Title: Data Market Design through Deep Learning
Venue: NeurIPS 2023
Accepted: yes
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

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 [7]. 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 [12]. 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 [32], we must learn signaling schemes rather than allocation rules and handle obedience constraints -these arising from modeling the downstream actions of buyers-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.

1 Introduction

Many characterize the current era as the Information Age. Companies such as Google and Meta (search and social media), Experian and TransUnion (credit agencies), and Amazon and American Express (commerce) 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 decisionmaking (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, 12] to replicate all known solutions from theory, expand to more complex settings, establish the optimality of new designs, and make conjectures in regard to 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 data [2, 11, 8, 5, 19, 33, 1].

In settings with a single buyer, Bergemann et al. [7] introduce a framework in which there is a data buyer who faces a decision problem under uncertainty, and has a payoff depending on 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 a menu of statistical experiments, each experiment offering a stochastic signal that reveals some (or all) of the seller's information and each of which is associated with a price. The buyer's willingness to pay for information is determined by their type , which defines their prior belief and their value for different outcomes in the world (these arising from the action that the buyer will choose to take).

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 a binary world state and a buyer with a 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 a 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 ([32, 34, 35, 38, 46, 47, 44, 45, 23, 25, 28, 36, 24, 29] etc.), we initiate 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 a 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 a 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 a 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.

In settings with a single buyer, we learn an explicit, parameterized representation of a 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 RochetNet architecture [32] that has been used successfully for optimal auction design with a 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-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-constrained auctions [34]). We give a 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 framework [42]). We see this as an important contribution, as ex post IC is a stronger notion of IC than BIC.

A third result is 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 multi-buyer setting where the prior information is uncertain. Despite the absence 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 in 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.

Related work. Conitzer and Sandholm [20, 21] introduced the use of automated mechanism design (AMD) for economic design and framed the problem as an integer linear program (or just a 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-facility 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 [10] 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 a 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 a 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. [18] extend their setting, considering budget-constrained buyers, formulating a linear program to find the solution of the problem for a discrete type space.

There exist several other models that study revenue-optimal mechanisms for selling data. For example, Liu et al. [40] characterize the revenue-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 a setting where the buyer can conduct their own costly experiment at a 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 a distribution and constitutes a part of the buyer's type. Agarwal et 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 is different in that the sender of information has preferences on the action taken by the receiver, setting up a game-theoretic problem of strategic misrepresentation by the sender. Dughmi and Xu [31] studied this from an algorithmic perspective, and Castiglioni et al. [16] also brought in considerations from mechanism design by introducing hidden types; see also [30], [27], and [14] for more work on information design and Bayesian persuasion.

2 Preliminaries

Model. We consider a setting with n data buyers, N = { 1 , . . . , n } , each facing a decision problem under uncertainty. The state of the world , ω , is unknown and is drawn from a finite state space Ω = { ω 1 , . . . ω m } . Each buyer i can choose an action , from a finite set A i . Let A = Π n i =1 A i . The ex post payoff to buyer i for choosing action a i ∈ A i under state ω is given by U i ( ω, a i ) . Unless otherwise specified, we consider the case of matching utility payoffs where the buyer seeks to match

the state and the action 2 . In such settings, we have | Ω | = |A i | for each i ∈ [ n ] , and the payoff is given by U i ( ω, a i ) = v i · 1 { ω = a i } where each v i is drawn independently from a distribution V i . Let V = Π n i =1 V i .

Each buyer i also has an interim belief, θ i ∈ ∆(Ω) , about the world state. Each buyer's belief, θ i , is drawn independently from a distribution Θ i . Let Θ = Π n i =1 Θ i . The type of a buyer is given by the tuple ( v i , θ i ) and is denoted by ρ i . If the buyers don't vary in v i or their interim beliefs θ i , then ρ i = θ i or ρ i = v i , respectively. Let P i ≜ V i × Θ i . The utility of a buyer i ∈ [ n ] is given by the utility function u i : A × Ω × P i → R . We assume that the utility of buyer i depends only on its own type ρ i but can depend on other players' actions as a negative externality. Additionally, we follow [12] and assume that this externality is separable , thus simplifying our utility u i ( a, ω, ρ i ) = v i · 1 { a i = ω } -E -i ( a -i , ω, ρ i ) . In this paper, we consider settings where the negative externality is given by E -i ( a -i , ω, ρ i ) = α n -1 ∑ j ∈ [ n ] \ i v i · 1 { a j = ω } where α ∈ R ≥ 0 where α captures the degree of competitiveness among buyers. Let ¯ α = α n -1

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 ∈ [ n ] , a set of signals S i and a signaling scheme σ i : Ω →△S i . If state ω ∈ Ω is realized, then the seller sends a signal drawn from the distribution σ i ( ω ) . Upon receiving a signal, the buyers update their prior beliefs and choose an optimal action accordingly. The signaling scheme σ i can also be represented as a matrix π i -a collection of m row vectors each of dimensions |S i | . The j -th row vector (for j ∈ [ m ]) specifies the likelihood of each signal when state ω j is realized. Thus we have σ i ( ω j ) = π i,j .

The Mechanism Design Problem. The mechanism design goal is to design a set of experiments 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 i ∈ [ n ] as σ i : Ω ×B → △S i and a payment function t i : B → R ≥ 0 . Given bids b = ( b 1 , . . . b n ) ∈ B , if state ω ∈ Ω is realized, then the seller sends a signal drawn from the distribution σ i ( ω, b ) to buyer i and collects payment t i ( b ) . The sequence of interactions between one or more buyers and the seller takes place as follows:

The seller commits to a mechanism, M = ( σ, t ) , where σ = ( σ 1 , . . . , σ n ) is a choice and t = ( t 1 , . . . , t n ) .

Each buyer i observes their type, ρ i . The seller observes the state, ω .

Each buyer reports a message b i .

The seller sends buyer i a signal, S i ∈ S i , generated according to the signaling scheme σ i ( ω, b ) , and collects payment t i ( b ) .

Each buyer i chooses an action a i ∈ A i , and obtains utility u i ( a, ω, ρ i ) -t i ( b ) , where a = ( a 1 , . . . , a n ) .

By the revelation principle for dynamic games [43, Section 6.3], as long as we consider incentivecompatible mechanisms, it 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 i ∈ [ n ] , |S i | = |A i | . In such mechanisms, the seller designs experiments where every signal leads to 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 ( σ, 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 ˆ E -i := E ρ -i ∼P -i denote the operator used for computing the interim representations.

For each ( ρ i , ρ ′ i ) ∈ P 2 i and for each deviation function δ : A i →A i , a BIC mechanism satisfies:

In particular, this insists that double deviations (misreporting the type and disobeying the recommendation) are not profitable.

We also consider the stronger notion of ex post incentive compatible (IC), which requires, for every agent i , and for each ρ ∈ P , and assuming that every other agent reports its type truthfully and follows the recommended action, then for each misreport ρ ′ i ∈ P i , and each deviation function, δ : A i →A i , the following condition:

Individual Rationality. A mechanism is interim individually rational (IIR) if reporting an agent's true type guarantees at least as much expected utility (in expectation over other agents' reports) as opting out of the mechanism. Let σ -i : Ω ×P -i →△A -i denote the recommendations to other participating buyers when buyer i opts out. For each agent i , and each ρ i ∈ P i , IIR requires

It is in the seller's best interest to instantiate the recommendation function to the other buyers when buyer i opts out, σ -i , to minimize the value of the RHS of this IIR inequality. For this reason, it is without loss of generality to rewrite the above equation as:

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 ρ ∈ P , we require:

3 Optimal Data Market Design in the Single-Buyer Setting

In this section, we formulate the problem of optimal data market design for a single buyer as an unsupervised learning problem. We study a parametric class of mechanisms, ( σ w , t w ) ∈ M , for parameters, w ∈ R d , where d > 0 . For a single buyer, the goal is to learn parameters w that maximize E ρ ∼P [ t w ( ρ )] , such that ( σ w , t w ) satisfy ex post IC and IR. 3 By adopting differentiable loss functions, we can make use of tools from automatic differentiation and stochastic gradient descent (SGD). For notational convenience, we 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 a menu of experiments. For this, we extend the RochetNet architecture to represent a menu of priced statistical experiments. Specifically, the parameters correspond to a menu of P choices, where each choice p ∈ [ P ] is associated with an experiment, with parameters γ p ∈ R | Ω |×| A | , and a price, β p ∈ R . Given this, we define a menu entry as,

The input layer takes ρ = ( v, θ ) as input and the network computes the P utility 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 a utility of v · max j ∈ [ m ] j j,k Pr [ s k ] = j ∈ [ m ] j j,k Pr [ s k ] . Taking an expectation over all signals, buyer i receives the following utility for the choice p ∈ [ P ] :

Pr [ ω = j | a k , θ ] = θ j π p j,k Pr [ s k ] . The best action deviation is thus δ ( k ) = arg max j ∈ [ m ] θ j π p j,k Pr [ s k ] , yielding θ π p v · max θ π p

We also include an additional experiment, π 0 , which corresponds to a null experiment that generates the same signal regardless of state, and thus provides no useful information. This menu entry has a price of t 0 = 0 , and thus h 0 ( ρ ) = v · (max j ∈ [ m ] θ j ) . In particular, this menu entry has the same utility as that of opting out.

Lemma 3.1. Let p ∗ ( ρ ) = arg max p ∈{ 0 ,...P } h p ( ρ ) denote the optimal menu choice. Then the mechanism M = ( σ w , t w ) where σ w ( ω, ρ ) = π p ∗ ( ρ ) ω and t w ( ρ ) = t p ∗ ( ρ ) satisfies IC and IR.

This mechanism is IC as it is agent optimizing, and is IR as it guarantees a buyer at least the utility of opting out.

Training Problem. The expected revenue of such an IC and IR mechanism that is parameterized by γ, β is given by E ρ ∼P [ β p ∗ ( ρ ) ] . Rather than minimizing a loss function that measures errors against ground truth labels (as in a supervised learning setting), our goal is to 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 γ and β , is thus given by L ( γ, β ) = E ρ ∼P [ -∑ P p =1 β p ˜ ∇ p ( ρ ) ] where ˜ ∇ p ( ρ ) = softmax ( h 1 ( ρ ) τ , . . . , h P ( ρ ) τ ) , and τ > 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 a trained network.

During training, this null experiment ( π 0 , t 0 ) remains fixed, while the parameters γ ∈ R p ×| Ω |× m and β ∈ R p are optimized through SGD on the empirical version of the loss calculated over ℓ i.i.d samples S = { ρ (1) , . . . ρ ( ℓ ) } 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.

4 Optimal Data Market Design in the Multi-Buyer Setting

In the multi-buyer setting, the goal is to learn parameters w ∈ R d that maximize E ρ ∼S [ ∑ n i =1 t i w ( ρ )] , for a parametric class of mechanisms, ( σ w , t w ) ∈ M , such that ( σ w , t w ) 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 [32].

Neural Network Architecture. The neural network architecture has two components: one that encodes the experiments and another that encodes payments. We model these components in a straightforward way as feed-forward, fully connected neural networks with Leaky ReLU activation functions. The input layer consists of the reported type profile, ρ , encoded as a vector. For both IC and BIC settings, the component that encodes experiments outputs a matrix π of dimensions n ×| Ω | × | Ω | , 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 is non-negative and sums up to 1, the neural network first computes an output matrix, ˜ π , of the same dimension. For all i, j, ω , we then obtain π i,ω,j by computing exp(˜ π i,ω,j ) / ∑ k exp(˜ π i,ω,k ) .

For the setting of IC (vs. BIC), we define payments by first computing a normalized payment , ˜ t w i ( ρ ) ∈ [0 , 1] , for each buyer i , using a sigmoid activation unit. The payment t w i ( ρ ) is computed as

follows:

Lemma 4.1. Any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) and t w satisfies Eqn 8 is ex post IR constraint for any w ∈ R d .

For the BIC setting we only need to compute the interim payment, as we can replace E ρ -i ∼P -i [ t i ( ρ i , ρ -i )] by t i ( ρ i ) . For this, we compute an interim normalized payment, ˜ t i ∈ [0 , 1] , for each buyer i by using a sigmoid unit. We compute the interim payment as:

Lemma 4.2. Any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) and t w satisfies Eqn 9 is interim IR for any w ∈ R d .

Training Problem. In order to train the neural network, we need to minimize the negated revenue subject to incentive constraints. Following Dütting et al. [32], we measure the extent to which a mechanism violates the IC (or BIC) constraints through the notion of ex post regret (or interim regret) and then appeal to Lagrangian optimization. The regret for an agent is given by the maximum increase in utility, considering all possible misreports and all possible deviations for a 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 a buyer i as RGT w i = E ρ ∈P [ max ρ ′ i ∈P i rgt w i ( ρ ′ i , ρ ) ] where rgt w i ( ρ ′ i , ρ ) is defined as:

Lemma 4.3. Any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is ex post IC if and only if RGT w i = 0 ∀ i ∈ [ n ] , except for measure zero events.

We define the interim regret for a buyer i as ̂ RGT w i = E ρ ∈P [ max ρ ′ i ∈P i ̂ rgt w i ( ρ ′ i , ρ i ) ] where ̂ rgt w i ( ρ ′ i , ρ i ) is defined as:

Lemma 4.4. Any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is interim IC if and only if ̂ RGT w i = 0 ∀ i ∈ [ n ] , except for measure zero events.

Given regret, we compute the Lagrangian objective on the empirical version of loss and regret over samples S = { ρ (1) , . . . , ρ ( L ) } . We solve this objective using augmented Lagrangian optimization.

The BIC setting also involves an inner expectation to compute the interim representations. There, for each agent i , we sample a separate subset S -i = { ρ (1) -i , . . . , ρ ( K ) -i } , 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 a dramatic speed-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 we 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 to the shape of optimal market designs.

Buyers with world prior heterogeneity. We first show that RochetNet can recover the optimal menu for all the continuous distribution settings considered in [7] where the input distributions are a continuum of types. We specifically consider the following settings with binary states and binary actions with payoffs v = 1 and interim beliefs drawn from:

an unit interval, i.e, θ ∼ U [0 , 1] .

an equal weight mixture of Beta (8 , 30) and Beta (60 , 30) .

The optimal menus for each of these settings are given by [7]. For the first setting, the optimal menu consists of a fully informative experiment with a price of 0 . 25 . For the second setting, the seller offers two menu options: a

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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.1.

We also give the results from additional experiments in Appendix C.2, where we demonstrate how the informativeness of the menu q changes, as we vary different properties of the economic environment.

Buyers with payoff and world prior heterogeneity. We are aware of no theoretical characterization of optimal data market designs when both v and θ vary. In such cases, we can use RochetNet to 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:

v ∼ U [0 , 1] and the interim beliefs are drawn from U [0 , 1] .

v ∼ U [ c, 1] and the interim beliefs are drawn from an equal weight mixture of Beta (8 , 30) and Beta (60 , 30) for c ∈ [0 , 1] .

For Setting C, RochetNet learns a menu consisting of a single fully informative experiment with a price of 0 . 14 . For Setting D, RochetNet learns a menu consisting of a single fully informative experiment when c < 0 . 55 . For higher values of c , RochetNet learns an additional partially informative menu option. In Figure 1, we show how the differential informativeness q of the partially informative menu changes with c .

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6 Experimental Results for the Multi-Buyer Setting

In this section, we show how we can use RegretNet to recover known existing results for the multibuyer, BIC setting. We also use RegretNet to conjecture the optimal solution for a multi-buyer problem in the ex post IC setting, and then prove the optimality of the design. Lastly, we give results for a setting where it is analytically hard to compute the optimal solution but RegretNet is used to understand how revenue varies with changes to the intensity of negative externality, again building economic intuition for the market design problem.

BIC settings. [12] study the multi-buyer market design problem with two buyers, binary actions, and fixed interim beliefs, the same for each agent. They show that a 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 are defined on the interim representation, the optimal mechanism is also able to sometimes recommend incorrect actions to a buyer (in effect sending recommendations that are opposed with the realized state).

We consider the following settings from [12] with interim beliefs are given by θ 1 = θ 2 = (0 . 5 , 0 . 5) and α = 0 . 5 and:

The payoffs v 1 , v 2 are sampled from the unit interval U [0 , 1] .

The payoffs v 1 , v 2 are sampled from the exponential distribution with λ = 1 .

The results for these settings with a BIC constraint are shown in Figure 2 (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 to recover the optimal revenue as well the optimal experiment design. The test revenue and regret along with additional results are in Appendix D.1.

Ex post IC settings. We apply RegretNet to the same three settings, but now adopting ex post IC constraints. This is of interest because there is no known analytical solution for the optimal mechanism for two buyers in the ex post IC setting. See Figure 2 (bottom). Based on this, and additional experiments (in Appendix D.2 ), we were able to conjecture that the structure of the optimal design in this ex post IC setting, and we prove its optimality by following Myerson'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 a common interim belief θ i = θ . The payoff v i for a buyer i is drawn from a regular distribution V i with a continuous density function. Define virtual value ϕ i ( v i ) = v i -1 -F ( v i ) f ( v i ) , for pdf f and cdf F of distribution V i . The revenue-optimal mechanism satisfying ex post IC and IR is a mechanism that sells the fully informative experiment to buyer i if ϕ i ( v i ) ≥ ¯ α ∑ i = j ϕ j ( v j ) . Otherwise buyer i receives an uninformative menu, where the signal corresponds to the most likely state based on the prior.

Studying the effect of varying the intensity of externalities. Following [12], we study the effect of the negative externality parameter α 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 consider the case where the payoffs are constant but the interim beliefs are independently sampled from U [0 , 1] . While there is no known analytical characterization for the latter, we show how easy it is to study the models learnt by RegretNet to analyze economic properties of interest.

Figure 3 shows the effect of α on the revenue. In the context of uncertain priors, we note that the effect on revenue is similar to the setting where payoffs are uncertain. As we enhance competition intensity via α , the revenue grows. Even though the buyers are not recommended the correct action, the seller manages to generate revenue by threatening to share exclusive information with a competitor. This effect becomes fiercer in the settings with uncertain priors.

7 Conclusion

We have introduced a new deep neural network architecture 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

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adapt to relatively complex scenarios and facilitate the discovery of new optimal designs. Note that while we have only considered matching utility payoffs in this paper, our approach can easily be extended to non-matching utility payoffs as well.

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 is 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 a 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 [22, 26, 15] that provides transformations between ϵ -BIC and BIC without revenue loss in the context of auction design, extending these transformations to 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 [1, 2, 5, 8, 11, 19, 33]. Our hope is that machine learning frameworks 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.

References

Daron Acemoglu, Ali Makhdoumi, Azarakhsh Malekian, and Asu Ozdaglar. Too much data: Prices and inefficiencies in data markets. American Economic Journal: Microeconomics , 14(4): 218-256, 2022.

Alessandro Acquisti, Curtis Taylor, and Liad Wagman. The economics of privacy. Journal of economic Literature , 54(2):442-492, 2016.

Anish Agarwal, Munther Dahleh, and Tuhin Sarkar. A marketplace for data: An algorithmic solution. In Proceedings of the 2019 ACM Conference on Economics and Computation , pages 701-726, 2019.

Anish Agarwal, Munther Dahleh, Thibaut Horel, and Maryann Rui. Towards data auctions with externalities. arXiv preprint arXiv:2003.08345 , 2020.

Susan Athey, Christian Catalini, and Catherine Tucker. The digital privacy paradox: Small money, small costs, small talk. Technical report, National Bureau of Economic Research, 2017.

Moshe Babaioff, Robert Kleinberg, and Renato Paes Leme. Optimal mechanisms for selling information. In Proceedings of the 13th ACM Conference on Electronic Commerce , EC '12, page 92-109, New York, NY, USA, 2012. Association for Computing Machinery. ISBN 9781450314152. doi: 10.1145/2229012.2229024. URL https://doi.org/10.1145/ 2229012.2229024 .

Dirk Bergemann, Alessandro Bonatti, and Alex Smolin. The design and price of information. American Economic Review , 108(1):1-48, January 2018. doi: 10.1257/aer.20161079. URL https://www.aeaweb.org/articles?id=10.1257/aer.20161079 .

Dirk Bergemann, Alessandro Bonatti, and Tan Gan. The economics of social data. The RAND Journal of Economics , 53(2):263-296, 2022. doi: https://doi.org/10.1111/1756-2171.12407. URL https://onlinelibrary.wiley.com/doi/abs/10.1111/1756-2171.12407 .

Dirk Bergemann, Yang Cai, Grigoris Velegkas, and Mingfei Zhao. Is selling complete information (approximately) optimal? In Proceedings of the 23rd ACM Conference on Economics and Computation , pages 608-663, 2022.

David Blackwell. Equivalent comparisons of experiments. The annals of mathematical statistics , pages 265-272, 1953.

Alessandro Bonatti. The platform dimension of digital privacy. Technical report, National Bureau of Economic Research, 2022.

Alessandro Bonatti, Munther Dahleh, Thibaut Horel, and Amir Nouripour. Selling information in competitive environments. arXiv preprint arXiv:2202.08780 , 2022.

Yang Cai and Grigoris Velegkas. How to Sell Information Optimally: An Algorithmic Study. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021) , volume 185 of Leibniz International Proceedings in Informatics (LIPIcs) , pages 81:181:20, Dagstuhl, Germany, 2021. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. ISBN 978-3-95977-177-1. doi: 10.4230/LIPIcs.ITCS.2021.81. URL https://drops.dagstuhl. de/opus/volltexte/2021/13620 .

Yang Cai, Federico Echenique, Hu Fu, Katrina Ligett, Adam Wierman, and Juba Ziani. Thirdparty data providers ruin simple mechanisms. Proceedings of the ACM on Measurement and Analysis of Computing Systems , 4(1):1-31, 2020.

Yang Cai, Argyris Oikonomou, Grigoris Velegkas, and Mingfei Zhao. An efficient ϵ -bic to BIC transformation and its application to black-box reduction in revenue maximization. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021 , pages 1337-1356. SIAM, 2021.

Matteo Castiglioni, Alberto Marchesi, and Nicola Gatti. Bayesian persuasion meets mechanism design: Going beyond intractability with type reporting. In Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems , page 226-234, 2022.

Junjie Chen, Minming Li, and Haifeng Xu. Selling data to a machine learner: Pricing via costly signaling. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research , pages 3336-3359. PMLR, 17-23 Jul 2022. URL https://proceedings.mlr.press/v162/chen22m.html .

Yiling Chen, Haifeng Xu, and Shuran Zheng. Selling information through consulting. In ACM-SIAM Symposium on Discrete Algorithms (SODA 2020). Presented in ACM/INFORMS Workshop on Market Design 2019 , 2020. URL https://arxiv.org/abs/1907.04397 .

Jay Pil Choi, Doh-Shin Jeon, and Byung-Cheol Kim. Privacy and personal data collection with information externalities. Journal of Public Economics , 173:113-124, 2019.

Vincent Conitzer and Tuomas Sandholm. Complexity of mechanism design. In Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence , UAI'02, page 103-110, San Francisco, CA, USA, 2002. Morgan Kaufmann Publishers Inc. ISBN 1558608974.

Vincent Conitzer and Tuomas Sandholm. Self-interested automated mechanism design and implications for optimal combinatorial auctions. In Proceedings of the 5th ACM Conference on Electronic Commerce , EC '04, page 132-141, New York, NY, USA, 2004. Association for Computing Machinery. ISBN 1581137710. doi: 10.1145/988772.988793. URL https: //doi.org/10.1145/988772.988793 .

Vincent Conitzer, Zhe Feng, David C. Parkes, and Eric Sodomka. Welfare-preserving ε -bic to bic transformation with negligible revenue loss. In Web and Internet Economics , pages 76-94, 2022.

Michael Curry, Ping-Yeh Chiang, Tom Goldstein, and John Dickerson. Certifying strategyproof auction networks. Advances in Neural Information Processing Systems , 33:4987-4998, 2020.

Michael Curry, Tuomas Sandholm, and John Dickerson. Differentiable economics for randomized affine maximizer auctions. In Edith Elkind, editor, Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI-23 , pages 2633-2641. International Joint Conferences on Artificial Intelligence Organization, 8 2023. doi: 10.24963/ijcai.2023/293. URL https://doi.org/10.24963/ijcai.2023/293 . Main Track.

Michael J Curry, Uro Lyi, Tom Goldstein, and John P Dickerson. Learning revenue-maximizing auctions with differentiable matching. In International Conference on Artificial Intelligence and Statistics , pages 6062-6073. PMLR, 2022.

C. Daskalakis and S. M. Weinberg. Symmetries and optimal multi-dimensional mechanism design. In Proceedings of the 13th ACM Conference on Electronic Commerce , pages 370-387, 2012.

Constantinos Daskalakis, Christos Papadimitriou, and Christos Tzamos. Does information revelation improve revenue? In Proceedings of the 2016 ACM Conference on Economics and Computation , pages 233-250, 2016.

Zhijian Duan, Jingwu Tang, Yutong Yin, Zhe Feng, Xiang Yan, Manzil Zaheer, and Xiaotie Deng. A context-integrated transformer-based neural network for auction design. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research , pages 5609-5626. PMLR, 17-23 Jul 2022. URL https://proceedings.mlr.press/v162/duan22a.html .

Zhijian Duan, Haoran Sun, Yurong Chen, and Xiaotie Deng. A scalable neural network for dsic affine maximizer auction design. In Thirty-seventh Conference on Neural Information Processing Systems , 2023. URL https://openreview.net/forum?id=cNb5hkTfGC .

Shaddin Dughmi. On the hardness of signaling. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science , pages 354-363. IEEE, 2014.

Shaddin Dughmi and Haifeng Xu. Algorithmic bayesian persuasion. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing , pages 412-425, 2016.

Paul Dütting, Zhe Feng, Harikrishna Narasimhan, David Parkes, and Sai Srivatsa Ravindranath. Optimal auctions through deep learning: Advances in differentiable economics. J. ACM , 2023. An earlier version appeared in Proceedings of the 36th International Conference on Machine Learning . ICML 2019.

Itay P Fainmesser, Andrea Galeotti, and Ruslan Momot. Digital privacy. Management Science , 2022.

Zhe Feng, Harikrishna Narasimhan, and David C Parkes. Deep learning for revenue-optimal auctions with budgets. In Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems , pages 354-362, 2018.

Noah Golowich, Harikrishna Narasimhan, and David C Parkes. Deep learning for multi-facility location mechanism design. In IJCAI , pages 261-267, 2018.

Dmitry Ivanov, Iskander Safiulin, Igor Filippov, and Ksenia Balabaeva. Optimal-er auctions through attention. In NeurIPS , 2022. URL http://papers.nips.cc/paper_files/paper/ 2022/hash/e0c07bb70721255482020afca44cabf2-Abstract-Conference.html .

Emir Kamenica and Matthew Gentzkow. Bayesian persuasion. American Economic Review , 101(6):2590-2615, 2011.

Kevin Kuo, Anthony Ostuni, Elizabeth Horishny, Michael J Curry, Samuel Dooley, Ping-yeh Chiang, Tom Goldstein, and John P Dickerson. Proportionnet: Balancing fairness and revenue for auction design with deep learning. arXiv preprint arXiv:2010.06398 , 2020.

Yingkai Li. Selling data to an agent with endogenous information. In Proceedings of the 23rd ACM Conference on Economics and Computation , EC '22, page 664-665, New York, NY, USA, 2022. Association for Computing Machinery. ISBN 9781450391504. doi: 10.1145/3490486. 3538360. URL https://doi.org/10.1145/3490486.3538360 .

Shuze Liu, Weiran Shen, and Haifeng Xu. Optimal pricing of information. In Proceedings of the 22nd ACM Conference on Economics and Computation , EC '21, page 693, New York, NY, USA, 2021. Association for Computing Machinery. ISBN 9781450385541. doi: 10.1145/ 3465456.3467551. URL https://doi.org/10.1145/3465456.3467551 .

Sameer Mehta, Milind Dawande, Ganesh Janakiraman, and Vijay Mookerjee. How to sell a dataset? pricing policies for data monetization. In Proceedings of the 2019 ACM Conference on Economics and Computation , EC '19, page 679, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450367929. doi: 10.1145/3328526.3329587. URL https://doi.org/10.1145/3328526.3329587 .

Roger B. Myerson. Optimal auction design. Mathematics of Operations Research , 6(1):58-73, 1981. ISSN 0364765X, 15265471. URL http://www.jstor.org/stable/3689266 .

Roger B Myerson. Game theory: analysis of conflict . Harvard university press, 1991.

Neehar Peri, Michael Curry, Samuel Dooley, and John Dickerson. Preferencenet: Encoding human preferences in auction design with deep learning. Advances in Neural Information Processing Systems , 34:17532-17542, 2021.

Jad Rahme, Samy Jelassi, Joan Bruna, and S. Matthew Weinberg. A permutation-equivariant neural network architecture for auction design. In Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2-9, 2021 , pages 5664-5672. AAAI Press, 2021. URL https://ojs.aaai.org/index.php/AAAI/article/view/16711 .

Sai Srivatsa Ravindranath, Zhe Feng, Shira Li, Jonathan Ma, Scott D Kominers, and David C Parkes. Deep learning for two-sided matching. arXiv preprint arXiv:2107.03427 , 2021.

Andrea Tacchetti, DJ Strouse, Marta Garnelo, Thore Graepel, and Yoram Bachrach. A neural architecture for designing truthful and efficient auctions. arXiv preprint arXiv:1907.05181 , 2019.

A Data Markets in the Single-Buyer Settings

A.1 RochetNet Architecture

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A.2 Setup and Hyper-parameters

Training: We train RochetNet with P = 1000 possible menu entries. We set the softmax temperature τ to 1 200 . We train RochetNet for 20 , 000 iterations with a minibatch of size 2 15 sampled online for every update.

We report all our results on a test-set of 20000 samples that are separate from the train set.

Time: For the settings studied in the paper, RochetNet take 7-8 minutes minutes to train on a single NVIDIA Tesla V100 GPU.

B Data Markets in the Multi-Buyer Setting

B.1 RegretNet Architecture

The neural network architecture for the multi-buyer ex post IC setting is depicted in Fig 5. For the BIC setting, the experiment network σ 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 ˆ E -i [ t i ( ρ i , ρ -i )] with just t i ( ρ i ) . 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 = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) and t w satisfies Eqn 8 is ex post IR constraint for any w ∈ R d .

Proof. For the ex post IC setting, we first compute a normalized payment , ˜ t w i ( ρ ) ∈ [0 , 1] , for each buyer i by using a sigmoidal unit. We scale this by the difference between utility achieved by the buyer when he is truthful and obedient and the utility of opting out. Thus, the payment t w i ( ρ ) is

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To establish ex post IR, we reason about the utility a buyer can obtain when not participating in the mechanism. As per the discussion in Section 2, in an optimal data market the seller will send optimal recommendations to participating buyers in the event that buyer i opts out, thus minimizing the utility of the buyer who opts out. In particular, the recommendation for any j = i , for an opting out buyer i , is such that E a ∼ σ -i ( ω ; ρ -i ) ω ∼ θ i [ 1 { a j = ω } ] = 1 . Given this, the utility of buyer i , when opting out, is

Taking this with the fact that ˜ t w i ( ρ ) ∈ [0 , 1] , we have

Rearranging, and substituting v i · (max θ i -α ) from Eqn 13, we get the IR constraint described in Eqn 5.

B.3 Proof of Lemma 4.2

Lemma 4.2. Any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) and t w satisfies Eqn 9 is interim IR for any w ∈ R d .

Proof. For the BIC setting, we work with the interim payment, and make use of a normalized interim payment, ˜ t i ( ρ i ) ∈ [0 , 1] , for each buyer i , by using a sigmoidal unit. In this case, we scale this normalized payment by the difference between the interim utility achieved when the buyer is truthful and obedient and the interim utility of opting out. Thus, the interim payment is

̸

The utility of the outside option is again computed similarly to the ex post IC setting, recognizing that the seller in the optimal mechanism will make recommendation ˆ E -i [ E a ∼ σ -i ( ω ; ρ -i ) , ω ∼ θ i [ 1 { a j = ω } ] ] = 1 to any agent j = i , in the case that i drops out. In particular, the utility of buyer i , when opting out, is

Taking this with the fact that ˜ t w i ( ρ i ) ∈ [0 , 1] , we have

Rearranging the above and substituting v i · (max θ i -α ) from Eqn 16, we get the IIR constraint described in Eqn 4.

B.4 Proof of Lemma 4.3

Lemma 4.3. Any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is ex post IC if and only if RGT w i = 0 ∀ i ∈ [ n ] , except for measure zero events.

Proof. We first prove the forward direction: if any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is ex post IC, then RGT w i = 0 , ∀ i ∈ [ n ] .

For the ex post IC setting, the incentive compatibility constraints requires that for every agent i , and for each ρ ∈ P , and assuming that every other agent reports its type truthfully and follows the recommended action, then for each misreport ρ ′ i ∈ P i , and each deviation function, δ : A i →A i , the following condition:

Note that the above hold for any deviation function δ . Therefore, we can write that

We can expand to get the following equation:

Pushing the max inside (since other terms don't involve δ ( k ) ), we note that:

Thus we have,

Note that by our definition of ex post regret in Equation 10, the above is equivalent to rgt w i ( ρ ′ i , ρ ) ≤ 0 . Furthermore, this holds for any deviating report ρ ′ i . Thus, we can write:

But note that

where the last step holds since max k ′ ∈ [ m ] { π w i,k ′ ,k ( ρ ) θ i,k ′ } ≥ π w i,k,k ( ρ ) θ i,k

Combining Equations 23 and 24, we have max ρ ′ i ∈P i rgt w i ( ρ ′ i , ρ ) = 0 . Taking the expectation over all profiles ρ ∈ P , we have E ρ ∈P [ max ρ ′ i ∈P i rgt w i ( ρ ′ i , ρ ) ] = 0 . Thus, RGT w i = 0 , ∀ i ∈ [ n ] . Thus, we've shown that if any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is ex post IC, then RGT w i = 0 , as desired.

Next, we consider the reverse direction: if RGT w i = 0 , ∀ i ∈ [ n ] , we want to show that any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is ex post IC. Starting with RGT w i = 0 , ∀ i ∈ [ n ] , we have that by definition:

We showed in Equation 24 that rgt w i ( ρ i , ρ ) ≥ 0 . Since ρ i ∈ P i , we have that max ρ ′ i ∈P i rgt w i ( ρ ′ i , ρ ) ≥ 0 . Together with the fact that E ρ ∈P [ max ρ ′ i ∈P i rgt w i ( ρ ′ i , ρ ) ] = 0 , we have that max ρ ′ i ∈P i rgt w i ( ρ ′ i , ρ ) = 0 = ⇒ rgt w i ( ρ ′ i , ρ ) ≤ 0 .

Plugging in our definition for ex post regret, we have that:

This can be transformed back to:

This is exactly the ex post IC constraint described in Equation 2

B.5 Proof of Lemma 4.4

Lemma 4.4. Any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is interim IC if and only if ̂ RGT w i = 0 ∀ i ∈ [ n ] , except for measure zero events.

Proof. We first prove the forward direction: if any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is interim IC, then ̂ RGT w i = 0 , ∀ i ∈ [ n ] .

For the BIC setting, for each ( ρ i , ρ ′ i ) ∈ P 2 i and for each deviation function δ : A i → A i , a BIC mechanism satisfies:

Note that the above hold for any deviation function δ , therefore we can write that

We can expand to get the following equations:

The above equations follow from the linearity of expectations and from pushing the max to the inside (since other terms don't involve δ ( k ) ).

Note that by our definition of interim regret in Equation 11, the above equation is equivalent to ̂ rgt w i ( ρ ′ i , ρ i ) ≤ 0 . Furthermore, this holds for any deviating report ρ ′ i . Therefore, we can write that

But note that

where the second step follows from linearity of expectation, and the last step holds since we are taking the max of all possible k ′ ∈ [ m ] , which includes k ′ = k .

Combining Equations 31 and 32, we have that max ρ ′ i ∈P i ̂ rgt w i ( ρ ′ i , ρ i ) = 0 . Taking the expectation over all profiles ρ , we have ˆ E ρ ∈P [ max ρ ′ i ∈P i ̂ rgt w i ( ρ ′ i , ρ i ) ] = 0 = ⇒ ̂ RGT w i = 0 , ∀ i ∈ [ n ] . Thus, we've showed that if any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is BIC, then ̂ RGT w i = 0 , as desired.

Next, we consider the reverse direction: if ̂ RGT w i = 0 , ∀ i ∈ [ n ] , we want to show that any mechanism M = ( σ w , t w ) where σ w i ( ω, ρ ) = ( π w i,ω ( ρ ) ) is BIC. Starting with ̂ RGT w i = 0 , ∀ i ∈ [ n ] , we have that by definition: ˆ E ρ ∈P [ max ρ ′ i ∈P i ̂ rgt w i ( ρ ′ i , ρ i ) ] = 0

We showed in Equation 32 that ̂ rgt w i ( ρ i , ρ i ) ≥ 0 . Since ρ i ∈ P i , we have that max ρ ′ i ∈P i ̂ rgt w i ( ρ ′ i , ρ i ) ≥ 0 . Together with the fact that ˆ E ρ ∈P [ max ρ ′ i ∈P i ̂ rgt w i ( ρ ′ i , ρ i ) ] = 0 , we have that max ρ ′ i ∈P i ̂ rgt w i ( ρ ′ i , ρ i ) = 0 = ⇒ ̂ rgt w i ( ρ ′ i , ρ i ) ≤ 0 . Plugging in our definition for interim regret, we have that:

This can be transformed back to:

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 a learning rate of 0 . 001 . The Lagrangian parameters ( λ 1 , . . . λ n ) are set to 10 . 0 . The coefficient of the penalty term is initialized to 1 . 0 . The Lagrangian updates are performed once every 100 iteration.

Testing: We report all our results on a test size of 20000 samples. For the ex post setting, we use 100 misreports as initialization to warm-start the inner maximization to compute regret. For the BIC setting, we use other samples of the minibatch to compute a defeating misreport as noted in the previous subsection. We then run 100 steps of gradient ascent with a learning rate of 0 . 005 to compute the defeating misreport and the regret more accurately.

Training time: RegretNet takes 11 - 12 minutes per experiment for training for all the settings studied in this paper. All our experiments were run on a single NVIDIA Tesla V100 GPU.

C Experimental Results for the Single Buyer Setting

C.1 Buyers with world prior heterogeneity

We present optimal menus, associated prices, revenue, and RochetNet revenue for the Single Buyer Setting A and B in Figure 6

C.2 Additional Experiments for Buyers with world prior heterogeneity

We design additional experiments where we use RochetNet to study how the informativeness of learned experiments vary when we change different properties of the prior distributions. For this, we consider the mixture of Beta distributions considered in setting B. This is a bimodal distribution with modes at θ L = ( 7 36 , 29 36 ) and θ H = ( 59 88 , 29 88 ) . We call the former 'low type" and the latter 'high type" following the analysis of [7] for the case of binary types, and order this way because

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| θ L -0 . 5 | ≥ | θ H -0 . 5 | . We also call the value ( | θ -0 . 5 | ) the precision of a type's prior belief. We design two experiments:

Single buyer, Binary State, and Binary Actions, where the payoffs v 1 = 1 and the interim beliefs are drawn from a mixture of Beta (8 , 30) and Beta (60 , 30) weighted by (1 -γ, γ ) for γ ∈ [0 , 1] .

Single buyer, Binary State, and Binary Actions, where the payoffs v 1 = 1 and the interim beliefs are drawn from an equal weight mixture of Beta ( a 1 , b 1 ) and Beta (60 , 30) . We vary a 1 , b 1 so that the mode of Beta ( a 1 , b 1 ) decreases (therefore the precision of the low type's prior belief | θ L -0 . 5 | increases) while ensuring that the ratio of values at the modes of these distributions stays constant.

In particular, the modes of the two distributions are θ L = ( a -1 a + b -2 , b -1 a + b -2 ) and θ H = ( 59 88 , 29 88 )

in the case if Experiment G. 4 . θ L and θ H are non-congruent types, i.e., without the supplemental information, a buyer with type θ L takes action 0 while the other takes action 1 . θ H also values a fully informative experiment more than θ L . In the first experiment, we change the likelihood of type θ H with respect to type θ L . In the second experiment, we change the precision of the belief of one type while keeping the other fixed (we vary values of a, b while numerically ensuring that the probability distribution function of all the plotted points have the same height at the mode).

The results are given in Figure 7. For Experiment G, increasing the high type's frequency decreases the low type experiment's informativeness. For Experiment H, increasing the precision of the low type's prior belief decreases the low type experiment's informativeness. [7] characterize the informativeness of the optimal experiment for similar settings, but for discrete distributions with two types. In particular, Proposition 3 from [7] states that informativeness of the low type π 1 , 1 , decreases when the frequency of the high type or with the precision of low type's prior belief | θ L -0 . 5 | . It is interesting, then, that we observe the same behaviour in the RochetNet results for a continuous analog of their discrete distribution set-up. This suggests a new target for economic theory.

D Experimental Results for the Multi Buyer Setting

We consider the following multi-buyer data market design problem with two buyers, binary action and fixed interim beliefs, adopting all possible combinations of the following choices in regard to the configuration of the economy:

θ = (0 . 5 , 0 . 5) or θ = (0 . 75 , 0 . 25)

α = 0 . 5 or α = 2 . 0

Both payoffs drawn from UNF, EXP, or Asym UNF

UNF is the uniform distribution over the unit interval U [0 , 1] . EXP is the exponential distribution with λ = 1 . For the Asym UNF, the payoff for buyer i is uniform over the interval [0 , i +1] . In the next two subsections, we present our results for the BIC and ex post IC settings, respectively.

D.1 BIC Settings

Figures 8, 9, and 10 show the optimal data market design and the data marker learned by RegretNet for the BIC settings described above, and Figure 11 gives the test revenue and regret obtained by RegretNet and the revenue of the optimal mechanism.

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D.2 Ex post IC Settings

Figures 12, 13, and 14 show the optimal data market design and the data market learned by RegretNet for the ex post IC settings described above, and Figure 15 gives the test revenue and test regret obtained by RegretNet along with the revenue of the optimal mechanism.

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D.3 Additional Results for the Irregular Distributions

We present additional results for both BIC and ex post IC settings for two buyers, binary actions, and interim beliefs θ = (0 . 5 , 0 . 5) , the same for each buyer. We set α = 0 . 5 , and the payoffs are drawn from the irregular distribution whose pdf f ( v ) is:

The optimal solutions for these problems make use of ironed virtual values as the payoff distribution is irregular (see Fig 16).

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=0.50

0.0

0.2

0.4

0.6

0.8

v

1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

v

2

0.0

0.2

0.4

0.6

0.8

v

1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

v

2

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

IRR,

=0.50,

=0.50

D.4 Proof of Theorem 6.1

Theorem 6.1. Consider the setting with Binary State and Binary Actions where the buyers have a common interim belief θ i = θ . The payoff v i for a buyer i is drawn from a regular distribution V i with a continuous density function. Define virtual value ϕ i ( v i ) = v i -1 -F ( v i ) f ( v i ) , for pdf f and cdf F of distribution V i . The revenue-optimal mechanism satisfying ex post IC and IR is a mechanism that sells the fully informative experiment to buyer i if ϕ i ( v i ) ≥ ¯ α ∑ i = j ϕ j ( v j ) . Otherwise buyer i receives an uninformative menu, where the signal corresponds to the most likely state based on the prior.

̸

Proof. We consider a setting with binary states, binary actions, and where each buyer has the same interim belief, set to ( θ, 1 -θ ) . Without loss of generality, let θ ≥ (1 -θ ) . Since the interim beliefs are fixed, we will just represent buyer types with v i instead of ρ i . Let x i ( v ) = E a ∼ σ ( ω ; ρ ) , ω ∼ θ [ 1 { a i = ω } ] .

i Let π i ( v ) be the matrix representation of the experiment assigned to buyer i i.e. π i,ω ( v ) = σ i ( ω, v )

We first show that a mechanism is obedient if and only if the experiment assigned to the buyer satisfies x i ( v ) ≥ θ . If a mechanism is obedient, then θ · π i, 1 , 1 ( v ) ≥ (1 -θ ) π i, 2 , 1 ( v ) and (1 -θ ) π i, 2 , 2 ( v ) ≥ θ · π i, 1 , 2 ( v ) . We have x i ( v ) = θ · π i, 1 , 1 ( v ) + (1 -θ ) π i, 2 , 2 ( v ) ≥ θ · π i, 1 , 1 ( v ) + θ · π i, 1 , 2 ( v ) = θ ( π i, 1 , 1 ( v ) + π i, 1 , 2 ( v )) = θ .

In the other direction, consider when x i ( v ) ≥ θ . In this case, we show that both θ · π i, 1 , 1 ( v ) ≥ (1 -θ ) π i, 2 , 1 ( v ) and (1 -θ ) π i, 2 , 2 ( v ) ≥ θ · π i, 1 , 2 ( v ) need to hold for obedience. Assume to the contrary, one of these fails to hold when x i ( v ) ≥ θ . We have one of,

(1 -θ ) π i, 2 , 2 ( v ) < θ · π i, 1 , 2 ( v ) , and we have x i ( v ) = θ · π i, 1 , 1 ( v ) + (1 -θ ) π i, 2 , 2 ( v ) < θ · π i, 1 , 1 ( v ) + θ · π i, 1 , 2 ( v ) = θ ( π i, 1 , 1 ( v ) + π i, 1 , 2 ( v )) = θ .

θ · π i, 1 , 1 ( v ) < (1 -θ ) π i, 2 , 1 ( v ) , and we have x i ( v ) = θ · π i, 1 , 1 ( v ) + (1 -θ ) π i, 2 , 2 ( v ) < (1 -θ ) π i, 2 , 1 ( v ) + (1 -θ ) π i, 2 , 2 ( v ) = (1 -θ )( π i, 2 , 1 ( v ) + π i, 2 , 2 ( v )) = (1 -θ ) ≤ θ .

In either case, we have a contradiction.

Since our payoff is linear, from Proposition 3.5 in [7], the IC constraints are satisfied only when the truthfulness and obedience constraints are satisfied. Denote ˜ x i ( v ) = x i ( v ) -α n -1 ∑ j ∈ [ n ] \ i x j ( v ) .

Thus, the optimal design problem to solve is:

The first constraint corresponds to truthfulness for the IC setting, the second is the IR constraint, and the third is the obedience constraint. We have thus reduced the data market design problem to that of Myerson's revenue maximizing single-item auction problem. Instead of ˜ x i ( v ) denoting the probability of allocating an item, it denotes expected payoff after accounting for the negative externalities. Also, rather than allocative constraints, we have obedience constraints on x i ( v ) which requires x i ( v ) ≥ θ . Thus, by Myerson's theory, the total expected revenue is equal to the expected virtual welfare minus some constant K (stemming from IR constraints) for some ˜ x i ( v i , v -i ) that is non-decreasing in v i . Thus, we have:

In order to maximize the revenue, we need to maximize the virtual welfare. Thus we can set x i ( v ) = 1 (a fully informative experiment) when ϕ i ( v i ) -∑ j ∈ [ n ] \ i ϕ j ( v j ) ≥ 0 and we x i ( v ) = θ (an uninformative experiment that always sends the same signal regardless of the state) otherwise. This is precisely the mechanism described in the theorem.

Note that such an ˜ x i ( v i , v -i ) is non-decreasing in v i (since x i ( v i , v -i ) is non-decreasing in v i and x j ( v i , v -i ) for j ∈ [ n ] \ i is non-increasing in v i for regular distributions V i and V j ). Moreover, x i ( v ) ≥ θ is also satisfied.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\n- 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\nQUESTIONS:\n- 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\nLIMITATIONS:\n- 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.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n1. 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.\n\nWEAKNESSES:\n1. 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.\n\nQUESTIONS:\n**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.\n\nLIMITATIONS:\nThe 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?", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nThe 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.\n\nQUESTIONS:\nCan the method in fact scale to larger (even just 3-5 buyers) settings, or not? This should be discussed.\n\nLIMITATIONS:\nSee questions.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-05"}

comments: 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.

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.

- 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.

-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.

QUESTIONS:
What are some baseline methods to compare with? For example, how does the standard rochetnet perform on the proposed market settings?

LIMITATIONS:
Yes.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n+ 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\nWEAKNESSES:\n+ 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\nQUESTIONS:\n+ 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\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-07"}