_{t-2}). \end{aligned} \end{aligned}$$
Hence, we have
$$\begin{aligned} \begin{aligned}&{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=1}^{\tau -1}\delta ^{s}\phi ^{a}_{s}({\tilde{\theta }}_{s})+ \delta ^{\tau }\xi ^{a}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{a}(\tau ) \Big ] - {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=1}^{\tau -1}\delta ^{s}\phi ^{b}_{s}({\tilde{\theta }}_{s}) + \delta ^{\tau }\xi ^{b}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{b}(\tau )\Big ]\\&\quad = {\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [\sum _{s=1}^{\tau -1}\delta ^{s}\phi ^{a}_{s}({\tilde{\theta }}_{s})+ \delta ^{\tau }\xi ^{a}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{a}(\tau ) \Big ] - {\mathbb {E}}^{\alpha |\theta _{t-1}}\Big [\sum _{s=1}^{\tau -1}\delta ^{s}\phi ^{b}_{s}({\tilde{\theta }}_{s}) + \delta ^{\tau }\xi ^{b}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{b}(\tau )\Big ]. \end{aligned}\nonumber \\ \end{aligned}$$
(107)
Induction gives the following
$$\begin{aligned}&{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=1}^{\tau }\delta ^{s}\phi ^{a}_{s}({\tilde{\theta }}_{s})+ \delta ^{\tau }\xi ^{a}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{a}(\tau ) \Big ] - {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=1}^{\tau }\delta ^{s}\phi ^{b}_{s}({\tilde{\theta }}_{s}) + \delta ^{\tau }\xi ^{b}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{b}(\tau )\Big ]\nonumber \\&\quad = {\mathbb {E}}^{\varXi _{\alpha }}\Big [\sum _{s=1}^{\tau }\delta ^{s}\phi ^{a}_{s}({\tilde{\theta }}_{s})+ \delta ^{\tau }\xi ^{a}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{a}(\tau ) \Big ] - {\mathbb {E}}^{\varXi _{\alpha }}\Big [\sum _{s=1}^{\tau }\delta ^{s}\phi ^{b}_{s}({\tilde{\theta }}_{s}) + \delta ^{\tau }\xi ^{b}_{\tau }({\tilde{\theta }}_{\tau }) +\rho ^{b}(\tau )\Big ]\nonumber \\&\quad =C_{\tau }. \end{aligned}$$
(108)
\(\square \)
Proof of Lemma 8
It is straightforward to see that
$$\begin{aligned} \begin{aligned} \frac{ \partial J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(\tau ,r|h^{\theta }_{t-1}) }{ \partial r } \Big |_{r= \theta _{t}}&=\frac{ \partial U^{\alpha ,\phi ,\xi ,\rho }_{t}(\tau ,r|h^{\theta }_{t-1}) }{\partial r }\Big |_{r=\theta _{t}}\\&= {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t}^{\tau } \frac{\partial u_{1,s}(r, \alpha _{s}({\tilde{\theta }}_{s})) }{\partial r }\Big |_{r={\tilde{\theta }}_{s}} G_{t,s}(h^{{\tilde{\theta }}}_{t,s}) \Big ]. \end{aligned} \end{aligned}$$
From Assumption 4, we have \(G_{t,s}(h^{{\tilde{\theta }}}_{t,s}) \ge 0\). Since \(u_{1,t}\) is a non-decreasing function of \(\theta _{t}\), \(\frac{ \partial J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(\tau ,r|h^{\theta }_{t-1}) }{ \partial r }\) \(\Big |_{r= \theta _{t}}\ge 0\). Therefore, \(J^{\alpha ,\phi ,\xi ,\rho }_{1,t}(\tau ,r|h^{\theta }_{t-1})\) is a non-decreasing function of \(\theta _{t}\), for all \(t\in {\mathbb {T}}\). \(\square \)
Proof of Proposition 6
From the construction of \(\phi \) in (38), we have, for any \(\tau ',\tau ''\in {\mathbb {T}}_{t+1}\),
$$\begin{aligned} \begin{aligned} \phi _{t}({\hat{\theta }}_{t}) - \phi _{t}(\theta _{t})&= \beta ^{\alpha }_{{\bar{S}},t}({\hat{\theta }}_{t}) - {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [\beta _{{\bar{S}},t+1}({\tilde{\theta }}_{t+1})\Big ] - u_{1,t}({\hat{\theta }}_{t}, \alpha _{t}({\hat{\theta }}_{t}))\\&\quad -\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t}) + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\beta _{{\bar{S}},t+1}({\tilde{\theta }}_{t+1})\Big ] + u_{1,t}(\theta _{t}, \alpha _{t}(\theta _{t}))\\&= \beta ^{\alpha }_{{\bar{S}},t}({\hat{\theta }}_{t}) - \beta ^{\alpha }_{{\bar{S}},t}(\theta _{t}) + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t}^{\tau ''-1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau ''-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau ''}({\tilde{\theta }}_{\tau ''}) \Big ]\\&\quad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau '-1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau '-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau '}({\tilde{\theta }}_{\tau '}) \Big ]. \end{aligned}\nonumber \\ \end{aligned}$$
(109)
From the definition of \(d^{\alpha }_{{\bar{S}}, t}\) in (64), we have for any \(\tau '\), \(\tau ''\in {\mathbb {T}}_{t}\),
$$\begin{aligned} \begin{aligned}&\text {R.H.S. of (109)} \le \sup _{\tau \in {\mathbb {T}}_{t}}\Big \{ {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}} \Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ] +\rho (\tau )\Big \}\\&\quad -\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ] +\rho (\tau ) \Big \}+d^{\alpha }_{{\bar{S}}, t}\\&\quad + {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t}^{\tau ''-1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau ''-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau ''}({\tilde{\theta }}_{\tau ''}) \Big ]\\&\quad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau '-1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau '-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau '}({\tilde{\theta }}_{\tau '}) \Big ]. \end{aligned} \end{aligned}$$
