SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We conclude the proof by showing an upper bound on the sender's expected rewards. In the following, we will exploit that the graph is cubic and  $|V| \geq k \geq \frac{|E|}{3}$ . Moreover, we assume w.l.o.g. that there are no unconnected vertices, which implies that  $|E| \geq \frac{|V|}{2}$  and thus that  $k \geq \frac{|V|}{6}$ . Hence, the sender's expected rewards when the initial state is  $s_0$  is at most:
$$
1 + \frac {1}{2} \frac {| \widehat {E} |}{| E |} + \delta \frac {| \widetilde {E} |}{| E |} = 1 + \frac {1}{2} \frac {n}{| E |} + \delta \left(1 - \frac {n}{| E |}\right) <   1 + \frac {\gamma}{1 8 0} + \delta .
$$
Moreover, the sender's expected rewards when the initial state is  $s_1$  is at most:
$$
\begin{array}{l} \delta \frac {| \widetilde {V} |}{| V |} + \frac {1}{2} \left(\frac {| V | - | \widetilde {V} |}{| V |}\right) \leq \delta \frac {(1 + \gamma) k - n}{| V |} + \frac {1}{2} \left(1 - \frac {(1 + \gamma) k - n}{| V |}\right) \\ = \frac {1}{2} - \left(\frac {1}{2} - \delta\right) \frac {(1 + \gamma) k}{| V |} + \frac {1}{2} \frac {n}{| V |} - \frac {\delta n}{| V |} \\ <   \frac {1}{2} - \left(\frac {1}{2} - \delta\right) \frac {(1 + \gamma) k}{| V |} + \frac {1}{2} \frac {n}{| V |} \\ \leq \frac {1}{2} - \left(\frac {1}{2} - \delta\right) \frac {(1 + \gamma) k}{| V |} + \frac {1}{2} \frac {\gamma | E |}{9 0 | V |} \\ \leq \frac {1}{2} - \left(\frac {1}{2} - \delta\right) \frac {(1 + \gamma) k}{| V |} + \frac {\gamma}{6 0}, \\ \end{array}
$$
where we used that  $|\widetilde{V}| \geq (1 + \gamma)k - n$ ,  $n \leq \gamma |E| / 90$  and that  $|E| / |V| \leq 3$ .
Thus, since we start from  $s_0$  or  $s_1$  with probability  $1/2$ , the sender's expected rewards is at most:
$$
\begin{array}{l} \frac {1}{2} \left(1 + \frac {\gamma}{1 8 0} + \delta + \frac {1}{2} - \left(\frac {1}{2} - \delta\right) \frac {(1 + \gamma) k}{| V |} + \frac {\gamma}{6 0}\right) \\ <   \frac {3}{4} - \frac {k}{4 | V |} + \frac {\gamma}{3 0} + \frac {\delta}{2} \left(1 + \frac {(1 + \gamma) k}{| V |}\right) - \frac {\gamma k}{2 | V |} \\ \leq \frac {3}{4} - \frac {k}{4 | V |} + \frac {\gamma}{3 0} + \frac {\delta}{2} \left(1 + \frac {3 k}{| V |}\right) - \frac {\gamma k}{2 | V |} \\ \leq \frac {3}{4} - \frac {k}{4 | V |} + \frac {\gamma}{3 0} + \frac {\delta}{2} (1 + 3) - \frac {\gamma k}{2 | V |} \\ \leq \frac {3}{4} - \frac {k}{4 | V |} + \frac {\gamma}{3 0} + \frac {\gamma}{5 0} - \frac {\gamma k}{2 | V |} \\ \leq \frac {3}{4} - \frac {k}{4 | V |} + \gamma \frac {8}{1 5 0} - \frac {\gamma k}{2 | V |} \\ \leq \frac {3}{4} - \frac {k}{4 | V |} + \gamma \frac {8}{1 5 0} - \frac {\gamma}{1 2} \\ \end{array}
$$
$$
\begin{array}{l} \leq \frac {3}{4} - \frac {k}{4 | V |} - \frac {3}{1 0 0} \gamma \\ \leq \frac {3}{4} - \frac {k}{4 | V |} - \frac {3}{1 0 0} \gamma \left(\frac {3}{4} - \frac {k}{4 | V |}\right) \\ \leq \left(1 - \frac {\gamma}{1 0 0}\right) \left(\frac {3}{4} - \frac {k}{4 | V |}\right), \\ \end{array}
$$
where we used that  $k \geq |V| / 6$  and  $\delta \coloneqq \gamma / 100$ .
