Data Market Design through Deep Learning
Authors: Sai Srivatsa Ravindranath
Yanchen Jiang
David C. Parkes
Venue: NeurIPS 2023 poster (2023)
Keywords: Data Markets
Information Design
Differentiable Economics
Economics
Deep Learning
Mechanism Design
Algorithmic Game Theory

Abstract: The  _data market design_ problem is a problem in economic theory to find a set of signaling schemes (statistical experiments) to maximize expected revenue to the information seller, where each experiment reveals some of the information known to a seller and has a corresponding price. Each buyer has their own decision to make in a world environment, and their subjective expected value for the information associated with a particular experiment comes from the improvement in this decision and depends on their prior and value for different outcomes. In a setting with multiple buyers, a buyer's expected value for an experiment may also depend on the information sold to others. We introduce the application of deep learning for the design of revenue-optimal data markets, looking to expand the frontiers of what can be understood and achieved. Relative to earlier work on deep learning for auction design, we must learn signaling schemes rather than allocation rules and handle  _obedience constraints_  &mdash; these arising from modeling the downstream actions of buyers &mdash; in addition to incentive constraints on bids.  Our experiments demonstrate that this new deep learning framework can almost precisely replicate all known solutions from theory, expand to more complex settings, and be used to establish the optimality of new designs for data markets and make conjectures in regard to the structure of optimal designs.

Data Market Design through Deep Learning
Sai Srivatsa Ravindranath" Yanchen Jiang" Harvard John A. Paulson School of Engineering and Applied Sciences {saisr, yanchen_jiang parkes) &g hw..haarrdedd.
David C. Parkes
Abstract
The data market design problem problem in economic theory to find a set of signaling schemes (statistical experiments) to maximize expected revenue the information seller, where each experiment reveals some of the information known to seller and has corresponding price [7]. Each buyer has their own decision to make in a world environment, and their subjective expected value for the information associated with particular experiment comes from the improvement in this decision and depends on their prior and value for different outcomes. In setting with multiple buyers, a buyer's expected value for experiment may also depend on the information sold to others [12.. We introduce the application deep learning for the design of revenue-optimal data markets, looking expand the frontiers of what can be understood and achieved. Relative to earlier work on deep learning for auction design (32), must learn signaling schemes rather than allocation rules and handle obedience constraints-thes arising from modeling the downstream actions of buyers- in addition incentive constraints on bids. Our experiments demonstrate that this new deep learning framework can almost precisely replicate all known solutions from theory, expand to more complex settings, and be used to establish the optimality of new designs for data markets and make conjectures in regard to the structure of optimal designs.
Introduction
Many characterize the current era the Information Age. Companies such as Google and Meta (search and social media), Experian and TransUnion (credit agencies), and Amazon and American Express (commeree) hold vast quantities of data about individuals. In turn, this has led data markets, where information about an individual can be purchased in real-time to guide decision- making (e.g., LiveRamp, Segment, Bloomreach). In this paper, we advance the design of market rules that govern the structuring and sale of this kind of information. Using machine learning, we leverage existing theoretical frameworks 7 [2] replicate all known solutions from theory, expand to more complex settings, establish the optimality of new designs, and make conjectures in regard the structure of optimal designs. Although this is not our focus, there are also important ethical questions in regard to data markets, in regard to privacy, ownership, informed consent, and the use of
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In settings with a single buyer, Bergemann et al. 7 introduce a framework in which there a data buyer who faces decision problem under uncertainty, and has payoff depending their choice of action and the underlying state of the world. There is a data seller who knows the world state, and can disclose information through menu of statistical experiments, each experiment offering stochastic signal that reveals some (or all) of the seller's information and each of which is associated with price. The buyer's willingness to pay for information is determined by their rype, which defines their prior belief and their value for different outcomes in the world (these arising from the action that the buyer will choose take).
Equal Contribution.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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The optimal data market design problem is to find a set of experiments and associated prices with which to maximize expected revenue, given a distribution over the buyer's type. Bergemann et al. 7 characterize the optimal design when there is binary world state and buyer with binary action. In some settings there may be multiple buyers and buyers may compete downstream, and the more informed one buyer is, the lower the payoff may become for others. Bonatti et al. [12] take up this problem, and characterize the optimal design for multiple buyers in the binary state/binary action setting, and further limited to buyers who each have common prior on the world state. It remains an open problem to obtain theoretical results for richer, multi-buyer settings, and this motivates the need for gaining new understanding and making progress through computational approaches.
Contributions. Inspired by the recent advances in differential economics (B32 335 38 46 47 44 45 23
36 24, 29] etc.), we imitiate the use of deep learning for the automated design of data markets. This market sells information rather than allocates resources, and the value of information depends on the way in which buyer will use the information- and this downstream action by a buyer needs to be modeled. Following the revelation principle for dynamic games 43 Section 6.3), it is without loss of generality to model an experiment as generating recommended action and insisting on designs in which the buyer will prefer to follow this action. This brings about new challenges, most notably to extend this framework to handle obedience. While the other aspect of incentive compatibility that we need to handle is more typical, i.e., that of achieving incentive alignment so that buyer will choose to truthfully report their prior beliefs and values for different outcomes, we also need to ensure there are no useful double deviations, where a buyer simultaneously misreports their type and acts contrary to the seller's ' recommended action.
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In settings with a single buyer, we learn an explicit, parameterized representation of menu of priced experiments to offer the buyer and with which to model the action choices available to the buyer. In this way, we extend the RochetNer architecture [32] that has been used successfully for optimal auction design with single buyer. This enables us to obtain exact incentive compatibility for the single buyer setting: a buyer has no useful deviation from a recommended action, no useful deviation in reporting their type, and no useful double deviation. In settings with multiple buyers, we seek to learn revenue- revenue-maximiziig maximizing designs while also minimizing deviations in disobeying action recommendations and misreporting types and including double deviations. This extends the RegretNet framework (32) that has been used successfully for optimal auction design with multiple buyers and gives approximate incentive alignment.
Our first experimental result is to show through extensive experiments that these new neural network architectures and learning frameworks are able to almost exactly recover all known optimal solutions from Bergemann et al. 7 and Bonatti et al. [12. Following the economic theory literature on data market design, we consider the notion of Bayesian incentive compatibility (BIC). To handle this, we use samples of a buyer's type to compute an interim experiment and interim payment, averaging over samples drawn from the type distribution (this builds on the BIC methods for deep learning for auctions that were used in the context of differentiable economics for budget- bunget-constrained auctions (34). We give training method that enables the efficient reuse of computed interim allocations and interim payments from other samples to swiftly calculate the interim utility of misreports, dramatically speeding up training.
Whereas analytical results are only available for the BIC setting, which is, in effect, lower-dimensional, and easier to analyze, we are able to study through computational techniques the design of data markets in the ex post IC setting, which is a setting without existing theory. A second result, in the setting of multiple buyers, is to use our framework to conjecture the structure of an optimal design and prove its optimality (for this, we make use of virtual values analogous to Myerson's 's framework [42)). We see this as important contribution, as ex post IC is stronger notion of IC than BIC.
third result to demonstrate how our framework extends its utility beyond empirical results and serves as a toolbox to guide economic theory. To illustrate this, we study how revenue varies with competition in the of existing theoretical results for this particular setting, our framework enables us to derive trends in revenue effortlessly. We also conjecture the structure of solutions for problems the single buyer setting with an enlarged type, where both the buyer payoffs and priors are uncertain. For this case, we again derive empirical results using our proposed framework and use it to conjecture the properties of the underlying theoretical solution.
setting where the prior information is uncertain. Despite the absence
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Related work. Conitzer and Sandholm 21] introduced the use of automated mechanism design (AMD) for economic design and framed the problem as an integer linear program (or just linear program for Bayesian design). Responding to challenges with the enumerative representations used in these early approaches, Dütting et al. [32] introduced the use of deep neural networks for auction design, attaining more representational flexibility. Since then, there has been a line of work on this so-called approach of differentiable economics, including to problems with budget-constrained bidders [34), for minimizing agents' payments [47), applying to multi-facilityy location problems (35), balancing fairness and revenue ( ((38), and applying to two-sided matching [46]. To the best of our knowledge, none of this work has considered the setting of the design of optimal data markets, which introduce new challenges in regard to handling agent actions (obedience) as well as incorporating negative externalities.
In regard to the data market problem, Bergemann et al. 7 build upon the decision-theoretic model pioneered by Blackwell [T0] and study a setting with a single buyer. Cai and Velegkas [13] also give computational results in this model, making use of linear programs to compute the optimal menu for discrete type distributions. They also investigate generalization of this model that allows multiple agents to compete for useful information. In this setting, at most, one agent receives the information. While this approach can be used for continuous type distribution by applying the LP to discretized valuation space, solving it for even a coarser discretization can be prohibitively expensive. Further advancing economic theory, Bergemann et al. [9] consider more general type distributions and investigate both the cardinality of the optimal menu and the revenue achievable when selling complete information. Bonatti et al. [12] also study the multi-buyer setting modeling competition through negative externality.
Babaioff et al. [6] give a related framework, with the key distinction that the seller is not required to commit to a mechanism before the realization of the world state. As a result, the experiments and prices of the data market can be tailored to the realized state of the world. Chen et al. [8] extend their setting, considering budget-constrained buyers, formulating a linear program to find the solution of the problem for discrete type space.
There exist several other models that study revenue-optimal mechanisms for selling data. For example, Liu et al. [40] characterize the revenuc-optimal mechanism for various numbers of states and actions, and considering general payoff functions. However, in their setup, the state only impacts the payoff of the active action taken by a buyer, which provides considerable simplification and may not be realistic. Li (39] investigates setting where the buyer can conduct their own costly experiment at cost after receiving the signal. Different from the model considered in this paper, after receiving the signal from the data broker, the agent can subsequently acquire additional information with costs. The model also assumed that the valuation function of the agent is separable and that the private type of the agent represents her value of acquiring more information, which is different from the single buyer model studied in this paper, where the prior belief is also drawn from distribution and constitutes a part of the buyer's 's type. Agarwal ct al. [3] explore data marketplaces where each of multiple sellers sells data to buyers who aim to utilize the data for machine learning tasks, and Chen et al. 17 consider scenarios where neither the seller nor the buyer knows the true quality of the data.
Mehta et al. [41 and Agarwal et al. 4 also incorporate buyer externalities into the study of data marketplaces
Another line of research studies problems of information design, for example, the problem of Bayesian Persuasion 37 There, the model different in that the sender of information has preferences on the action taken by the receiver, setting up gaee-rhe problem of strategic misrepresentation by the sender. Dughmi and Xu (3I) studied this from an algorithmic perspective, and Castiglioni et al. [16] also brought in considerations from mechanism design by introducing hidden types; see also and [4] for more work on information design and Bayesian persuasion.
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Preliminaries
Model. We consider setting with data buyers, N [l,...,m]. n.. under uncertainty. The state of the world, w. is unknown and is drawn from finite state space Ω = (wi,- .......... Wm). Each buyer i can choose an action, from a finite set A. Let A= 1 A, The ex post payoff to buyer for choosing action a; E A, under state w is given by Ui(w, az). Unless otherwise specified, we consider the case of matching utility payoffss where the buyer seeks match
each facing a decision problem
3

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the state and the action In such settings, we have [92] = |A,1 for each i E [n], and the payoff is given by 24, (w. a,) = 0 1w = at} where each U; is drawn independently from distribution V,. Let V 1.
Each buyer i also has an interim belief, 0, E A(?), about the world state. Each buyer's belief, 0,, is drawn independently from distribution O Let θ = 1, The type of a buyer is given by the tuple (v,, 0.) and is denoted by P:. If the buyers don't vary in es or their interim beliefs 8, then P: 9g or p: Ug respectively. Let P. - V, x O The utility of a buyer i E [n] is given by the utility function u, A x Ω x P. + R. We assume that the utility of buyer i depends only on its own type Pi but can depend on other players' actions as a negative externality. Additionally, we follow [2] and assume that this externality is separable, thus simplifying our utility uy (a w, Pi) = V/ 1 aa; w) FF pi). In this paper, we consider settings where the negative externality is given by (_ n-I Ejelniy 1/- 1aaj w) where a Rgo
where captures the degree of competitiveness among buyers. Let ā -
Statistical Experiments. There is a data seller who observes the world state and wishes to sell information to one or more buyers to maximize expected revenue. The seller sells information through signaling schemes, where each scheme is called an experiment. The seller chooses, for each buyer i [r]]. set of signals S; and signaling scheme o, Ω + L.₁ If state S2 is realized, then the seller sends a signal drawn from the distribution o (w). Upon receiving signal, the buyers update their prior beliefs and choose an optimal action accordingly. The signaling scheme ag can also be represented as a matrix π; collection of m row vectors each of dimensions |S,]. The j-th row vector (for j E (ml) specifies the likelihood of each signal when state wj realized. Thus we have 0; (wr)) Ti.j.
The Mechanism Design Problem. The mechanism design goal is to design and corresponding prices to maximize the expected revenue of the seller. Let B denote a message (bid) space. Define the signaling schemes for a buyer [n] as o; 52 B + AS; and payment function 4; B - Rz-- Given bids b = (b,,... bn) E B, if state w E Ω is realized, then the seller sends a signal drawn from the distribution og (w. b) buyer and collects payment (b). The sequence of interactions between one or more buyers and the seller takes place as follows:
of experiments
1. The seller commits to a mechanism, .M = (a, t), where (a on) is choice and (p))
2. Each buyer i observes their type, Pt. The seller observes the state, w.
3. Each buyer reports a message b,.
4. The seller sends buyer signal, St Si, generated according to the signaling scheme aw b), and collects payment 4, (b).
5. Each buyer chooses an action a; E A., and obtains utility us us(a, w, pt) ((bb), where
a = (a1....,an)) a.).
By the revelation principle for dynamic games 43 Section 6.3], as long as we consider incentive- compatible mechanisms, is without loss of generality for the message space to be the type space of buyers, and for the size of the signal space to be the size of the action space, i.e., for each E [m]] [S| = [A,]. In such mechanisms, the seller designs experiments where every signal leads a different optimal choice of action. Following 7 we can then replace every signal as an action recommended by the seller.
Incentive Compatibility. A mechanism (ar t) is Bayesian incentive compatible (BIC) if the buyer maximizes their expected utility (over other agents' reports) by both reporting their true type as well as by following the recommended actions. For the sake of notational convenience, let È == Ep_ denote the operator used for computing the interim representations.
For each (pi, pQ) E PP? and for each deviation function 8 A - A., BIC mechanism satisfies:
E anda (w) wmit.
(u:la,w;,)) [x))
> i
(wap w
It is easier extend approach non-matching utilities as well

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In particular, this insists that double deviations (misreporting the type and disobeying the recommen- dation) are not profitable.
We also consider the stronger notion of ex post incentive compatible (IC), which requires, for every agent i, and for each p P, and assuming that every other agent reports its type truthfully and follows the recommended action, then for each misreport p E Pi, and each deviation function, ò : AA + A.. the following condition:
E (w-d(wwp)) wmith
(uela, [x(p)]
(u:(0(a:), lildop-i
awo wm/)
(2)
mechanism is interim individually rational (IIR) reporting an agent's true type guarantees least as much expected utility (in expectation over other agents' reports) as opting out of the mechanism. Let a-i x P_i + AA_; denote the recommendations to other participating buyers when buyer opts out. For each agent and each Pi E P., IIR requires 1)
Individual Rationality,
B_;
E
[us(a;w, tz(p)]
2
E
[uz a- W. p)]
(3)
(wip) wnlf;
a:EA:
- (w)p-1), w
is in the seller's best interest to instantiate the recommendation function to the other buyers when buyer opts out, a-i, minimize the value of the RHS of this IIR inequality. For this reason, is without loss of generality to rewrite the above equation as:
[us a a-w:ww)))) p
E_
E (-
[u;(a; p) - t(() 2 min max O,EA;
Ê_
(4)
ama clurp w0d
In particular, and recognizing that a buyer's utility decreases the more informed other buyers are, the seller can achieve this by sending optimal recommendations to participating buyers in order to minimize the utility of a non-participating buyer.
We also consider a stronger version of IR, namely ex post individual rationality (or simply IR). In this case, for each agent i, and each p P, we require:
E anve(wnp)) wmB
p)) ((()) 2 min max a-x a:EA;
[us aa;;;; p)]
enva (wop-(())-)) w~0
Optimal Data Market Design in the Single-Buye Setting
In this section, we formulate the problem of optimal data market design for single buyer as an unsupervised learning problem. We study parametric class of mechanisms, (o",, ,²" E M, for parameters, w E Rdd, where d > O. For a single buyer, the goal is to learn parameters we that maximize Ep-p [t" (p)), such that (o",t") satisfy ex post IC and IR By adopting differentiable loss functions, we can make use of tools from automatic differentiation and stochastic gradient descent (SGD). For notational convenience, drop the subscript in this case, as there is only one buyer.
Neural Network Architecture. For the single buyer setting, any IC mechanism can be represented as menu of experiments. For this. we extend the RochetNet architecture to represent a menu of priced statistical experiments. Specifically, the parameters correspond to menu of P choices, where each choice E [P] is associated with an experiment, with parameters 7 E x [[]], and a price, 3P E R. Given this, define menu entry as,
exp(12
P
1 -== 3
(6)
Eee(m exp(1
For the single bidder settings, ex post IC and IR are the same BIC and IIR.
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The input layer takes p (v, 0) input and the network computes the P Putility values corresponding to each menu entry. We do not impose obedience constraint explicitly. However, while computing the utility of a menu, we take into consideration the best possible deviating action an agent can take. From the perspective of buyer, if action k is recommended by an experiment choice p. then 4 Pr[su]"
yielding 0, 7 mela Og T. Taking an expectation over all signals, buyer Prisk] receives the following utility for the choice E [P]:
Pr(w- jlaz, 0]
The best action deviation is thus ((k) arg maXjejm]
Pr[+k]
utility of maxye[m] Prisk]
hp( = =
Σ ke[ml
me 10 BP
We also include an additional experiment, n², which corresponds to a null experiment that generates the same signal regardless of state, and thus provides useful information. This menu entry has a price of t = 0, and thus h (p) = u (max]e[m] 02). In particular, this menu entry has the same utility as that of opting out.
Lemma 3.1. Let p² = arg maxpe(O....PP (p) denote the optimal menu choice. Then the
mechanism M = (a"", t"") where o"(ww, p) = nu
=p'(1) (p) and (p) =p" satisfies IC and IR.
This mechanism is IC as it is agent optimizing, and is IR as it guarantees buyer at least the utility of opting out.
Training Problem.
The expected revenue of such an IC and IR mechanism that is parameterized by 3. is given by Ep-p|6e"((>
[e)]. Rather than minimizing a loss function that measures errors against ground truth labels (as in supervised learning setting), our goal is minimize expected negated revenue. To ensure that the objective is differentiable, we replace the argmax operation with a softmax during training. The loss function, for parameters y and B, is thus given by [(y, B)
Ep-P [-E (p) where (p(p) softmax (o) & ""(e)), and T 0 controls the quality of approximation. Note that, at test time, we revert to using the hard max, so as to guarantee exact IC and IR from trained network.
7
During training, this null experiment (77, ,t)) remains fixed, while the parameters y E RPx[[[|x and B E R optimized through SGD the empirical version of the loss calculated over / i.id samples S = (),,,... p(0) drawn from P. We report our results on a separate test set sampled from P. We refer to the Appendix A.2 for more details regarding the hyperparameters.
Optimal Data Market Design in the Multi-Buyer Setting
In the multi-buyer setting, the goal is to learn parameters E Rd that maximize E Di=l 1 (p)) for a parametric class of mechanisms, (o",, t²r) E M, such that (a",, tt) satisfy IC (or BIC) and IR (or IIR). By restricting our loss computations to differentiable functions, we can again use tools from automatic differentiation and SGD. For the multi-buyer setting, we use differentiable approximations to represent the rules of the mechanism and compute the degree to which IC constraints are violated during training adopting an augmented Lagrangian method 33
Neural Network Architecture. The neural network architecture has two components: one that encodes the experiments and another that encodes payments. We model these components a straightforward way as forwadd fully connected neural networks with Leaky ReLU activation functions. The input layer consists of the reported type profile, p. encoded as a vector. For both IC and BIC settings, the component that encodes experiments outputs matrix of dimensions n x |2]] x [.] which represents an experiment that corresponds to each of the n agents. In order to ensure feasibility, i.e., the probability values of sending signals for each agent under each state realization neg-negative and sums up to 1, the neural network first computes an output matrix, स, of the same dimension. For all i, w. we then obtain Ti...j by computing exp( (tww))/ Σx (
For the setting of IC (vs. BIC), we define payments by first computing a normalized payment, i." (p) E [0, 1). for each buyer i, using sigmoid activation unit. The payment t. (p) is computed as

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follows:
Training Problem. In order to train the neural network, we need to minimize the negated revenue subject to incentive constraints. Following Dütting et al. [22], we measure the extent to which mechanism violates the IC (or BIC) constraints through the notion of ex post regret (or interim (p)0sR (
(p)
Σ kE[m]
Tikk
- Σ Σ T),k,(()0ss, -mmax 0, -a)
(8)
ke(m) jEIn/ia
Lemma 4.1. Any mechanism M = (o0,10 where of (w, p) = (T (p)) and 1 satisfies Eqn ex post IR constraint for any w 12d.
For the BIC setting we only need to compute the interim payment, as we can replace p.p..>pPPlllt p-x)] by (((p)). For this, we compute interim normalized payment, & [0, 1]. for each buyer by using a sigmoid unit. We compute the interim payment as:
(p:) = 22th (p:)
Σ REim]
E E
T))x(p)00, (max 0; -a)
(9)
ke(m) jE(nlKi
Lemma 4.2. Any mechanism M = (ott, ,t") where o. (w. p) = (p)) and t satisfies Eqn is interim IR for any E IRd.
regret) and then appeal to Lagrangian optimization. The regret for an agent is given by the maximum increase utility, considering all possible misreports and all possible deviations for given misreport in consideration while fixing the truthful reports of others (or in expectation over truthful reports of others for the BIC setting) when the others are truthful and obedient.
We define the ex post regret for buyer i as RGT, EpEp [max rgt," (d. where rgt," (ph, p) is defined as:
rgl (p4.p) = U1
Σ km
(P ā Σ E'E(m)]
max
Σ
π 10 T P-:.p-))00.
RE(m] JE(NJV
0/4
(p) kE(m]
Σ
Σ
Σ
THe T)kkk(1)88.)
(t.t (p:.p-s) - (p)) (10)
kE[m] jEIn]Ys
Lemma 4.3. Any mechanism M = (a tt") where of (w, p) = (mi)(p) is ex post IC if and only
if RGT. = 0 E [n], except for measure zero events.
We define the interim regret for a buyer as RGT, EpEp max rgt, (p's Pa) where rilh (W/, pc) is defined as:
rgth (pE
Σ max R. K'[mm
w (P
-
Σ
kE|m
jE(m]V
- - B
Σ kE[m]
T (Pssppisp->00.. - ā Σ Σ
T. (P - -(4(0)) (((1))
kE(m) JE(NJ
(11)
Lemma 4.4. Any mechanism = (0", (1)) where a (w, = (p) interim IC if and only if RGT, = =0 vi E [n), except for measure zero events.
Given regret, compute the Lagrangian objective the empirical version of loss and regret over samples S (1) (L)) We solve this objective using augmented Lagrangian optimization.
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The BIC setting also involves an inner expectation to compute the interim representations. There, for each agent we sample a separate subset S-, {o ip_i..... ................................ p_x J. and replace the inner expectations with an empirical expectation over these samples. We sample several misreports, compute the best performing misreport among these samples, and use this as a warm-start initialization for the inner maximization problem. For the BIC setting, rather than sampling fresh misreports and computing the interim experiments and payments, we re-use the other samples from the minibatch and their already computed interim values to find the best initialization. This leads to dramatic speed-up up as the number of forward passes required to compute the regret reduces by an order magnitude of the number of initialized misreports. This, however, works only for the BIC setting, as the constraints are only dependent on the interim values.
We report all our results on a separate test set sampled from P. Please refer to Appendix B.6 for more details regarding the hyperparameters.
5 Experimental Results for the Single Buyer Setting
In this section, we demonstrate the use of RochetNet to recover known existing results from the economic theory literature. Additionally, we also show how can use this approach to conjecture the optimal menu structure for settings outside the reach of current theory. Further, we give in the Appendix C.2 representative examples of how we can characterize the differential informativeness of the menu options when we change different properties of the prior distribution. This builds economic intuition in regard the shape of optimal market designs.
Buyers with world prior heterogeneity. We first show that RochetNet recover the optimal menu for all the continuous distribution settings considered in (7] where the input distributions a continuum of types. We specif- ically consider the following settings with binary states and binary actions with payoft's drawn from:
0.00
-0.05
and interim beliefs
0.10
0.15
partial
info
A. unit interval, i.e, 0 U10, 1].
-0220
full
info
B. equal weight mixture of Beta 30) and (60,
0.00
0.25
0.50
1.00
The optimal menus for each of these settings are given by (7]. For the first setting, the optimal menu consists of fully informative experiment with a price of 0.25. For the second setting, the seller offers two menu options: a partially informative experiment and a fully informative experiment. In each case, RochetNet recovers the exact optimal menu. We describe the optimal menus, associated prices, revenue, and RochetNet revenue in the Appendix C.I
Figure 1: The differential informativeness q of the menu(s) learned by RochetNet for Setting D varying the parameter that in- stantiates the economic environment.
We also give the results from additional experiments in Appendix C.2 where we demonstrate how the informativeness of the menu g changes, as we vary different properties of the economic environment.
Buyers with payoff and world prior heterogencity. tion of optimal data market designs when both v and 0 vary. In such cases, we can use RochetNet conjecture the structure of an optimal solution. For this, we consider the following settings with enlarged buyer types with binary states and binary actions with:
We are
of no theoretical characteriza-
C. u U 1] and the interim beliefs drawn from U(, 1).
D. v ~~ U(e, 1] and the interim beliefs are drawn from an equal weight mixture of Bela 30) and Bela 30) for [0, 1].
For Setting[C] RochetNet learns menu consisting of a single fully informative experiment with price of 0414. For Setting D RochetNet learns a menu consisting of single fully informative experiment when e < 0.55. For higher values of c, RochetNet learns an additional partially informative menu option. Figure [ we show how the differential informativeness q of the partially informative menu changes with c.

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Unilom, a 0,50, 6 0,50
Exponenda
222
200
52
08
124
575
C21
02-
02-
520
635
62
00
05
14
2.5
Uniform 2,00.6
Exponendia
0.50,
325
08
325
64
64
03
215
5.0
Figure 2: Experiments Iearned for Settings and for BIC constraints (top) and ex post IC constraints (bottom). The plot shows the probability of recommending the correct action to buyer 1 (left subplot) and buyer 2 (right subplot) for varying values of (x-axis) and vi2 (y-axis). For each setting, the theoretically optimal experiments represented by solid lines separating the regions.
Experimental Results for the Multi-Buyer Setting
In this section, we show how we can use RegretNet to recover known existing results for the multi- buyer, BIC setting. We also RegretNet conjecture the optimal solution for multi-buyer problem in the ex post IC setting, and then prove the optimality of the design. Lastly, we give results for setting where understand how revenue varies with changes to the intensity of negative externality, again building economic intuition for the market design problem.
analytically hard compute the optimal solution but RegretNet is used to
12] study the multi- bulli--byyerrr market design problem with two buyers, binary actions, and fixed interim beliefs, the same for each agent. They show that deterministic signaling scheme is optimal, and thus characterize their solution in terms of recommending a correct action (selling a fully informative menu). Since the obedience constraints defined on the interim representation, the optimal mechanism also able to sometimes recommend incorrect actions to a buyer (in effect sending recommendations that are opposed with the realized state).
BIC settings.
We consider the following settings from I2 with interim beliefs are given by 0₃ = 002 = (0.5, 0.5) and 0.5 and:
E. The payoffs U1, U2 sampled from the unit interval U00, 1].
F. The payoffs 01, Ui are sampled from the exponential distribution with À = 1.
The results for these settings with BIC constraint are shown in Figure (top). The theoretically optimum solution contains two kinds of recommendations, those recommending the correct action (the optimal action to take given the realized state), and those recommending incorrect actions, separated by the black solid lines. The heatmap showed in the plots denotes our computational results, which show the probability of recommending the correct action. In particular, we can confirm visually (and from its expected revenue) that RegretNet is able recover the optimal revenue as well the optimal experiment design. The test revenue and regret along with additional results are in Appendix D.I.
Ex post IC settings. We apply RegretNet to the same three settings, but now adopting ex post IC constraints. This of interest because there no known analytical solution for the optimal mechanism for two buyers in the ex post IC setting. See Figure (bottom). Based on this, and additional experiments (in Appendix D.2] ), we were able conjecture that the structure of the optimal design in this ex post IC setting, and we prove its optimality by following Myerson's 's framework for single item auctions. We defer the proof to the appendix [D.4

---

Theorem 6.1. Consider the setting with Binary State and Binary Actions where the buyers have common interim belief 0 = 8. The payoff v, for a buyer i is drawn frow a regular distribution V; with a continuous density function. Define virtual value (1(1)) = U; 1-F(y.) f(w)) for pdf f and cdft
F of distribution V.. The revenue-optimal mechanism satisfying ex post IC and IR is a mechanism that sells the fully informative experiment buyer if (v(v)) 2 ā Litj receives an uninformative menu, where the signal corresponds the most likely state based on the prior.
itj 05(2,) Otherwise buyer i
Studying the effect of varying the intensity of externalities. Following 12 we study the effect of the negative externality parameter on the revenue for the BIC, multi-buyer setting with a common and known prior for each buyer and payoffs sampled from U|0 1]. We also con- sider the case where the payoff's are constant but the interim beliefs are independently sampled from U[O, 1]. While there is no known analytical characterization for the latter, we show how easy it is study the models learnt by RegretNet to analyze economic properties of interest. Figure shows the effect of a on the revenue. In the con- text of uncertain priors, we note that the effect revenue is similar to the setting where payoffs are uncertain. As we enhance competition intensity via a, the revenue grows. Even though the buyers are not recommended the correct action, the seller manages to generate revenue by threaten- ing to share exclusive information with competitor. This effect becomes fiercer in the settings with uncertain priors.
2.5
Uncertain 0
Uncertain
2.0
1.5
1.0
Conclusion
0.5
0.0
0.5
1.0
1.5
2.0
2.5
We have introduced a new deep neural network architec- ture and learning framework to study the design of optimal data markets. We have demonstrated through experimental work the flexibility of the framework, showing that it can adapt relatively complex scenarios and facilitate the discovery of new optimal designs. Note that while we have only considered matching utility payoff's in this paper, our approach can easily be extended to non- non-matching utility payoffs as well.
0
Figure 3: Efffcc of a on revenue
We also point out some limitations. First, for our approach to continue to provide insights into the theoretically optimal design for larger problems (e.g., with more buyers, more signals, more actions), it will be important to provide interpretability to the mechanisms learned by RegretNet (designs learned by RochetNet, on the other hand, are immediately interpretable). Second, while our approach scales well with the number of buyers or states in the ex post IC setting, it does not scale as easily with the number of buyers in the BIC setting. The challenge in the BIC setting comes from the interim computations involving conditional expectations over reports of others and scaling beyond what studied in this paper will require new techniques, for example, exploiting symmetry. Third, we are making use of gradient-based approaches, which may suffer from local optima in non-convex problems. At the same time, deep learning has shown success in various problem domains despite non-convexity. The experiments reported here align with these observations, with our neural network architectures consistently recovering optimal solutions, when these are known, and thus optimality can be verified, and despite non-convex formulations. Fourth, we attain in the multi-buyer setting only approximate, and not exact, incentive alignment, and this leaves the question of how much alignment is enough for agents to follow the intended advice of market design (there is little practical or theoretical guidance in this regard). Moreover, we do not know for a given solution whether there is an exact IC solution nearby. While there is some recent guiding theory 222 26 [5] that provides transformations between e-BIC and BIC without revenue loss in the context of auction design, extending these transformations approximate IC settings and to problems with both types and actions presents an interesting avenue for future research.
Lastly, we return to where we started, and underline that markets for trading data about individuals raise substantive ethical concerns [ 2 such as those introduced here can be used to strike new kinds of trade-offs, for example allowing individuals to also benefit directly from trades on data about themselves and to embrace privacy and fairness constraints.
19 33.. Our hope is that machine learning frameworks
10

---

Acknowledgements
We would like to thank Zhe Feng and Alessadro Bonatti for the initial discussions. The source code for all experiments is available from Github at https://g/ thub. wtt http//sisivivtaaa deep-data-markes
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14

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A Data Markets in the Single-Buyer Settings
A.1 RochetNet Architecture
IR Option
h
h
max
u(p)
Figure 4: RochetNet: Neural network architecture for learning the optimal for single bidder setting. neural network takes in as inputs
the buyer's payoff v and interim belief 0 distribution on world state) and computes the utility to the buyer for each of P menu options, where option 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 matrix TP with dimensions [S]] x |S2], along with an associated price fº, Instead of explicitly enforcing the obedience constraint, consider all potential actions 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
T to 200 We train RochetNet for 20, 000 iterations with a minibatch of size 2 sampled online for every update.
1
We report 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 For BIC setting, the experiment network 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 E [t: p-i)] with just 6; (p:).. We also have separate payment networks for each buyer if their type distributions are different.
B.2 Proof of Lemma 4.1
Lemma 4.1. Any mechanism M = (o",(") where (w, (IT and 1 satisfies Eqn 8 ex post IR constraint for any t IRd
Proof. For the post IC setting, we first compute normalized payment, (p) E [0, 1]. for each buyer by using 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 E (p) is
15

---

Experiment Network o
Payment Network
7mm
res, Pgtt.... rof.
Figure 5: RegretNet: Neural network architecture for the multi-buyer ex post IC setting. The inputs the reported types p of each buyer. The experiment network outputs an experiment size [[] x [St] for each buyer, where a row-wise softmax operation ensures the experiments are well defined. The payment network outputs the normalized payment ?," (p) E [0, 1] through sigmoid activation unit that is used to compute payment via Equation g
computed as follows:
& (p) = (p) (p).u) Us
to Σ kE[m)
Tpk -
Σ
"h ())x())00
(max 0i, - a)) (12)
&
kE[m] jE(N]yi
To establish ex post IR, reason about the utility buyer can obtaín when not participating in the mechanism. As per the discussion 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 # i, for an opting out buyer i, Given this, the utility of buyer when opting out, is
such that Eavo- w
[1j w)]
min max G.E.A;
[un(ās, ans
araa ((app-e)) w
1)
max G. E,E.;
Eava- (wip-r) Whit;
Di l(ū; = ---
- Uza
Σ
lfā; =w]
(13)
jENnJ
max n, E,A.;
(Euro [v; l[ā; = w/ - vvae))
= (max 0g - a)
Note that Exe(m) T. (p)0s.t (p)00
max 0.. and ā (Eeejm) Ejetni (1),0(0)000) s
(Ejetmny 1) Thus have Ebeiml Th T. ā (EEEm)) Ejetniss T. 33), (max 0i,6 a) 0.
16

---

Taking this with the fact that (p) E [0, 1], we have
(p) 2.' (p) us
Σ REim]
E
E
(max Bi a.) TT)x(P)00b
ke[m) jE(n]li
S Ui
1(a; w/ Σ
1aa = w/ =w))
max 02,0 - -a))
ama (wmp)) writs
jE(m]}
(14)
]]
E and (wip), whb,
Us
1(a; = - - Σ
1fa w/
[ 01, a))
jEln/ (max
E anva " (wmp), wn6B;
[ui(a;w;p) p] -mnxx - aa)
Rearranging, and substituting uy (max 0; - (x) from Eqn 13 we get the constraint described in
Eqn 5
B.3 Proof of Lemma 4.2
Lemma 4.2. Any mechanism M (0 th) where (w, (p) and t satisfies Eqn interim IR for any w E Rdd.
Proof. For the BIC setting, payment, (p:) E [0, 1]. for each buyer by using a sigmoidal unit. 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
work with the interim payment, and make use of normalized interim
f.th (p:) 1." (p:) B.,
(.M ā Σ Σ km(m)
Σ
T
(p)x (max 0; - a)
kE[m) jELMJ
(15)
The utility of the outside option is again computed similarly to the ex post IC set- ting, recognizing that the seller in the optimal mechanism will make recommendation E_; Euvo--(up-)). (1 w)]] 1 to any agent j * i, in the case that i drops out. In par- ticular, the utility of buyer i, when opting out, is
min max a-s &,EA;
E_i
Vul(āi,ass p)]
ana ilwip i): wolls
11) (16)
Pi E Pana-ilupp-i))
lfū; w/ vz Σ Ila; =m)
a.EAA,
jE(njii
max (Eawo (w: ftās = w/ - -1;aa) a.EAA,
"i (max 0: aa)
Note that i_ EnE(m) T. (p)0..))
2 max 0i and B_, aa (Ene(m) T5
ā [Es(ta 1)
= Thus we have
E
Σ
W A. (p)0.. k - Σ
Σ
w (p)0 - -(max (max00n
20
kE (m]
kE[m) jEinjKi
17

---

Taking this with the fact that E [0, 1], we have
(p.) E," (p.) 24 B_,
Σ
Σ
Σ
th (.kx -mmax 02 -a)
kE(m]
kE/mJ jEln]ia
v B
Σ
Wi,k, (p)0c,
Σ E beim]jE(n])
v. (maxx 0; - -a]) T))a(1)000
kc(m
]]
E_
E
1fa, = à Σ
1aa w}
us (max 0; -a)
anvo (wip). wnl.
jEIN/V
< E_
E amai (wsp). wmll
[usla;w, p)l - -04 (max 0; - -a))
(17)
Rearranging the above and substituting es (max 0g - a) from Equ 16 we get the IIR constraint described in Equ 4
B.4 Proof of Lemma 4.3
Lemma 4.3. Any mechanism M (t" where of (w, p) (xK ex post IC if and only if RGT. vi E [n], except for meusure zero events.
Proof. We first prove the forward direction: if any mechanism M (t"),"") where of (w. (TT) is ex post IC, then RGTT 0, Vi [n].
For the ex post IC setting, the incentive compatibility constraints requires that for every agent and for each p E P, and assuming that every other agent reports its type truthfully and follows the recommended action, then for each misreport pi E P,, and each deviation function, 5 ::A4 Ar - A,, the following condition:
[u.(a,w,p) t<(p)] amar(w))); woi
E (wop wrdd;
(u((6(a); ((pp
(18)
Note that the above hold for any deviation function d. Therefore, we can write that
E awo(w.)), wm0B;
[urla, w, p) tx(p)) 2 max à
E [u:(6(a),, a..w,,) t t. p-1)l
anda (wip(ap-s). wrB
(19)
We can expand to get the following equation:
U:
Σ (
Σ Σ
2h (p), tx((p)
km(m]
JeE ke[mm] jE(n|Ks
max Up
Σ kE[m]
á Σ Σ
T.e.k T (p': -460b.p1))
DEIm)JElMJ
(20)
Pushing the max inside (since other terms don't involve 5(k)),, note that:
max Σ I kE(m]
Σ
max
[T ( p0.
(21)
kE(m
18

---

Thus we have,
Σ
w Ti
a Σ
Σ
J. (i(()
[m(m
[m(m) jE(n]Ki
2 U;
Σ kE[m]
max (0 K'[mm]
E ke|m] jEinjAa
Σ
4(d.pps))
(22)
Note that by our definition of ex post regret in Equation 10, the above is equivalent to rgt,"(p(. p) S 0. Furthermore, this holds for any deviating report p.. Thus, we can write:
max rgl (ps 0o dEP:
(23)
But note that max rgt," (p., p) 2 rgt (p)sp) MEP.
Uj
Σ max
1 (()
- Σ Σ
wo Ted (e)
& [e(m]
kE[mJ
kE(mJ jEjn
w T.) - E Σ kelm)jj[nn)
s J.kkk T. (0)0s,
(24)
02
km[m)
0;
(Ti (p))att Th (p)x(()0s. E'Emm]
max
kE/mJ
where the last step holds since max { TT (T 2 T. (p)a.,
Combining Equations 23 and 24 we have maxper, rgt (ph, P) = O. Taking the expectation over all profiles p E P, we have EpEP |maxp/EP. rgt (p).P) = 0. Thus, RGT," 0, vi E [n]. Thus, we've shown that any mechanism M (o" (t)) where o."¹ (w, (xt, (p)) is ex post IC, then RGT, = 0, as desired.
Next, consider the reverse direction: if RGT." 0, Vi E [n]] we want show that any mechanism M (a" t² where (w, (p)) is post IC. Starting with RGT." 0, Vi e [nl], have that by definition:
E
EPEP max rgth (Ph.0) =0 dEP,
(25)
We showed in Equation 24 that rgt," (pr.p) 2 0. Since P: E Pi, we have that max rgth (P), 2 0. Together with the fact that EpEr |max rgt (Pop)) = O. we have that ma rgt," (P),P) = = rgt," (ph.p) s 0.
Plugging in our definition for ex post regret, we have that:
Σ
max T K'Elm]
(e's p-i)0 ( ā Σ
Σ
[m
ke ml] Je(n|Vi
Σ
Th T
Σ Σ
T
- -t( S 0 (26)
kE[m]
(EIm)JE[M]]
19

---

This can be transformed back to:
E
(W, w. p) - -) 1[(() 2 max
E
[u<(b(a)); a.;,;; (1;,11))) ((pp.p-))
anv((w,))
(noppipp anll
wrll;
(27)
(n=(o(as), p-i)
wmlll;
This is exactly the ex post IC constraint described in Equation
B.5 Proof of Lemma
Lemma 4.4. Any mechanism M (at), where of (w, = (Tl interim IC if and only if RGT, = E [n]. except for measure zero events.
Proof. We first prove the forward direction: if any mechanism M (ott, tt)) where o (w. (p)) is interim IC, then RGT; 0, Vi [v]].
For the BIC setting, for each (ps. p4) E P. and for each deviation function 8 ::A, - Ai, BIC mechanism satisfies:
E
E
(u, tx(p)]
2 E
[u, a t((ph.p-))
anda (w,p)
and (wipl.pp-) w00:
wmith
(28)
Note that the above hold for any deviation function 5, therefore write that
(u,la,w,p) 4<(0)l
max
E (wap wrl.
[u((8(a)) (000)-4(0000000
anvo(ur)))
8
wmBi
(29)
We expand to get the following equations:
0/ R_, Σ
uh Klink P-) p-))0 1)1-kkkk - Σ Σ
( (pipp(piii))... 1((p))
AE(m)
REim JE[M]Ki
T ( )1
max Uy Bit
Σ
Σ
(((s))
kE|mJ
kE|m] JEMNI
max U2
Σ i_t
TT (..61k),, (1 á Σ
J. P=:)0ii
ke(m)
jElnjii
max B_
(00 - ā Σ
P
π The
t((p))
kE[m]
jEInj/s
Σ max
TiKk - & Σ
T.k P P000
[m)
jE(n]M
(30)
The above equations follow from the linearity of expectations and from pushing the max to the inside (since other terms don't involve 8(k)).
Note that by our definition of interim regret in Equation the above equation is equivalent to
rgt, (p's P.) < 0. Furthermore, this holds for any deviating report P/. Therefore, write that
20

---

max rgt; (p S PEEP.
(31)
But note that
max rgt, (d).P) 2 rgt, (c). P.P:
max Bi FE(m kE/ma
Σ
P ā Σ
(((p))
U/
jE(M]M
Us E_
Σ kE(ml
Th (ps: p-i)0s,A PPo
E
Σ
" (P.,
+t((p))
ke(m) JEIn/Ks
E_ [ V. Le (=:s B. [ (eis p=))0cn)]
max [E(m)
kE|m
20 0
(32)
where the second step follows from linearity of expectation, and the last step holds since taking the max of all possible k' E [m]] which includes K' =.
Combining Equations 31 and 32 we have that max rgt. (P..Pt) = 0. Taking the expectation over all profiles p, we have EpEP maxMEp, rgt; (P.,
RGT, 0, Vi E [n]. Thus,
we've showed that if any mechanism M (o", t²)) where o (w, (=."., is BIC, then
RGT; = 0, as desired.
Next, we consider the reverse direction: if RGT, = 0, Vi E [n], we want to show that any mechanism
M = (a", (t") where o. (w, = (p) is BIC. Starting with RGT, = 0, E [n]. we have that by definition: EpEP max,(EP, rgt, (p4 (p) 0
We showed in Equation 32 that rgl, (pi,
0. Since Pi ma rgt, (P)Pt) 2 0. Together with the fact that BpEP max 194, (opii) 0,
P., have that
have that max rgt, (6% = = rggh" (s) & O. Plugging in our definition for interim regret, have that:
)) tx(p))
Σ max B_g EEEmm [E(m
The Me (-:.p-:)0... ā Σ
Th JR (p', P-:)0 (p:
0i
jE(m/Vi
< U3
Σ
Te p-i)0s, P=)
Σ Σ
2 Tps p- - tt((p)) (33)
kE(m)
AE ke(m] jE(n]\i
This can be transformed back to:
E_
E anol(w))) wn0i
[ug w. [(()
2 max Ri_
E and (waphip wri.
[u,(5(a)) 4(6.
E_
E
E
(us(o(as), (.;(,)).-1;,,pp) - ((
awoolui wri.
(34)
21

---

This is exactly the interim IC constraint described in Equation
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 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 learning rate of 0.001. The Lagrangian parameters (A1 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.
We report all our results on a test size of 20000 samples. For the ex post setting, we use 100 misreports initialization 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.
Testing:
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 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
Rochet Menu
Rochet rev
Distribution
Opt reV
Experiment
Price
[1.00
0.00
Setting A
0.25
0.125
0.125
0.00
1.00
0.78
0.22
0.14
0
1.00
Setting B
0.167
0.166
1.00
0.00
0.26
0.00
1.00
Figure Menu(s) and associated prices learned by RochetNet, RochetNet revenue and Optimal revenue on test data in Settings A and 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 Bela distributions considered in setting B This is a bimodal distribution We call the former "low type" and the latter
with modes at 0r.- (35 36
29 and Of
59
88
ag
"high type" following the analysis of [7] for the case of binary types, and order this way because
22

---

0.0
0.0
-0.2
0.4
0.4
-0.6
-0.6
-0.8
-1.0
-1.0
0.00
0.25
0.50
0.75
1.00
0.25
0.30
0.35
0,40
Precision of low type's prior belief
Figure 7: Results for Experiment G (Left) and H (Right). For both experiments, the vertical axis is the differential informativeness g of the experiment used 0z. For Experiment G (left), the horizontal axis is the frequency of the high type, and for Experiment (right), the horizontal axis precision of the low type's prior belief 10z 0.5/.
00 0.5| 2 10 0.5|. We also call the value (lf 0.5l) the precision of a type's prior belief. We design two experiments:
G. Single buyer, Binary State, and Binary Actions, where the payoffs 2 = 1 and the interim beliefs drawn from mixture of Beta(8, 30) and Beta (60, 30) weighted by (1 7>
for 7 E [0, 1].
H. Single buyer, Binary State, and Binary Actions, where the payoffs 01 = 1 and the interim beliefs are drawn from an equal weight mixture of Beta(a1, b1) and (60, 30). We vary 01, bi so that the mode of (eta(a1, by) decreases (therefore the precision of the low type's prior belief 0 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 0, 1-1 1++b a+b-2 and 0,
b-1
in the case if Experiment G 0, and 0μ are non-congruent types, i.e., without the supplemental information, buyer with type 0r, takes action while the other takes action 1. 0H also values a fully informative experiment more than 0z, In the first experiment, we change the likelihood of type Oh with respect to type O,, 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 '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, for discrete distributions with two types. In particular, Proposition 3 from 7 states that informativeness of the low type IT,1, decreases when the frequency of the high type or with the precision low type's prior belief [0, 0.5/. is interesting, then, that we observe the same behaviour in the RochetNet results for 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 the configuration of the economy:
(0.5, 0.5) or 0 = 0075,0.25
*This also requires e-1
69 0.5), otherwise Beta(a, becomes high type instead.
a=b-2
our experiments, we only consider data points at which this condition holds, in other words, we only consider combinations of 6 such that Bete(a, 6) is the low type with mode at 0z and Beta(60, (60, 30) is the high type with mode 0z.
23

---

a 0.5 or = 2.0
Both payofts drawn from UNF, EXP, or Asym UNF
UNF is the uniform distribution over the unit interval U|, 1J. EXP the exponential distribution with X = L. For the Asym UNF, the payoff for buyer is uniform over the interval [0, 10,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 9 and 10 show the optimal data market design and the data marker learned by RegretNet for the BIC settings described above, and Figure [1 gives the revenue and regret obtained by RegretNet and the revenue of the optimal mechanism.
Unifom, -
08
58
04
03
02
12
Unifom, 2.000
Uniform, 2.00, 0.75
1.0
0.4
C4
54
38
52
Figure 8: Experiments learned for BIC constraints when the payoff's are drawn from UNF
Exponential, 0.300
Exponentia
744
2.00
2.2
133
123
032
030
215
Exponential, 2,00, 0.50
Expenentia 2.000.
178
3.
3.5
3:7
122
2 1.0
54
642
255
0.50
038
220
C21
Figure 9: Experiments learned for BIC constraints when the payoffss 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 settings described above, and Figure [5] gives the test revenue and test regret obtained by RegretNet along with the revenue of the optimal mechanism.

---

Asym Uniferm, a-0050 9
Asym Uniferm, d 0.50, e 0.75
138-
190
159
075
0.5
675
8.5
80
80
04
0.3
00
82
Asym Uniform, d=2.00, 0.50
Asym Uniffrmm 2.00, 0.75
14t
2.2
2.5
192-
350
50 140
033
025
275
945
50
24
84
02
08
12
C25
60
44
235
64
24
Figure 10: Experiments learned for BIC constraints when the payoff's drawn from Asym UNF
RegretNet
Opt
Distribution
rgt
(0.5, 0.5)
0.400
0.395
0.5
(0.75, 0.25)
0.237
0.237
UNF
<0.001
(0.5, 0.5)
1.056
1.042
2.0
(0.75, 0.25)
0.763
0.75
(0.5, 0.5)
0.632
0.632
0.5
(0.75, 0.25)
0.448
0.447
EXP
<0.001
(0.5, 0.5)
1.603
1.613
2.0
(0.75, 0.25)
1.390
1,415
(0.5, 0.5)
0.614
0.609
0.5
(0.75, 0.25)
0.366
0.369
Asym UNF
<0.001
(0.5, 0.5)
1.599
1.594
2.0
(0.75, 0.25)
1.139
1.135
Figure 11: Test Revenue and Test Regret obtained by RegretNet for the BIC Settings.
Unifom, a=5.00 0.50, 8- 0.500
Uniform, 0.50 50.0000000.. 0:75
02
02
1.-
ca
04
64
62
Figure 12: Experiments learned for ex post TC constraints when the payoffs are drawn from UNF
25

---

Expenential, 0.50, 0
Exponential d= 0.50. 0.75
158
190
150
073
075
675
875
60
15
15
023
00
Expenential, 2.000, 0.50
Exponential g- 2,00,00 0,75
145
222
2.5
198
355
140
033
C25
275
945
50
18
C29
2.5
Figure 13: Experiments learned for post IC constraints when the payoff's drawn from EXP
Uniferm,
Uniform.
244
3:40
2.09
2:00
359
128
321
525
620
50
0:00
02
08
14
000
40
04
Unifarm,
Asym Uniffrrm
242
1.2
125
325
325
058
6.5
655
14
Figure 14: Experiments learned for ex post constraints when the payoff's are drawn from Asym UNF
RegretNet
Opt rev
Distribution
rEV
ryl
(0.5, 0.5)
0.277
0.27
0.5
(0.75, 0.25)
0.14
0.135
UNF
<0.001
(0.5, 0.5)
0.553
0.541
2.0
(0.75, 0.25)
0.278
0.270
(0.5, 0.5)
0.405
0.405
0.5
(0.75, 0.25)
0.204
0.202
EXP
<0.001
(0.5, 0.5)
0.801
0.809
2.0
(0.75, 0.25)
0.418
0.405
(0.5, 0.5)
0.426
0.423
0.5
(0.75, 0.25)
0.421
0.21
Asym UNF
<0.001
(0.5, 0.5)
0.84
0.841
2.0
(0.75, 0.25)
0.423
0.42
Figure 15: Test Revenue and Test Regret for RegretNet for the ex post IC Settings.
26

---

D.3 Additional Results for the Irregular Distributions
We present additional results for both BIC and ex post settings for two buyers, binary actions, and interim beliefs f = (0.5, 0.5), the same for each buyer. We set = 0.5, and the payoff's are drawn from the irregular distribution whose pdf f()) is:
2.5 if S v
f(u) = 0.5
(35)
0.3 - 0.8
The optimal solutions for these problems make use of ironed virtual values as the payoff distribution is irregular (see Fig I6
Ir, 2 0.50, 6 0.50
IRR, a= 0,50, 0:500
I6
2
00
66
Figure 16: Experiments learned for BIC and ex post IC constraints when the payoff's are drawn from irregular distribution whose pdf is given by Eqn 35 For both these settings, we the ironed virtual value functions 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 common interim belief 0, = 8. The payoff u, for a buyer i is drawn frownfrom a regular distribution V. with a continuous density function. Define virtual value (1;1)) = U; f(()]
F-F(v.) for pdf f and cdf
F of distribution V.. The revenue-optimal mechanism satisfying ex post IC and IR is a mechanism that sells the fully informative experiment buyer if (v:))) 2 Ents 02(2,). Otherwise buyer i receives uninformative menu, where the signal corresponds the most likely state based on the prior.
Proof. We consider setting with binary states, binary actions, and where each buyer has the same interim belief, set to (0, 1 -). Without loss of generality, let 0 2 (1 (1-0), 0). Since the interim beliefs are fixed, will just represent buyer types with v, instead of p₁. Let x, En-o(wwp)) [1 w}].
woih;
Let n. (v) be the matrix representation of the experiment assigned to buyer i.e. (e, (v) (w v)
We first show that a mechanism is obedient if and only if the experiment assigned to the buyer satisfies
(v) 2 9. If mechanism is obedient, then 0 (T) 2 0) ())))( and (1 - ()) 2 0 (v). We have x; (v) 0 (v) + 8 T.22(() 2 0 (V) + 0 TT122(D) =
((T,LI(1) (v))
In the other direction, consider when (v) 2 0. In this case, we show that both 0 TT,1. (w) 2 (1 m 2 (v) and (1 2)) 2 0 T1)2((v) need hold for obedience. Assume to the contrary, one of these fails to hold when (u) 2 9. We have one of,
- ()T222(() < 0 m.,, and we have az(v) = 0 TT,1, + 0)T,22(1) <
0 (1) + (,,),(()) = ( T,,12(1)) = 0.
0 (.) (v) <
0 ()), and we have x1(1) = 8. T. + 22) <
11-0)=22100 (v) + (1 - - (1 (T:2, +i222 = (1-)) -) <
In either case, have contradiction.
Since our payoff linear, from Proposition 3.5 in (7) the IC constraints are satisfied only when the truthfulness and obedience constraints satisfied. Denote (,()) zz(v) ب E
27

---

Thus, the optimal design problem to solve is:
Exwy
Σ ((ws))
s.t. (1) - t())) 2 u. - -((/000)))
WE E V, [n]
(1) - t,) 04 a)
E WeVii Vi [n] Ve E [n]
x2(v)= 2
The first constraint corresponds truthifulness for the IC setting, the second is the IR constraint, and the third the obedience constraint. We have thus reduced the data market design problem that of Myerson's 's revenue maximizing single-item auction problem. Instead of #:(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 2;(4) which requires (v) 2 0. Thus, by Myerson's 's theory, the total expected revenue is equal to the expected virtual welfare minus some constant K (stemming from IR constraints) for some (1;; v-4) that is non-decreasing in v;-- Thus, we have:
E
Σ (e:() iE(m)
= EOny
Σ ic|n]
d((u))=(())
(36)
E
di(v;)
I2(0)
a
Σ 2y(U)
K
(37)
iE(m)
jE(n/li
Enpay
(1(())
(5(45)) 22(())
K
(38)
n
iE
jE(N]\i
In order to maximize the revenue, need to maximize the virtual welfare. Thus d; (c) = (a fully informative experiment) when dv(w;) Ejejnny oy (1²) 2 0 and we (v(v) = (an uninformative experiment that always sends the same signal regardless the state) otherwise. This is precisely the mechanism described in the theorem.
set
Note that such an 7;(0, u-s) is non-decreasing in v (since (v., v-4) is non-decreasing in e and Iy (o,, v-1) for j [n]] \ i is non-ine non-increasing in e; for regular distributions V; and V;). Moreover, De (v) 2 8 is also satisfied.
28


Reviews:
[{"review_id": "h0HGiJ7jlm", "reviewer": "Reviewer_zvsA", "summary": "The authors present a novel approach to the problem of data market design, which seeks to find a set of signaling schemes, each revealing some of the information known to a seller and having a corresponding price, where the goal is to maximize expected revenue. Then, the authors introduce the application of a deep learning framework to the automated design of the data market. The paper discusses the importance of data market design and its potential applications in real-world settings, such as data marketplaces where sellers sell data to buyers for ML tasks. The authors demonstrate that their new learning framework can replicate known solutions from theory, expand to more complex settings, and establish the optimality of new designs. The paper also highlights some limitations of the approach, such as the need for interpretability of the mechanisms learned by the RegretNet approach for larger problems, the potential for local optima in non-convex problems, and the challenge of achieving exact incentive alignment in multi-buyer settings.", "questions": "+ Could you provide more details on how the RegretNet approach can be made more interpretable for larger problems? Are there any specific techniques or methods that could be used to achieve this?\n\n+ Have you considered using other optimization techniques besides gradient-based approaches to address the potential for local optima in non-convex problems? If so, what are some alternative approaches that could be used?\n\n+ What are some potential ways to provide more practical or theoretical guidance on how much alignment is enough for agents to follow the intended advice of a market design? Are there any existing frameworks or approaches that could be used to address this issue?\n", "limitations": "The authors acknowledge the ethical concerns raised by markets for trading data about individuals and suggest that machine learning frameworks such as those introduced in this paper can be used to strike new kinds of trade-offs, such as allowing individuals to benefit directly from trades on data about themselves. This shows that the authors are aware of the broader implications of their work and are thinking critically about its potential impact.", "rating": 6, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "+ The paper presents a novel approach to the problem of data market design, which uses deep learning to automate the design of data markets.\n\n+ The authors demonstrate that their new learning framework can almost precisely replicate all known solutions from theory, which shows that the approach is effective and reliable.\n\n+ The paper shows that the new learning framework can be used to establish the optimality of new designs and conjecture the structure of optimal designs, which is a significant contribution to the field.\n\n", "weaknesses": "+ The paper acknowledges that for the approach to provide insights into the theoretically optimal design for larger problems, it will be important to provide interpretability to the mechanisms learned by the approach. However, the RegretNet approach used in the paper is not immediately interpretable, which limits its usefulness in this regard.\n\n+ The paper notes that the approach uses gradient-based approaches, which may suffer from local optima in non-convex problems. This suggests that the approach may not always find the global optimum and may be limited in its ability to handle more complex problems.\n\n+ The paper attains in the multi-buyer setting approximate and not exact incentive alignment, which leaves the question as to how much alignment is enough for agents to follow the intended advice of a market design. This suggests that the approach may not be able to achieve exact incentive alignment in all settings, which could limit its effectiveness.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Interpretability**  \n\nWe can interpret the designs learned by RegretNet in a few ways.\n- We can plot the probability of recommending the correct action for different agent types. We adopt this approach and visualize these as heatmaps in the main paper (Figure 2) and Appendix (Figures 8, 9, and 10) to show different optimal designs. This kind of visualization borrows from the existing economics literature and can be extended for larger designs.\n- We can compare the similarity of outputs learned by RegretNet to well-known baselines and analyze how economic properties of interest vary with different design choices. For instance,  [1, 2] study how externalities affect the optimal design of data markets. In our paper, as an illustrative example, we show how we can understand revenue patterns for new settings, studying here the effect of varying the intensity of negative externality on revenue.\n- Another interesting direction, not taken here but interesting for future work, is to initialize the mechanism to a known baseline, use our framework to optimize it further and see whether known mechanisms can be improved.\n- Another interesting direction for future work is to consider techniques such as distillation to “compile” a design into something more interpretable. Generally, we like that this agenda is “future proof,” meaning that advances in machine learning, including distillation and interpretability, can be applied here. \n\n---\n**Non-Convex Formulations and Alternate Optimization Methods**  \n\nWe've addressed this in the global comment [here](https://openreview.net/forum?id=sgCrNMOuXp&noteId=OyHh7ioeDB).\n\n---\n**Approximate Incentive Alignment** \n\nFirst, we emphasize that we can achieve very small empirical regret violations (see Fig. 11 and Fig. 15 in the Appendix). We find this encouraging. Another useful observation, in support of the effectiveness of approximate incentive alignment, comes from the machine learning pipeline almost exactly recovering optimal designs for all known settings (as per the above comments). \n\nThat said, it remains an empirical question as to how much alignment “is enough.” We would expect this to depend on context, e.g., how complex it is to deviate from truthful reporting vs. satisficing behavior, how high the stakes are, and how well-informed the participants are. An interesting path forward in this regard would be to develop simulators of behavior as an additional way to test performance. There is also a growing literature on modeling complexity in the context of strategic paper, e.g., [3].\n\nIn the context of auction design, there is also some recent guiding theory [4, 5, 6] that provides transformations between $\\epsilon$-BIC and BIC without revenue loss. Interesting avenues for future work include extending these transformations to general class problems — problems with both types and actions, as in the data market settings. It also remains open to extending these results to obtain $\\epsilon$-DSIC to DSIC transformations for general distributions. \n\n---\n\n*References*  \n\n[1] Agarwal, A., Dahleh, M., Horel, T., and Rui, M. (2020). Towards data auctions with externalities. arXiv preprint arXiv:2003.08345 \n\n[2] Bonatti, A., Dahleh, M., Horel, T., & Nouripour, A. (2022). Selling information in competitive environments. arXiv preprint arXiv:2202.08780.\n\n[3] Modibo K. Camara. 2022. Computationally Tractable Choice. In Proceedings of the 23rd ACM Conference on Economics and Computation (EC '22)\n\n[4] Daskalakis, C., & Weinberg, S. M. (2012, June). Symmetries and optimal multi-dimensional mechanism design. In Proceedings of the 13th ACM conference on Electronic commerce (pp. 370-387).\n\n[5] Cai, Y., Oikonomou, A., Velegkas, G., & Zhao, M. (2021). An Efficient∊-BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA) (pp. 1337-1356). Society for Industrial and Applied Mathematics.\n\n[6] Conitzer, V., Feng, Z., Parkes, D.C., Sodomka, E. (2022). Welfare-Preserving $\\epsilon$-BIC to BIC Transformation with Negligible Revenue Loss. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds) Web and Internet Economics. WINE 2021. \n"}, {"review_id": "q5m7T6FQlQ", "reviewer": "Reviewer_QSAL", "summary": "This paper introduces a deep learning framework for the automated design of data markets, a novel and timely application in the field of economics. The authors address the data market design problem, which involves designing a set of signaling schemes to maximize expected revenue. The paper extends previous work on deep learning for auction design by learning signaling schemes and handling obedience constraints that arise from the actions of agents.", "questions": "What are some baseline methods to compare with? For example, how does the standard rochetnet perform on the proposed market settings?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Innovative Application: The paper introduces a novel application of deep learning to the data market design problem, expanding the scope of machine learning in the field of economics.\n\n- The paper is well-written overall.", "weaknesses": "-Incremental work: It seems that the core contribution, the proposed neural network architecture, is a simple extension of existing model called RochetNet, by slightly modifying the loss function.\n\n-Lack of comparison with baselines: mechanism design for information acquisition is a long standing problem. I was surprised to see no baseline comparison in the experiments, and no discussion on how/why existing approaches may not work in the methodology. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Novelty**  \n\nWe've addressed this in the global comment [here](https://openreview.net/forum?id=sgCrNMOuXp&noteId=OyHh7ioeDB).\n\n---\n\n**Comparison with other baselines**\n\nWe are unaware of other computational baselines, and we already compare them with the theoretically optimal designs from [1, 2] where they are available.  In terms of prior computational results, we primarily deal with continuous distributions, whereas [3] provides computational results for discrete types. Handling continuous distributions through discretization using linear programs (LPs) is understood to lead to significant scalability issues, as demonstrated for optimal auction design [4].\n\nWe also discuss some alternate data market formulations in the related work section. But these are different problems, and solution approaches to these problems do not provide baselines for the problem formulation that we take up. \n\nRochetNet cannot be used for the data market setting, since this departs from optimal auction design in several substantive ways. The incentive compatibility constraints under the data market setting are more complex, comprising both the truth-telling part (reporting one’s type truthfully) and the obedience part (taking the recommended action instead of deviating to another action). If we adopt the standard RochetNet idea from the auction setting, we will miss the obedience part, assuming agents act according to our action recommendation when they might deviate to other actions strategically instead.\n\nProblems of information design, particularly “Bayesian Persuasion,” have been studied in both economics and computer science, but this is a quite different setting from that of the present paper. In particular, it does not model problems of revenue, bidding, or multiple receivers (or competition across multiple receivers).  \n\nProblems of optimal auction design are also widely studied in computer science and economics, but very different in structure to the data market problem; e.g., much of the auction literature considers the sale of one or more “rival good” (i.e., a good that cannot be allocated to multiple people, as is the case with information). Moreover, the auction design literature does not deal with posterior beliefs, arising from information sale, and effect of beliefs on downstream actions. \n\n\n---\n\n*References*  \n\n[1] Bergemann, D., Bonatti, A., & Smolin, A. (2018). The design and price of information. American economic review, 108(1), 1-48.\n\n[2] Bonatti, A., Dahleh, M., Horel, T., & Nouripour, A. (2022). Selling information in competitive environments. arXiv preprint arXiv:2202.08780.\n\n[3] Cai, Y., & Velegkas, G. (2021). How to sell information optimally: An algorithmic study. 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), pages 81:1--81:20.\n\n[4] Dütting, P., Feng, Z., Narasimhan, H., Parkes, D., & Ravindranath, S. S. (2019, May). Optimal auctions through deep learning. In International Conference on Machine Learning (pp. 1706-1715). PMLR.\n"}, {"review_id": "IpMGoqN0KQ", "reviewer": "Reviewer_wNRV", "summary": "The authors are concerned with a problem of \"data market design\". In such a setting, a mechanism designer with access to an unknown world state interacts with buyers who have private types, and need to take actions whose payoffs vary depending on the world state. These buyers purchase (in the single buyer case) or bid on (in the multi-buyer case) access to a signaling scheme which, given reports from the agents and the world state, sends a signal to the buyers (which without loss of generality can just be a recommended action). This mechanism, treated as a direct-revelation mechanism, needs to be both truthful (incentivizing honest reporting by the buyers) and obedient (once the buyers receive their signal, they should be incentivized not to deviate from the recommendation). Subject to those constraints (either Bayesian or ex post), the mechanism designer wants to maximize their revenue.\n\nThis problem shares some similarities to truthful revenue-maximizing auction design. In that domain, there has been recent progress using the tools of \"differentiable economics\" to approximately learn high-performing (and sometimes even provably optimal) auctions, in both single- and multi-bidder settings.\n\nThe authors apply very similar techniques to this data market problem. In single-buyer settings (as in auctions) they are able to ensure exact IC; for multi-buyer settings they use a Lagrangian during training to approximately enforce IC constraints. They experiment on a relatively wide variety of problem instances, reproducing known results, finding new optimal mechanisms, and conjecturing optimal mechanisms where they cannot find them.", "questions": "Can the method in fact scale to larger (even just 3-5 buyers) settings, or not? This should be discussed.", "limitations": "See questions.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "The paper comprehensively shows how to successfully apply differentiable economics to a new domain where it has not previously been applied. The authors are able to reproduce optimal mechanisms and find new ones, showing that their adaptation of these technique is in fact useful in producing novel results. This helps to further push these techniques towards being practically helpful tools for theorists and modelers.", "weaknesses": "The network architectures here are essentially the same as those used in previous work for auctions, only adapted slightly for the data market setting. This is fine, but it does mean that from the perspective of differentiable economics, there is no novel methodological contribution.\n\nThe experiments appear to consider at most 2 buyers. While (as in the case of multi-parameter auctions) even selling to just two buyers may be a very challenging case, it would be more interesting to consider a slightly larger number of buyers.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the review and the feedback. We've discussed concerns regarding the weaknesses - especially regarding **Methodological Contributions** and **Scaling up** in the global comment [here](https://openreview.net/forum?id=sgCrNMOuXp&noteId=OyHh7ioeDB).\n\n"}, {"review_id": "IRscCthjNa", "reviewer": "Reviewer_u4po", "summary": "This paper introduces a deep learning application to the data market designs that find optimal signaling schemes to maximize the revenue of data sellers. The proposed method is designed to handle truthfulness and obedience (i.e., buyers following recommendations). The overall approach follows the prior frameworks of RochetNet and RegretNet for auction design. The authors are able to demonstrate the method’s ability to recover existing analytical optimal solutions and extend to cases where analytical results are not available. Some experimental results are provided for both single-buyer and multiple-buyer settings.", "questions": "**Major**\n\n1. Are there any references to support the assumptions made in the preliminaries section? For example, why is the matching utility payoff reasonable in data market design? How do you interpret that in the binary-state setting in the real world? How about a more complex non-binary setting?\n2. For the single buyer setting Lemma 3.1, it is claimed that the mechanism is Incentive Compatible as it is agent optimizing. Why is it agent optimizing when the objective is to maximize the payment by the agents?\n3. How to access the validity of the results from the networks when there is no analytical solution (more complex settings)? For example, for the price of 0.14 outputted for setting C, how do you know whether it is close to optimal? Also, could you provide a more intuitive interpretation of the price and results?\n4. What are the challenges in conducting experiments on binary states, actions? Also, can you perform experiments on more than two buyers? Can the method be extended to much more complex scenarios with a large number of players, actions and states?\n\n**Minor**\n\n5. Grammar. Lines 80, 103, 242. Punctuations and formats: Lines 146, 153-160, 239.\n6. Some notations can be confusing, especially the subscripts, superscripts and brackets.\n7. What is $\\Delta$ in Line 129, never explained before.", "limitations": "The authors have sufficiently discussed the limitations of the approach in the limitation section. Additionally, I wonder how well this framework applies in real-world scenarios. Could the author clarify the limitations of adopting the method in real life for data pricing, or provide a practical example/application?", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper applies deep learning to the new domain of data market design, illustrating the feasibility of learning solutions to optimal data market design.\n2. It considers the obedience of data buyers in the design. This makes the approach more practical.\n3. The paper provides a sound analysis of Individual Rationality for the mechanism and payments.", "weaknesses": "1. The writing could be improved. Preliminaries could be better structured to explain essential terms like menu entry, signaling, state of the world, how the mechanism works, etc. Interpretations could be added after Lemmas and computation equations (e.g., (10)) to improve clarity.\n2. The scales of the experiments are not large enough to be convincing. If larger experiments are not possible, challenges and limitations should be clearly stated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the feedback regarding the clarity. We will make the necessary edits and add interpretations to our lemmas and equations to make them more readable. The response to the questions including concerns regarding the scale are addressed below\n\n---\n\n**1. Matching Utilities under binary states**  \nOur focus in this paper is on the model formulated by Bergemann et al. (2018) and Bonatti et al. (2022), where the assumption of matching utility under binary states is widely used. Under this assumption, the buyer faces *binary* payoffs in each state where the outcomes can be nicely categorized into ‘right’ and ‘wrong’. For this case, the restriction is without loss of generality relative to a general payoff matrix. There, for every state (say state 1), we can always subtract the (state-dependent) constant of the utility of taking the non-matching action (action 2) under the current state (state 1). This linear transformation normalizes the payoffs of the data buyer by setting off-diagonal utility to 0 without affecting the optimality conditions of the data buyer's decision problem.  For more details, see the matching utility section in Bergemann et al. (2018). We also discuss a practical example in the last subsection of this comment.\n\nFurthermore, our choice to primarily focus on matching utility is because this is assumed in the results of Bergemann et al. (2018) and Bonatti et al. (2022), which we use as baselines for a comparison of learned and analytically optimum results. The proposed RochetNet and RegretNet formulation only needs a properly specified utility function to work and is readily able to handle custom utility values. The framework can also be easily extended, without new challenges, to a non-binary state problem.\n\n**2. Clarification regarding agent-optimizing**  \nWe apologize for the lack of clarity. This concept of “agent optimizing” is from the economic theory of mechanism design, and we will add a reference. In particular, it requires that each agent is presented with a menu of options that does not depend on their report, and where their report is – in effect – used to pick an option that maximizes their utility. This is a theoretical framing that allows one to reason about incentive properties. Given this as a design constraint, we can then seek to maximize expected revenue, subject to this property of “agent optimality.”  In the present paper, this “agent picking the best option” is hard-coded in the RochetNet architecture in a differentiable way. \n\n**3. Validity and Interpretation of results**  \nFor the *ex post IC* setting for the multi-bidder case with uncertain payoffs, we conjecture the structure of the solution and prove its optimality. \n\nFor other settings, we conjecture but do not prove the structure of the optimal solution. For Setting C, the conjectured optimal design is a menu of size one that offers a fully informative experiment. For Setting D, we conjecture that the optimal design is a menu of size two with one fully informative experiment and one partially informative experiment. Both these results provide a new target for economic theory. \n\n**4. Scaling up**  \nWe've addressed this in the global comment [here](https://openreview.net/forum?id=sgCrNMOuXp&noteId=OyHh7ioeDB).\n\n**5, 6, 7. Typos and Clarifications**  \nThanks for pointing this out, and we will address it. The triangle denotes the probability simplex (over states in Ln 129)\n\n---\n\n**Practical Example and Challenges**   \nWe adapt the following real-world example from Bonatti et al. (2022)\nConsider a platform similar to Amazon that wishes to monetize selling information about a consumer, like their shopping history, with retailers. The success of each retailer is influenced by two main things: (i) how accurately they can target their ads to consumers, meaning they show the right products to the right people; and (ii) how unique their ads are compared to what other competitors are offering, based on what consumers like.\n\nRetailers decide how much they're willing to pay for extra information based on how much profit they make from each sale, considering their costs. As this is only privately known to the merchant, the platform must elicit it through its choice of mechanism. This reduces to designing a menu of (experiment, payment) pairs, each corresponding to an advertising campaign.\n\nIn this setting, there are two states $\\Theta$ = {0, 1} representing consumer preference. There are 2 retailers (indexed by $i$ and $j$) whose goal is to match their product to a consumer’s preference. Their action sets are thus given by $a_i = a_j$ = {0, 1}. The payoffs for retailer $i$ is given by $1_{\\theta = a_i}  - \\alpha \\cdot 1_{\\theta = a_j}$  where $\\alpha$ controls the competition factor. The competitor $j$ having correct information about a consumer increases the competition and imposes a negative externality of $\\alpha$ in this case.\n\n*Challenges* - As discussed in Bonatti et al. (2022), this model abstracts away certain details and dynamics of online advertising. For example, the model does not capture the cost of transmitting data, and nor does the model explain a practical style by which partially correct information can be communicated.\n\n---\n\n*References*  \n\n[1] Bergemann, D., Bonatti, A., & Smolin, A. (2018). The design and price of information. American economic review, 108(1), 1-48.\n\n[2] Bonatti, A., Dahleh, M., Horel, T., & Nouripour, A. (2022). Selling information in competitive environments. arXiv preprint arXiv:2202.08780.\n"}, {"review_id": "DUP5FWRjA8", "reviewer": "Reviewer_nspR", "summary": "This paper studies the market design problem, specifically for data markets. In particular, different from existing analytic approaches, the proposed approach is based on (deep) learning to recover/discover market designs. They adopt and extend an existing RochetNet architecture to both single- and multi-buyer setting and empirically demonstrate the effectiveness of the approach in recovering/discovering the market design.", "questions": "- In lines 20-22, \"... hold vast quantities of data about individuals. In turn, this has led to data markets, where information about an individual can be purchased in real-time to guide decision-making (e.g., LiveRamp, Segment, Bloomreach).\" This seems to hint at that the aforementioned companies are selling data about individuals, is it what it means?\n\n- In lines 60-62, \"Further, we give a training method that enables the efficient reuse of computed interim allocations and payments from other samples to swiftly calculate the interim utility of misreporting, dramatically speeding up training.\" Is this empirically or theoretically demonstrated, specifically about \"dramatically speeding up training\"? What is it comparing against, in terms of speed of training?\n\n- In line 122, \"The state of the world, $\\omega$, is unknown and is drawn from a finite state space ... \" Is there an assumption on the distribution of this?\n\n- In line 127, \"where each $v_i$ is drawn independently from a distribution $\\mathcal{V}_i$\". What is the interpretation of $v_i$ and what does the distribution $\\mathcal{V}_i$ depend on?\n\n- In lines 137-138, it seems that the negative externality is in the form of decreasing payment for one buyer $i$ as the gain for some other buyers. In other words, if another buyer $j$ gains (in ex post payoff), this buyer $i$ \"loses\" (i.e., has a lower utility), is this correct? How should this be interpreted in an example? \n\n- In line 139, \"There is a data seller who observes the world state ... \" How to justify or realize this assumption that the actual world state is exactly known by the data seller?\n\n- In line 159 (5-th bulletin point), \"$u_i(a,\\omega, V_i, \\theta_i)$\", is it meant to be $V_i$ or $v_i$?\n\n- In line 192, \"... an unsupervised learning problem.\" Is it referring to optimizing the softmax version of Equation (9)? If so, it looks more like an optimization problem (i.e., parametric fitting) instead of a learning problem. Often, unsupervised learning is to learn about the inter or intra structure of the data instead of to fit a functional form. Please help interpret why the loss function in line 222 is an unsupervised learning problem.\n\n", "limitations": "- Typically in an optimization approach, if the objective is non-convex (or more complex), it is difficult to establish theoretical guarantees in terms of the optimality or quality of the final solution obtained. This is also mentioned by the authors in lines 374 - 375. The implication is that, it is difficult to obtained a principled understanding of how good the solution (i.e., learnt market design) is, obtained from the gradient-based optimization.\n\n- With regard to lines 378-380, \"we return to where we started, and underline that markets for trading data about individuals raise a number of ethical concerns.\" In light of the potential ethical concerns of data trading, a (deep) learning-based approach potentially makes it even more difficult to manage and parse the working mechanism of the data trading. As a result, such an approach can make it even more difficult to reliably/verifably address those concerns.\n", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper studies the problem of market design and it is relevant for data market.\n- The proposed learning-based approach is interesting in that it can recover some analytic solutions.\n- There are relatively extensive empirical results.", "weaknesses": "- The motivation and justification of a (deep) learning-based approach can be made stronger.\n    \n    In lines 40-42, \"The difficulty of using analytical tools for this problem of data market design is highlighted by this example, and it remains an open problem to obtain theoretical results for richer multi-buyer settings. This motivates the need for computational approaches.\" While it is perceived that analytic solutions are difficult, and computational approaches seem a viable alternative. Is it really necessary to use deep learning? In other words, are there less complex computational approaches that can be tried first or reasons why they would not work as well? \n\n    In particular, (how) can the assumption of i.i.d. samples from $\\mathcal{P}$ for training the deep learning model be satisfied? It requires the type of the buyer (i.e., both belief and the $v$) to remain fixed throughout observing the signals. Does this assumption have conflicts with \"Upon receiving a signal, the buyers update their prior beliefs and choose an optimal action accordingly\" (lines 143-144)?\n\n\n- The inline equations in the paper can break the flow of the writing and make it more difficult for the reader to catch the most important points.\n\n    For instance, equations (1)-(4) are used to discuss (different variants of) incentive compatbility. It is not so clear which equation the reader should pay most attention to. Furthermore, it seems that equation (4) (i.e., ex post incentivie compatible) is not interpreted after the equation.\n\n- Some experimental results can be difficult to interpret (or understand their significance), due to the lack of (existing) analytic characterization of optimum solution. \n\n    For instance, in lines 294-296, \"We are aware of no theoretical characterization of optimal data market designs when both $v$ and $\\theta$ vary. In such cases, we can use RochetNet to conjecture the structure of an optimal solution.\" As a result, it is not clear to the reader how to understand whether the proposed method is effective. It further goes to the first point regarding the motivation/justification of a learning based approach: There lacks a solution or ground truth (i.e., analytic optimum or approixmate optimum) to evaluate the approach. Hence, it seems appealing to first establish such a solution before a computational approach, otherwise, how to effectively evaluate the proposed computational approach?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Justification for using Deep Learning:** Please note that the goal here is to learn a differentiable _function_ that represents an entire mechanism, i.e., a complete mapping from all possible inputs to all possible outputs. The goal is not to simply find a pointwise output for a given input because incentive considerations of mechanism designs rely on reasoning about the global properties of a learned function.  Tools from deep learning are very well suited to these kinds of tasks since they are highly configurable and can learn complex and non-linear patterns. In our setting, once we have a differentiable representation of a mechanism and loss function, we can use standard machine learning pipelines to optimize these representations easily (“computation as learning”). As noted, the other added advantage of using a deep learning based approach is the flexibility in accommodating different design requirements.\n\nIn regards to the other computational approaches, the most relevant, existing computational method for this data market design problem is Cai and Velegkas (2021), but this focuses on discrete inputs and makes use of linear programming, and does not consider continuous inputs. Handling continuous inputs through discretization and then optimizing for discretized inputs through linear programs (LPs) has been demonstrated to lead to significant scalability issues in the context of computational approaches to the design of optimal auctions Dütting et al. (2019).\n\nIn this work, we follow the agenda of differentiable economics and leverage the expressive power of deep neural networks to discover designs that are approximately optimizing and approximately incentive aligned. These approaches have worked well for the settings of auction design, and also two-sided matching problems and problems of social choice, and in this work, we demonstrate that they also perform very well for the design of data markets – with suitable extensions to handle the new behavioral considerations.\n\n**i.i.d assumption:** In this model, all the buyers have a prior belief over which state is likely to occur, and the prior and the payoff for taking the correct action constitutes a buyer’s type. Each buyer then reports their type to the data seller. \n\nFrom the revelation principle for dynamic games (Myerson 1991), it's sufficient to restrict our attention to mechanisms that are IC (where players report their true types). It’s these types that constitute the training samples from our mechanisms, and thus the assumption of iid samples is satisfied.\n\nA buyer then updates its beliefs if they choose to buy the experiment: they’re sent a signal based on the experiment that they purchase, and this signal is used to update their prior belief. This happens after they’ve taken part in the mechanism, and after the statistical experiments have been sold. Please refer to Ln 152-160 for an overview of the timing of the mechanism.\n\n**Inline equations:** Thanks for pointing this out. We will make the necessary changes to make it more readable.\n\n**Motivation for a learning based approach:** We think about this a little differently – we’re inspired by the challenges with developing theoretical solutions in these multidimensional settings and are looking to the use of computational tools to attack this problem. Can we use this kind of framework to generate interesting conjectures that can then be proved? \n\nThis has been a successful pipeline across a number of discovery-based, scientific domains, with in silico design leading to tests in the lab (here, through “the lab” would be “proved” or “tested in the experimental economics lab,” and perhaps “in simulation with AI participants” in the future). One of the main motivations of our work is to provide economists with a new computational tool with which to test conjectures and analyze the structural properties of optimal designs.\n\n---\n**Companies and Data:**  Indeed, these companies are all customer data platforms which unify first-party customer data from multiple sources to build a single coherent consumer profile. Liveramp, for example, onboards data by matching offline files, names, addresses to digital identifiers so that companies can target users with online ads. Based on this, it aggregates information for a person’s activities from different browsers, devices, and channels that represent a  consumer’s digital footprint and provides access to these data via its marketplace.\n\n**Regarding Speedup:** We will add a note about this. In existing BIC networks, for a batch size B with K samples for computing interim values and M misreports, we will have to compute B x K x M forward passes. However, in our approach, we don’t sample new misreports but rather re-use other data points from the minibatch as misreports, thereby doing only B x K forward passes.\n\n**Assumptions and Explanations:** Some of the assumptions made here are standard in the mechanism design and information design literature. For better clarity, please refer to the practical example discussed in the response to Reviewer u4po [here](https://openreview.net/forum?id=sgCrNMOuXp&noteId=KNtPhdCieC)\n\n**Typos:**  Thanks for pointing this out. It is $v_i$\n\n**Unsupervised learning problem:**  We think of this as unsupervised learning because there are no “ground truth labels,” which in our setting would be examples of how to allocate and price information coming from an optimal mechanism. In fact, the optimal mechanism is a priori unknown in many of the settings studied, and this is a problem of discovery of new mechanisms. Instead, the pipeline that we formulate is used to optimize for revenue subject to IC constraints, and we look to interpret the optimized designs, and introduce new conjectures. \n\n---\n**Non-Convexity:** Please refer the global comment [here](https://openreview.net/forum?id=sgCrNMOuXp&noteId=KNtPhdCieC).  \n**Ethical Concerns:** Please refer to the subcomment.\n\n"}]

Author rebuttal:
We thank all the reviewers for their helpful feedback. Three main comments are regarding non-convexity of our formulation, novelty and scaling up. We address these concerns here. The remaining questions and concerns are addressed in the individual responses.

--- 
**Non-Convex Formulations and Alternate Optimization Methods** 

Although local optima can be a concern when optimizing non-convex objective functions, gradient-based methods have demonstrated remarkable success, with extensive empirical evidence showing convergence to high-quality solutions and even in the presence of non-convexity. There is also theoretical support for a “no local optima” phenomenon in the ML theory literature. In addition, a recent paper [1] also shows how the local optima are connected by a path, where the revenue along the path is at least as much as one of the endpoints, justifying the empirical success of RochetNet for auction design. Although open, we conjecture similar theoretical results for the adapted version of RochetNet to the present paper.

Regarding our specific experiments, one way in which we validate the effectiveness of the proposed method is by looking at settings in which theoretical results are available. In this way, we demonstrate that we can reliably recover optimal solutions for all prior settings in which analytical solutions are known (see the continuum of types setting in Section 5 and the BIC settings in Section 6)

Considering alternate computational approaches for the problem that we study, while [2] propose using LPs for settings with discrete types, we don’t expect this to extend to the continuous settings of the present paper since introducing discretization to enable the application of LPs leads to significant scalability challenges, as demonstrated in [3] in application to revenue-optimal auction design.

---

**Novelty**  

We respectfully disagree that attaining the ability to train models in this pipeline is “just about modifying the loss function” from earlier work. Rather, the crucial aspect is to capture a new kind of strategic behavior within the learning pipeline.  A significant innovation is to be able to handle both _obedience_ and _truthfulness_ constraints on behavior, including _double deviations_ (misreporting, along with deviating from suggested actions). To our knowledge, this is novel in the differentiable economics literature. 

We also introduce new sampling techniques for achieving BIC, which are able to reuse interim allocations and interim payment calculations to compute IC violations. In existing BIC networks [4], for batch size, B with K samples for computing interim values and M misreports, we will have to compute B x K x M forward passes. By contrast, in our approach, we don’t sample new misreports but rather reuse other data points from the minibatch as misreports, thereby doing only B x K forward passes. This substantially speeds up the training process. 

Another contribution is the new results we provide in the application domain itself, which is an application domain garnering considerable interest in economics and computer science and one with substantial societal importance, given the multifaceted nature of the design problem.  Whereas analytical results are only available for the BIC setting, which is, in effect, lower-dimensional, and easier to analyze, we are able to study through computational techniques the design of data markets in the _ex post_ IC setting, which is a setting without existing theory. In Section 6, for the _ex post_ IC setting, we conjecture the structure of the optimal designs and prove their optimality through Myerson's framework (proof in the Appendix). We see this as an important contribution, as _ex post_ IC is a stronger notion of IC than BIC.

In addition, we provide an illustrative example to showcase the framework's versatility as a toolbox for economists. For instance, in Section 6, we study how the revenue varies as we vary the intensity of negative externality, thereby varying the competition. This hasn’t been studied before for the setting with uncertain priors, and we show economically meaningful variation, as no analytical solution is known.

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**Scaling up**  

We choose to study the case of two buyers because it is easier to visualize the results and develop conjectures. We would also like to emphasize that Theorem 6.1, which is stated for the *ex post* IC setting with uncertain payoffs, holds for any number of bidders. 

The described approach can scale to more buyers and states, especially in the ex post IC settings. For instance, in the case of uncertain priors, for 10 agents and 10 states, our approach takes 62 min to run for 20000 iterations on a single NVIDIA Tesla V100 GPU.

Scaling is harder for the BIC setting, however. As we increase the number of agents to $n$, obtaining the interim values accurately involves computing the marginal values over $n  - 1$ dimensions. This can be quite expensive.

We will add a discussion on scaling to the paper.

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*References*

[1] Hertrich, C., Tao, Y., & Végh, L. A. (2023). Mode Connectivity in Auction Design. arXiv preprint arXiv:2305.11005.

[2] Cai, Y., & Velegkas, G. (2021). How to sell information optimally: An algorithmic study. 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), pages 81:1--81:20.

[3] Dütting, P., Feng, Z., Narasimhan, H., Parkes, D., & Ravindranath, S. S. (2019, May). Optimal auctions through deep learning. In International Conference on Machine Learning (pp. 1706-1715). PMLR.

[4] Feng, Z., Narasimhan, H., & Parkes, D. C. (2018, July). Deep learning for revenue-optimal auctions with budgets. In Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems (pp. 354-362).

Meta-review: The submitted paper considers a deep learning framework for automated data market design that can handle obedience and truthfulness constraints and experimentally shows that the framework can replicate known solutions from theory but also extend to novel designs. The paper was reviewed by 5 knowledgeable reviewers, all of whom recommended acceptance (3x borderline, 2x weak). They appreciated the novel application and the proposed deep-learning framework. A discussion among reviewers and authors took place, and the authors were able to clarify many of the raised concerns and provide additional details regarding raised questions. The discussions revealed that the submitted paper has shortcomings in clearly explaining certain aspects of the proposed framework (scalability, novelty relative to some existing works, parts of the experimental evaluation) but the authors provided clear and convincing responses therefor. Hence, despite the shortcomings, I am recommending acceptance of the paper but strongly encourage the authors to improve their paper in line with the reviewers' comments, the discussions, and the clarifications provided by the authors themselves.
Decision: Accept (poster)