SlowGuess/NeurIPS_2023_OCR huggingface dataset:
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A Data Markets in the Single-Buyer Settings
A.1 RochetNet Architecture
Figure 4: RocketNet: Neural network architecture for learning the optimal menu for a single bidder setting. The neural network takes in as inputs the buyer's payoff  $v$  and interim belief  $\theta$  (a distribution on world state) and computes the utility to the buyer for each of  $P + 1$  menu options, where option 0 corresponds to the uninformative experiment.
The neural network architecture for the single buyer settings is depicted in Fig 4. In this architecture, each menu entry  $p$  is represented by a matrix  $\pi^p$  with dimensions  $|\Omega| \times |\Omega|$ , along with an associated price  $t^p$ . Instead of explicitly enforcing the obedience constraint, we consider all potential actions an agent may take to deviate from the recommended action to compute the utility. This utility for the menu option  $p$  is given by Eqn 7 in the main paper.
A.2 Setup and Hyper-parameters
Training: We train RochetNet with  $P = 1000$  possible menu entries. We set the softmax temperature  $\tau$  to  $\frac{1}{200}$ . We train RochetNet for 20,000 iterations with a minibatch of size  $2^{15}$  sampled online for every update.
We report all our results on a test-set of 20000 samples that are separate from the train set.
Time: For the settings studied in the paper, RochetNet take 7-8 minutes minutes to train on a single NVIDIA Tesla V100 GPU.
B Data Markets in the Multi-Buyer Setting
B.1 RegretNet Architecture
The neural network architecture for the multi-buyer ex post IC setting is depicted in Fig 5. For the BIC setting, the experiment network  $\sigma$  remains the same and the interim experiments are computed through sampling. They payment network only takes the type corresponding to the buyer as input as we can replace  $\hat{\mathbb{E}}_{-i}[t_i(\rho_i,\rho_{-i})]$  with just  $t_i(\rho_i)$ . We also have separate payment networks for each buyer if their type distributions are different.
B.2 Proof of Lemma 4.1
Proof. For the ex post IC setting, we first compute a normalized payment,  $\tilde{t}_i^w (\rho)\in [0,1]$ , for each buyer  $i$  by using a sigmoidal unit. We scale this by the difference between utility achieved by the buyer when he is truthful and obedient and the utility of opting out. Thus, the payment  $t_i^w (\rho)$  is
Figure 5: RegretNet: Neural network architecture for the multi-buyer ex post IC setting. The inputs are the reported types  $\rho$  of each buyer. The experiment network outputs an experiment of size  $|\Omega| \times |\Omega|$  for each buyer, where a row-wise softmax operation ensures the experiments are well defined. The payment network outputs the normalized payment  $\tilde{t}_i^w(\rho) \in [0,1]$  through a sigmoid activation unit that is used to compute payment via Equation 8.