C Proofs omitted from Section 4
Lemma 1. Given a promise-form signaling scheme  $\sigma \coloneqq \{(I_h,\varphi_h,g_h)\}_{h\in \mathcal{H}}$ , for every  $h\in \mathcal{H}$  and history  $\tau = (s_1,a_1,\ldots ,s_{h - 1},a_{h - 1},s_h)\in \mathcal{T}_h$  up to step  $h$ , the following holds:
Proof. The statement can be easily proved by induction on the steps  $\mathcal{H}$ .
The base case of the induction is  $h = H$ . By definition of  $V_H^{\mathsf{R},\phi^\sigma}(a,\tau)$ , for every action  $a \in \mathcal{A}$  and history  $\tau = (s_1, a_1, \ldots, s_{H-1}, a_{H-1}, s_H) \in \mathcal{T}_H$ , the following holds:
$$
V _ {H} ^ {\mathsf {R}, \phi^ {\sigma}} (a, \tau) = \sum_ {\theta \in \Theta} \mu_ {H} (\theta | s _ {H}) \phi_ {\tau} ^ {\sigma} (a | \theta) r _ {H} ^ {\mathsf {R}} (s _ {H}, a, \theta).
$$
Since by construction  $\phi_{\tau}^{\sigma}(\theta) = \varphi_{H}(s_{H},\iota_{\tau}^{\sigma},\theta)$  for every  $\theta \in \Theta$ , we have that:
$$
\begin{array}{l} V _ {H} ^ {\mathsf {R}, \phi^ {\sigma}} (a, \tau) = \sum_ {\theta \in \Theta} \mu_ {H} (\theta | s _ {H}) \phi_ {\tau} ^ {\sigma} (a | \theta) r _ {H} ^ {\mathsf {R}} (s _ {H}, a, \theta) \\ = \sum_ {\theta \in \Theta} \mu_ {H} (\theta | s _ {H}) \varphi_ {h} (a | s _ {H}, \iota_ {\tau} ^ {\sigma}, \theta) r _ {H} ^ {\mathsf {R}} (s _ {H}, a, \theta) \\ = \mathcal {V} _ {H} ^ {\mathrm {R}, \sigma} (a, s _ {H}, i _ {\tau} ^ {\sigma}). \\ \end{array}
$$
Now, take  $h < H$  and assume that the statement holds for  $h + 1$ . By definition of  $V_h^{\mathsf{R},\phi^\sigma}(a,\tau)$ , for every  $a \in \mathcal{A}$  and  $\tau = (s_1,a_1,\ldots ,s_{h - 1},a_{h - 1},s_h) \in \mathcal{T}_h$ , it holds:
$$
V _ {h} ^ {\mathsf {R}, \phi^ {\sigma}} (a, \tau) = \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \phi_ {\tau} ^ {\sigma} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s _ {h}, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} (s ^ {\prime} | s _ {h}, a, \theta) V _ {h + 1} ^ {\mathsf {R}, \phi^ {\sigma}} (\tau \oplus (a, s ^ {\prime}))\right).
$$
Moreover, for every  $s' \in S$ , the relation  $V_{h+1}^{\mathsf{R},\phi^{\sigma}}(\tau \oplus (a,s')) = \mathcal{V}_{h+1}^{\mathsf{R},\sigma}(s',\iota_{\tau \oplus (a,s')}^{\sigma})$  holds by induction since  $\tau \oplus (a,s')$  belongs to  $\mathcal{T}_{h+1}$ . Given how Algorithm 2 is designed, it holds:
$$
\iota_ {\tau \oplus (a, s ^ {\prime})} ^ {\sigma} = g _ {h} \left(s _ {h}, a, \iota_ {\tau} ^ {\sigma}, s ^ {\prime}\right),
$$
which, together with  $\phi_{\tau}^{\sigma}(\theta) = \varphi_h(s_h,\iota_\tau^\sigma ,\theta)$  for every  $\theta \in \Theta$ , gives  $V_{h}^{\mathsf{R},\phi^{\sigma}}(a,\tau) = V_{h}^{\mathsf{R},\sigma}(a,s_{h},\iota_{\tau}^{\sigma})$
Similar inductive arguments show that  $V_h^{\mathbb{S},\phi^\sigma}(\tau) = \mathcal{V}_h^{\mathbb{S},\sigma}(s_h,\iota_\tau^\sigma)$  for every step  $h \in \mathcal{H}$  and history  $\tau = (s_1,a_1,\ldots ,s_{h - 1},a_{h - 1},s_h) \in \mathcal{T}_h$  up to  $h$ , concluding the proof.
Proof. We prove the statement by induction.
The base case of the induction is  $h = H$ . Since  $\sigma$  is  $\eta$ -honest, for every  $s \in S$  and  $\iota \in I_H(s)$ :
$$
\mathcal {V} _ {H} ^ {\mathrm {R}, \sigma} (s, \iota) = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {H} (\theta | s) \varphi_ {H} (a | s, \iota , \theta) r _ {H} ^ {\mathrm {R}} (s, a, \theta) \geq \iota - \eta .
$$
Now, take  $h < H$  and assume that the statement holds for  $h + 1$ . For every  $s \in S$  and  $\iota \in I_h(s)$ :
$$
\begin{array}{l} \mathcal {V} _ {h} ^ {\mathbb {R}, \sigma} (s, \iota) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} | s, a, \theta\right) \mathcal {V} _ {h + 1} ^ {\mathsf {R}, \sigma} \left(s ^ {\prime}, g _ {h} \left(s, a, \iota , s ^ {\prime}\right)\right)\right) \\ \geq \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} | s, a, \theta\right) \left(g _ {h} (s, a, \iota , s ^ {\prime}) - \eta (H - h)\right)\right) \\ \geq \iota - \eta - \eta (H - h) \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} | s, a, \theta\right) \\ = \iota - \eta (H - h + 1), \\ \end{array}
$$
where the first inequality holds by induction, the second one by  $\eta$ -honesty, while the last equality holds since  $\sum_{\theta \in \Theta} \mu_h(\theta | s) \varphi_h(a | s, \iota, \theta) \sum_{s' \in S} p_h(s'| s, a, \theta) = 1$ . This concludes the proof.
$$
\begin{array}{l} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a, \theta\right) g _ {h} (s, a, \iota , s ^ {\prime})\right) \geq \\ \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a ^ {\prime}, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a ^ {\prime}, \theta\right) \widehat {V} _ {h + 1} ^ {\mathsf {R}} \left(s ^ {\prime}\right)\right). \tag {2} \\ \end{array}
$$
Proof. We recall that, for  $\epsilon \geq 0$ , the promise-form signaling scheme  $\sigma$  is  $\epsilon$ -persuasive if the history-dependent signaling scheme  $\phi^{\sigma} := \{\phi_{\tau}^{\sigma}\}_{\tau \in \mathcal{T}}$  induced by  $\sigma$  is  $\epsilon$ -persuasive according to Definition 1.
As a consequence, in order to prove the statement, we have to show that, for every step  $h \in \mathcal{H}$ , history  $\tau = (s_1, a_1, \ldots, s_{h-1}, a_{h-1}, s_h) \in \mathcal{T}_h$  up to step  $h$ , and pair of actions  $a, a' \in \mathcal{A}$ , it holds:
$$
V _ {h} ^ {\mathsf {R}, \phi^ {\sigma}} (a, \tau) \geq \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \phi_ {\tau} ^ {\sigma} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s _ {h}, a ^ {\prime}, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} (s ^ {\prime} | s _ {h}, a ^ {\prime}, \theta) \widehat {V} _ {h + 1} ^ {\mathsf {R}} (s ^ {\prime}) - \epsilon\right).
$$
By Lemma 1, we have that  $V_h^{\mathsf{R},\phi^\sigma}(a,\tau) = \mathcal{V}_h^{\mathsf{R},\sigma}(a,s_h,\iota_\tau^\sigma)$ . Moreover, the following holds:
$$
\begin{array}{l} \mathcal {V} _ {h} ^ {\mathsf {R}, \sigma} (a | s _ {h}, v _ {\tau} ^ {\sigma}) \\ = \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \varphi_ {h} (a | s _ {h}, \iota_ {\tau} ^ {\sigma}, \theta) \left(r _ {h} ^ {\mathsf {R}} \left(s _ {h}, a, \theta\right) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} | s _ {h}, a, \theta\right) \mathcal {V} _ {h + 1} ^ {\mathsf {R}, \sigma} \left(s ^ {\prime}, g _ {h} \left(s _ {h}, a, \iota_ {\tau} ^ {\sigma}, s ^ {\prime}\right)\right)\right) \\ \geq \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \varphi_ {h} (a | s _ {h}, \iota_ {\tau} ^ {\sigma}, \theta) \left(r _ {h} ^ {\mathsf {R}} (s _ {h}, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} | s _ {h}, a, \theta\right) \left(g _ {h} \left(s _ {h}, a, \iota_ {\tau} ^ {\sigma}, s ^ {\prime}\right) - \eta (H - h)\right)\right) \\ \geq \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s _ {h}, \iota_ {\tau} ^ {\sigma}, \theta) \left(r _ {h} ^ {\mathsf {R}} \left(s _ {h}, a ^ {\prime}, \theta\right) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s _ {h}, a ^ {\prime}, \theta\right) \widehat {V} _ {h + 1} ^ {\mathsf {R}} \left(s ^ {\prime}\right) - \eta (H - h)\right) \\ \end{array}
$$
where the first inequality holds by Lemma 2, while the second one holds thanks to Equation 2. Finally, the statement is readily proved by noticing that  $\eta (H - h)\leq \eta H$
Proof. The proof works by showing that, given any persuasive history-dependent signaling scheme, one can always build an  $\eta$ -honest and persuasive promise-form signaling scheme whose sender's expected reward is at least as good. This, together with Lemma 1 clearly proves the statement.
Given any persuasive history-dependent signaling scheme  $\phi = \{\phi_{\tau}\}_{\tau \in \mathcal{T}}$ , we build a promise-form signaling scheme  $\sigma = \{(I_h,\varphi_h,g_h)\}_{h\in \mathcal{H}}$  as follows:
-  $I_h(s) \coloneqq \left\{ V_h^{\mathsf{R},\phi}(\tau) \mid \tau = (s_1, a_1, \ldots, s_{h-1}, a_{h-1}, s_h) \in \mathcal{T}_h \wedge s_h = s \right\}$  for all  $h > 1, s \in S$ .
-  $\varphi_h(s,\iota ,\theta) = \phi_{\tau_{h,s,\iota}}(\theta)$  for all  $h\in \mathcal{H}$ ,  $s\in \mathcal{S}$ ,  $\iota \in I_h(s)$ , and  $\theta \in \Theta$ , where:
$$
\tau_{h,s,\iota}^{\star}\in \operatorname *{argmax}_{\substack{\tau = (s_{1},a_{1},\ldots ,s_{h - 1},a_{h - 1},s_{h})\in \mathcal{T}_{h}:\\ s_{h} = s\wedge V_{h}^{\mathsf{R},\phi}(\tau) = \iota}}\left\{V_{h}^{\mathsf{S},\phi}(\tau)\right\} ,
$$
which is guaranteed to exist given how  $I_h(s)$  is defined.
-  $g_{h}(s,a,\iota,s') = V_{h+1}^{\mathsf{R},\phi}(\tau_{h,s,\iota}^{\star} \oplus (a,s'))$  for all  $h \in \mathcal{H}$ ,  $s \in \mathcal{S}$ ,  $a \in \mathcal{A}$ ,  $\iota \in I_{h}(s)$ , and  $s' \in \mathcal{S}$ .
As a first step, we prove that, if  $\phi$  is persuasive, then the promise-form signaling scheme  $\sigma$  that we have just built is persuasive as well. This can be easily proved by exploiting Lemma 3.
First, we prove that  $\sigma$  is an  $\eta$ -honest for  $\eta = 0$ . Formally, for every  $h \in \mathcal{H}$ ,  $s \in S$ , and  $\iota \in I_h(s)$ :
$$
\begin{array}{l} \iota = V _ {h} ^ {\mathsf {R}, \phi} \left(\tau_ {h, s, \iota} ^ {\star}\right) (6) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \phi_ {\tau_ {h, s, \iota} ^ {\star}} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a, \theta\right) V _ {h + 1} ^ {\mathsf {R}, \phi} \left(\tau_ {h, s, \iota} ^ {\star} \oplus \left(a, s ^ {\prime}\right)\right)\right) (7) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \phi_ {\tau_ {h, s, \iota} ^ {\star}} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a, \theta\right) g _ {h} (s, a, \iota , s ^ {\prime})\right) (8) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a, \theta\right) g _ {h} (s, a, \iota , s ^ {\prime})\right), (9) \\ \end{array}
$$
where Equation (6) holds by definition of  $\tau_{h,s,\iota}^{\star}$ , Equation (7) holds by definition of  $V_h^{\mathsf{R},\phi}(\tau_{h,s,\iota}^{\star})$ , Equation (8) holds by definition of  $g_h(s,a,\iota,s')$ , while Equation (9) holds by definition of  $\varphi_h(a|s,\iota,\theta)$ . This proves that  $\sigma$  is an  $\eta$ -honest promise-form signaling scheme, for  $\eta = 0$ .
In order to apply Lemma 3, we also need to prove that  $\sigma$  satisfies the conditions in Equation (2). By applying definitions, it is easy to check that, for every  $h\in \mathcal{H}$ $s\in S,\iota \in I_h(s)$ , and  $a\in \mathcal{A}$ :
$$
\begin{array}{l} V _ {h} ^ {\mathsf {R}, \phi} (a, \tau_ {h, s, \iota} ^ {\star}) = \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \phi_ {\tau_ {h, s, \iota} ^ {\star}} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} (s ^ {\prime} | s, a, \theta) V _ {h + 1} ^ {\mathsf {R}, \phi} \left(\tau_ {h, s, \iota} ^ {\star} \oplus (a, s ^ {\prime})\right)\right) \\ = \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \phi_ {\tau_ {h, s, \iota} ^ {*}} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a, \theta\right) g _ {h} (s, a, \iota , s ^ {\prime})\right) \\ = \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a, \theta\right) g _ {h} (s, a, \iota , s ^ {\prime})\right). \tag {10} \\ \end{array}
$$
Moreover, since  $\phi$  is persuasive we have that, for every  $h\in \mathcal{H}$ $s\in S,\iota \in I_h(s)$  , and  $a,a^{\prime}\in \mathcal{A}$  ..
$$
\begin{array}{l} V _ {h} ^ {\mathsf {R}, \phi} (a, \tau_ {h, s, \iota} ^ {\star}) = \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \phi_ {\tau_ {h, s, \iota} ^ {\star}} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} (s ^ {\prime} | s, a, \theta) V _ {h + 1} ^ {\mathsf {R}, \phi} \left(\tau_ {h, s, \iota} ^ {\star} \oplus (a, s ^ {\prime})\right)\right) \\ \geq \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \phi_ {\tau_ {h, s, \iota} ^ {*}} (a | \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a ^ {\prime}, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a ^ {\prime}, \theta\right) \widehat {V} _ {h + 1} ^ {\mathsf {R}, \phi} \left(s ^ {\prime}\right)\right) \\ = \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a ^ {\prime}, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s, a ^ {\prime}, \theta\right) \widehat {V} _ {h + 1} ^ {\mathsf {R}, \phi} \left(s ^ {\prime}\right)\right), \tag {11} \\ \end{array}
$$
where the last equality holds by definition of  $\varphi_h(a|s,\iota ,\theta)$ . Then, by combining Equation (10) and Equation (11), we get that, for every  $h\in \mathcal{H}$ ,  $s\in S$ ,  $\iota \in I_h(s)$ , and  $a,a^{\prime}\in \mathcal{A}$ :
$$
\begin{array}{l} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} (s ^ {\prime} | s, a, \theta) g _ {h} (s, a, \iota , s ^ {\prime})\right) \geq \\ \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s) \varphi_ {h} (a | s, \iota , \theta) \left(r _ {h} ^ {\mathsf {R}} (s, a ^ {\prime}, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} (s ^ {\prime} | s, a ^ {\prime}, \theta) \widehat {V} _ {h + 1} ^ {\mathsf {R}, \phi} (s ^ {\prime})\right), \\ \end{array}
$$
which means that  $\sigma$  satisfies Equation (2). Thus, by Lemma 3 we can conclude that  $\sigma$  is persuasive.
In order to conclude the proof, it remains to show that  $\sigma$  achieves a sender's expected reward at least as good as that obtained by  $\phi$ . Formally, we prove that  $V_h^{\mathsf{S},\phi^\sigma}(\tau) \geq V_h^{\mathsf{S},\phi}(\tau)$  for every step  $h \in \mathcal{H}$  and history  $\tau \in \mathcal{T}_h$  up to  $h$ . We will prove such a result by induction.
The base case of the induction is  $h = H$ . For every  $\tau = (s_1, a_1, \ldots, s_{H-1}, a_{H-1}, s_H) \in \mathcal{T}_H$ :
$$
\begin{array}{l} V _ {H} ^ {\mathbf {S}, \phi} (\tau) \leq V _ {H} ^ {\mathbf {S}, \phi} \left(\tau_ {H, s _ {H}, \iota_ {\sigma}} ^ {\star}\right) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {H} (\theta | s _ {H}) \phi_ {\tau_ {H, s _ {H}, \iota_ {\tau} ^ {\sigma}} ^ {\star}} (a | \theta) r _ {H} ^ {\mathbb {S}} (s _ {H}, a, \theta) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {H} (\theta | s _ {H}) \varphi_ {H} (a | s _ {H}, \iota_ {\tau} ^ {\sigma}, \theta) r _ {H} ^ {\mathbb {S}} (s _ {H}, a, \theta) \\ = \mathcal {V} _ {H} ^ {\mathbf {S}, \sigma} \left(s _ {H}, \iota_ {\tau} ^ {\sigma}\right) \\ = V _ {H} ^ {\mathbf {S}, \phi^ {\sigma}} (\tau), \\ \end{array}
$$
where the last equality holds by Lemma 1.
Now, let us take  $h < H$  and assume that the statement that we want to prove holds for  $h + 1$ . Then, for every  $\tau = (s_1, a_1, \ldots, s_{h-1}, a_{h-1}, s_h) \in \mathcal{T}_h$ , it holds:
$$
\begin{array}{l} V _ {h} ^ {\mathbf {S}, \phi} (\tau) \leq V _ {h} ^ {\mathbf {S}, \phi} \left(\tau_ {h, s _ {h}, \iota_ {\tau} ^ {\sigma}} ^ {\star}\right) (12) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \phi_ {\tau_ {h, s _ {h}, \iota_ {\tau} ^ {\sigma}} ^ {\star}} (a | \theta) \left(r _ {h} ^ {\mathbf {S}} (s, a, \theta) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} (s ^ {\prime} | s, a, \theta) V _ {h + 1} ^ {\mathbf {S}, \phi} \left(\tau_ {h, s _ {h}, \iota_ {\tau} ^ {\sigma}} ^ {\star} \oplus (a, s ^ {\prime})\right)\right) \\ = \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \varphi_ {h} (a | s _ {h}, \iota_ {\tau} ^ {\sigma}, \theta) \left(r _ {h} ^ {\mathbf {S}} \left(s _ {h}, a, \theta\right) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s _ {h}, a, \theta\right) V _ {h + 1} ^ {\mathbf {S}, \phi} \left(\tau_ {h, s _ {h}, \iota_ {\tau} ^ {\sigma}} ^ {\star} \oplus (a, s ^ {\prime})\right)\right) (13) \\ \leq \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \varphi_ {h} (a | s _ {h}, \iota_ {\tau} ^ {\sigma}, \theta) \left(r _ {h} ^ {\mathbf {S}} \left(s _ {h}, a, \theta\right) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s _ {h}, a, \theta\right) V _ {h + 1} ^ {\mathbf {S}, \phi} \left(\tau_ {h + 1, s ^ {\prime}, g _ {h} \left(s _ {h}, a, t _ {\tau} ^ {\sigma}, s ^ {\prime}\right)} ^ {\star}\right)\right) (14) \\ \leq \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \varphi_ {h} (a | s _ {h}, \iota_ {\tau} ^ {\sigma}, \theta) \left(r _ {h} ^ {\mathsf {S}} \left(s _ {h}, a, \theta\right) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s _ {h}, a, \theta\right) V _ {h + 1} ^ {\mathsf {S}, \phi^ {\sigma}} \left(\tau_ {h + 1, s ^ {\prime}, g _ {h} \left(s _ {h}, a, \iota_ {\tau} ^ {\sigma}, s ^ {\prime}\right)} ^ {\star}\right)\right) (15) \\ \leq \sum_ {a \in \mathcal {A}} \sum_ {\theta \in \Theta} \mu_ {h} (\theta | s _ {h}) \varphi_ {h} (a | s _ {h}, \iota_ {\tau} ^ {\sigma}, \theta) \left(r _ {h} ^ {\mathbb {S}} \left(s _ {h}, a, \theta\right) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {h} \left(s ^ {\prime} \mid s _ {h}, a, \theta\right) \mathcal {V} _ {h + 1} ^ {\mathbb {S}, \sigma} \left(s ^ {\prime}, g _ {h} \left(s _ {h}, a, \iota_ {\tau} ^ {\sigma}, s ^ {\prime}\right)\right)\right) (16) \\ = \mathcal {V} _ {h} ^ {\mathbf {S}, \sigma} \left(s _ {h}, \iota_ {\tau} ^ {\sigma}\right) \\ = V _ {h} ^ {\mathbf {S}, \phi^ {\sigma}} (\tau). (17) \\ \end{array}
$$
where Equation (12) holds by definition of  $\tau_{h,s_h,\iota_\tau^\sigma}^\star$  for the promise  $\iota_{\tau}^{\sigma}\in I_{h}(s_{h})$ , Equation (13) holds by definition of  $\varphi_h(s_h,\iota_\tau^\sigma ,\theta)$ , Equation (14) holds by definition of  $\tau_{h + 1,s',g_h(s_h,a,\iota_\tau^\sigma ,s')}^\star$  together with the fact that  $g_{h}(s_{h},a,\iota_{\tau}^{\sigma},s^{\prime}) = V_{h + 1}^{\mathsf{R},\phi}(\tau_{h,s_h,\iota_\tau^\sigma}^\star \oplus (a,s'))$ , Equation (15) holds by induction, while Lemma 1 proves Equation (16) and Equation (17).
D Proofs omitted from Section 5
Lemma 4. Let  $\sigma \coloneqq \{(I_h,\varphi_h,g_h)\}_{h\in \mathcal{H}}$  be returned by Algorithm 3 instantiated with any oracle  $\mathcal{O}_{h,s,\iota}$  as in Definition 3. For every  $h\in \mathcal{H}$ ,  $s\in S$ ,  $\iota \in I_h(s)$ , it holds that  $\mathcal{V}_h^{\mathrm{S},\sigma}(s,\iota)\geq M_h^\delta (s,\iota)$ .
Proof. Let  $\sigma = \{(I_h, \varphi_h, g_h)\}_{h \in \mathcal{H}}$  be the promise-form signaling scheme returned by Algorithm 3 instantiated with an approximate oracle  $\mathcal{O}_{h,s,\iota}$  as in Definition 3, and let  $(\kappa, q, v) \gets \mathcal{O}_{h,s,\iota - \delta}(M_{h + 1}^{\delta})$ . We prove the statement by induction. The base case for  $h = H$  of the induction holds trivially. Then, for every  $h < H$ ,  $s \in S$ , and  $\iota \in I_h(s)$ , assuming the statement holds for  $h + 1$  we have that: