SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} v _ {i} \cdot \hat {\mathbb {E}} _ {- i} \left[ \sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho_ {i}, \rho_ {- i}) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho_ {i}, \rho_ {- i}) \theta_ {j, k} \right] - t _ {i} (\rho_ {i}) \\ \geq \max  _ {\delta} v _ {i} \cdot \hat {\mathbb {E}} _ {- i} \left[ \left(\sum_ {k \in [ m ]} \pi_ {i, \delta (k), k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, \delta (k)} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k}\right) \right] - t _ {i} \left(\rho_ {i} ^ {\prime}\right) \\ = \max  _ {\delta} v _ {i} \cdot \left(\sum_ {k \in [ m ]} \hat {\mathbb {E}} _ {- i} \left[ \pi_ {i, \delta (k), k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, \delta (k)} - \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k} \right]\right) - t _ {i} \left(\rho_ {i} ^ {\prime}\right) \\ = v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {\delta} \hat {\mathbb {E}} _ {- i} \left[ \pi_ {i, \delta (k), k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, \delta (k)} - \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k} \right]\right) - t _ {i} \left(\rho_ {i} ^ {\prime}\right) \\ = v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {k ^ {\prime} \in [ m ]} \hat {\mathbb {E}} _ {- i} \left[ \pi_ {i, k ^ {\prime}, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, \delta (k)} - \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k} \right]\right) - t _ {i} \left(\rho_ {i} ^ {\prime}\right) \tag {30} \\ \end{array}
$$
The above equations follow from the linearity of expectations and from pushing the max to the inside (since other terms don't involve  $\delta(k)$ ).
Note that by our definition of interim regret in Equation 11, the above equation is equivalent to  $\widehat{rgt_i^w} (\rho_i',\rho_i)\leq 0$ . Furthermore, this holds for any deviating report  $\rho_{i}^{\prime}$ . Therefore, we can write that
$$
\max  _ {\rho_ {i} ^ {\prime} \in \mathcal {P} _ {i}} \widehat {r g t} _ {i} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {i}\right) \leq 0 \tag {31}
$$
$$
\begin{array}{l} \max _ {\rho_ {i} ^ {\prime} \in \mathcal {P} _ {i}} \widehat {r g t} _ {i} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho\right) \geq \widehat {r g t} _ {i} ^ {w} \left(\rho_ {i}, \rho_ {i}\right) \\ = v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {k ^ {\prime} \in [ m ]} \hat {\mathbb {E}} _ {- i} \left[ \pi_ {i, k ^ {\prime}, k} ^ {w} (\rho_ {i}, \rho_ {- i}) \theta_ {i, k ^ {\prime}} - \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho_ {i}, \rho_ {- i}) \theta_ {j, k} \right]\right) - t _ {i} (\rho_ {i}) \\ - v _ {i} \cdot \hat {\mathbb {E}} _ {- i} \left[ \sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} (\rho_ {i}, \rho_ {- i}) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} (\rho_ {i}, \rho_ {- i}) \theta_ {j, k} \right] + t _ {i} (\rho_ {i}) \\ = v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {k ^ {\prime} \in [ m ]} \hat {\mathbb {E}} _ {- i} \left[ \pi_ {i, k ^ {\prime}, k} ^ {w} \left(\rho_ {i}, \rho_ {- i}\right) \theta_ {i, k ^ {\prime}} \right] - \hat {\mathbb {E}} _ {- i} \left[ \pi_ {i, k, k} ^ {w} \left(\rho_ {i}, \rho_ {- i}\right) \theta_ {i, k} \right]\right) \\ \geq 0 \tag {32} \\ \end{array}
$$
where the second step follows from linearity of expectation, and the last step holds since we are taking the max of all possible  $k' \in [m]$ , which includes  $k' = k$ .
Combining Equations 31 and 32, we have that  $\max_{\rho_i' \in \mathcal{P}_i} \widehat{rgt}_i^w(\rho_i', \rho_i) = 0$ . Taking the expectation over all profiles  $\rho$ , we have  $\hat{\mathbb{E}}_{\rho \in \mathcal{P}}\left[\max_{\rho_i' \in \mathcal{P}_i} \widehat{rgt}_i^w(\rho_i', \rho_i)\right] = 0 \implies \widehat{RGT}_i^w = 0, \forall i \in [n]$ . Thus, we've showed that if any mechanism  $\mathcal{M} = (\sigma^w, t^w)$  where  $\sigma_i^w(\omega, \rho) = (\pi_{i,\omega}^w(\rho))$  is BIC, then  $\widehat{RGT}_i^w = 0$ , as desired.
Next, we consider the reverse direction: if  $\widehat{RGT}_i^w = 0$ ,  $\forall i \in [n]$ , 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 BIC. Starting with  $\widehat{RGT}_i^w = 0$ ,  $\forall i \in [n]$ , we have that by definition:  $\hat{\mathbb{E}}_{\rho \in \mathcal{P}}\left[\max_{\rho_i' \in \mathcal{P}_i} \widehat{rgt}_i^w(\rho_i', \rho_i)\right] = 0$
We showed in Equation 32 that  $\widehat{rgt_i^w} (\rho_i,\rho_i)\geq 0$ . Since  $\rho_{i}\in \mathcal{P}_{i}$ , we have that  $\max_{\rho_i'\in \mathcal{P}_i}\widehat{rgt}_i^w (\rho_i',\rho_i)\geq 0$ . Together with the fact that  $\hat{\mathbb{E}}_{\rho \in \mathcal{P}}\left[\max_{\rho_i'\in \mathcal{P}_i}\widehat{rgt}_i^w (\rho_i',\rho_i)\right] = 0$ , we have that  $\max_{\rho_i'\in \mathcal{P}_i}\widehat{rgt}_i^w (\rho_i',\rho_i) = 0\Rightarrow \widehat{rgt}_i^w (\rho_i',\rho_i)\leq 0$ . Plugging in our definition for interim regret, we have that:
$$
\begin{array}{l} v _ {i} \cdot \left(\sum_ {k \in [ m ]} \max  _ {k ^ {\prime} \in [ m ]} \hat {\mathbb {E}} _ {- i} \left[ \pi_ {i, k ^ {\prime}, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {i, k ^ {\prime}} - \bar {\alpha} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \theta_ {j, k} \right]\right) - t _ {i} \left(\rho_ {i} ^ {\prime}\right) \\ \leq v _ {i} \cdot \hat {\mathbb {E}} _ {- i} \left[ \sum_ {k \in [ m ]} \pi_ {i, k, k} ^ {w} \left(\rho_ {i}, \rho_ {- i}\right) \theta_ {i, k} - \bar {\alpha} \sum_ {k \in [ m ]} \sum_ {j \in [ n ] \backslash i} \pi_ {j, k, k} ^ {w} \left(\rho_ {i}, \rho_ {- i}\right) \theta_ {j, k} \right] - t _ {i} \left(\rho_ {i}\right) \tag {33} \\ \end{array}
$$
$$
\begin{array}{l} \hat{\mathbb{E}}_{-i}\left[ \begin{array}{c}\mathbb{E}_{\substack{a\sim \sigma (\omega ,\rho)\\ \omega \sim \theta_{i}}} [u_{i}(a,\omega ,\rho) - t_{i}(\rho)] \end{array} \right]\geq \max_{\delta}\hat{\mathbb{E}}_{-i}\left[ \begin{array}{c}\mathbb{E}_{\substack{a\sim \sigma (\omega ;\rho^{\prime}_{i},\rho_{-i})\\ \omega \sim \theta_{i}}} [u_{i}(\delta (a_{i}),a_{-i},\omega ,\rho) - t_{i}(\rho^{\prime}_{i},\rho_{-i})] \end{array} \right] \\ \geq \hat {\mathbb {E}} _ {- i} \left[ \underset { \begin{array}{c} a \sim \sigma \left(\omega ; \rho_ {i} ^ {\prime}, \rho_ {- i}\right) \\ \omega \sim \theta_ {i} \end{array} } {\mathbb {E}} \left[ u _ {i} \left(\delta \left(a _ {i}\right), a _ {- i}, \omega , \rho\right) - t _ {i} \left(\rho_ {i} ^ {\prime}, \rho_ {- i}\right) \right] \right] \tag {34} \\ \end{array}
$$
B.6 Setup and Hyper-parameters
Training: For the multi-buyer setting, all our neural networks consist of 3 hidden layers with 200 hidden units each. For the IC setting, we sample a minibatch of 1024 samples online to update the parameters of the neural networks. We sample 100 misreports for every sample to compute the best initialization. For the BIC setting, we use a minibatch of 128 samples. We compute the interim values wherever required over 512 samples drawn from  $\mathcal{P}_{-i}$ .
We observe that sampling alone is sufficient to approximate the regret while training and do not perform any additional gradient descent steps, as this does not improve the training performance (however we anticipate this step would be necessary for larger settings).
We train the neural networks for 20000 iterations and make parameter updates using the Adam Optimizer with a learning rate of 0.001. The Lagrangian parameters  $(\lambda_1, \dots, \lambda_n)$  are set to 10.0. The coefficient of the penalty term is initialized to 1.0. The Lagrangian updates are performed once every 100 iteration.
Testing: We report all our results on a test size of 20000 samples. For the ex post setting, we use 100 misreports as initialization to warm-start the inner maximization to compute regret. For the BIC setting, we use other samples of the minibatch to compute a defeating misreport as noted in the previous subsection. We then run 100 steps of gradient ascent with a learning rate of 0.005 to compute the defeating misreport and the regret more accurately.
Training time: RegretNet takes 11 - 12 minutes per experiment for training for all the settings studied in this paper. All our experiments were run on a single NVIDIA Tesla V100 GPU.
C Experimental Results for the Single Buyer Setting
C.1 Buyers with world prior heterogeneity
We present optimal menus, associated prices, revenue, and RochetNet revenue for the Single Buyer Setting A and B in Figure 6
Distribution	Rchet Menu	Rchet rev	Opt rev
Experiment	Price
Setting A	[1.00 0.00]	0.25	0.125	0.125
[0.00 1.00]
Setting B	[0.78 0.22]	0.14	0.167	0.166
[0 1.00]
[1.00 0.00]	0.26
[0.00 1.00]
Figure 6: Menu(s) and associated prices learned by RochetNet, RochetNet revenue and Optimal revenue on test data in Settings A and B. RochetNet recovers the optimal menu's for both these settings
C.2 Additional Experiments for Buyers with world prior heterogeneity
We design additional experiments where we use RochetNet to study how the informativeness of learned experiments vary when we change different properties of the prior distributions. For this, we consider the mixture of Beta distributions considered in setting B. This is a bimodal distribution with modes at  $\theta_{L} = \left(\frac{7}{36},\frac{29}{36}\right)$  and  $\theta_{H} = \left(\frac{59}{88},\frac{29}{88}\right)$ . We call the former "low type" and the latter "high type" following the analysis of [7] for the case of binary types, and order this way because
This is exactly the interim IC constraint described in Equation 1
Figure 7: Results for Experiment G (left) and H (Right). For both experiments, the vertical axis is the differential informativeness  $q$  of the experiment used by  $\theta_{L}$ . For Experiment G (left), the horizontal axis is the frequency of the high type, and for Experiment H (right), the horizontal axis is the precision of the low type's prior belief  $|\theta_{L} - 0.5|$ .
$|\theta_L - 0.5| \geq |\theta_H - 0.5|$ . We also call the value  $(|\theta - 0.5|)$  the precision of a type's prior belief. We design two experiments:
G. Single buyer, Binary State, and Binary Actions, where the payoffs  $v_{1} = 1$  and the interim beliefs are drawn from a mixture of  $Beta(8,30)$  and  $Beta(60,30)$  weighted by  $(1 - \gamma ,\gamma)$  for  $\gamma \in [0,1]$ .
H. Single buyer, Binary State, and Binary Actions, where the payoffs  $v_{1} = 1$  and the interim beliefs are drawn from an equal weight mixture of  $Beta(a_{1}, b_{1})$  and  $Beta(60, 30)$ . We vary  $a_{1}, b_{1}$  so that the mode of  $Beta(a_{1}, b_{1})$  decreases (therefore the precision of the low type's prior belief  $|\theta_{L} - 0.5|$  increases) while ensuring that the ratio of values at the modes of these distributions stays constant.
In particular, the modes of the two distributions are  $\theta_{L} = \left(\frac{a - 1}{a + b - 2},\frac{b - 1}{a + b - 2}\right)$  and  $\theta_{H} = \left(\frac{59}{88},\frac{29}{88}\right)$  in the case if Experiment G.4.  $\theta_{L}$  and  $\theta_{H}$  are non-congruent types, i.e., without the supplemental information, a buyer with type  $\theta_{L}$  takes action 0 while the other takes action 1.  $\theta_{H}$  also values a fully informative experiment more than  $\theta_{L}$ . In the first experiment, we change the likelihood of type  $\theta_{H}$  with respect to type  $\theta_{L}$ . In the second experiment, we change the precision of the belief of one type while keeping the other fixed (we vary values of  $a,b$  while numerically ensuring that the probability distribution function of all the plotted points have the same height at the mode).
The results are given in Figure 7. For Experiment G, increasing the high type's frequency decreases the low type experiment's informativeness. For Experiment H, increasing the precision of the low type's prior belief decreases the low type experiment's informativeness. [7] characterize the informativeness of the optimal experiment for similar settings, but for discrete distributions with two types. In particular, Proposition 3 from [7] states that informativeness of the low type  $\pi_{1,1}$ , decreases when the frequency of the high type or with the precision of low type's prior belief  $|\theta_L - 0.5|$ . It is interesting, then, that we observe the same behaviour in the RochetNet results for a continuous analog of their discrete distribution set-up. This suggests a new target for economic theory.
D Experimental Results for the Multi Buyer Setting
We consider the following multi-buyer data market design problem with two buyers, binary action and fixed interim beliefs, adopting all possible combinations of the following choices in regard to the configuration of the economy:
$$
\bullet \theta = (0. 5, 0. 5) \text {o r} \theta = (0. 7 5, 0. 2 5)
$$
This also requires  $\left| \frac{a - 1}{a + b - 2} - 0.5 \right| \geq \left| \frac{59}{88} - 0.5 \right|$ , otherwise  $Beta(a, b)$  becomes the high type instead. For our experiments, we only consider data points at which this condition holds, in other words, we only consider combinations of  $a, b$  such that  $Beta(a, b)$  is the low type with mode at  $\theta_L$  and  $Beta(60, 30)$  is the high type with mode at  $\theta_H$ .
-  $\alpha = 0.5$  or  $\alpha = 2.0$
- Both payoffs drawn from UNF, EXP, or Asym UNF
UNF is the uniform distribution over the unit interval  $U[0,1]$ . EXP is the exponential distribution with  $\lambda = 1$ . For the Asym UNF, the payoff for buyer  $i$  is uniform over the interval  $[0,i + 1]$ . In the next two subsections, we present our results for the BIC and ex post IC settings, respectively.
D.1 BIC Settings
Figures 8, 9, and 10 show the optimal data market design and the data marker learned by RegretNet for the BIC settings described above, and Figure 11 gives the test revenue and regret obtained by RegretNet and the revenue of the optimal mechanism.
Figure 8: Experiments learned for BIC constraints when the payoffs are drawn from UNF
Figure 9: Experiments learned for BIC constraints when the payoffs are drawn from EXP
D.2 Ex post IC Settings
Figures 12, 13, and 14 show the optimal data market design and the data market learned by RegretNet for the ex post IC settings described above, and Figure 15 gives the test revenue and test regret obtained by RegretNet along with the revenue of the optimal mechanism.
Figure 10: Experiments learned for BIC constraints when the payoffs are drawn from Asym UNF
Distribution	α	θ	RegretNet	Opt
rev	rgt	rev
UNF	0.5	(0.5, 0.5)	0.400	<0.001	0.395
(0.75, 0.25)	0.237	0.237
2.0	(0.5, 0.5)	1.056	1.042
(0.75, 0.25)	0.763	0.75
EXP	0.5	(0.5, 0.5)	0.632	<0.001	0.632
(0.75, 0.25)	0.448	0.447
2.0	(0.5, 0.5)	1.603	1.613
(0.75, 0.25)	1.390	1.415
Asym UNF	0.5	(0.5, 0.5)	0.614	<0.001	0.609
(0.75, 0.25)	0.366	0.369
2.0	(0.5, 0.5)	1.599	1.594
(0.75, 0.25)	1.139	1.135
Figure 11: Test Revenue and Test Regret obtained by RegretNet for the BIC Settings.
Figure 12: Experiments learned for ex post IC constraints when the payoffs are drawn from UNF
Exponential,  $\alpha = 2.00$ $\theta = 0.50$
Exponential,  $\alpha = 2.00,\theta = 0.75$
Figure 13: Experiments learned for ex post IC constraints when the payoffs are drawn from EXP
Figure 14: Experiments learned for ex post IC constraints when the payoffs are drawn from Asym UNF
Distribution	α	θ	RegretNet	Opt
rev	rgt	rev
UNF	0.5	(0.5, 0.5)	0.277	<0.001	0.27
(0.75, 0.25)	0.14	0.135
2.0	(0.5, 0.5)	0.553	0.541
(0.75, 0.25)	0.278	0.270
EXP	0.5	(0.5, 0.5)	0.405	<0.001	0.405
(0.75, 0.25)	0.204	0.202
2.0	(0.5, 0.5)	0.801	0.809
(0.75, 0.25)	0.418	0.405
Asym UNF	0.5	(0.5, 0.5)	0.426	<0.001	0.423
(0.75, 0.25)	0.421	0.21
2.0	(0.5, 0.5)	0.84	0.841
(0.75, 0.25)	0.423	0.42
Figure 15: Test Revenue and Test Regret for RegretNet for the ex post IC Settings.
D.3 Additional Results for the Irregular Distributions
We present additional results for both BIC and ex post IC settings for two buyers, binary actions, and interim beliefs  $\theta = (0.5, 0.5)$ , the same for each buyer. We set  $\alpha = 0.5$ , and the payoffs are drawn from the irregular distribution whose pdf  $f(v)$  is:
$$
f (v) = \left\{ \begin{array}{l l} 2. 5 & \text {i f} 0 \leq v <   0. 3 \\ 0. 5 & 0. 3 \leq v <   0. 8 \end{array} \right. \tag {35}
$$
The optimal solutions for these problems make use of ironed virtual values as the payoff distribution is irregular (see Fig 16).
Figure 16: Experiments learned for BIC and ex post IC constraints when the payoffs are drawn from an irregular distribution whose pdf is given by Eqn 35. For both these settings, we use the ironed virtual value functions to compute the optimal data market design.
D.4 Proof of Theorem 6.1
Theorem 6.1. Consider the setting with Binary State and Binary Actions where the buyers have a common interim belief  $\theta_{i} = \theta$ . The payoff  $v_{i}$  for a buyer  $i$  is drawn from a regular distribution  $\mathcal{V}_i$  with a continuous density function. Define virtual value  $\phi_i(v_i) = v_i - \frac{1 - F(v_i)}{f(v_i)}$ , for pdf  $f$  and cdf  $F$  of distribution  $\mathcal{V}_i$ . The revenue-optimal mechanism satisfying ex post IC and IR is a mechanism that sells the fully informative experiment to buyer  $i$  if  $\phi_i(v_i) \geq \bar{\alpha} \sum_{i \neq j} \phi_j(v_j)$ . Otherwise buyer  $i$  receives an uninformative menu, where the signal corresponds to the most likely state based on the prior.
Proof. We consider a setting with binary states, binary actions, and where each buyer has the same interim belief, set to  $(\theta, 1 - \theta)$ . Without loss of generality, let  $\theta \geq (1 - \theta)$ . Since the interim beliefs are fixed, we will just represent buyer types with  $v_{i}$  instead of  $\rho_{i}$ . Let  $x_{i}(v) = \mathbb{E}_{\substack{a \sim \sigma(\omega; \rho), \\ \omega \sim \theta_{i}}} [1 \{a_{i} = \omega\}]$ .
Let  $\pi_i(v)$  be the matrix representation of the experiment assigned to buyer  $i$  i.e.  $\pi_{i,\omega}(v) = \sigma_i(\omega ,v)$
We first show that a mechanism is obedient if and only if the experiment assigned to the buyer satisfies  $x_{i}(v) \geq \theta$ . If a mechanism is obedient, then  $\theta \cdot \pi_{i,1,1}(v) \geq (1 - \theta)\pi_{i,2,1}(v)$  and  $(1 - \theta)\pi_{i,2,2}(v) \geq \theta \cdot \pi_{i,1,2}(v)$ . We have  $x_{i}(v) = \theta \cdot \pi_{i,1,1}(v) + (1 - \theta)\pi_{i,2,2}(v) \geq \theta \cdot \pi_{i,1,1}(v) + \theta \cdot \pi_{i,1,2}(v) = \theta (\pi_{i,1,1}(v) + \pi_{i,1,2}(v)) = \theta$ .
In the other direction, consider when  $x_{i}(v) \geq \theta$ . In this case, we show that both  $\theta \cdot \pi_{i,1,1}(v) \geq (1 - \theta)\pi_{i,2,1}(v)$  and  $(1 - \theta)\pi_{i,2,2}(v) \geq \theta \cdot \pi_{i,1,2}(v)$  need to hold for obedience. Assume to the contrary, one of these fails to hold when  $x_{i}(v) \geq \theta$ . We have one of,
-  $(1 - \theta)\pi_{i,2,2}(v) < \theta \cdot \pi_{i,1,2}(v)$ , and we have  $x_{i}(v) = \theta \cdot \pi_{i,1,1}(v) + (1 - \theta)\pi_{i,2,2}(v) < \theta \cdot \pi_{i,1,1}(v) + \theta \cdot \pi_{i,1,2}(v) = \theta (\pi_{i,1,1}(v) + \pi_{i,1,2}(v)) = \theta$ .
-  $\theta \cdot \pi_{i,1,1}(v) < (1 - \theta)\pi_{i,2,1}(v)$ , and we have  $x_{i}(v) = \theta \cdot \pi_{i,1,1}(v) + (1 - \theta)\pi_{i,2,2}(v) < (1 - \theta)\pi_{i,2,1}(v) + (1 - \theta)\pi_{i,2,2}(v) = (1 - \theta)(\pi_{i,2,1}(v) + \pi_{i,2,2}(v)) = (1 - \theta) \leq \theta$ .
In either case, we have a contradiction.
Since our payoff is linear, from Proposition 3.5 in [7], the IC constraints are satisfied only when the truthfulness and obedience constraints are satisfied. Denote  $\tilde{x}_i(v) = x_i(v) - \frac{\alpha}{n - 1}\sum_{j\in [n]\backslash i}x_j(v)$ .
Thus, the optimal design problem to solve is:
$$
\mathbb {E} _ {v \sim \mathcal {V}} \left[ \sum_ {i \in [ n ]} t _ {i} (v _ {i}) \right]
$$
$$
\begin{array}{l} \text {s . t .} v _ {i} \tilde {x} _ {i} (v) - t _ {i} (v) \geq v _ {i} \tilde {x} _ {i} \left(v _ {i} ^ {\prime}; v _ {- i}\right) - t _ {i} \left(v _ {i} ^ {\prime}; v _ {- i}\right) \quad \forall v \in \mathcal {V}, v _ {i} ^ {\prime} \in \mathcal {V} _ {i}, i \in [ n ] \\ v _ {i} \tilde {x} _ {i} (v) - t _ {i} (v) \geq v _ {i} (\theta - \alpha) \quad \forall v \in \mathcal {V}, i \in [ n ] \\ x _ {i} (v) \geq \theta \quad \forall v \in \mathcal {V}, i \in [ n ] \\ \end{array}
$$
The first constraint corresponds to truthfulness for the IC setting, the second is the IR constraint, and the third is the obedience constraint. We have thus reduced the data market design problem to that of Myerson's revenue maximizing single-item auction problem. Instead of  $\tilde{x}_i(v)$  denoting the probability of allocating an item, it denotes expected payoff after accounting for the negative externalities. Also, rather than allocative constraints, we have obedience constraints on  $x_{i}(v)$  which requires  $x_{i}(v)\geq \theta$ . Thus, by Myerson's theory, the total expected revenue is equal to the expected virtual welfare minus some constant  $K$  (stemming from IR constraints) for some  $\tilde{x}_i(v_i,v_{-i})$  that is non-decreasing in  $v_{i}$ . Thus, we have:
$$
\begin{array}{l} \mathbb {E} _ {v \sim \mathcal {V}} \left[ \sum_ {i \in [ n ]} \left[ t _ {i} \left(v _ {i}\right) \right] \right] = \mathbb {E} _ {v \sim \mathcal {V}} \left[ \sum_ {i \in [ n ]} \phi_ {i} \left(v _ {i}\right) \tilde {x} _ {i} (v) \right] - K (36) \\ = \mathbb {E} _ {v \sim \mathcal {V}} \left[ \sum_ {i \in [ n ]} \phi_ {i} (v _ {i}) \left(x _ {i} (v) - \frac {\alpha}{n - 1} \sum_ {j \in [ n ] \backslash i} x _ {j} (v)\right) \right] - K (37) \\ = \mathbb {E} _ {v \sim \mathcal {V}} \left[ \sum_ {i \in [ n ]} \left(\phi_ {i} (v _ {i}) - \frac {\alpha}{n - 1} \sum_ {j \in [ n ] \backslash i} \phi_ {j} (v _ {j})\right) x _ {i} (v) \right] - K (38) \\ \end{array}
$$
In order to maximize the revenue, we need to maximize the virtual welfare. Thus we can set  $x_{i}(v) = 1$  (a fully informative experiment) when  $\phi_{i}(v_{i}) - \sum_{j\in [n]\setminus i}\phi_{j}(v_{j})\geq 0$  and we  $x_{i}(v) = \theta$  (an uninformative experiment that always sends the same signal regardless of the state) otherwise. This is precisely the mechanism described in the theorem.
Note that such an  $\tilde{x}_i(v_i,v_{-i})$  is non-decreasing in  $v_{i}$  (since  $x_{i}(v_{i},v_{-i})$  is non-decreasing in  $v_{i}$  and  $x_{j}(v_{i},v_{-i})$  for  $j\in [n]\setminus i$  is non-increasing in  $v_{i}$  for regular distributions  $\mathcal{V}_i$  and  $\mathcal{V}_j$ ). Moreover,  $x_{i}(v)\geq \theta$  is also satisfied.