computed as follows:
$$
t _ {i} ^ {w} (\rho) = \tilde {t} _ {i} ^ {w} (\rho) \cdot v _ {i} \cdot \left(\sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k} - \left(\max  _ {k} \theta_ {i, k} - \alpha\right)\right) \tag {12}
$$
To establish ex post IR, we reason about the utility a buyer can obtain when not participating in the mechanism. As per the discussion in Section 2, in an optimal data market the seller will send optimal recommendations to participating buyers in the event that buyer  $i$  opts out, thus minimizing the utility of the buyer who opts out. In particular, the recommendation for any  $j \neq i$ , for an opting out buyer  $i$ , is such that  $\mathbb{E}_{a \sim \sigma_{-i}(\omega; \rho_{-i})}[\mathbf{1}\{a_j = \omega\}] = 1$ . Given this, the utility of buyer  $i$ , when opting out, is
$$
\begin{array}{l} \min_{\sigma_{-i}}\max_{\tilde{a}_{i}\in \mathcal{A}_{i}}\left(\underset { \begin{array}{c}a\sim \sigma_{-i}(\omega ;\rho_{-i}),\\ \omega \sim \theta_{i} \end{array} }{\mathbb{E}}[u_{i}(\tilde{a}_{i},a_{-i};\omega ,\rho)]\right) \\ = \max  _ {\tilde {a} _ {i} \in \mathcal {A} _ {i}} \left(\mathbb {E} _ {\substack {a \sim \sigma_ {- i} (\omega ; \rho_ {- i}) \\ \omega \sim \theta_ {i}}} \left[ v _ {i} \cdot \mathbf {1} \{\tilde {a} _ {i} = \omega \} - v _ {i} \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \mathbf {1} \{\tilde {a} _ {i} = \omega \} \right]\right) \tag{13} \\ = \max _ {\tilde {a} _ {i} \in \mathcal {A} _ {i}} \left(\mathbb {E} _ {\omega \sim \theta_ {i}} \left[ v _ {i} \cdot \mathbf {1} \{\tilde {a} _ {i} = \omega \} - v _ {i} \alpha \right]\right) \\ = v _ {i} \cdot (\max  \theta_ {i} - \alpha) \\ \end{array}
$$
Note that  $\sum_{k\in [m]}\pi_{i,k,k}^{w}(\rho)\theta_{i,k}\geq \max_k\theta_{i,k}$  and  $\bar{\alpha}\left(\sum_{k\in [m]}\sum_{j\in [n]\setminus i}\pi_{j,k,k}^{w}(\rho)\theta_{j,k}\right)\leq$ $\bar{\alpha}\left(\sum_{j\in [n]\setminus i}1\right) = \alpha$ . Thus we have  $\sum_{k\in [m]}\pi_{i,k,k}^{w}(\rho)\theta_{i,k} - \bar{\alpha}\left(\sum_{k\in [m]}\sum_{j\in [n]\setminus i}\pi_{j,k,k}^{w}(\rho)\theta_{j,k}\right) - (\max_k\theta_{i,k} - \alpha)\geq 0.$
Taking this with the fact that  $\tilde{t}_i^w (\rho)\in [0,1]$ , we have
$$
\begin{array}{l} t _ {i} ^ {w} (\rho) = \tilde {t} _ {i} ^ {w} (\rho) \cdot v _ {i} \cdot \left(\sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k} - \left(\max  _ {k} \theta_ {i, k} - \alpha\right)\right) \\ \leq v _ {i} \cdot \left(\underset { \begin{array}{c} a \sim \sigma_ {i} ^ {w} (\omega ; \rho), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} \left[ \left(\mathbf {1} \left\{a _ {i} = \omega \right\} - \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \mathbf {1} \left\{a _ {j} = \omega \right\}\right) \right] - \left(\underset {k} {\max } \theta_ {i, k} - \alpha)\right)\right) \tag {14} \\ = \underset { \begin{array}{c} a \sim \sigma_ {i} ^ {w} (\omega ; \rho), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} \left[ v _ {i} \cdot \left(\mathbf {1} \{a _ {i} = \omega \} - \bar {\alpha} \sum_ {j \in [ n ] \setminus i} \mathbf {1} \{a _ {j} = \omega \}\right) \right] - v _ {i} \cdot \left(\max  _ {k} \theta_ {i, k} - \alpha\right) \\ = \underset { \begin{array}{c} a \sim \sigma_ {i} ^ {w} (\omega ; \rho), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (a; \omega , \rho) ] - v _ {i} \cdot (\max  \theta_ {i} - \alpha)) \\ \end{array}
$$
Rearranging, and substituting  $v_{i} \cdot (\max \theta_{i} - \alpha)$  from Eqn 13, we get the IR constraint described in Eqn 5.