(110)
From the condition (43), \(\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t})\ge \beta ^{\alpha }_{S,t}(\theta _{t})\), for all \(\theta _{t}\in \varTheta _{t}\), \(t\in {\mathbb {T}}\). Hence, \(\beta ^{\alpha }_{{\bar{S}},t}(\theta _{t})\ge \) \( \xi _{t}(\theta _{t}) + \) \( u_{1,t}(\theta _{t},\) \(\alpha _{t}(\theta _{t}))\), for all \(\theta _{t}\in {\mathbb {T}}\), \(t\in {\mathbb {T}}\). Then,
$$\begin{aligned} \begin{aligned}&\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{ {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}} \Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ]\\&\qquad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau -1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau }({\tilde{\theta }}_{\tau }) \Big ]\Big \}\\&\quad \le \sup _{\tau \in {\mathbb {T}}_{t}}\Big \{ {\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}} \Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ]\\&\qquad -{\mathbb {E}}^{\alpha |{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ]\Big \}\\&\quad =\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{ \rho (\tau )\Big \}. \end{aligned}\nonumber \\ \end{aligned}$$
(111)
Hence, from (111), we have, for any \(\tau '\in {\mathbb {T}}_{t}\)
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (110) }&\le {\mathbb {E}}^{\alpha |\theta _{t}}\Big [ \sum _{s=t}^{\tau '-1}\delta ^{s}u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau '-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\beta ^{\alpha }_{{\bar{S}},\tau '}({\tilde{\theta }}_{\tau '}) \Big ] + \sup _{\tau \in {\mathbb {T}}_{t}}\Big \{\rho (\tau ) \Big \}+d^{\alpha }_{{\bar{S}}, t}\\&\quad -\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) \Big ] +\rho (\tau ) \Big \}. \end{aligned}\nonumber \\ \end{aligned}$$
(112)
From the construction of \(\rho \) in (40) and Lemma 2, we have, for some \(\tau '\in {\mathbb {T}}_{t}\)
$$\begin{aligned} \begin{aligned} \text {R.H.S. of (112)}&\le {\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau '}\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau '-1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau '}\xi _{\tau '}({\tilde{\theta }}_{\tau '}) +\rho (\tau ')\Big ]+d^{\alpha }_{{\bar{S}}, t}\\&\quad +\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{\rho (\tau ) \Big \}\\&\quad -\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ] \Big \}\\&= \sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ]\Big \}\\&\quad -\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ] \Big \}\\&\quad +d^{\alpha }_{{\bar{S}}, t}+\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{\rho (\tau ) \Big \}, \end{aligned} \end{aligned}$$
which is equal to
$$\begin{aligned} \begin{aligned}&\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t}}\Big [\sum _{s=t}^{\tau }\delta ^{s} u_{1,s}({\tilde{\theta }}_{s},\alpha _{s}({\tilde{\theta }}_{s}))+ \sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ]\Big \}\\&\quad -\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{{\mathbb {E}}^{\alpha |\theta _{t},{\hat{\theta }}_{t}}\Big [ \sum _{s=t}^{\tau }\delta ^{s}\big [u_{1,s}({\tilde{\theta }}_{s}, \alpha _{s}({\tilde{\theta }}_{s}))\big ] +\sum _{s=t+1}^{\tau -1}\delta ^{s}\phi _{s}({\tilde{\theta }}_{s}) +\delta ^{\tau }\xi _{\tau }({\tilde{\theta }}_{\tau }) +\rho (\tau )\Big ] \Big \}\\&\quad +d^{\alpha }_{{\bar{S}}, t}+\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{\rho (\tau ) \Big \}. \end{aligned} \end{aligned}$$
Hence, from the definition of \(U^{\alpha ,\phi ,\xi ,\rho }_{{\bar{S}},t}\), we have
$$\begin{aligned} U^{\alpha ,\phi ,\xi ,\rho }_{{\bar{S}},t}(\theta _{t}|h^{\theta }_{t-1})+d^{\alpha }_{{\bar{S}}, t}+\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{\rho (\tau ) \Big \}\ge U^{\alpha ,\phi ,\xi ,\rho }_{{\bar{S}},t}(\theta _{t}, {\hat{\theta }}_{t}|h^{\theta }_{t-1}). \end{aligned}$$
Following the similar way, we can prove the following
$$\begin{aligned} U_{S,t}^{\alpha ,\phi , \xi , \rho }(\theta _{t}|h^{\theta }_{t-1})+ d^{\alpha }_{S,t}\ge U_{S,t}^{\alpha ,\phi , \xi , \rho }(\theta _{t}, {\hat{\theta }}_{t}|h^{\theta }_{t-1}), \end{aligned}$$
where \(d^{\alpha }_{S,t}\) is defined in (63).
Therefore, we can conclude that the mechanism is \(\Big \{d^{\alpha ^{\circ }}_{S,t}, d^{\alpha ^{\circ }}_{{\bar{S}},t}+\sup _{\tau \in {\mathbb {T}}_{t}}\Big \{\rho ^{\circ }(\tau ) \Big \}\Big \}\)-DIC if \(d^{\alpha }_{S,t}>0\) and \(\Big [d^{\alpha }_{t}+ \sup _{\tau \in {\mathbb {T}}_{t}}\) \(\Big \{\rho (\tau ) \Big \}>0\); otherwise, the mechanism is DIC. We can prove the case when the payment rule \(\rho \) is constructed according to (51) by following the similar way. \(\square \)