B.3 Proof of Lemma 4.2
Proof. For the BIC setting, we work with the interim payment, and make use of a normalized interim payment,  $\tilde{t}_i(\rho_i) \in [0,1]$ , for each buyer  $i$ , by using a sigmoidal unit. In this case, we scale this normalized payment by the difference between the interim utility achieved when the buyer is truthful and obedient and the interim utility of opting out. Thus, the interim payment is
$$
t _ {i} ^ {w} \left(\rho_ {i}\right) = \tilde {t} _ {i} ^ {w} \left(\rho_ {i}\right) \cdot v _ {i} \cdot \hat {\mathbb {E}} _ {- i} \left[ \sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k} - (\max  \theta_ {i} - \alpha) \right] \tag {15}
$$
The utility of the outside option is again computed similarly to the ex post IC setting, recognizing that the seller in the optimal mechanism will make recommendation  $\hat{\mathbb{E}}_{-i}\left[\mathbb{E}_{a\sim \sigma_{-i}(\omega ;\rho_{-i})},[\mathbf{1}\{a_j = \omega \} ]\right] = 1$  to any agent  $j\neq i$  , in the case that  $i$  drops out. In particular, the utility of buyer  $i$  , when opting out, is
$$
\begin{array}{l} \min  _ {\sigma_ {- i}} \max  _ {\tilde {a} _ {i} \in \mathcal {A} _ {i}} \left(\hat {\mathbb {E}} _ {- i} \left[ \begin{array}{c} \mathbb {E} _ {a \sim \sigma_ {- i} (\omega ; \rho_ {- i}),} \\ \omega \sim \theta_ {i} \end{array} [ u _ {i} (\tilde {a} _ {i}, a _ {- i}; \omega , \rho) ] \right]\right) \\ = \max  _ {\tilde {a} _ {i} \in \mathcal {A} _ {i}} \left(\hat {\mathbb {E}} _ {- i} \left[ \underset {\omega \sim \theta_ {i}} {\mathbb {E}} _ {a \sim \sigma_ {- i} (\omega ; \rho_ {- i})}, \left[ v _ {i} \cdot \mathbf {1} \{\tilde {a} _ {i} = \omega \} - v _ {i} \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \mathbf {1} \{a _ {j} = \omega \} \right] \right]\right) \tag {16} \\ = \max  _ {\tilde {a} _ {i} \in \mathcal {A} _ {i}} \left(\mathbb {E} _ {\omega \sim \theta_ {i}} \left[ v _ {i} \cdot \mathbf {1} \{\tilde {a} _ {i} = \omega \} - v _ {i} \alpha \right]\right) \\ = v _ {i} \cdot (\max  \theta_ {i} - \alpha) \\ \end{array}
$$
Note that  $\hat{\mathbb{E}}_{-i}\left[\sum_{k\in [m]}\pi_{i,k,k}^{w}(\rho)\theta_{i,k}\right]\geq \max_k\theta_{i,k}$  and  $\hat{\mathbb{E}}_{-i}\left[\bar{\alpha}\left(\sum_{k\in [m]}\sum_{j\in [n]\backslash i}\pi_{j,k,k}^{w}(\rho)\theta_{j,k}\right)\right]\leq$ $\bar{\alpha}\left(\sum_{j\in [n]\backslash i}1\right) = \alpha$  . Thus we have
$$
\hat {\mathbb {E}} _ {- i} \left[ \sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k} - \left(\max  _ {k} \theta_ {i, k} - \alpha\right) \right] \geq 0
$$
Taking this with the fact that  $\tilde{t}_i^w (\rho_i)\in [0,1]$ , we have
$$
\begin{array}{l} t _ {i} ^ {w} (\rho_ {i}) = \tilde {t} _ {i} ^ {w} (\rho_ {i}) \cdot v _ {i} \cdot \hat {\mathbb {E}} _ {- i} \left[ \sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k} - (\max  \theta_ {i} - \alpha) \right] \\ \leq v _ {i} \cdot \hat {\mathbb {E}} _ {- i} \left[ \sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k} \right] - v _ {i} \cdot (\max  \theta_ {i} - \alpha)) \\ = \hat{\mathbb{E}}_{-i}\left[\underset { \begin{array}{c}a\sim \sigma_{i}^{w}(\omega ;\rho),\\ \omega \sim \theta_{i} \end{array} }{\mathbb{E}}\left[v_{i}\cdot \left(\mathbf{1}\{a_{i} = \omega \} -\bar{\alpha}\sum_{j\in [n]\setminus i}\mathbf{1}\{a_{j} = \omega \}\right)\right]\right] - v_{i}\cdot (\max \theta_{i} - \alpha)) \\ \leq \hat {\mathbb {E}} _ {- i} \left[ \underset { \begin{array}{c} a \sim \sigma_ {i} ^ {w} (\omega ; \rho), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (a; \omega , \rho) ] \right] - v _ {i} \cdot (\max  \theta_ {i} - \alpha)) \tag {17} \\ \end{array}
$$
Rearranging the above and substituting  $v_{i} \cdot (\max \theta_{i} - \alpha)$  from Eqn 16, we get the IIR constraint described in Eqn 4.
B.4 Proof of Lemma 4.3
Proof. We first prove the forward direction: if any mechanism  $\mathcal{M} = (\sigma^w, t^w)$  where  $\sigma_i^w(\omega, \rho) = (\pi_{i,\omega}^w(\rho))$  is ex post IC, then  $RGT_i^w = 0, \forall i \in [n]$ .
For the ex post IC setting, the incentive compatibility constraints require that for every agent  $i$ , and for each  $\rho \in \mathcal{P}$ , and assuming that every other agent reports its type truthfully and follows the recommended action, then for each misreport  $\rho_i' \in \mathcal{P}_i$ , and each deviation function,  $\delta: \mathcal{A}_i \to \mathcal{A}_i$ , the following condition:
$$
\underset { \begin{array}{c} a \sim \sigma (\omega , \rho), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (a, \omega , \rho) - t _ {i} (\rho) ] \geq \underset { \begin{array}{c} a \sim \sigma (\omega ; \rho_ {i} ^ {\prime}, \rho_ {- i}), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (\delta (a _ {i}), a _ {- i}, \omega , \rho) - t _ {i} (\rho_ {i} ^ {\prime}, \rho_ {- i}) ] \tag {18}
$$
Note that the above hold for any deviation function  $\delta$ . Therefore, we can write that
$$
\underset { \begin{array}{c} a \sim \sigma (\omega , \rho), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (a, \omega , \rho) - t _ {i} (\rho) ] \geq \max  _ {\delta} \underset { \begin{array}{c} a \sim \sigma (\omega ; \rho_ {i} ^ {\prime}, \rho_ {- i}), \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (\delta (a _ {i}), a _ {- i}, \omega , \rho) - t _ {i} (\rho_ {i} ^ {\prime}, \rho_ {- i}) ] \tag {19}
$$
We can expand to get the following equation:
$$
\begin{array}{l} v _ {i} \cdot \left(\sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k}\right) - t _ {i} (\rho) \\ \geq \max  _ {\delta} v _ {i} \cdot \left(\sum_ {k \in [ m ]} \left\{\pi_ {i, \delta (k), k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, \delta (k)} \right\} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k}\right) - t _ {i} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \tag {20} \\ \end{array}
$$
Pushing the max inside (since other terms don't involve  $\delta(k)$ ), we note that:
$$
\max  _ {\delta} \sum_ {k \in [ m ]} \left\{\pi_ {i, \delta (k), k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, \delta (k)} \right\} = \sum_ {k \in [ m ]} \max  _ {k ^ {\prime}} \left\{\pi_ {i, k ^ {\prime}, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, k ^ {\prime}} \right\} \tag {21}
$$
Thus we have,
$$
\begin{array}{l} v _ {i} \cdot \left(\sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k}\right) - t _ {i} (\rho) \\ \geq v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {k ^ {\prime} \in [ m ]} \left\{\pi_ {i, k ^ {\prime}, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, k ^ {\prime}} \right\} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k}\right) - t _ {i} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \tag {22} \\ \end{array}
$$
Note that by our definition of  $ex$  post regret in Equation 10, the above is equivalent to  $rgt_i^w (\rho_i',\rho)\leq 0$ . Furthermore, this holds for any deviating report  $\rho_{i}^{\prime}$ . Thus, we can write:
$$
\max  _ {\rho_ {i} ^ {\prime} \in \mathcal {P} _ {i}} r g t _ {i} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho\right) \leq 0 \tag {23}
$$
But note that
$$
\begin{array}{l} \max _ {\rho_ {i} ^ {\prime} \in \mathcal {P} _ {i}} r g t _ {i} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho\right) \geq r g t _ {i} ^ {w} \left(\rho_ {i}, \rho\right) \\ = v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {k ^ {\prime} \in [ m ]} \left\{\pi_ {i, k ^ {\prime}, k} ^ {w} (\rho) \theta_ {i, k ^ {\prime}} \right\} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k}\right) \\ - v _ {i} \cdot \left(\sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k}\right) \tag {24} \\ = v _ {i} \cdot \left(\sum_ {k \in [ m ]} \left[ \max  _ {k ^ {\prime} \in [ m ]} \left\{\pi_ {i, k ^ {\prime}, k} ^ {w} (\rho) \theta_ {i, k ^ {\prime}} \right\} - \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} \right]\right) \\ \geq 0 \\ \end{array}
$$
where the last step holds since  $\max_{k' \in [m]} \left\{ \pi_{i,k',k}^w (\rho) \theta_{i,k'} \right\} \geq \pi_{i,k,k}^w (\rho) \theta_{i,k}$ .
Combining Equations 23 and 24, we have  $\max_{\rho_i' \in \mathcal{P}_i} rgt_i^w(\rho_i', \rho) = 0$ . Taking the expectation over all profiles  $\rho \in \mathcal{P}$ , we have  $\mathbb{E}_{\rho \in \mathcal{P}}\left[\max_{\rho_i' \in \mathcal{P}_i} rgt_i^w(\rho_i', \rho)\right] = 0$ . Thus,  $RGT_i^w = 0$ ,  $\forall i \in [n]$ . Thus, we've shown that if any mechanism  $\mathcal{M} = (\sigma^w, t^w)$  where  $\sigma_i^w(\omega, \rho) = (\pi_{i,\omega}^w(\rho))$  is ex post IC, then  $RGT_i^w = 0$ , as desired.
Next, we consider the reverse direction: if  $RGT_{i}^{w} = 0, \forall i \in [n]$ , we want to show that any mechanism  $\mathcal{M} = (\sigma^{w}, t^{w})$  where  $\sigma_{i}^{w}(\omega, \rho) = (\pi_{i,\omega}^{w}(\rho))$  is ex post IC. Starting with  $RGT_{i}^{w} = 0, \forall i \in [n]$ , we have that by definition:
$$
\mathbb {E} _ {\rho \in \mathcal {P}} \left[ \max  _ {\rho_ {i} ^ {\prime} \in \mathcal {P} _ {i}} r g t _ {i} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho\right) \right] = 0 \tag {25}
$$
We showed in Equation 24 that  $rgt_{i}^{w}(\rho_{i},\rho) \geq 0$ . Since  $\rho_{i} \in \mathcal{P}_{i}$ , we have that  $\max_{\rho_{i}^{\prime} \in \mathcal{P}_{i}} rgt_{i}^{w}(\rho_{i}^{\prime},\rho) \geq 0$ . Together with the fact that  $\mathbb{E}_{\rho \in \mathcal{P}}\left[\max_{\rho_{i}^{\prime} \in \mathcal{P}_{i}} rgt_{i}^{w}(\rho_{i}^{\prime},\rho)\right] = 0$ , we have that  $\max_{\rho_{i}^{\prime} \in \mathcal{P}_{i}} rgt_{i}^{w}(\rho_{i}^{\prime},\rho) = 0 \Rightarrow rgt_{i}^{w}(\rho_{i}^{\prime},\rho) \leq 0$ .
Plugging in our definition for ex post regret, we have that:
$$
\begin{array}{l} v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {k ^ {\prime} \in [ m ]} \left\{\pi_ {i, k ^ {\prime}, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, k ^ {\prime}} \right\} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k}\right) \\ - v _ {i} \cdot \left(\sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho) \theta_ {j, k}\right) - \left(t _ {i} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) - t _ {i} ^ {w} (\rho)\right) \leq 0 \tag {26} \\ \end{array}
$$
This can be transformed back to:
$$
\begin{array}{l} \mathop{\mathbb{E}}_{\substack{a\sim \sigma (\omega ,\rho)\\ \omega \sim \theta_{i}}} [ u_{i}(a,\omega ,\rho) - t_{i}(\rho)]\geq \max_{\delta}\mathop{\mathbb{E}}_{\substack{a\sim \sigma (\omega ;\rho^{\prime}_{i},\rho_{-i}),\\ \omega \sim \theta_{i}}} [ u_{i}(\delta (a_{i}),a_{-i},\omega ,\rho) - t_{i}(\rho^{\prime}_{i},\rho_{-i})] \\ \geq \underset { \begin{array}{c} a \sim \sigma \left(\omega ; \rho_ {i} ^ {\prime}, \rho_ {- i}\right) \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} \left[ u _ {i} \left(\delta \left(a _ {i}\right), a _ {- i}, \omega , \rho\right) - t _ {i} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \right] \tag {27} \\ \end{array}
$$
This is exactly the ex post IC constraint described in Equation 2
B.5 Proof of Lemma 4.4
Proof. We first prove the forward direction: if any mechanism  $\mathcal{M} = (\sigma^w, t^w)$  where  $\sigma_i^w(\omega, \rho) = (\pi_{i,\omega}^w(\rho))$  is interim IC, then  $\widehat{RGT}_i^w = 0, \forall i \in [n]$ .
For the BIC setting, for each  $(\rho_i,\rho_i^{\prime})\in \mathcal{P}_i^2$  and for each deviation function  $\delta :\mathcal{A}_i\to \mathcal{A}_i$ , a BIC mechanism satisfies:
$$
\hat {\mathbb {E}} _ {- i} \left[ \underset { \begin{array}{c} a \sim \sigma (\omega , \rho) \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (a, \omega , \rho) - t _ {i} (\rho) ] \right] \geq \hat {\mathbb {E}} _ {- i} \left[ \underset { \begin{array}{c} a \sim \sigma (\omega ; \rho_ {i} ^ {\prime}, \rho_ {- i}) \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (\delta (a _ {i}), a _ {- i}, \omega , \rho) - t _ {i} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) ] \right] \tag {28}
$$
Note that the above hold for any deviation function  $\delta$ , therefore we can write that
$$
\hat {\mathbb {E}} _ {- i} \left[ \underset { \begin{array}{c} a \sim \sigma (\omega , \rho) \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (a, \omega , \rho) - t _ {i} (\rho) ] \right] \geq \max  _ {\delta} \hat {\mathbb {E}} _ {- i} \left[ \underset { \begin{array}{c} a \sim \sigma (\omega ; \rho_ {i} ^ {\prime}, \rho_ {- i}) \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} [ u _ {i} (\delta (a _ {i}), a _ {- i}, \omega , \rho) - t _ {i} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) ] \right] \tag {29}
$$
We can expand to get the following equations: