An Instrumental Value for Data Production and its Application to Data Pricing
Authors: Rui Ai
Boxiang Lyu
Zhaoran Wang
Zhuoran Yang
Haifeng Xu
Venue: ICML 2025 poster (2025)
Keywords: Instrumental Value
Data Production Process
Data Pricing
Data Customization
Primary area: theory->game_theory

Abstract: We develop a framework for capturing the instrumental
value of data production processes, which
accounts for two key factors: (a) the context of
the agent’s decision-making; (b) how much data
or information the buyer already possesses. We
"micro-found" our data valuation function by establishing
its connection to classic notions of signals
and information design in economics. When
instantiated in Bayesian linear regression, our
value naturally corresponds to information gain.
Applying our proposed data value in Bayesian linear
regression for monopoly pricing, we show that
if the seller can fully customize data production,
she can extract the first-best revenue (i.e., full surplus)
from any population of buyers, i.e., achieving
first-degree price discrimination. If data can
only be constructed from an existing data pool,
this limits the seller’s ability to customize, and
achieving first-best revenue becomes generally
impossible. However, we design a mechanism
that achieves seller revenue at most $\log(\kappa)$ less
than the first-best, where $\kappa$ is the condition number
associated with the data matrix. As a corollary,
the seller extracts the first-best revenue in the
multi-armed bandits special case.

An Instrumental Value for Data Production and its Application to Data Pricing
Rui Ai Boxiang Lyu? Zhaoran Wang? Zhuoran Yang* Haifeng Xu 5
Abstract
ing (Goldsmith & Freiden, 2004) data-hungry applica- tions such natural language processing (Kang et al., 2020) and self-driving (Ni et al., 2020). These applications high- light the demand for high-quality labeled data, demand that a growing number of data sellers wish to meet. Start-up companies such as Scale AI, Parallel Domain, Dataloop, and V7 compete with established companies, such as Acxiom, Oracle, and Nielsen, in this growing arena, underlying the importance of formal analyses of buying and selling data.
We develop a framework for capturing the instru- mental value of data production processes, which accounts for two key factors: (a) the context of the agent's decision-making: (b) how much data information the buyer already possesses. We "miero-foundd our data valuation function by es- tablishing its connection to classic notions of sig- nals and information design in economics. When instantiated in Bayesian linear regression, our value naturally corresponds to information gain. Applying our proposed data value in Bayesian lin- ear regression for monopoly pricing, we show that if the seller can fully customize data production, she can extract the first-best revenue (i.e., full sur- plus) from any population of buyers, i.e., achiev- ing first-degree price discrimination. If data can only be constructed from an existing data pool, this limits the seller's ability to customize, and achieving first-best revenue becomes generally impossible. However, we design a mechanism that achieves seller revenue at most log(k) less than the first-best, where K is the condition num- ber associated with the data matrix. As a corol- lary, the seller extracts the first-best revenue in the cenii- bandits special case.
Like physical goods and systems, data has both instrumen- tal value and intrinsic value at the same time. For example, the New York Times (NYT) as a news source has its own intrinsic value, containing major up-to-date news. However, serving as a Corpus for a particular machine learning task like retrieval augmented generation (RAG) (Lewis al., 2020), its value towards improving the performance of the task is instrumental. The fundamental difference between instrumental vs. intrinsic value is widely studied in philoso- phy. Citing the widely accepted definitions from Beardsley (1965): X has instrumental value means X conducive to something that has intrinsic value. In the above example of NYT data, its instrumental value a RAG task depends on the extra values it adds and, crucially, depends what data the buyer already has. For instance, when the RAG system already has the data from other substitutable news sources (e.g.. the Washington Post, Wall Street Journal, etc.), the NYT data's instrumental value will become smaller, despite holding the same intrinsic value. This is also the reason that most economic activities, including the sale of data, are based items' instrumental values, as illustrated by Foster (1981): "The instrumental efficiency of the economic pro- cess is the criterion of judgment in terms of which, and only in terms of which, we may resolve economic problems". To our knowledge, there is no notion of such instrumental data value in current literature; this is what we embark on studying in this work.
1. Introduction
Trading data forms an ever-growing segment in today's digital economy. Advances in machine learning motivate companies to apply data-driven approaches to solve grow- ing variety of problems, ranging from personalized market-
Instittte for Data, Systems, and Society, MIT, Cambridge, MA, USA 'Booth School of Business, University of Chicago. Chicago, II, USA 'Department Management Sciences, Northwestern University, Evanston, II, USA Depparmment of Statistics and Data Science, Yale University, New Haven, CT, USA Department of Computer Science, Univer- sity of Chicago, Chicago, IL. USA. Correspondence to: Rui Ai <ruiai @eii@@iittcc edu>, Haifeng Xu <haifengxu uuchucco..cdcccc.......
Industrial Engineering and
Data sold by commercial companies falls into situ- ations, which we dub perfect customization and limited customization. Perfect customization can offer data labels for any feature directions specified by the buyer, whereas limited customization can only curate data based on some pre-collected dataset, hence has limited flexibility. For ex- ample, a company that wishes to develop a model for image classification can either [perfect customization] submit its
Proceedings of the nnnd International Conference Machine Learning, Vancouver, Canada, PMLR 267, 2025. Copyright 20255 by the author(s).

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An Instrumental Value for Data Production and its Application to Data Pricing
most desirable set of unlabelled images and buy labels for these images from the data seller, or [limited customiza-
also hints the potential concerns of highly-screwed (to- wards seller) surplus distribution data markets and calls for future government regulation.
tion] purchase a labeled dataset of useful, though possibly not the most desirable images.
Approximate optimality for selling data under limited customization. When the seller has limited customiza- tion ability and can only curate data derivatives from a pre-collected dataset, we exhibit a novel data-selling mecha- nism based on the SVD decomposition of the data matrix and show that it almost extracts full welfare, up to small constant which relates to the condition number of the data matrix of the to-be-sold dataset. Therefore, under this mech- anism, buyers can only enjoy at most a constant amount of surplus.
1.1. Our Contributions
In the paper, we first characterize valid valuations in Sec- tions 2 and 3 and then present concrete application, say, mechanism design for data pricing, in Section 4. We sketch our contributions as follows.
new notion for data's instrumental value. We first introduce a general framework to quantify the instrumental value of data which closely relates generalized notion of Bregman Divergence, inspired by Frankel & Kamenica (2019). Our notion rests on a deep duality relation between the measure of parameter uncertainty and the measure of data value. We show that, when instantiated in Bayesian linear regression, our value corresponds precisely to the nat- ural notion of information gain of the new data, relative to existing data. This special form of data instrumental value offers a useful basis for practitioners since the Bayesian linear model is a widely studied foundational model, and has been proven to be a good approximation when we do not have specific knowledge about the underlying model structure. Quoting Besbes Zeevi (2015), linearity offers the greatest robustness under misspecification. Following this spirit, we use this partieular valuation form demon- strate our insights. Notably, our instrumental data value crucially differs from Data Shapley (Ghorbani & Zou, 2019) and its variants. Data Shapley captures a data point's ex- pected contribution to an ML task, whereas our data value captures a dataset's is marginal contribution beyond existing data, hence more naturally captures the dataset's 's economic value. Another strong advantage is that our new data val- uation can be computed with only one model retraining, hence much more practical to compute in reality (see more detailed discussions in Section 2.3).
1.2. Related Works
Our work is closely related to the literature on valuing data and information (Ghorbani & Zou, 2019; Jia et al., 2019; Frankel Kamenica, 2019; Schoch et al., 2022), and selling data and information (Admati & Pfleiderer, 1988; Babaioff et al., 2012; Xiang & Sarvary, 2013; Bergemann & Bonatti, 2015; Hörner & Skrzypacz, 2016; Kastl et al., 2018; Chen et al., 2019; et al., 2022). Due to the space limit, we discuss these and more related works thoroughly in Appendix A.I.
2021; Liu et al., 2021a; Chen
2. An Instrumental Value of Data Production and its Micro-foundation
In this section, we introduce a new way to quantify the instrumental value of a Data Production Process (DPP), which captures DPP's expected value for certain task. Its formulation is fundamentally different from the widely studied Data Shapley (Roth, 1988; Ghorbani & Zou, 2019; Jia et al., 2019), which quantifies the value of realized dataset, hence can usually be evaluated only by accessing the realized data. In contrast, our developed value for DPPs can be evaluated with just the knowledge about how the data are produced, but without the need to truly access the realized data. Additionally, the developed valuation function is relative prior information as well as the user's context, hence it is possible in our definition that high-quality data may have low value to certain user if the user already has sufficient prior knowledge or if the user cares about a context that is not reflected in the data.
The power of perfect data customization. With the in- troduced measure of data instrumental value, we next turn to its application to selling data. When the seller can cus- tomize data production for buyers, show that there exists pricing mechanism that achieves first-best revenue; that is, it extracts full welfare, hence leaves no surplus for buy- ers. This result highlights a fundamental difference between selling data and selling physical goods. Specifically, when selling goods, the power of customization is limited, at most through randomized allocation, leading to buyers' utility linearity (in allocation probabilities). However, for selling data, there is significantly more power of customization by producing various (high-dimensional and non-linear) derivatives of data. Such rich "data allocation" space is the fundamental reason for the first-best revenue extraction. It
Our new data valuation function is rooted in few classic subjects, including Bayesian regression, decision-making and the economic value of information, which we now elab- orate on.
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An Instrumental Value for Data Production and its Application to Data Pricing
2.1. The Context: Bayesian Regression and Data Production
salient feature of data sale that is intrinsically different from classic pricing problems of physical goods (Myerson, 1981). As we show later, it gives data sellers the power to tailor their data production to be useful only for one particular buyer but of little use to others, hence increasing sellers' power of price discrimination.
Our valuation function for DPP is based on the context of fundamental statistical learning problem - i.c., estimating parameters of generic regression problem y = fx) + c, where fa(-) is a function parameterized by B. Here, x E IRd is the feature vector and is some zero-mean random noise, capturing the measuring error. Let tuple (a, y) denote a generic data point whereas {(1., y6)) denote dataset with n points. A core challenge in studying the value of data production procedure' is that the value needs to be well-defined not only for the process of simply producing data records {z,, 31)) 1i=1. but also for any post post-processing of the data. For instance, instead of directly giving away all data records {xx, y:) y=) the seller could also just produce a single averaged feature sponding label y = 1.
In principle, a DPP g can be an arbitrary data production process and does not even need to be related to the under- lying regression model information about / (c.g.. a fully informative DPP could even directly reveal (9). However, throughout this paper and similar to Example 1, we will mostly think of g as being cu- rated from certain data records {22; y: y:)] (==>> for example, either the full partial list of {(zi, 34 w)> or as some statistics (e.g., mean, variance, moments, etc.) computed using these data records. This more realistically captures how data is generated and stored in most ML problems, and also is more intuitive to think about. In these situations, the randomness of the data production process g inherits from the model's noise and possibly extra randomness the data owner may add (see Example 4 for more details). Our valu- ation will be able to capture the value of all these variants. Throughout the paper, we assume the data-generating dis- tribution ga( D) is publicly known (though 33 is unknown); the DPPs in Example are all described by varied feature directions or their functions.
[a(#) + e so long as it carries
Ei=1 24 and its corre- 2 Ei= Vi- More general statisties could include weighted data combinations, some moments of the data, or even beyond, which all carry information about the underlying parameter 3. This motivates us to consider the following general notion of a data production process.
Definition 2.1 (Data Production Process (DPP)). A data production process (DPP) is described by a data generating distribution gø( D), which produces realized data D with probability g () under parameter 33 When 3 is clear from context, we simply use to denote this DPP.
As in standard Bayesian regression, we assume the data buyer possesses a prior distribution q over the parameter 3 and aims to further improve his prior q by purchasing additional data. In reality, can be a strong prior estimated from much data that the buyer already has, or can be a weak prior as an uninformative distribution. In either situation, additional data can help to refine the prior q. forming the posterior p. Hence, data buyer can use the realized data D produced by DPP g update his belief about B as follows?
Example 1 (Examples of DPPs). Consider a linear model y = (x. () + with uninformative Gaussian prior B ~ (0, 12). A learner would like to purchase data to predict (r. () for x= = 11. A simple example of DPPs
de- scribed by two feature directions x1 = (1,3), 3), X2 = (3, 1) simply produces response variables via model y = (z. B) + along feature direction 2 #2: for example, 3/1 -1 and y/ = 1. In this case, I2 specified DPP whereas the realized y1, 1/2 are the data D with data generating distribu- tion ga (131, (1)) = E(y - (m1, 8)) - (x2, B)), where E()) is the noise distribution.
Distribution of data: P(D) = Ss ((() go = g o q
(1)
Posterior updates from realized data D: p(3 q4 D) x 4(3) (D) for each B.
Besides directly producing data yı, 3/2 for directions æ1, 22, one could also produce the data y for feature direction J 22122 using DPP, which illustrates the rich space of DPPs. Perhaps more interestingly, we could also de- sign DPP that produces data y² = !1 3/2 for direction x = I1 - Iz. This DPP is of no interest to the learner above since x is orthogonal to the learner's 's interested direc- tion x. Nevertheless, the realized y does carry information about 3 and hence may be of interest to other users with a different context x'. Such careful curation of DPPs is a
(2)
For instance, how the data producing distribution gs( (D) and posterior update p(3 q. D) lead variance reduction in Example 1 can be calculated as follows.
Example 2 (Example I Continued). If we produce data T. the variance of the to-be-predicted variable (x,, (3) reduces from 2 to 3T same as fully revealing original data records ( 91) (222, y2)]. However, if we only reveal (x+t ,t)). the variance remains 2. Notably, these improvements can
Data production has the same meaning as data generation or curation. We choose to use the term "production" mainly to em- phasize the active generation of data, especially for the economic purpose of data sales.
"This is also known as the convolution of the parameter distri- bution g(3) and model ga-
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An Instrumental Value for Data Production and its Application to Data Pricing
all be calculated without knowing the realized data. This will be useful for selling DPPs since it means the value of DPP can be quantified without seeing the realized data. This resolves the concern that data will lose its value after being seen, say the seller can reveal Tt or x- for advertisement.
Note that CBDM simply enriches standard BDM by allow- ing the decision depend on some particular context x; which then allows the decision maker to have more fine- grained distribution estimation y|æ] of the random state y. For instance, still in the above production decision prob- lem, the context x could be the demographic feature of a certain population, and y|æ]] is the to-be-predicted price tar- geting this particular population. Similarly, we may predict E [[]]]]] fx) for some other tasks in an average manner.
For brevity, refer to the posterior p when q: D are clear from the context. We sometimes consider the situation with null data production, i.e., g = 0, where no Bayes update happens and p( q. 9) = q.
Knowing the context the data buyer can improve his decision-makinn by purchasing data produced by DPP to refine his estimation of 3 (hence refine the prediction of y) from his prior belief ((5) to posterior p( q). In a fully Bayesian world, such data should always increase the decision maker's utility, and this utility increase in expec- tation is a natural candidate for the value of the DPP g for the given CBDM problem. This motivates us to study the following valuation function of data.
Finally, for our theory, we need a natural generalization of Bregman divergence to concave finnctionals over (continu- ous) distributions in order to accommodate the continuous parameter space for 3 (Cilingir al., 2020).
Definition 2.2 (Concave Functionals and Generalized Breg- man Divergence (cf. Definition B. B.1)). We say functional F(-) is concave if for any À E [0. 1), we have
F(Ap + (1 ))g)= 2 (p + (1 X)F(g).
Definition 2.4 (Valuation Functions of a DPP). Suppose the regression model y = f2(#) + and data production process (D) are public. Then a valuation function for realized data D has format val(D; g, x)) that depends on the buyer's prior belief q of /3 and the decision context x.
Moreover, we say VF( is its (functional) superdifferential
F(g)= (VF(g). p q) 2 F()).
for any probability distributions p and The generalized Bregman divergence induced by functional F is defined as
Consequently, V(g: q, 2) = ED q. is called the instrumental valuation function for DPP g. Moreover, we call cost(q, x) = val(V; x) the cost of uncertainty where g = V denotes the fully informative DPP that directly produces data D = 3 (hence before sceing the data, the belief about the to-be-seen D's distribution is the prior q = g o q).
(p.). = F()) - F(p) + (VF(g).p-o)
2.2. The Value for a Data Production Process and its Foceo-Founnation
Throughout, we assume the Bayesian regression model y = =f(())+ + and any designed DPP g are publicly known. Our goal is to develop and characterize class of valua- tion function that captures the instrumental value of g to particular task.
A few remarks are worthwhile to mention about Defini- tion 2.4. For simplicity, we wish to minimize the valuation functions val's 's dependence on various quantities. However, its dependence on g, is essential. First, val's 's dependence on prior q is natural because what the buyer originally knows affects the value of newly acquired data. More valuable data are those that significantly shift the buyer's prior, whereas the data that does not change it much is less valuable. Sec- ond, the dependence on the decision context is also natural since if the data isn't informative to the particular decision context x, it won't 't have much value for the buyer regardless of how informative it may be to estimate the directions of 33 that is not useful for improving prediction of fa()).² Finally, the cost function is simply the value of the most informa- tive DPP, which helps the buyer to pin down parameter 3 precisely so that the only uncertainty in his decision-making is the inevitable noise from nature.
Our proposed valuation is rooted in a fundamental micro- economic problem of Bayesian decision-making (BDM). Recall that a standard BDM problem is described by a utility function u: A x y + IR where u(a, y) the utility under action AA and random state of the world y E y that captures uncertainty in the decision making. As an example, in simple production decision problem, u(a, y) could be ay- - a² where a is the number of products to be produced, y is the price of the products (often a random variable that needs to be predicted), and a² captures the production costs for producing a amount. Our following definition of con- textual variant of the above problem simply says that the random variable y can be context-deppnndet.
Definition 2.3 (Contextual Bayesian Decision Making (CBDM)). A CBDM problem is a tuple (u, y|ズ| = fa(x) + <) where the utility function u : A x y + R represents a standard BDM problem and y|х] follows a distribution
Now that we propose format of the valuation function
'orr instance, if fs(x) = (x,,)), then accurate estimation of B alone any subspace orthogonal to a will not be useful for the buyer's 's decision-making with context z.
specified by context x under response model y = f0(2) + c.
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An Instrumental Value for Data Production and its Application to Data Pricing
V(ys 2, However, obviously, not all such functions should be considered "valid" valuation functions. For instance, negative valued ones or those which decrease g become more informative are clearly not reasonable candidates for data valuation. Hence, the true scientific question is what kind of functions V(g: q, 2) could be considered as "valid" data valuation functions. To answer this question, we resort to Bayesian decision-makinn in order to "microfound" valid V(g; 2).
and (3) Invariance to data acquisition orders."
Our first main result establishes connection between val (hence also V) and cost.
Theorem 2.7. A valid data valuation val q, z) and cost of uncertainty cost(q, x) are coupled if and only if val is a generalized Bregman divergence (Definition 2.2) of cost in the following sense
va, q, xr) cost(q, x) (Vcost((,)))) p- --))-
Definition 2.5. [Valid Value Functions and Cost of Uncertainty] A valuation function V(g: q, En-gog D; q, x) is valid if and only if its correspond- ing (a: function can be expressed as follows for some contextual Bayesian decision-making problem (u. y/2| = fx(x) + e):
where p(-lq, D) defined in Equation (2) is the posterior of 8 and V-ost(-) any superdiiferentiall of cost defined in Definition 2.2.
The theorem highlights the fundamentality of Bregman di- vergence, a commonly used concept in statistical learning and online learning (Banerjee al., 2005; Gutmann & Hi- rayama, 2011; Raskutti & Mukherjee, 2015), in the valu- ation of data and cost of uncertainty. That is, any valid candidate of data valuation functions must correspond to the Bregman divergence of some concave function that captures the cost of uncertainty. Notably, since machine learning training typically involves minimizing some divergence, our results to some extent justify the choice of the loss function. As we will see below, the selection of entropy reduction is not only due to its ease of optimization but also because it corresponds to a certain preference.
E va:(D: B-o( -oo
[u(a" (p)= y) ula" (4): v),
(3)
where B ~ p(-1g, D) + fz(xt) + denote process of producing y by first drawing B ~ p and then drawing y ~ [a(()) + e; a" (q) is the optimal action when the decision maker's distribution belief about parameter /3 is q. or formally optimal action under parameter distribution q: a" (4) = arg max.e.A Sgga-vo-(s(a)+tlll y)-
Analogously, we can define coupled valid cost of uncer- tainty function, and we postpone it to Appendix B.2 for interested readers.
In other words, a valuation function V(g: 2) is valid if there exists a CBDM problem (u, y[z] = fa(x)= + c) that "microfounds" it in the sense that its corresponding val(D: q. x) can be written as the utility difference between the more informed (by extra data D) optimal action a" (p) and original uninformed action a (g)) in expectation over the randomness of the state y [z(²)) + where parameter 3 is generated by posterior p. In a fully Bayesian world, this is precisely how much the realized data D helps to increase the decision maker's ' expected utility.
2.3. Comparisons to Data Shapley
Data Shapley (Ghorbani & Zou, 2019; Wang et al., 2024) is another common valuation method, yet often overstates valuation. Originating from cooperative game theory, Data Shapley allocates value to each datum in an existing dataset. It computes an expectation over exponentially many (hence intractable generally) possible scenarios as follows: Φ; x ΣscD-()) val(SU(())ggr) (S1g..)..) Our previous result (1011)
characterizes what valuation function val is "valid". Theo- rem 2.7 proves that any natural val is the Bregman diver- gence of some concave function, which also matches most choices in practice. More importantly, our valuation only relies on val q. 2) and val(D U {i];9, (since it is the marginal value) hence is much easier to compute in prac- tice since it only requires to evaluate val D U {i); g, by retraining the model one more time with the additional data i. Moreover, our valuation method can be applied to any transformations of the dataset as well (so long as the DPP allows us to update parameters), so it could be applicable more broadly than Data Shapley.
Definition 2.5 captures valid data valuation functions from micro-economic perspective. However, these definitions are not very helpful for to verify whether any given V, val, cost functions are valid since it may be gener- ally difficult to uncover the underlying CBDM problem (u,yla y|œ] = ff(()= + c). Next, we offer some results that offer alternative yet equivalent characterizations of these functions as well as their connections. These characteriza- tions are properties of these functions themselves, hence are much easier to verify. We leave analogous proposition for cost(.,)) -) to Proposition B.4 in Appendix B.2.
We now illustrate our motivation for considering marginal contribution, while not averaging over all possible data coali-
Proposition 2.6 (Characterization of Valid Valuation Func- tions (see Proposition B.3 for details)). A valuation function val is valid if and only if it satisfies the following properties simultaneously: (1) No value for null data, (2) Positivity
'It means the total expected value of data is invariant to the order of data acquisition.
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An Instrumental Value for Data Production and its Application to Data Pricing
tions. Suppose a company hopes to buy a new confidential dataset fine-tune its large language model; it only needs to consider the status before and after fine-tuning, How- ever, Data Shapley gives some weight to the value of a virtual model trained by part of the original data and the new dataset. Considering many of these hypothetical situ- ations will usually overestimate the instrumental value of the new dataset and twist the corresponding data pricing. This inconsistency arises from ignoring the timeline when each dataset is collected. The following example illustrates why the Shapley Value usually overestimates. In fact, in its extreme, suppose a seller resells a repetitive data/dataset which is already in the buyer's 's training data pool, this same dataset will nevertheless have a non-zero Shapley value, whereas it will have 0 instrumental value in our setting as it's useless for updating the prior.
regression problem is due to multiple reasons. First, the linear model is arguably the most fundamental model and has been proven to be very useful under uncertainties and misspecification (Besbes & Zeevi, 2015). Second, multiple recent works have shown that data's valuations are often "transferable" in the sense that when the same sets of data are evaluated under different learning models (e.g., linear regression or ML methods), their relative value order of- ten does not change much though the absolute value may change (Schoch et al., 2022; Jía et al., 2021). Finally, in domains such as causal inference and clinical healthcare, lin- ear models serve as effective approximations of real- world dynamics and provide valuable insights for future method- ological developments. Given these, the valuation under the linear regression model serves as a useful basis for any future analysis of data valuations, hence, we term them the canonical valuation function.
Example 3 (The comparison between v(::.-) and Data Shapley). Assuming 3 has prior (0, 1) and data fol- lows ((8, 1), we hope to formulate the entropy reduction associated with the second datum which is 2 log(3 ( Here, the first datum reduces the posterior variance from 1 to 1 and the second one reduces from 1 to 1 By choosing val as negative entropy, V = 1/ log(3 8 Data Shapley overestimates valuation in this scenario.
Let us start by recalling the classic Bayesian linear re- gression problem, which assumes that label is produced by the following linear model y = (a. B) + c, where ~ NN(00 is zero-mean Gaussian noise.
2 3'
The Bayesian linear regression framework shares a similar sentiment with Thompson sampling (Agrawal & Goyal, 2013). It not only holds high value in machine learning theory but also finds extensive practical applications in real life, like drug monitoring ( Hammad-Ud-Din et al., 2017: van den Elsen et al., 2019).
②) but Φ 3og showing
Here, we use entropy reduction associated with KL di- vergence as a valid -.. KL divergence is com- monly applied in training machine learning models, like VAE (Kingma, 2013) and GAN (Goodfellow et al., 2014). Nonetheless, our instrumental value is also legitimate for some other widely-used loss functions like L2 loss.
Notably, here we allow the variance (xx) to also depend on the context æ. We also assume 3 follows Gaussian prior N (H, Mg). One of the most widely adopted cost of uncer- tainty functions is the Shannon entropy (Shannon, 1948), or differential entropy for continuous distributions. We con- veniently refer to both as "entropy" and defer its standard definition to Appendix C. It is easy to verify that entropy is a valid cost of uncertainty over (z, B) and measures the confidence in the estimation. One natural question, hence, is what is the associated data valuation function coupled with entropy? Our following result provides closed-form characterization.
Data Shapley corresponds to a uniform distribution (Ghor- bani& & Zou, 2019), and some paper tries other distributions like Beta distributions (Kwon& & Zou, 2021) to capture vari- ous facets of data valuation. Our instrumental value can be regarded a kind of limitation of Shapley value that we use a Dirac delta distribution, merely focusing on S = D {i). It precisely describes the realistic gains from new data with computational tractability and has potential applications in the data market, like valuing data used for fine-tuning, We visualize the advantages of instrumental value with some numerical experiments in Appendix B.3.
With the structure of Bayesian linear regression, we may write out exactly v(::..) in the Bayesian linear regression setting in the following theorem.
3. The Canonical Data Valuation Function for Linear Regression & Entropy
Theorem 3.1. [The Canonical Valuation of DPPs] In clas- sic Bayesian linear regression, the valuation function for any data production process g coupled with Entropy is the following function:
Section 2 introduced a general characterization for the value of data and its associated cost of uncertainty. In this section, we instantiate this framework in a fundamental Bayesian regression problem (i.c., linear regression) with perhaps the most widely used cost of uncertainty function (i.e., entropy), ), and develop closed-forn characterization for the DPP val- uation function ((:;:)) in this case. This focus on a linear
V(g:
E Dingog
1 1log(="
1 Mx²)
2 log (x" Ep(l)))))
(4)
where o q is the data distribution as in Equation (1). p is the posterior distribution over 33 E Rdd as in Equation (2),
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An Instrumental Value for Data Production and its Application to Data Pricing
and mp = Eg - mp) - 1yp) is its covariance matrix. (3ppp))--
ence (Liu et al., 2021b) and clinical trials (Berry, 2006). As salient realistic feature of data pricing, we always assume the seller can only generate data of form ÿ (z, B) + where e <~ (0,, o that is, data are noisy responses for examined feature direction T. Moreover, as a natural statistical assumption, the larger the length ||>||| of x is, the larger the noise's 's magnitude is. Hence we assume the noise's variance scales with ||z||, or formally, (x) = o ( (T |||=]|| as widely adopted in statistical literature (Linnet, 1990; Kumar & Klefsjö, 1994; Bland & Altman, 1996; Wang & Mauldon, 2006). 6 Certainly, data are limited (which is why it has value). To capture its scarcity, we assume the seller is lim- ited to producing at most data points of the above form, and the production has no cost." ?
Theorem 3.1 offers a closed-form expression (coinciding with information gain) for the valuation of any data produc- tion process 9, associated with entropy in linear regression due to their fundamentality, we call the V(g: 9, x) function in Equation (4) the canonical valuation function for DPPs.
So far, we have been working with the most general data production process g. Next, we're re going to instantiate our characterization in Theorem 3.1 with a concrete, perhaps the most straightforward. data production process that is, a collection of data records { y/ )²= produced by the model y = (z. B) + c. In this case, the data production procedure is completely determined by the data matrix X E Raxd with its i-th row as x².'. hence denoted as g*. The realized data is the realized response variable values Y = (V1..... )r) This concrete DPP allows us to derive a closed-form valuation function for these data records.
Prior-free mechanism design and performance bench- mark. While q is public knowledge, the decision context x Rd is the buyer's private information, also conven- tionally referred to the buyer rype, which is unknown to the seller. Unlike Bayesian mechanism design, we study prior-free mechanism design, which is known to be noto- riously more challenging (Hartline et al., 2020; Hartline & Johnsen, 2021) but much more practical due to being free of assumptions on the seller's prior knowledge and on how buyer type is generated (Goldberg & Hartline, 2001; Devanur & Hartline, 2009). In this case, the performance benchmark is the "first-best" revenue, i.c., the revenue when perfectly knowing buyer type and producing n data points based on x.
Corollary 3.2. [The Canonical Valuation of Producing Data Records] For any data records {(Ia: Vim) gen- erated by the linear regression model y = (xr, (8) + 6, let X E ignd denote the design matrix. The canoni- cal valuation of this DPP procedure, denoted as gx, is churacterized in closed form as follows: Viyx; = log(x" E,yz) -3 log(z" x= (E + +XTE-IX) -x)rr,, where Σ = diag (a (x1)²
)2 ,.... (Ix)) )2).
4. Optimal Pricing of Data Production under Canonical Valuations
It turns out that solutions to this mechanism design question crucially hinge on the seller's power of customizing data production. We consider the following two natural settings.
In this section, we use the developed closed-form charac- terization for canonical valuations of DPPs in Section 3 to study a natural optimal pricing problem for selling DPPs. While the choice of valuation functions generally may be domain-dependent, the goal of this section is to showcase natural economic application of the newly developed valua- tion functions of DPPs. We study how to sell data produc- tion processes (DPPs) to a machine learner (henceforth, the buyer).
Perfect Customization: here, the seller has the flexi- bility to produce data for any direction Given such flexibility, the benchmark of the first-best is to pro- duce n responses for the buyer's type x (assumed to be known in the benchmark) and then charge the buyer his value for this DPP. We denote this particular DPP as g [ax] which produces responses y/s = (at, B) + e, for i = 1,... ,n. Hence, the benchmark revenue here is simply the buyer type x's value for g" [x]] which is V(g" [x]:
We consider the following standard mechanism design framework, albeit being instantiated in our DPP pricing setup with fundamentally different utility functions and al- location rules. Specifically, the buyer would like estimate E|y[a]] = (z, B). The 3 here is the to be-estimated pa- rameter by the buyer. For simplicity, we assume the buyer initially has no knowledge about the model parameter 3; formally, the buyer's initial belief y = AO, Ia) about / is a standard d-dimensional Gaussian." The buyer would like to buy data from the seller to refine his Bayesian posterior belief about B, which is quite common in management sci-
Limited Customization: here the seller has an ex- isting set of data records ((1,, B Regardless of which buyer type x is, the highest possible buyer value the seller can produce is to give all the data to the buyer. We denote this particular DPP x which is deter-
"Otherwise, we only need to consider the length corresponding to the minimum relative noise in the direction, and this direction collapses to such a point.
$ *Our results can he generalized general Gaussian prior (same for every buyer), but with more complex notations.
Another interpretation is that the data scarcity, i.e., most data points, reflects the production cost.
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An Instrumental Value for Data Production and its Application to Data Pricing
mined by their data matrix X and simply produces all their labels {= Hence, the benchmark revenue for buyer type is ;(g";g,))
Algorithm I An Optimal Mechanism under Perfect Data
Customization
Input: buyer's report ã.
(2) a()) Seller produces n responses for the same 2: D. = (f, B3 + ci and reveals [ūr Hi=1 to the buyer.
Seller charges buyer ((T) = 2 log(
Convenient Notations. If any mechanism achieves ex- pected revenue that is additively within a of the benchmark described above for every buyer type I, we say this is an a-regret mechanism. $ A O-regret mechanism is simply the mechanism that achieves the in-hindsight optimality. In this section, we always have q = (0, La). For notational convenience, we will instead write the above value function as V(g. xr (only) in this section by ignoring q. Moreover, only the buyer knows his own type v, whereas the seller has no information at all about æ. Hence, naturally, the buyer may misreport his type as some R. not necessarily equal to x, whenever this is more beneficial. We will always use g[ to denote the seller-designed DPP for a buyer report G. Like classic mechanism design, we say mechanism is prentive-compatibee (IC) if each buyer with type x finds it optimal to report x truly. A mechanism is individually rational (IR) if no buyer type has a negative expected utility.
conclude that the payment rule for allocation g|H] must have the following form that t(x) = S v.V(glul:s) y=s ds + t(2o).
Compared with the well-known results of single-dimension in Myerson (1981), there are two main differences consid- ering multi-dimensional type. First of all, need to generalize the derivation of a one-dimensional function to the gradient computation in the case of higher dimensions. Second, when calculating the payment rule t(-), the integral result is required to be independent of the initial starting point and the integral path we choose. In our situation, means that matter what zn and the path from it the final choose, the result of the integral should be the same. These conditions constrain the application scenarios of the extended method for dineii-dmmesionall mechanism design, as we only need monotonicity for the single-dimensional situation. Nevertheless, the proof will elaborate that our definition of instrumental value satisfies all these conditions endogenously, highlighting the importance of our frame- work of instrumental value and suggesting its broad range of applications.
4.1. The Power of Perfect Data Customization: Achieving the First Best
Our main result under perfect data customization is a sur- prisingly positive result showing that is possible to have a O-regret mechanism for selling DPPs that achieves the first-best in hindsight. Notably, such a full surplus charge has been shown to be impossible in classic item allocation problems (Myerson, 1981; Hartline et al., 2020; Hartline & Johnsen, 2021). This is fundamentally due to the different valuation functions in the two settings, as a function of al- located data. Our result demonstrates the surprising power of price discrimination brought by the special valuation of data as well as the flexibility of data customization.
4.2. Selling Existing Data Records with Limited Customization
We now turn to the situation when the seller has n data points {(I;, bs)) at hand and can only process this ex- isting dataset and sell it to the buyer, where Σ(X) = diag(o(<))) (x,))) and Y = (y1,....,,,)) yn, The de- sign matrix X is public knowledge, but the buyer does not observe the corresponding responses Y. In practice, for ex- ample, in clinical trials, information about the subject group is usually known, but the evaluation of the drug's 's effective- ness needs to be purchased. Sellers in real-world scenarios usually disclose data quality. We model it by X(X). which represents the measurement error or the specifications of the experimental instruments, such as their precision. We first note that while our benchmark here is the buyer's maximum possible value (2)) this benchmark impossible to obtain due to not knowing the buyer's 's type, since if the seller indeed reveals all these data records to every buyer, then every buyer type would want to misreport his type to be some 0 that has the smallest V(gxx ;)). Hence, some data processing based on {(x,, y4 ) is necessary to achieve price discrimination and get the mechanism close to the
The announcus mochanism.
The buyer with type reports R.
seller produces, gives out data and then charges i(f).
Figure 1. Timing line for perfect data customization.
Theorem 4.1. With perfect data customization, the follow- ing Mechanism 1 with DPP gاāو (² and payment (T) = 1/2 log( a(z)2+n) for any buyer reported type Tt is IC, IR and O-regres.
Theorem 4.1 shows that the optimal allocation is to produce responses for type x. The key challenge is to find the optimal payment rule. As type x is high- high-dimensional, tradi- tional methods (Myerson, 1981) have become agnostic. We
'In online algorithm design, additive loss from the optimal in hindsight is often denoted "regret" whereas multiplicative loss called "competitive ratio".
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An Instrumental Value for Data Production and its Application to Data Pricing
benchmark v for every æt Indeed, it turns out that in this case, perhaps expected, there is no O-regret mech- anism. However, surprisingly, we nevertheless show that there is an IC and IR mechanism with small constant regret.
we find a new information structure called mulli-armed bandits setting (cf. Appendix D.2) which also leads to O-regret for the seller. Therefore, we know that whether O-regret is achievable depends on the information leakage. In Corollary 4.4, since every direction confers equal value, it discloses the buyer's willingness to pay. In the MAB setting, the intuition is that other directions provide no information, discouraging the buyer from submitting untruthful reports and naturally leading to price discrimination.
The seller announces mechanism and produces dataset D.
The buyer with type x reports R.
The seller gives out a(R] and dharges t(R).
Figure 2. Timing line for limited data customization.
Theorem 4.2. When the type space is continuous, there is no O-regret mechanism under limited customization.
5. Conclusion and Discussion
Compared to a dataset's intrinsic value, the instrumental value better captures its economic value the market. This paper first introduces a principled way to quantify a dataset's instrumental value, and then studies how to apply this valu- ation data pricing problem. The designed mechanism illustrates the surprising power of data customization, which may help data sellers to do first-degree price discrimination, leaving zero surplus to buyers. Even under limited data customization, the flcxibility of selling any derivatives or statistics from an existing dataset already gives the seller significant power discriminate against buyers. These re- sults hint at potential need for regulations in order foster sustainable data market. On the technical side, our data valuation's 's connection to the value of information (Frankel & Kamenica, 2019) may be of independent interest. More- over, our designed mechanisms for multi-dimensional buyer preference, particularly the mechanism based on the sin- gular value decomposition, may be of both theoretical and practical interest.
Theorem 4.3. The regret of the following SVD mech- anism under limited data customization is at most log(n((V/X)) (EV/(()->>)))))) -X)), where a((v/(X)))=X) is the square root of the ratio between the largest and the smallest singular values of x" (X)x())-XX))
Algorithm/ 2 The SVD Mechanism
Input: buyer's 's report R. seller's 's design matrix X
Normalize (X, Y)= + ((vE(X)) X. (.(v E(X)) 'Y)
and announce normalization
S
Perform SVD over X, obtaining X XU O(n-d)
V
where diag ,A .a)
S--17
Let denote the left inverse of X where =
V
Define the mapping b(.) : Rd + R" where b(z)
L
|
Output:
g
(((T)"XXX()))"
((3)
1/ log( aT 2) lo X²
Several open questions are worth future investigation. Is it feasible to explicitly resolve the mechanism design chal- lenge under dissimilar information flows? Our data valua- tions are derived from linear regression; how transferable are they to data valuations under other learning models? Previous works have shown that Data Shapley often has similar order when computed using different ML mod- els (Jia et al., 2021; Schoch et al., 2022). Moreover, given that we have demonstrated the potentially high-skewed sur- plus allocation between the seller and buyer, how should regulatory authorities step in (Mas- Colell et al., 1995; Jehle, 2001) to sustain market efficiency? What would be the im- plications of multiple sellers operating in the market, or if it were perfectly competitive market? We leave these in- triguing follow-up questions potential avenues for future research.
Theorem 4.3 gives an illustration of the ability of price dis- crimination under limited customization. Unlike perfect customization, the seller cannot achieve the first-best rev- enue in all cases. In other words, the worst-case distance between optimal revenue and the first-best revenue is strictly positive, which implies that limited customization limits the level of market unfairness. However, even though the buyer could get a non-zero consumer surplus, it is still small, and he still suffers inequality against the seller.
The following results further show that for certain X, the seller can achieve the first-best revenue and O-regret, high- lighting the potential unfairness in the data market even under limited customization.
Corollary 4.4. When the seller has isotropic data, i.e., all singular values of X" ((X)-'X X are the same, there exists a mechanism that satisfies both IC and IR, and achieves O-regret.
At the same time, if the private types of the buyer, which are public knowledge, have certain structures, it's also possible for the seller to achieve the first-best revenue. For example,
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An Instrumental Value for Data Production and its Application to Data Pricing
Acknowledgements
Beck, J., Dia, B. M., Espath, L. F., Long, Q., and Tempone, R. Fast bayesian experimental design: Laplace-base based importance sampling for the expected information gain. Computer Methods in Applied Mechanics and Engineer- ing, 334:523-553, 2018.
Zhaoran Wang acknowledges National Science Foundation (Awards 2048075, 2008827, 2015568, 1934931). Simons Institute (Theory of Reinforcement Learning), Amazon, J.P. Morgan, and Two Sigma for their supports. Zhuoran Yang acknowledges support from NSF DMS 2413243. Haifeng Xu is supported by the A12050 program at Schmidt Sci- ences (Grant G-24-66 104), Army Research Office Award W91 1-0030, and NSF Award CCF-2303372. The authors would like to thank the reviewers for their valuable comments and suggestions, which have greatly improved the article.
Bergemann, D. and Bonatti, A. Selling cookies. Ameri- Economic Jourmal: Microeconomics, 7(3):259-294
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Bergemann, D. and Bonatti, A. Markets for information: An introduction. Annual Review of Economics, 11:85-107, 2019.
Bergemann, D., Bonatti, A., and Smolin, A. The design and price of information. American economic review, 108(1):
Impact Statement
1-48, 2018.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Bergemann, D., Cai, Y., Velegkas, G., and Zhao, M. Is selling complete information (approximately) optimal? In Proceedings of the 23rd ACM Conference on Economics and Computation, pp. 608-663, 2022.
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An Instrumental Value for Data Production and its Application to Data Pricing
Appendix for "An Instrumental Value for Data Production and its Application to Data Pricing"
A. Omitted Details in Section 1
A.I. Related Works
Valuating data and information: The paper is related to existing works for valuating data and information. The existing literature has been extensively focused on Shapley value (or variants thereof) for valuing data (Schoch et al., 2022: Ghorbani & Zou, 2019; Jia et al., 2019), which is a principled way to "fairly" attribute (Roth, 1988) the contribution of each data point/sct to an ML task. This is why the Shapley value is an average contribution, averaged over all other possible training situations. Later works also realize that equal-weigh average may not be ideal for ML tasks, hence proposed variants of Data Shapley that prioritize some training situation in the average (e.g., beta-Shapley (Kwon & Zou, 2021)). Recently, Guo et al. (2025) considers how to value data in human-Al decision-making. In contrast, our data value is instrumental and is to quantify a dataset's s marginal contribution. Hence, the meaning of this value is not to average over many possible training situations but instead to look at marginal increase over the previous best training situation. Contemporaneousll. Choe et al. (2024) uses a gradient-based data valuation for large language models, and stem from Bayesian regression.
Selling data: For existing works on selling data, Regura-Rodriguee (2021) studies selling data to a buyer who wishes to minimize the quadratic loss of an estimate. The biggest difference that separates our work from this work is the buyer's private type. In our case, we assume the features of the observations that the buyer wishes to obtain labels for are unknown to the seller. On the other hand, of label the buyer wishes to estimate is the unknown private type. For instance, if a production company purchases data from Nielsen, our work corresponds the setting that the TV show the company is producing is its private type and is unknown Nielsen, whereas eeguraaRodriguez (2021) assumes the TV show itself is known, but Nielsen does not know if the production company wants to predict the show's ratings the show's 's audience demographics. The difference in the information structure leads to fundamental differences between our work and theirs, both in terms of the proposed mechanisms and in terms of the theoretical analyses. Chen et al. (2022) studies the problem of selling data to a machine learner, using a costly signaling step to ensure that the buyer can accurately evaluate the quality of the data purchased. Our work eschews the step but leverages the properties of instrumental value in the Bayesian linear regression setting. Chen et al. (2019) values the data based on the model that the buyer learns from the data, Agarwal et al. (2019) studies a two-sided data market with multiple data buyers and sellers, in which the buyers compete with one another by bidding for data. The latter two works, however, assume that the buyer's learned model is observable by the seller, which avoids the complication introduced by the private type considered in our work.
(2021) assumes the buyer's 's features are known a priori, but the type
Selling information: Our work is also related to the long line of work on selling information (Babaioff et al., 2012; Hörner & Skrzypacz, 2016; Admati & Pfleiderer, 1988; Kastl et al., 2018; Xiang & Sarvary, 2013; Bergemann & Bonatti, 2015; Liu et al., 2021a) and we refer interested readers to Bergemann & Bonatti (2019) for a survey of these works. Of these works, Babaioff et al. (2012) studies the optimal one-round revelation mechanisms for selling signals, assuming that the buyer's type space is discrete and finite, and Chen et al. (2020) extends the problem setting one where the buyer also has finite budget. Bergemann al. (2018) and later work Bergemann et al. (2022) focuses the optimal sales of Blackwell experiments under the assumption that the buyer's 's type space is finite. Hartline et al. (2020): Hartline & Johnsen (2021) study optimal type-prior independent approximation multiplicative factor, whereas we focus on the corresponding addictive factor and pave the way for prior-independent mechanism design in an addictive manner.
There is also a rich body of literature on active learning (Freund et al., 1997; Settles, 2009), studying the effect of data sequence. However, active learning studies how to select sequential data and train models to maximize accuracy, whereas our paper studies how to define valid data valuation function that captures downstream users' utilities. We approach this study from a utilitarian perspective, and our value corresponds to information gain only in a very canonical special case.
B. Omitted Details in Section 2
B. Omitted Details Section 2.1
We now give omitted algebra details of Example 1.
Example 4 (Examples of DPPs). Consider a linear model y y= (x., + c. A learner would like purchase data to predict
y = (z. B3 for x = (1, A simple example of DPPs - described by two feature directions Φ1 = (1, 3). 22 = (3, 1) simply produces response variables corresponding to the given æn, 02: y1 = (21, + 6 = -1 and y/2 = ( 8) + (2 = 1.
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An Instrumental Value for Data Production and its Application to Data Pricing
In this case, D [y1 123 is the realized data which is produced by data production process go(D) = Pe (v1 (#1, 88) Pe(u2 (222 ). Hence the DPP here is fully described by two feature directions 11; 222- The realized data D {V1, y2). together with the knowledge of DPP g, can help the learner to refine the estimation of [3.
Instead of directly producing data y1 3/2 for directions 21: 22, one could also produce the data D = y = M for feature direction ã = == using DPP with (D) = IP .e (ū- (ā, (8)) where the probability density is over the randomness of both noise terms E1, e2. Notably, this DPP is subtly different from directly generating the response y/ = (ž, + e because y has a smaller variance than y, despite having the same mean, hence is strictly more informative. Indeed, to produce data
D = y = (z. 3) + e, the data owner would need to add extra noise to y yn +12 in order to match its distribution. This illustrates the rich space of DPPs.
Perhaps more interestingly. we could also design a DPP that produces data y- y1 1/2 -2 for direction xt T1 22 = 23. This DPP is of no interest to the buyer above since F 1 2, but nevertheless its realized data D = y carries information about 33 and hence may be of interest to other buyers.
Suppose the learner has an uninformative Gaussian prior ~ NO T) and the noise in response variable follows the standard Gaussian (00 1). Then the data generating function go gy(y) for a DPP(() that produces data y = {x, B) +e is gy) = 2 cxp(-(y (z. /2). If we produce data ū 21= as the average response variable for 21, 02,
9/17
-8/17
9/17 }>
the posterior of 3 prescribed in Equation (2) can be calculated as No
and the variance of the
8/17
to-be-predicted variable (zz, 3) reduces from 2 to 1.4 same as fully revealing original data records {(11, (22, v2)}.
3/5
2/5
However, if we only reveal (x y the posterior is (V2/5, -2/5),
3/5 ) and the variance remains 2. Notably,
2/5
these improvements can all be calculated without knowing the realized data. This will be useful for selling DPPs since it means the value of a DPP can be quantified without seeing the realized data. This resolves the concern that data will lose its value after being seen.
We here give a rigorous definition of concave functionals and generalized Bregman divergence in Definition 2.2.
Definition B.1. Let B denote the domain of parameter B and consider functional F (B) + E that maps the space of probability distributions over B to a real value. We say functional F-)) is concave if for any pair of distributions p, q A(B) and any X E [0, 1J. we have
F(Ap + (1 A)4) 2 (p + (1 - -))((g)
Moreover, say VF(()) is its (functional) superdifferential if for any p, q E A(B) we have
F(g) + (VF(g). d) F(p).
The generalized Bregman divergence induced by functional F is defined as
Dp(p.g) F(4) F(p) + (VF()),) 9)-
B.2. Omitted Details in Section 2.2
Next, we are ready to give more details on the characteristics of val(::.)) and cost(-,)).
Definition B.2 (Valid Cost of Uncertainty (Definition 2.5 Continued)). Relatedly, the corresponding cost(g, x) = val(V: q, .r) is called a valid cost of uncertainty function. Moreover, in this case, we say the val (V; q, x) and cost x) are coupled (since they correspond to the same CBDM).
Proposition B.3 (Characterization of Valid Valuation Functions). A canuatiionaacctin val is valid if and only if satisfies the following properties simultaneously
1. No value for null data: val(O: q. x) = 0 for any prior q:
2. Positivity: val q, x:) 2 for any realized data D and any prior q over parameter (3;
3. Invariance to data acquisition orders: The expected value of data is invariant to the order of data acquisition. Formally, given any prior over 3 and any two data production process 91 and g2. let p1 = Dt) and P2 = p(- D2) be the posteriors updated by realized data D2 and D2 respectively. Then, we have
ED, p. (val (Dvaaalliiiaa) q, 2)) + val p1, x)) =
+ val (D1:P,))
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An Instrumental Value for Data Production and its Application to Data Pricing
0.58
Fandor Shapley
Daty Shadkty stsply
0.20
0.04
0.00
0.78
number Daa
of Detarttt
(a) Scatter plot between instrumental value (b) Correlation of instrumental value and (c) Computation time of instrumental value, and Data Shapley.
Random Shapley with Data Shapley. Data Shapley and Random Shaplcy.
Figure 3. Comparison between instrumental value, Data Shapley and Random Shapley,
The first two properties are both natural. The third property indicates that the order of obtaining data from two data production processes should not change the total expected value of these two DPP. This is also a natural property for any valid valuation function. Interestingly, it turns out that these three properties suffice to guarantee the validity of val in the sense of Definition 2.5 and Equation (3).
The following proposition offers a characterization for the validity of cost of uncertainty function cost(g,
Proposition B.4. A cost of uncertainty function cost(q, x) is valid if and only if it satisfies the following properties
simultaneously
1. 0 cost under certainty: when the prior q degenerates to a Dirac delta distribution over B, the cost of uncertainty is O, i.e., cost(ō3, .)) foror any x.
2. Concavity: cost(q,z) x)) is concave in for any x.
The first property is obvious from the definition. To see the second property intuitively, g Aqı + (1- A)g2 for some E [o, 1] and beliefs 91, 92. and consider the following situation. Suppose there is some data informing the buyer that is drawn either from distribution Q1 (with probability X) or from 92 (with probability
A). If the buyer must make decision prior to observing the signal, his cost of uncertainty is cost(Aqı + (1 A)g2). If the buyer allowed make a decision after observing the signal, his expected cost of uncertainty would be Acost(q)) + (1 A)cost((2), which by concavity is no greater than that for when he is not allowed to observe the signal. Thus, concavity is natural formalization of the intuition that having more information on the parameter distribution will not increase the cost of uncertainty (in this case, the additional information is the signal pinning down whether 3 is drawn from or q2). What is surprising, however, is that these two natural properties suffice to pin down any valid functions for the cost of uncertainty.
B.3. Omitted Details in Section 2.3
B.3.1. NUMERICAL STUDY
We conducted numerical experiments using Python 3.9.15 on server equipped with dual Intel(R) Xcon(R) Platinum 8260 CPUs. We use entropy reduction as an illustrative example of val(:: )). We first show that the instrumental value has a high correlation with Data Shapley, hinting at consistent underlying economic intuition between them. Since the time complexity of computing Data Shapley is exponential the size of the dataset, we choose
10 and the dimension d 5. We randomly generate D with some mean and variance. Then, we randomly sample of interest with some other mean and variance. We assume 9 = (O, 1). We sample 1000 trials in parallel and observe in Figure 3(a) that Data Shapley is generally larger (around twice) than the instrumental value which agrees with our theory. In the meanwhile, they share a high correlation, say 0.8650, implying consistent economic intuition. This phenomenon inspires us to study whether we can use instrumental value to approximate Data Shapley. When we calculate the empirical Data Shapley using the mean of 102 samples, we call it Random Shapley. In this case, we obtain almost the same correlation of 0.8645. However, the average computing time is 031380, 0.0002 and 0.0218 for Data Shapley, instrumental value and Random Shapley
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An Instrumental Value for Data Production and its Application to Data Pricing
respectively, showing the great potential of instrumental value for computational traceability and offering new perspective for caleulating Data Shapley.
Next, we empirically study the time complexity of instrumental value, Data Shapley and Random Shapley, well as the promising potential of using instrumental value to approximate Data Shapley instead of Random Shapley. We now range size n from to 13. For Random Shapley, we set the sample size to 2" and has a similar correlation with Data Shapley as instrumental value as shown Figure 3(b). Nonetheless, the instrumental value has a great advantage computation time. We detail the average computing time Table 1. It enjoys polynomial time complexity with respect size while Data Shapley and Random Shapley suffer from exponential curse. See Figure 3(c) for reference. Therefore, instrumental value has promising application prospects in the industrial sector. It is also of independent interest to implement our valuation real-world data platforms.
Size
Instrumental Value
Data Shapley
Random Shapley
0.000129
0.003475
0.001 134
0.000130
0.007387
0.002147
0.000116
0.012579
0.003002
0.000180
0.030939
0.006260
0.000202
0.060909
0.010930
10
0.000202
0.130378
0.020986
0,000195 11
0.273053
0.039111
12
0.000215
0.560476
0.072644
13
0.000240
1,120840
0.134825
0.000227
2.280680
0.253490
15
0.000239
4.566204
0.474172
Table I. Average computing time among three valuations.
C. Omitted Details in Section
We would like to give the concrete definition of "entropy" here.
Definition C.1 (Differential Entropy (Thomas & Joy, 2006)). The differential entropy h(Z) of a continuous random variable Z with density (==) is (()) = - Sses (1(2) log((())d)) where is the support of the random variable Z.
D. Omitted Details in Section 4
D.1. Omitted Details in Section 4.1
V(x, represent V(g (r) when easy infer from context. Let's consider wider range of DPPs that the seller can add man- made noise c' ~ N (O, d)))²) when revealing y. That is say, the seller will reveal (1r, y c') rather than (z, v). After that, are going to show that adding noise, i.e. 8(x) 0, can lead to O-regret and it's definitely what the seller will choose.
For simplicity,
Concerning the definition of ()))))) in Equation (4), we can establish a lemma exhibiting the concrete instrumental value of data in detail.
Lemma D.1. the seller gives the buyer n data points, it holds that the instrumental value of data for the buyer
v(z = [(0 = log (x (xxx)- log x" (1+ ( + 8(3)2
NET
By revelation principle (Gibbard, 1973; Holmström, 1978; Myerson, 1979), we know that any social choice function with the same implemented by an arbitrary mechanism can be implemented by an creettve-cooppaibilit (IC) constraint and at least equilibrium outcome. Therefore, we focus on mechanisms that satisfy incentive one optimal mechanism belongs to this subclass. In the context of our model, incentive sxsentivecoompatbblii constraint means that the buyer has the motivation to report his own private type truthfully considering the allocation rule g[- and payment rule
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An Instrumental Value for Data Production and its Application to Data Pricing
(()). To ensure the buyer is motivated to purchase data, we further impose individual rationality (IR) constraints. means that rational buyer only participates in transaction when his expected revenue is no less than zero. Formally, the IC and IR constraints for mechanism (gl-), (()) are
IR
v(a, x) 2 t(x) for all x, V(z, x) - ((x) 2 Vzz 2) (()) for all x
IC
In other words, IC constraint means that truthful reporting is dominant strategy for the buyer and he has no incentive to report wrong type there no benefits by doing IR constraint gives the buyer reason to join the market and motivates the transaction as his expected utility is non-negative
|| it is evident that we could solely consider the direction of the type, rather than its Since we assume (x) क
length. This can be achieved by substituting x and E with " and
|| respectively. We then have
T
V(z, T) V
|
We thus restrict our attention to types with a unit norm, i.c. Sd-1 = f-= ||||| = 1}, which is equivalent to considering
a general space in Rd-1 Inspired by the method in Myerson (1981), we have the following lemma when æ has multiple dimensions.
Lemma D.2. For any payment rule satisfying the IC constraint, it holds that
vi(x) V,,(o,
Therefore, it leads to the following form of payment rule that
t(x) = [" V,,(s,u) ds t(xo)) = 2 + + C, t(xo)
log( xxx)2+((x))
+ 5(x)) +n
and it's independent of the selection of so and the path of integration.
Therefore, for any (xx), we have the corresponding optimization problem for optimal mechanism, that is
log( xxx) 5(x)² + n )] 1 holds),
max Ex((x) = EzIC + 2 olx)2 + 8(x)2 c,5(-)
where
IIR holds) 1C log( (()))+((x)) 5(x)² + n a(r)² + 5(x)2 ") + [og(x7(1+ alx)² + 8(x)) )''x) S 5))
NET. T
Though information gain is a widely used metric, our analysis method can be extended to a wider family of instrumental values as long as keeping the concavity property as what log(-) does. We add the following remark, inspired by f- divergence (Rényi, 1961), to show how to generalize our results.
Remark D.3. The method can be easily extended to any V(...) function following the form
V(r, 3) = 1( 2T
(a(3)) 486)))16)))
where f(.) is a concave function, and the results remain valid when t(x) is of the form
t(x) C (x)² + +(x))+(())++++)))) +n,
(x)²+ "
With the characteristics of t(-), we can conclude that when the seller is embedded with the ability of perfect customization, she can obtain the birst-best revenue and leave zero consumer surplus to the buyer, say Theorem 4.1. It reflects the price discrimination scenario in the data market and shows corresponding unfaimeess which quite different than the situation in other markets.
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D.2. Omitted Details in Section 4.2
Let us recall what data the seller has. The seller produces n data points { yi) y:) consisting dataset (DPP resp.) D ( resp.). Here, yi = x. / + e, where follows normal distribution with mean 0 and variance (x;)). For simplicity, let X E Xdx denote the design matrix of the whole dataset and Y the corresponding responses. We use (XX) to represent the noise covariance matrix, i.e. Σ(X)- diag (iag(o(())) (01))) ....., (
In this particular type of information flow, the buyer will not experience as much unfairness as the one in the situation described in Section 4.1. This implies that offering customization options may have negative consequences for society, and raises questions about how to regulate market power in the data market. However, it is important to note that the buyer's s surplus will be bounded from above by a constant unrelated to the buyer's type x;, which means that in monopolized market, buyers will likely find it difficult to attain high surplus.
Similar to Section 4.1, we only consider mechanisms that satisfy both IC and IR constraints, that is,
IR
V(a, x)) 2 t(x) for all x,
IC
V(x, x) - ((x) 2 ((z, σ) (()) for all x,
As we discussed before, if it's impossible to achieve the first-best revenue, i.e., O-regret, in this subclass of mechanisms, there won't exist a mechanism achieving the first-best revenue among all unconstrained mechanisms.
However, as demonstrated by Theorem 4.3, even though the seller cannot achieve the first-best revenue, the buyer's surplus is also upper bounded by a constant relative only the pre-produced X, rather than his personal type x. It is important to note that the buyer may have varying consumer surpluses for different personal types, while the union of upper bounds holds for all types æ. The proof of Theorem 4.3 leads directly to the following corollary about the seller that she can achieve within an additive factor of log(k) of the first-best revenue.
Corollary D.4. For any pre-produced datasel X, there exists mechanism in which the expectation of the seller's revenue is at most (og((<((/)))))))) ((((//((-))))))))) ()) less than the first-best revenue.
Now, let's give the definition of multi-armed bandits setting in our data pricing scenario.
Definition D.5. We call a setting having form of arme bandits if every type vector x is member of the standard basis of a d-dimensional Euclidean space R4, In other words, only one component of x is one and the others are all zeros.
It is closely related to multi-armed bandits, the famous machine learning model. In multi-armed bandits, observations of rewards of one arm contain no information about rewards of other arms. In the data pricing, the multi-armed bandits setting means that data points associated with type x involve no information about other types. Using a geometric perspective, illustrates that the information corresponding to different types is orthogonal. Now, we have a corollary in which the seller can obtain the first-best revenue.
Corollary D.6. In the multi-armed bandits (MAB) setting, there exists a mechanism that satisfies both IC and IR, and achieves the first-best revenue, i.e., O-regret.
Together with Theorem 4.3, Corollaries 4.4 and D.6 show that the quantity of the consumer surplus not only depends on the concrete properties of the dataset X, but probably related information contained in the structure of the buyer's private types well. In light of the universal Theorem 4.3, insofar as the diverse directions of data hold distinct values for the buyer, the seller harbors a motivation to induce the buyer, possessing greater willingness to pay, to disclose his authentic type. To devise a truthful mechanism, the seller must relinquish some charge in specific instances to attain the maximum expected revenue. Notwithstandding as per Corollary 4.4, if each direction is isotropic, then it will confer equal value to the buyer. Hence, it discloses the buyer's willingness to pay, which is pivotal for the seller to implement price discrimination. illuminates the crucial point of mechanism design in the data market, where the seller's focus not on the buyers' individual type, but rather on their willingness pay. Corollary D.6 however highlights the role of information leakage. In the context of multi-armed bandits, data from other arms provides no information then no benefit to the buyer, dissuading the buyer from untruthful reports, creating natural case of price discrimination.
E. Omitted Proof in Section 2
The proofs for Proposition 2.6, Proposition B.4, and Theorem 2.7 are closely related.
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For the only if direction, i.e. showing that the desiderata imply the existence of some corresponding decision-making problem, all three results share the same underlying constructed instance. As such, we organize our proof by first introducing series of propositions and lemmas that will be essential for the proof, and then justifying the results by combining these results.
We introduce two concepts, measure of information and measure of uncertainty, which serve as the statistical analog to Definition 2.5 and are crucial to formally establishing the results in Section 2.
Definition E.1. Any functional d(Hu, v/ A()) x A())) + R is a measure of information and any H()): : A()) + R measure of uncertainty. We say d and H are coupled if for any signaling scheme 8, we have E[d(p(s)) = (n) - (x(s)),, where p(s) denotes the posterior distribution given by Bayes rule after observing the signal s.
The conditions imposed in Propositions 2.6 and B.4 can then be similarly defined as follows.
Definition E.2. We say a measure of information d satisfies no value for the null data dubbed null-information if d(p, jo) = 0 for all μ E A()'). positivity if d(js, v) 2 for all μr, v E A(Y), and invariance data acquisition dubbed order-invariance if for any pair of signaling schemes Ta and Tap the expected information gain is independent of the order in which the signals 8 and $2 are observed, i.e.
Es sz [d(p(s)), p) + d(u(s1 n $2), p(s1))] = (d(((()))))) R)) + d(p(s2 ns1), ((22))
Definition E.3. We say a measure of uncertainty H satisfies 0 cost under certainty dubbed null-uncertainty if H(gs) = 0 for any 3 and concavity H is a concave functional over A()).
For the rest of the proof. we heavily make use of the notions d and H, for they directly measure the distance between distributions or the uncertainty within some distribution. Compared with the economic formulation in Section 2, these notions are easier to manipulate mathematically. We begin by first showing d should be the Bregman divergence of H.
We use μ. in Lemma E.4 to emphasize that the result holds for generic distributions μ., E A(Y),, not necessarily those induced by some belief over the parameter space and a suitable model g.
Lemma E.4. Let d(u, v) denote some measure of distance between distributions μ, E A()). Let (V)) denote measure of information that satisfies mull-uncertainty and concavity. Then the following two statements are equivalent
1. d satisfies null- informaifonm.nn. positivity, order-invariance, and is coupled with H.
2. d a Bregman divergence of H.
We can further refine the results by focusing on distributions that can be expressed as some 9g which is the convolution of the prior q and DPP gæ- In other words, q is some belief over 3 and gB the model itself. Proposition E.5. For any I and measure of information d that satisfies mull-information and order-invariance, let H (gq) = ((3)d(gs. 9q d3.. Then H is coupled with d and satisfies mull-uncertaint.
Before discussing the implications, we strengthen Proposition E.5 via the following corollary.
Corollary E.6. For any d that satisfies mull-information, positivity, and onder-invariance, there is unique measure of uncertainty (gg)) = ((3)d My 93 that satisfies null-uncertainty, concavity, and is coupled with d.
These results show that once the measure of information d is determined, not only can the parametric form of can be expressed explicitly, but the form itself is further unique, even in Bayesian regression settings where the response is continuous. With the properties on d and H laid out, we take step back and demonstrate that the value of information val and the cost of uncertainty cost share similar properties, despite their original motivations as decision decision-making problems.
Proposition E.7. Let val be the value of information and cost the cost of uncertainty of contextual decisiionmakinng problem. We know (1) val satisfies null- null-infornation, positivity, and inver-invariance, (2) cost satisfies null- mall-uncertainty and concavity, and (3) val and cost are coupled.
Proposition E.7 combined with Lemma E.4 show that the value of information is the Bregman divergence defined with respect to the cost of uncertainty in any contextual decision decision-making problem associated with A. u, and x, showing that Bregman divergences are intrinsically linked to -aaiing.
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An Instrumental Value for Data Production and its Application to Data Pricing
Proposition E.8. For any model g and measure of uncertainty H that satisfies null-uncertainty, and concavity, letting d be the Bregman divergence H induces, there is some decision-maken problem in the direction where the cost of uncertainty c) H (9g) and the value of information val(D; q, d(g 9a), where p is the posterior induced by D.
Proposition E.8 stipulates that as long the concave functional H satisfies the conditions in Definition E.3, then the functional and its Bregman divergence correspond to the value of information and the cost of uncertainty in decision- making problem.
We now discuss the proofs for Proposition 2.6, Proposition B.4, Theorem 2.7.
Proof of Proposition 2.6. The if direction, that is, any val must satisfy null-information, positivity, and order-invariance is proven by Proposition E.7. We then consider when val satisfies the properties. Since the definition does not limit the choice of possible data D, for any p that can be induced by 9. we can devise a corresponding dataset that induces p. Consequently, we can construct some measure of information d via value of data val. By Corollary E.6, we can explicitly write out corresponding H coupled with d and, by extension, corresponding cost that is coupled with val. By Proposition E.8, cost and val are then the cost of information and value of uncertainty for some decision decision-making problems.
Proof of Proposition B.4. The if direction, that is, any cost must satisfy Proposition E.7. We then consider when cost satisfies both properties. Note that cost can be written some measure of uncertainty H and, by Proposition E.8, there exists corresponding d and, consequently, a corresponding val.
and concavity is proven by
Proof of Theorem 2.7. The only if direction, that is, val must be Bregman divergence of H, is implied by Lemma E.4 and the fact that val and cost are coupled by Proposition E.7. The if direction is established by Corollary E.6, which shows the uniqueness of cost that couples with val, and Proposition E.8, which constructs a suitable decision eerision-mmaing problem for which val and cost are valid.
E.1. Proof Lemma E.4
We begin by showing the first statement implies the second.
Let Q denote the space of probability distributions and т E A(2).. Well-known result in Bayesian persuasion tells us that if Eper [x] v, then there must be signaling scheme क that induces the distribution 79.
(v) + H()] 0 for all distributions 7 such that Since E. [d(((ss)) v)] [x(H(r(s)) [v)], Eu-rld(p,))
E [14] = V.
We now show d(u, v) - H(v) + (1)) is linear in y for any v. For convenience, we use the following shorthand notation
(1uD) d(u, H(V) + H()).
Let μ be arbitrary distribution that can be induced from by some signaling scheme and assume there exists some constant C' and measure A such that μ + CA, where C' is an arbitrary yet sufficiently large constant that ensures + A are both valid distributions. Note that C' must exist for any A as can be written as the posterior of v under some signaling scheme, implying that y must be absolutely continuous with respect to v. Consider then two possible posteriors. Consider first 7ı, which assigns probability mass 1 to both v + and v Since Eper, [n] = v, we have
7(1+ + A:v) = -- A;v).
Then consider 72, which assigns probability 1 and 1
to + CA and v A, respectively. Since the expectation of the 14C
C
distribution is again v, We have
Cy(v-A:P) + 7(v -C; = = 7( CC,r) CC(( Aiv) = Cy(1 + Azv).
Take || ||| 0 and know (1 + CA:,) Cr(x+ 1; for all C> 0 and all directions A may take. Therefore, 7(4 v) must be linear in and thus can be written as an affine function. Moreover, note that (1; 0 by direct calculation.
Since Ey-r [n] and prohability measures are non-negative, a must be absolutely continuous w.r.t. v. We then take g, which well defined by absolute continuity and integrates by the fact that E [p] =
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An Instrumental Value for Data Production and its Application to Data Pricing
There then must exist functional f(V) such that
d(p, v) (v) + H()) ((u;)) = (f(v). = =(p,v)= H()) + (f(1), v).
Moreover, by positivity, we know that
d( - H(v) - H()) + ()
for all pairs of distributions. By definition superdifferentials, the bound necessarily implies f(v) must superdifferential of H, and hence d(js, 2) the Bregman divergence of H. In other words, the first statement implies the second.
We now show the second statement implies the first. Let V denote arbitrary and fixed superdifferential of H and d(u, v) H(v) (p) + (VI()), μ v). By the property of superdifferential and the fact that H is concave, d satisfies positivity. Note that for any signaling scheme y + S, we have E. [u()]] v, and thus
[x]d(v(s), ] E. (M) (v()))) (V)
n)) (v( - H(((s)))
showing d and H are coupled. Additionally, d casily satisfies null- mulliinforaatin,, d(v, H()) H(v) + (VH()),
v) = O. Thus, all that remains is to show d satisfies order invariance. Recalling we have already shown d and H are coupled,
for any signals s and s', we have
E (d(v(s), v) U()(s v(s)] + H(v(s)) -()))-H(------- = E, ((vb)))))
(5)
=
thereby completing the proof.
E,2. Proof of Proposition E.5
Consider two direct signals Tsy and usz where with slight abuse of notation we let Tsz denote the fully informative signaling scheme, that is,
Ta (8)) 1[3 B'),
which directly tells the buyer the underlying parameter B. We thus have
Eng (d(gg(e2), 94) = E(d(gg, 94)1 = / (19))(92.9))d) (
Consider first the case where $1 is observed after s2- As Ts. cannot further alter the probability distribution, due to a2 being fully informative, we know q(81 ( (1)2) ((s2). Using the fact that d satisfies null-information, have E.g (ggterrne))) 9e(sz))] for all possible realizations of the signals $1, $2, which in turn implies
B (d( 94) nex)) Po(e2))) H(gy)-
We now consider the case where the fully informative 82 is observed after 8t. Again exploiting the fact that g(s) O 82) q( ((52). for any realized signal 81
(((l)) = E,, (dgg 9q(s1))] = H (9
As d satisties grder-invariance we have
Es, .9 (d(gg((s)) 9q) d[gy(sanss)) E.,,, [dggt( + d(gg(sinez)
E., H (g,)) H(g)).
By direct expansion and linearity of expectation, we also know that
Es, 22.9 [d(g)(=1)) 94) + (alo))+((gggaaaa d(gg(suriz)) Iq(ss))] = E., (.)) + E., (9g(s)))
In other words, for any signaling scheme usz and signal 81, have
Es, [d(()()))) 04)1 E., [H(g.) H9g((n)))
which shows that d and H are coupled.
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An Instrumental Value for Data Production and its Application to Data Pricing
E.3. Proof of Corollary E.6
We first show that the proposed H/ satisfies concavity. Note that for any parameter B have
H(gs) d(gs. gs) O,
since d satisfies null-information. For concavity, let yi and d2 be two arbitrary and fixed measures over the distribution of 3 and let E (0, 1) be an arbitrary scalar. Let q eqi + (1 - c)g3 be the convex combination of qi and q2. It is easy to see that there exists signaling scheme Ra, that induces 9i with probability and d2 with probability 1 - Applying Proposition E.5, we know H is coupled with d, and thus
s, (H,)) H (gn(s))) Es, [d(gg(s)) 9q)1 20
by positivity of d. We then know H is concave by noting E,,, [H(((())))] (g)) + (1 H) Applying Proposition E.5 again, we know that H further satisfies null-uncertainty and is coupled with d.
All that remains is to show that H is unique. Again consider some fully informative signal Te2 Let H be an arbitrary measure of information that satisfies null null-uncertainty. concavity, and is coupled with d. By null- sull-uncertainty of H, for any distribution !
(ggg) - E., [HI (gg(s2)) = (gg))
Moreover, as H and d are coupled
(gg) 1., (0 = Eaz [d([g(2))) 9q)1 ((3)((92.9)))) (g.).
In other words, the proposed H is unique as (gn)) Hgg)) for arbitrary q, completing the proof.
E.4. Proof of Proposition E.7
We begin by proving the second statement. Showing cost satisfies cull-uncertainty is straightforward as we directly plug in definition. Let y and y/ denote two arbitrary and fixed distributions over and let = oq + (1 - c)ą² be an arbitrary convex combination of the two. Directly writing out the expectations, we know Eg, (u(:u v)] ClEy,, [u(a; y)] + (1 E) [u(a; y)] for any action and thus
BBa,, [u(a" (g)): + (1 - c) c) [u(a" (g2): + (1 c) (4):v)) -Eg, (u)aa()):)) 8)]-
ZcBgo
Moreover, also by directly writing out the expectations.
ci iu(a" (5g): y)] + (1 - E))
y)] Eg, [u(a" (5g): y)].
Combining the two shows that cost(q) is concave y, completing the proof for the second property.
We then show the third property holds. Let q be some arbitrary belief over and, for convenience, D) an arbitrary dataset given. Let p be the posterior induced by the dataset. We have
E Enccost (q, 2) cost(p,x))
Ep Eg, [u((a((5))
(53): -
Eg, (8s):)) u(a" (p):
Notice that Ep(p] q by Bayes' theorem. Therefore
Ed [Eg, [u(a" (6s):v) - Eg, [u(a" [u(a"(());))]] O,
and thus
En(cost(g, 2) cost(p, x)) ED [ (p):))))) E.g, (u(""((a"())
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An Instrumental Value for Data Production and its Application to Data Pricing
We now turn our attention back to val. Notice that
E 2D] Ep
[ [a(aaaa(aa) (p): u(a"(a))))))))) -(d"(())
Focus on the term u()"(g);)). As the optimal action is taken over the distribution its value for any specific is independent the realized value of D. Therefore, again using Bayes' rule to derive, we have
En (u(a" (q)s y)ll E. [u([a (4):
recalling that is the prior distribution. Noting that taking the expectation of the term above D does not affect its value. Therefore
Ep[val(D:q.x)] = Ep [B [u(a" (p)i (1):v)l
= -Ep [g (u(a"()): v)] - Eg [u(a" (g):
= Ep EEp[cost(g x) - cost(p,
completing our proof of the third property, as we have now shown val and cost are coupled.
Finally we focus on the first property. Null-information and positivity are straightforward by definition of val. Furthermore, val and cost coupled, similar to Equation (5), we know that val also satisfies order invariance, completing the proof of the first property.
E.5. Proof of Proposition E.8
Let the action set A be subset of all possible distributions over y and consider the following utility function
u(a;y) y) = H(aa - (VH(a), + (VH(a), oÿ) if a strictly positive at y
otherwise,
where 6y is the Dirac measure defined y and V follows its definition in Definition 2.2. An immediate consequence of the construction is that for any model | and parameter 3
gy [u(a; v)] / ulaa y)g (v)dy
-((()) (VH(a), a)) / go(v)dy + / (VH((a)).) ( go(y)dy
= -((a) (VH(a), + (VH(O).
(((a) (VH( a) H(g))
d(ga,a),
where for the second to last equality we use the fact that H satisfies null mull-uncertainty and the last equality leverages the fact that di is the Bregman divergence with respect to H. We then know that for any
Ey-g, [u(a; v) = f u(a: (M)
I ( ((()))(()43) ula; v)dy
= f a(अ) (f (a d3
= ((3)d(gs.
Eg(d(gs, a)).
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An Instrumental Value for Data Production and its Application to Data Pricing
Consider then two beliefs over V, induced by beliefs over (3, denoted p and g, such that q is absolutely continuous with respect p. The actions p and q then satisfy
Eg [u(p; 3) - -,)
( H(g) (V 9)
(gpp) + H(gs) (VH p)]
(g) - H(g))) (VH())).p-- q)]
= H((g)) (gp)) + (9g). = d(p.9) 9) 2 0,
where for the third equality other words, for any belief p, the buyer's 's utility can only be harmed by moving from p to some belief q that is absolutely continuous with respect p. Moreover, by the construction of u(a; v). we know that it is impossible for any q that is not absolutely continuous with respect to p to be near optimal, as otherwise there exists some y for which u(g: y) = -00.
the linearity of (gp) 9g - pp) in Js and recall that p = [y by definition. In
Consequently, for any belief p. the action p maximizes the expected utility Eg [u(a; y)]. Recalling the definition of the value of information, we thus have
val val(D;g,x) = R.g
[ (u(aa(()))
- ulaa v)) = d(p, 9),
where p the posterior induced by D. In other words, d(p, q) a value of information for the contextual decision-making problem associated with A, x, and By Proposition E.7, it is then coupled with the cost of uncertainty of the same decision-makinn problem. Since H is coupled with d and satisfies null-uncertainty and concavity, by Corollary E.6 it is unique, and thus H is the cost of uncertainty for the problem with A, x and u.
F. Omitted Proof in Section 3
F.1. Proof of Theorem 3.1
We first state that entropy leads to qualified candidate of the value function, and then we derive its closed form,
We focus E[y[=]]] (x. B). There are two reasons use it rather than y. First, the composition of u(::)) and the expectation operator won't 't influence the existence of u and then the existence of v(:v..) as is white noise. Second, in reality, people always focus on fa(r) rather than the one with uninformative noise So, the entropy of predicting fa(r), namely, (z, B) makes more sense in practice.
We are now going to show that entropy can generate a valid value function. Note that we can view g as a signal transforming our belief over B. The model for [F[[<=]] is then Ely[a]] (8, x). Consider the measure of information d(p, q) (plg).
By Corollary E.6, the unique measure of uncertainty corresponding to is
((gg)) (gn
In this case, letting 9n denote the prior belief over B[y[=]] = (x, B) and associating with the posterior p p= p(- (1,1 D), by definition of coupling between H and d, we have
Enwwg [d(pps 9q)1 =EDng (H(gg) Hggr)]
/ ((3)) (gs, 9,
I ((8))Duk(osss.)dd
-EDgg
E D
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An Instrumental Value for Data Production and its Application to Data Pricing
We focus the terms involving q and Dq. the case for p and 9p is similar. Notice that
f ((3) DKL (ga 9o)d3
- -- / (()) -b (oona//a.....llooooleleeeledLAa0
+ +1ʃ 10 (1) (B)
=x)) + Egng / 2) (By
where the last line comes from the direct observation that
h(f(())) - =- /
B) (b
Similarly, recalling the definition of (z)),(z)), we know that
Epig, [d(pp. 9q)1 -Epng, [h(fg(z)) h(fp(=))]
/ [(B|yle]]: log((( (B)
+ Bag ~a
[ (0
- Epp
log 3)
Observe that in the model defined, f [(Ely|=]]]: x, 3) log(((E[y[]]]) ( [x]]]) is independent of the value of B, and thus the latter two terms cancel out. Moreover, as will soon discuss in the proof of Theorem 3.1, in the special case of Gaussian distribution, h(fp(z) is affected by only the feature matrix, but not necessarily the realized labels. In other words, when D = (1 ys) 2 RDD, ())p(z)) only depends on {f1,...,Ta and we have that
h()((z)) h(fz()) = d(gps9hb)
where d(gp: 94) is the measure of information defined in Definition E. that serves as the value of information of some decision-making problem. Therefore, we can have the following formal definition which coincides with information gain in Bayesian regression.
Definition F.1 (Information Gain). For buyer type 2t, valid value, i.e., the information gain from g is
V(g: q. x)) Ep (p(h(f, (zf())))).
where (x)) and hxfp(x)) are the entropy of his prior and posterior distributions over Ely[æ]] and depends on both q and D.
Note that v is exactly the expected value of D to the buyer when he makes the purchase. The fact that V(:::; :) is the buyer's ex-ante value of data is crucial, as it impossible for the buyer to ascertain exactly the data's value to him prior to observing and updating his belief. It justifies the choice of information gain and we list its detailed concept and closed form under Bayesian linear regression which provides some convenience for optimal mechanism design in Sections 4. and 4.2.
Information gain is a concept commonly used in statistical learning literature, whose applications range from experiment design (Beck et al., 2018), to bandits (Vakili et al., 2021), and to feature selection (Azhagusundari et al., 2013). As we show in the sequel, it also has the desirable property that it is invariant to scaling of the feature vector x, ensuring that the value of data depends on x only through direction.
As we have shown differential entropy yields valid value function, it's time derive the concrete form Theorem 3. We first state the following lemma.
Lemma F2 (Label's Prior Entropy). The entropy of the buyer's prior distribution over E|y[=]] is
h(f((x)) = 1og(1>><00")) 1
d ( + log(2x)),
where we use E.g to denote the covariance of the prior distribution over
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An Instrumental Value for Data Production and its Application to Data Pricing
Proof. We begin by stating the following basic facts the entropy of Gaussian random vectors.
Fact E3 (Entropy of Gaussian Random Vectors (Example 8.1.2 Thomas & Joy (2006))). For any Gaussian random vector 7 E Rd where Z ~ (m,, Σ), it holds that
d 1 log( + 2 (1 + log(2x))
h(2) 2
We know that [[y[a]]] - (a, B) where enjoys prior variance 2g. As we recall that Ely[a]] integrates over a zero-mean random noise .V (0, )² ) the prior belief that the buyer holds over [By[a=]] is that it has variance xE Applying Fact F.3 completes the proof.
With Lemma F22 and Fact F.3 in hand, we can similarly derive h(fp(z)) by substituting the prior variance with posterior variance. Then Definitions C.1 and F.1 lead Theorem 3.1 directly.
F.2. Proof of Corollary 3.2
Since the prior is (Hgg Eg), after observing {<2,, y4 y)) with calculations using Equation (2), we know the variance matrix of the posterior of 3 (E,²) x To be specific,
(x)_
p( q, D) x exp(- - 3 3) exp( 1 - -x Xx - -33)).
Note that it's independent of {81....,yhi yn so we can get rid of the expectation over them. Then, Equation (4) yields Corollary 3.2 immediately.
Note that : here only depends on the design matrix X but not the realized y. Hence, we can get rid of I Enggo in such a case.
G. Omitted Proof in Section 4
G.1. Omitted Proof in Section 4.1
Readers may wonder why we use [(g"[=]: q) as the benchmark revenue. We assume without loss of generality, the buyer a+o(2)2
with type x prefers the associated data mata-generating process, ceteris paribus. It equals to n+o(z)] 2 (a,,))² Taking union bound over n, a sufficient condition is x( revealed after data production processes. Therefore, there is motivation for buyers to misreport based a. Note that this assumption is not necessary but only for concise presentation. Otherwise, there exists a mapping yp Idd - R such that (xx)) is type-z buyer's favorite direction. Composing sp() with the instrumental value function V(:.,) yields an equivalent mechanism design problem.
a(x) 2 (xr,)) Besides, in practice, buyers may lack information on a and it's only
Additionally, from Corollary 3.2, we know that only the design matrix will influence V rather than response variable values.
G. 1.1.1. I. PROOF OF LEMMA D.I
Since have Theorem 3.1, we only need to calculate the posterior distribution of B. For data g (Ø, ( H==), using Bayesian Formula, it holds that
P(3, (0 Ia). (7, () (O Ia) P(2, (8:) (3))1199)
119)
exp( 2 11113) * exp( Σ
1
(Tr -
20 + 5 1=1
expl 1 (8
- (1a
o + 5 ( )3))
n+aT
considering only the quadratic terms of 3. is because that e and are independent and Var(e + = -2- 88. From now on, we omit the prior g = (0, Ta) when easy to infer from the context for simplicity.
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An Instrumental Value for Data Production and its Application to Data Pricing
Therefore, have that the variance of the posterior distribution of is
T22E
Var [ = Ep = -Ia +
and Theorem 3.1 leads to what we need.
G.1.2. PROOF OF LEMMA D.2 Now, we going prove Lemma D.2. With IC constraint, it holds that
V(z, x) t(x) 2 v(z, - t(z) for all x;,
Rearranging the algebra items leads to t(T) f(x:) 2 V(z, 2) V(z, x). Now, we change the placement of and T. it holds that (()) - (x) < V(ū, G) v(o, x). Combining the two parts, we have
V(z, ē) V(z, x) S ((T) t(x) s v( v( Then, dividing by ã x and letting ã converge to x results in Vt(x) V,V(z. y=E" Now, we calculate the gradient of v(.,)). We have the following equations.
1 V, 1olo0("
nyy 01y) + 8(v)2 )'2x) xx)
V (Vz 1F= =
(I+
2
1
x² vyy x
V. log(1 (o(w)² + (1))²)/ n + y=
2
x² "y x
2x yo
2 20
Vo 5(4) Vō(y)
2
+y))++60)²
+-8))+)))))
0())++8())= + 1)2
y=
1 (x)) +5(2)
2x
20 (0(x)/ + +6 1)²
2 (f +.
The first equation comes from the definition v(:)) )) while the third equation comes from the derivation of multi-variant functions. When we replace y with we have the last equation since ||=1|² = x" σ 1. As for the second equation, we need the following lemma.
Lemma G.1 (Sherman-Morrison Mordison-Wodduury Formula (Sherman & Morrison, 1950).. Let A and B be nonsingular m X m and n X n matrices, respectively, and let U be m x n and V be n x m. Then
(L+UBVV)T A-1 A UB + BVA - BVA
Taking I, y. V = and B -())= in Lemma G.1 leads to the second equation.
Since E (x: ||>|| 1). holds that dr Then, it holds that
(xx) = C+
VyV(s, g-m. ds
m0
1 1 120
CC+
2 (0))))+6()22)/ (lo()))+dd))))nn+++ ds
Ja
1 C+ log(
(o(a)3+
(((x))6(x))))))
with little abuse of the constant C.
The first equation is straightforward however we need to check if it's independent of the path of integration. The second and third equations are some calculations. It's casy to see that the construction of t(z) satisfies the condition in (Jehiel et al., 1999) that the field is conservative. Therefore, the value irrelevant to the path of integration and the payment rule is well-defined. The proof suggests that our method has greater verifiability compared to other methods, such as cycle monotonicity (Lavi Swamy, 2007), as well.
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An Instrumental Value for Data Production and its Application to Data Pricing
G. 1.3. PROOF OF THEOREM 4.1
From Lemma D.2, we know that the form of optimal payment rule is ((x) C' + 1 log( +00)
[(())+((5)*
Now, we need to calculate the indicator function of the IR constraint that
((a)2+0(00)) + =" " + log(=" nXx o + 0(x))
((C + log(
(2)) x) S 50 1(C s 5).
oxx) d(x)²
is because that
nxT
(1 o(r)² 0())
1 = x²(1 (x)² 2 + 8(x) + xx Ja
+n
ox2) + 0(x)² 0x0)² + +(z)) +n
The first equation comes from Lemma G.1 with A =, and B a())+8(2)) The second equation comes from | = while the last equation is from simple algebra calculation.
Therefore, the optimal C' zero for any x and the payment rule is exactly
1 log( (x)²² + 5(x)) +n
((x) 2 axx)² + 8(x))²
As for 8()), we will show that the optimal solution is too. Since log( is a decreasing function with respect to æ for any fixed n, taking 5()) leads to the optimality.
" are the optimal solution to the optimization problem D. Then, for any type x, adding no noise and charging 1 log( aiz)²
the distribution of type is 6- where 6() is the Dirac delta function. However, since the solution is optimal to any type x, it's also the optimal solution to the general optimization problem D.1 as long as satisfying the IC constraint which guaranteed by Lemma D.2.
Combining the above parts leads our theorem and finishes proof.
G.2. Omitted Proof in Section 4.2
G.2.1. PROOF OF THEOREM 4.2
Recall that the value of the data g [ is
V(x, 7) = log(k Var(3))=*1) log (/x Var(Sp (ar(3))977))x" 1)()))=²
Similar to the argument presented in Section 4.1, we make the assumption that the variance of the prior belief regarding Ia. without loss of generality. We also assume that the seller can attain the first-best revenue and will demonstrate contradiction in due course. To achieve the first-best revenue, since the buyer reveals their report after the seller produces the data, the seller must disclose all data in some condition for any report ž in order to prevent loss of information and charge the corresponding information gain. For instance, let us assume that the type represented by a vio-dimensional vector denoted by x = (a,b)T, where a, E [1, 2]. Hence, each data point assists predicting each component of 3. The first-best revenue implies that the payment rule is precisely ((7) V( 7) and the buyer will derive no consumer surplus.
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An Instrumental Value for Data Production and its Application to Data Pricing
Now, we ready calculate the exact form of ((-). We first calculate the posterior distribution of B. It holds that
[[
x (0, Ia) * I | (8)
1 (33-X3) X8))
exp(
|1313) exp(
1
1
exp(
2
[|311²) exp(
2
x exp(
2 (T+X"=))))3)
The first line comes from the Bayesian formula and the following lines are simple algebra calculations. Recall that Σ is a mapping from X. Therefore, holds that Var 1 11)) is
Var(3p19* (T+ -1+X"
With the posterior distribution in hand, it holds that
1 log(2" 3) - 10
i 2
Besides, the value of data is
V(x, x) log(o" 2) (I+
1
Therefore, the extra surplus of reporting X
V(z, 2) (T)) = [lo((="") x:) loo(2
log( 3) +
In order satisfy the IC constraint, we need prove that ((T) is constant for any T. However, since the type space continuous, it holds that - should be constant times identity matrix. However, X can be determined by the following optimization problem
X = argmax = argmax Eg log(a" T x) log(x (I++X --X-XXX'---- x)))
Therefore, there exists some prior distribution for type Dirac delta distribution for example, that XT X isn't 't constant times identity matrix which leads to contradiction. With the continuity of V(--). there exists continuous distribution leading to contradiction as well.
Therefore, means that our assumption that the seller can achieve the first-best revenue is wrong and ends our proof.
G.2.2. PROOF OF THEOREM 4.3
For any design matrix X, the proof of Theorem 4.2, the total surplus of the buyer and the seller is no larger than V(z, Φ) log(x"x> x) log(x" (1 x Therefore, we only need to construct mechanism satisfying More
IC and IR constraints that for any type x, the seller's revenue is no smaller than V(x,, =) - log((((( (<(((<((X))))>)))))))) generally, we allow the seller to manipulate her data by adding some noise as in Section 4.1 or doing linear combination as in Algorithm 2. However, the data- getiraainnntngggggg process must be pellucid and known by both the seller and the buyer which is demanded by mechanism design problems. The result can directly lead to Theorem 4.3.
Since x(() <((vVV(/ ((XX) is the square root of the condition number or the ratio between the largest and the smallest singular Xx is not full-rank, Theorem 4.3 is right trivially. So, we only need to consider the
values of 7 situation that X(X) is full-rank. As X is a n x d matrix, it means that n 2 d for sure. It means that in order to separate buyers with different personal types, the seller needs least some amount of data to achieve price discrimination.
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An Instrumental Value for Data Production and its Application to Data Pricing
For simplicity, we use X replace (VE(X))-XX. For any dataset, can transform (X,Y) ((VE(X))-XX
"y) so that the noise distribution is indeed (o0,T) I). It's just for readers' understanding and transforming back leads to the original Theorem 4.3.
The key challenge our setting caused by the fact that æ, the buyer's private type, is continuous. A difficulty caused by continuity is ensuring incentive compatibility. While it is easy to deter buyers from scaling their feature vector when reporting (e-g. rescale all reported features on the seller's end prior to allocation and pricing), stopping buyers from rolating the reported feature vector is much harder. As the data design matrix X is not equally informative in all directions in the feature space, by rotating the reported feature vector, the buyer could act as he did not gain much information from purchasing the data. At the same time, slightly rotating the reported direction could still lead to significant information gain from the buyer due the small angle between his true private type and his reported type. We visualize the impact of rotating the reported type in Figure
actual x
reported
the amount of information on each direction IR²
Figure 4. Visualization of the effect of reporting the rotated private type in R², The ellipse represents the amount of information that X contains each direction in B. Long-dash-dot-dot line denotes the buyer's private type and dash line the buyer's report S.
Intuitively, the radius of the ellipse in Figure roughly measures the amount of information that contains on each direction in the parameter space R22 While the buyer's true type, depicted he were to report truthfully, by reporting ū, visualized by the dash line, the buyer could lead the seller believe that he gains less information from the data purchased, thereby potentially reducing the price the seller charges. Moreover, when the angle between T and is sufficiently small, the buyer would still gain sufficient information from untruthful reporting, while potentially paying less.
line Figure 4, benefits the most if
To deter buyers from rotating their report, we propose the SVD mechanism, whose hallmark is performing Singular Value Decomposition (SVD) over the buyer's 's feature matrix X to obtain
S
X =
V
0 Oln-dixd
where S = diag(Ai..... deh. d(ndd E (nd-d) is matrix filled with zeros, and U E Raxn V E Rdxd are unitary matrices. particular, (2 )a) are singular values of X, we define X's condition number as =(X)
max, We also
min, A;
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An Instrumental Value for Data Production and its Application to Data Pricing
5-1
obtain the left-inverse of x, - U
V. such that
[-1 V-1. [
L -T==V-(--
0 OUT
Geometrically speaking, U and V are rotation matrices that help orthogonalize different components, The entries in the diagonal matrix S, )s are the singular values that dictate the amount of information contained in a specific direction in R Particularly, for the multi-armed bandits setting we deferred to Corollary D.6, X equals to ynz where n, the number of observations the i-th component. Intuitively speaking, the larger A is, the more information contains the particular direction.
Recall that we have the SVD mechanism in Algorithm 2 and let's provide high-level description of the algorithm. Performing SVD over X recovers the amount of information the seller's data contains on directions in Rd. In particular, these directions are given by the row vectors of V, and form an orthonormal basis of the Idd parameter space. Using the results of SVD, the mechanism constructs projection operator L and normalizes the projection of T. 6(T). The allocation and payment rules are constructed using the projection ((T) according to the penultimate line Algorithm 2.
As the payment rule never charges more than the buyer's information gain whenever he is truthful, the mechanism is individually rational, which we formalize in the following lemma.
Lemma G.2. The mechanism satisfies individual rationality (IR).
Proof. For any truthful buyer, the payment (()) = =1(x) = V(z, æ) is exactly his value of the received data g[z]].
Proving that the mechanism is incentive-compatible is more involved. Intuitively, projecting the reported type T along L as opposed to X normalizes the amount of information gained in each direction. Recall that performing SVD over X yields U,S-1. V, from which T. is constructed. Previously, we showcased that the diagonal entries in S correspond to the amount of information X has in particular direction. Conversely, the entries in S normalizes the amount of information X has along the direction. Projecting & over L then uses the values in S to normalize the information the buyer may gain in particular direction. As opposed to the ellipse better resembles a circle, and the buyer is thus discouraged from untruthful reports, as doing so would not alter his payment significantly. We formalize the intuition in the following lemma.
observe in Figure 4, the information gf:] provides for each direction now
Lemma G.3. The mechanism satisfies incentive compatibility (IC).
Proof. We need to prove that the buyer's utility is maximized when he reports truthfully, namely for all F E X
Var, 3) (TT) V(x, (xx).
Recall by construction of f(-) that V(z, x) t(x) and t(3) = v(G,, T). Thus, we only need to show that v(ā, S v(ā,8). We first note that the payment rule t(-) is scale-invariam. Indeed, for any 2 X and 2 O, we have
co
log
T T 3 +
log
-1 x) - = ((7),
2
as we note that b(7) is also scale invariant. It is then without loss of generality to assume that ||>||| |.||| Expanding both v(ē, x) and v(i, f), we know that proving IC is equivalent to showing
x²
XT (I (I+ X ((2)(()
as we recall the logarithmic function log(:) (-))))))))))))))))) is monotonic and note that the matrix I + X ((F)((T) X is positive definite.
Define the shorthand (x, = (T x) which is a strongly convex quadratic function. Observe that proving the inequality holds for any pair x, 2 can be reduced to showing that for any G, the induced quadratic function fzz 2) is minimized by 2 over the set far e Rd ||ж||| = ||]]]], }, as we recall is without loss of generality to focus the set of vectors with the same length as T.
31

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An Instrumental Value for Data Production and its Application to Data Pricing
By first order condition of strongly convex functions, we need to show that the set far TR ||=||| |[]]], only need to prove that &e (T X xr Sherman-Morrison (Lemma G.I), we have
(f(z
Oi == O. Since we restrict ourselves to
x) -
0 for any y such that F y 0. By
x=b(x)((x)XX
(I + XI
1+b(a) XX)
Clearly 1y T" 0. Additionally, we know that
Te x²)b(2)(z) Xy
2x"
Lax
L Xy =
| x Lax = 0,
|
as -. y= 0 and X = 7. We then know that
T argmin
ZE [|=])]||=||-()))
for any a and consequently, V(ž,z) x) S V(i, 2) completing the proof.
Finally, we show that the mechanism achieves near-optimal revenue with at most log(k(X)) loss. Note that the maximum revenue is no greater than the social welfare for any IR mechanism. Moreover, selling all data maximizes information gain, and by extension, social welfare. Then, for any private type 2, a trivial upper bound over the revenue V(g X q, I), which holds for all possible choices of g[-]. Recall that we dub this revenue, V(gr"; q, x)), the first-best revenue, as it the maximum revenue that any potential non-IC mechanism can extract from a buyer with private type x.
We now show that the gap between the revenue achieved by mechanism and this theoretical upper bound is always no greater than
constant.
Theorem G.4. The SVD mechanism achieves revenue less than v - log(n(X)) for any buyer with private feature where we recall xxX) = >mmxx (X)/ Aminn is the ratio between the largest and the smallest singular values of X.
Proof. Recall that we use X to replace (VE(X))- for simplicity. By Lemma G.3, our mechanism is truthful, and any buyer with private type would report truthfully. Recalling the construction of ((-), our goal is to then bound the difference between V(z, I) and V(gx:q :q,z) which, by definition of V(,,), ), is equivalent to showing
log(z (T X (a))((()) 1x) log(s (T+ log()(X)).
2
Moreover, examining Inequality (6), we know it is further without loss of generality to assume that ||>||| 1. By construction, b(z)) is invariant to the scaling of æ. The effect of scaling then cancels out in the left-hand side.
Let us begin with the first term on the left-hand side of Inequality (6). Expanding (I + xxb( by Sherman- Morrison, we have
IT (=b((=)((x) X) IJ Xx
1+ b() ++b())=XX666) XXTb(x)
Focus on the enumerator first. Recalling b(x) = ||a where L is the left inverse of X, for the enumerator we have
x" xT xTxT ((=)((())) Xx
1
x x Lax LaxTLTX Xx
Txx
x T xxx T
|
|La|
||
As for the denominator, again recalling b(x)
Lx ||
we
have
1 + 6(x))( xx [(()) = 1+
T L 1T
La
|La| (1 + IJ xxxxXLLX Lx)
|
| T (IIT L T)r.
32

---

min, X3 and the An Instrumental Value for Data Production and its Application to Data Pricing
[--1]
Additionally, note that L = U
[0]
V with U and V being orthonormal matrices, we have
[-- [ v - VT VT diag ( 1 1 27"1.!! 23
L-L=V2 0| U
V,
as we recall& S = diag(A
Ad) and is diagonal. We then easily know
1 1+ X? A
1 1 X
x" (L" L L + 1)x = x" diag
Vr
X
and therefore
+
x VVVddiag
1+X x
1+1
Vx
A
We now turn our attention (I X Applying Woodbury matrix identity (i.e. Lemma G.I), we know
(L + x=xx== = X + XX") Since x² Xx") = V diag
x² (+
x
V as we recall the
T+x7
SVD for X, we have
X2 I+ X7
X X
x² 2" + (1+
V diag
Vx.
+
Combining Equation (7) and Equation (8), we have
TT + (()))(<))
1
x1 X
1 1- > i=l
(V Vx)²
nd 1+ (V2)²
riil (V
T+X?
Σ X2
(Vx)²
E
(1+ X
(V x)}
Ei= कमズ (Vx);)
Σ x=1 X? (Vx);
x (Vz)))² x(())
where (Vx); denotes the i-th component of the d-dimensional vector Vx. Here, the first inequality holds due to the Cauchy-Schward inequality while the second inequality holds because the numerator is no larger than 1 denominator no less than
Plugging Equation (9) back into Inequality (6) completes the proof.
With the argument that X is full column-ramn rank, holds that x(X) Amax (X) Amin (X) as same as the square root of the condition number or the ratio between the largest and the smallest singular values of x Therefore. Theorem G.4 leads to the following results that there exists mechanism that the seller achieve least V(g qg x) log(n(X)) for any type which leads to what we need directly. Transforming X back to [XX]) X ends our proof.
means that for any mechanism if the seller's exceptional revenue is less than E, V(g q, x) log(k(( (((X))-1x] or the seller's surplus is larger than log(n( / - the seller will have the motivation to change to the SVD mechanism in order to gain more revenue. Therefore, since the seller rational, Theorem 4.3 holds.
We conclude Appendix G.2.2 with the following remarks. Firstly, we note that there exists sharper, yet harder interpret bound, for Theorem G.4. If we define
Σ
V y ) (Ed= T43 v?
x
f(A Aa) max V
1+X X
:.. x) - of ..., Aa)) for any buyer with private feature F, as Equation (9) exactly bounded by the function. On the other hand, means that the buyer can enjoy
then the SVD mechanism achieves revenue no less than v(gx
33

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An Instrumental Value for Data Production and its Application to Data Pricing
only 3 log( Aa)) surplus most. Besides, the condition number of the seller's feature matrix is commonly found in statistical learning literature and is closely related to concentration bounds in the Bayesian linear regression setting (Wasserman 2004), thus we believe the unfairness between seller and buyers characterized in Theorem 4.3 is easier to interpret.
G.2.3. PROOF OF COROLLARY 4.4
Now, we are ready to show that the seller can achieve the first-best revenue even the buyer reports his type later when the data is isotropic.
Since every singular value of XXE is the same, it holds that x(X* with the definition of s((). Therefore, Theorem 4.3 tells us that the buyer can only enjoy no more than log ((Xxx(X))1X surplus which is 0 under this situation.
This once again demonstrates that the crucial factor is the buyers' willingness to pay, rather than their personal types. Given isotropic data, the seller possesses precise information regarding every buyer's willingness to pay. Accordingly, it is straightforward for her to implement price discrimination, which manifested by the first-best revenue.
G.2.4. PROOF OF COROLLARY D.6
To better interpret the proposed SVD mechanism, we show that in the Mult-Armed Bandit (MAB) setting, it reduces to an intuitive mechanism with the first-best revenue. Consider a d-armed bandit setting where for any j E [d]. arm j's reward is a random variable independent of other arms' rewards.
The seller's 's data consists of measurements on each of the d-arms and the buyer's private type x E Id is a member of the standard basis of the d-dimensional Euclidean space RRd namely x E X' = (e1...., ea). Similarly, the seller's dataset's features are all members of d-dimensional standard basis.
We can thus demonstrate that the SVD mechanism can be simplified to straightforward and comprehensible mechanism, wherein the seller conveys to the buyer the average reward for the arm in question. It is not arduous to establish that this mechanism attains the first-best revenue, as the average reward constitutes the most accurate estimate for the reward of the buyer's queried arm, given the seller's s dataset. We provide a formal articulation of this proposition in the ensuing lemma. Lemma G.5. In the MAB setting, the SVD mechanism reduces to the following
X
Σ d.
t(3) = V 9 V(2.
(10)
Vaz x. = T]
dE(
where we use X(- to denote the part of the design matrix the seller will keep and the seller forms gl-l correspondingly; Moreover, the mechanism is IC, IR, and achieves the first-best revenue.
Proof. We can manually perform SVD over X show that in this case, suitable choice for V is the d-dimensional identity matrix la and S = diag VHx; z, = eiJ,.... Vzz 2; ea]l). namely, S is the diagonal matrix where each diagonal entry corresponds the number of measurements on that particular arm. Here U is conforming matrix ensuring that = USV, and the SVD mechanism reduces to Equation (10).
We divide the rest of our proof into three parts, beginning with IR, then IC, and ending with showing that the mechanism achieves the first-best revenue.
IR
The individual rationality is directly implied by Lemma G.2.
IC
The incentive compatibility is directly implied by Lemma G.3.
First-best revenue
We note that
v(g :g.z) hxf((x)) - - hxf((=)) h(fp(())o() -kf
= V(g X(=); 9.
by chain rule of conditional entropy and the fact that any data point different from x is a mcasurement a random variable independent of the buyer's private y which the label corresponding to x and hence leads to an information gain of zero.
34

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An Instrumental Value for Data Production and its Application to Data Pricing
Note that the posterior (2) may come from different datasets and explicitly give the conditions X and X [2r] to make it clarified. From the inequality, know that for any possible private type the information gain that the buyer obtains from receiving g g|a equals the information gain that he obtains from observing the entire dataset, gx Moreover, when the buyer reports his type truthfully, he is charged exactly that amount. Thus, the revenue achieved by the mechanism is the highest among all IR (and potentially non-IC) mechanisms.
As Lemma G.5 shows, in the multi-armed bandits (MAB) setting, the SVD mechanism can achieve the first-best revenue which means that the consumer surplus is 0. Therefore, we finish our proof.
This highlights fundamental distinction between the data market and traditional markets for tangible goods. In the data market, the value of dataset varies significantly among buyers. As in the multi-armed bandit setting, buyers derive no utility from obtaining data about different arm. Consequently, buyers have incentive to misreport their valuations, which enables the seller to effortlessly attain the highest possible revenue.
35


Reviews:
[{"review_id": "XrnEBH8Ryc", "reviewer": "Reviewer_TrgW", "summary": "This paper introduces a framework for quantifying the instrumental value of data production processes (DPPs) under the bayesian linear model. The authors focus on how much additional benefit (or marginal contribution) new data brings to a decision‐maker’s task. The proposed data value is mathematically equivalent to information gain. They then leverage this valuation to study optimal pricing in data markets. Two key selling scenarios are explored: (a) perfect customization—where the seller can generate data tailored exactly to the buyer’s needs, allowing for full (first-best) surplus extraction; and (b) limited customization—where the seller can only curate data from an existing pool, yielding a mechanism that, while not fully optimal, achieves revenue within a bounded regret.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed and constructive comments. We will answer your questions according to logical order.\n\n \n\n**Comparison with Other Data Valuation Methods**: Compared to the fairness-based Shapley value, our instrumental value accounts for sequential data settings where Shapley can underestimate value (cf. Example 4). Theoretically, instrumental value avoids the exponential complexity of Shapley, making it more scalable for large ML datasets. The acceleration methods you mentioned just approximate the Shapley value, and similar ideas may apply to instrumental value. Our experiments show the conceptually significant advantages of instrumental value in terms of computation and sequential settings. Defining a universally better metric is unrealistic and not our intention. Meanwhile, Shapley value in fact also requires a suitable choice of valuation function, while we, under the Bayesian framework, rigorously characterize what valuations are valid for decision making (Sec 2).\n\n \n\nAs for https://arxiv.org/abs/2405.13954, it also stems from Shapley's inefficiency but takes a different angle—focusing on gradient-based LLMs. In contrast, we take an economics perspective (hence the game theory track), aiming for decision-oriented valuation (Sec 2.2) with theoretical guarantees, which they do not pursue. Thanks for your suggestions—we’ll discuss these works in the final version.\n\n \n\n**Discussing Our Assumptions**: Regarding the Bayesian assumption, we can use either strong or uninformative beliefs (see right, lines 120–130), just to cover the general case. Bayesian updating is also very common—recent work (https://arxiv.org/pdf/2503.04722) even shows that LLMs exhibit Bayesian behavior. The paper you mentioned (https://pubsonline.informs.org/doi/10.1287/msom.2020.0915) also adopts a standard Bayesian setting, but we focus on what constitutes a valid valuation under this framework (Sec 2), while it assumes a specific utility function. We then explore the mechanism design problem under valid valuations, which is also entirely new. \n\n \n\nRegarding linearity, some studies show that data valuation is transferable—value order under linear models often holds under non-linear ones as well (see right side, lines 253–260). Moreover, when the true model is unknown, linear models are the most robust (lines 65–70). Finally, linearity enables tractable theoretical results, which help us analyze the underlying economic intuition. \n\n \n\nAs for the DPP assumption, one example is mechanism design in clinical trials, where experimenters actively select samples and often update beliefs in a Bayesian manner—see *Bayesian Clinical Trials* published in *Nature* (https://www.nature.com/articles/nrd1927). Building an experimental platform based on instrumental value is a promising direction with high scientific and economic potential.\n\n \n\n**Utility vs Valuation**: Our data valuation measures how much utility the posterior distribution (after observing data) brings over the prior—for example, more precise estimates (with lower variance or entropy) yield higher monetary revenue. This mapping is common in practice; in clinical trials, for instance, pharmaceutical companies often have clear estimates of the value of improved precision.\n\n \n\n**Difference from Active Learning**: While data sequence has effects on both our instrumental data value and active learning, however these two areas are completely different. Active learning studies how to select sequential data and train models to maximize **accuracy**, whereas our paper studies how to define a valid data valuation function that captures downstream users’ utilities. We approach this study from a utilitarian perspective, and our value corresponds to information gain only at a very canonic special case. We will clarify this point in the camera-ready version.\n\n \n\nWe hope our rebuttal has clarified the reviewer’s confusion and respectfully hope that the reviewer would consider re-evaluating the merit of our work accordingly.", "claims_and_evidence": "**Claims:**\n\nInstrumental Value Framework: It argues that the value of data should be measured in terms of its marginal improvement on a decision-maker’s utility rather than by an average contribution (as in Data Shapley).\n\nMicrofoundations via Bayesian Decision-Making: The authors claim that by grounding the valuation in a contextual Bayesian decision-making problem, one can rigorously justify the use of Bregman divergence to capture data’s value.\n\nOptimal Pricing Mechanisms: For perfect customization, they claim that there exists an incentive-compatible and individually rational mechanism that extracts full surplus (zero-regret). Under limited customization, they provide a mechanism based on singular value decomposition (SVD) that achieves revenue nearly as high as the first-best benchmark, with regret bounded by a term related to the condition number of the data matrix.\n\n**Evidence:**\n\nThe evidence provided is largely theoretical under strong assumptions such as Bayesian Linear Model and . The authors back their claims through a series of definitions, propositions, and theorems (e.g., Theorem 3.1, Theorem 4.1, and Theorem 4.3), complete with mathematical derivations. Although the paper includes references to numerical experiments in an appendix B.3.1, the investigation is mainly showing it is faster and close to DataShapley, while there are existing works to speed up shapley value calculation (e.g., https://arxiv.org/abs/2107.07436) which the authors did not consider. \n\nI find it weird that the only numerical evaluation is to demonstrate the instrumental value is close to data shapley and there is no numerical evidence that the instrumental value is a better metric.", "methods_and_evaluation_criteria": "The methodology is primarily theoretical:\n\n* Framework Development: The authors define a data production process (DPP) and introduce a valuation function that depends on the buyer’s prior beliefs and decision context.\n\n* Microfoundations: By invoking a contextual Bayesian decision-making framework, the paper ties data valuation to improvements in expected utility. I have concerns on the strong assumption of the bayesian linear model and the DPP assumption. I find the data shapley's framework that replies on the realized data much easier to understand. In practice, it is quite difficult for the data buyer to formulate their prior belief and how their belief will be updated after acquiring the data. If the practical scenario is more complex than a simple bayesian linear regression update, how should I apply this paper's method and why it is better than data shapley? I think the authors should include more realistic examples to strengthen the paper. \n\n* the numerical experiments in B.3.1 only shows the instrumental value is comparable to data shapley, which I don't think bring much value to the paper. Ideally I expect to see how in practice the proposed method should be used and why it is better than the data shapley.", "theoretical_claims": "Overall the proofs seem correct. I have some issues: \n\n*  it seems the authors are considering the data utility and the customer valuation as the same thing, but I do not think they are equivalent. The valuation describes how much the data can improve the posterior update, but valuation describes how much the customer is willing to pay in the monetary value. How can the authors find the mapping in practice? The paper should also be significantly changed in terms of this. \n\n* I think the whole theoretical proof until theorem 3.1 is a re-invention of bayesian active learning in the name of ``instrumental value''. The value of data is commonly defined as information gain in the active learning literatures. There is nothing new here. The paper also did not cite any active learning papers.", "experimental_designs_or_analyses": "See my comments on numerical analysis above.", "relation_to_broader_scientific_literature": "The paper is closely related to data shapley and data attribution methods. \n\nI think most methods in the literature focus on the instrumental value, so the claims that it is novel is weak. The claim that the data buyer can update prior knowledge part I feel is widely studied in theoretical work such as information sharing between retailers with the similar theoretical framework that authors did not discuss (e.g., https://pubsonline.informs.org/doi/10.1287/msom.2020.0915).", "essential_references_not_discussed": "The authors did not cite most data attribution methods after 2022, which seems neglected many related works. Also, I think most these works study values similar to the ``instrumental value'' that considers additional knowledge gains (https://arxiv.org/abs/2405.13954). \n\nActive learning is a widely-studied field that measures the value of additional data. It is very established that information gain can be used to quantify the additional ``instrumental value'' of new data. The paper did not cite active learning papers.", "other_strengths_and_weaknesses": "NA.", "other_comments_or_suggestions": "NA.", "questions_for_authors": "NA.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "NA"}}, {"review_id": "ptlKpvcEYE", "reviewer": "Reviewer_84M6", "summary": "The paper studies the mechanism design problem in pricing and designing data generating processes in the context of Bayesian regression. Concretely, the buyer first has a prior $q$ of the regression parameter, reports his feature $x$ that he wants prediction on, and then upon obtaining data (including the data points $D$ and knowledge of the data generating process $g$) at some price $t$, forms a posterior $q$. Buyer's utility in the setup is the Bregman divergence between p and q corresponding to the entropy cost. The paper consider the mechanism design problem of (D,g,t), such that certain nice mechanism design properties are achieved, such as IC, IR, and regret. The paper separately considers perfect customization and limited customization settings, the second one being the harder setting to analyze.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your kind remarks and questions, and we would now like to answer your questions in logic order. Below, we first address the reviewer's major questions, and then clarify a few minor comments. We are happy to engage with any further questions.\n\n\n\n**Re Corresponding Utility Function in Thm 3.1**: In Theorem 3.1, the corresponding utility function reflects the confidence in the estimation, which is captured by the entropy. As our confidence in the estimation increases, the corresponding entropy decreases.\n\n\n\n**Re Design Matrix in the Limited Customization**: In the limited customization setting, we assume that the design matrix $X$ is public knowledge, but the buyer does not observe the corresponding responses $Y$. In practice, for example, in clinical trials, information about the subject group is usually known, but the evaluation of the drug’s effectiveness needs to be purchased.\n\n\n\n**Re Realized Data and Noise**: By \"realized data,\" we refer to the realized response $Y$, while the variance function $\\sigma$ is assumed to be known. In real-world scenarios, sellers typically disclose the measurement error or the specifications of the experimental instruments, such as their precision. When the buyer has no information about data quality, pricing in our setting becomes infeasible. \n\n\n\n **Re Other Comments Or Suggestions:**\n\n1. Thanks for your suggestion. Our idea is to first introduce the characteristics of valid valuations (Sections 2 and 3), and then present a concrete application from game theory, say, mechanism design (Section 4). We'll include a roadmap in the final version to outline the content of each section.\n2. Due to the page limit, we have deferred the calculation of the posterior distribution and variance to the appendix. Please refer to Lines 667–677.\n3. Thank you for pointing out the typo. We will correct it in the camera-ready version.\n4. We introduced the definition of $g^n[x]$ in Lines 334–343, which produces $n$ responses for the buyer’s type $x$. Thank you for your suggestion — we will highlight its definition again around Theorem 4.1. \n\n\n\nWe hope our rebuttal has clarified your concerns and would greatly appreciate your re-evaluation of our work’s merits if these clarifications address your doubts.", "claims_and_evidence": "All claims are supported by proofs.", "methods_and_evaluation_criteria": "The paper discusses classical mechanism properties such as IC, IR and regret.", "theoretical_claims": "I checked the proof of Theorem 4.1 which relies on a close-form expression of the valuation function and a generalization of results from Myerson saying that the pricing rule takes the form of derivative of the valuation.", "experimental_designs_or_analyses": "The main text only contains theories.", "relation_to_broader_scientific_literature": "The paper cited sufficient related works.", "essential_references_not_discussed": "The paper cited sufficient related works.", "other_strengths_and_weaknesses": "Strengths\n\n1. The problem of pricing additional data nowadays is crucial given that large language models require new data to perform alignment/ fine-tuning/RAG etc. The paper provides a theoretical discussion of pricing and mechanism design of incremental value of data in the Bayesian regression setting and devises several mechanisms. \n\n2. Overall the paper is well written and provides sufficient explanation of concepts. I also like Example 1 and 2 to explain the setup of selling data generating processes and changes in variances.\n\n3. The paper contains a few interesting results for pricing data, such as the equivalence between valid valuation functions and Bregman divergence, and also the results that in the limited customization setting, there is not 0-regret generally but if the design matrix is isotropic then there is. \n\nWeakness\n\nTo me there are a few clarifying problems to be addressed to make the problem setup clearer, eg concretely what are exactly known to buyer when the mechanism is announced (see below). Based on the above I recommend weak accept, and I'm happy to adjust the score if my concerns are addressed.", "other_comments_or_suggestions": "Some comments on writing. \n1. In terms of writing, the main paper only really considers the entropy setting, while the section 2 discusses settings of a general cost functions. The Characterization of Valid Valuation Functions is interesting, but for streamlining the paper I feel the main mechanism design problem should maybe be described first in a self-contained way. \n\n2. In example 2 the derivation of the variance is missing. I assume the computation of posterior is omitted?\n\n\n\nLine 275 notably.\n\nNortation $g^n$ in Theorem 4.1 needs to be explained near the theorem.", "questions_for_authors": "A few clarifying questions.\n\n1. What is the corresponding utility function for the valuation function in Theorem 3.1?\n2. In the limited customization setting, are the design points also announced as to the buyer?\n3. In section 2.1 it says \"This will be useful for selling DPPs since it means the value of a DPP can be quantified without seeing the realized data\". But in section 4 the seller has a variance function $\\sigma$. So is $\\sigma$ known to buyer?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6cKgsHhtCH", "reviewer": "Reviewer_PAUD", "summary": "The paper analyzes how to quantify a datasets' instrumental value by taking into account the prior knowledge/data sources the buyer has. The authors argue that switching from intrinsic value (as in data Shapley) to this instrumental value is useful for avoiding overestimating the data value.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed and constructive comments. We will answer your questions according to logical order.\n\n \n\nWe respectfully disagree with the reviewer’s comment on relevance to the machine learning community. Below, we provide our detailed explanations:\n\n \n\n1. In many scenarios, such as experimental design and clinical data, sellers can effectively control the subjects and collect data. Even in the field of deep learning, data annotation companies like Scale AI can provide labeling for data with specific features. These are examples of scenarios where perfect customization can be applied.\n\n2. Similarly, the Bayesian setting is also widely used in machine learning. For example, please refer to the *Bayesian Clinical Trials* article published in *Nature* (https://www.nature.com/articles/nrd1927). Also, Bayesian updating is also very common in deep learning literature—recent work (https://arxiv.org/pdf/2503.04722) even shows that LLMs exhibit Bayesian behavior.\n\n3. In Section 2, we consider the general Bayesian setting of valuation, which does not rely on a known data-generating distribution. Then, in Section 3, we examine a special linear case and Gaussian noise. The reasons are as follows: Gaussian noise is the most common type of noise in nature. Regarding linearity, some studies show that data valuation is transferable—the value ranking under linear models often holds under nonlinear ones as well (see right side, lines 253–260). Moreover, when the true model is unknown, linear models are the most robust (lines 65–70). Finally, linearity allows for tractable theoretical analysis, which helps us better understand the underlying economic intuition. Extending our framework and tailoring it to specific scenarios is out of the scope of this paper but a promising direction for future work.\n\n4. For potential machine learning applications, especially in industries such as clinical healthcare, our instrumental value can effectively evaluate the value of sequential data, whereas the Shapley value tends to overestimate it. Moreover, as shown in Section 2.3, despite certain assumptions possibly being violated in large-scale datasets, instrumental value reduces the computational complexity from exponential to polynomial. Appendix B.3 further demonstrates through numerical experiments that instrumental value exhibits a certain degree of correlation with the widely used Shapley value. Using instrumental value as an empirical approximation for data valuation—and potentially for pricing—in large-scale datasets could be a promising direction for future applications.\n\n5. ICML features many fundamental machine learning papers, such as those in statistical learning, and is not solely focused on deep learning or empirical work. This is also why the theory track—including areas like game theory—exists. These theoretical contributions provide the foundation and guidance for future large-scale applications. The mutual reinforcement between theory and application is what makes machine learning a broad and inclusive field. Therefore, we believe that the problem we study, categorized in Theory-> Game Theory, is not only of interest to the ML community but also meaningful and impactful.\n\n \n\nWe hope our rebuttal has clarified the reviewer’s confusion and respectfully hope that the reviewer would consider re-evaluating the merit of our work accordingly.", "claims_and_evidence": "The paper's theory builds on some models that are not very realistic for machine learning applications. But for the studied models, the claims and results seem convincing.", "methods_and_evaluation_criteria": "- The data generation process described in the paper is a bit too far from what's happening in practice in machine learning. The data sellers typically do not have control over the data they collect, so the perfect customization in the way the authors present seems unrealistic. \n\n- The analysis and its use in estimating data value rely on data generation distribution. However, such a distribution would be very complex and unknown in practice, especially in machine learning.\n\n- The Bayesian regression model oversimplifies the problem without any justification or intuition into why it might be a good model for practical data-utility relation. So, I am not sure how much of the analysis in the paper is actually relevant in practice. Or how straightforward is it to extend to other possible relations?\n\n- The weaknesses above limit the paper's practical relevance to data valuation in machine learning. Perhaps, the authors didn't necessarily think of data used for machine learning. But then, I am not sure if ICML is the right venue for this study.", "theoretical_claims": "I checked the theorems and they seem correct to me.", "experimental_designs_or_analyses": "There are no experimental results in the paper.", "relation_to_broader_scientific_literature": "Due to impractical assumptions and models used in the paper, I don't think the results would be very useful for the machine learning community. They could be more useful in other areas where the assumptions around the data and its value are more realistic.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "I think it's great to think about the instrumental value of datasets, especially for applications like training foundation models on a mix of many datasets. However, the current analysis makes strong assumptions about the data generation process and how its value would behave, which are not very realistic for machine learning datasets.", "other_comments_or_suggestions": "-", "questions_for_authors": "Do the authors have any ideas on how to extend their results to better suit for machine learning applications?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces a framework to quantify the instrumental value of data within a Bayesian linear model. The main idea is that the value of data should be measured in terms of its sequentially updated utility instead of using a static notion such as Shapley value. Based on the instrumental value, the authors propose mechanisms for data pricing in two scenarios: Perfect customization and Limited customization. The main results are: (1) The instrumental value naturally captures the information gain in a Bayesian setting, and (2) When designing a data-pricing mechanism under these assumptions, one can achieve high revenues with appropriate incentive-compatible rules.

This paper is clearly motivated and delivered, with theoretical claims rigorously proved in the appendix. The concepts of mechanism design integration such as IC and IR are crucial, and is also novel in the ML community. Since there contains the category of “game theory”, I think ICML should be a suitable value for this paper as they are solving ML problems (i.e. data evaluation from the perspective of sequential information gain).

However, there are still a few concerns among the reviewing boards. Firstly, whether the instrumental value is significantly different from Shapley value is unclear and lack of evidence. As pointed out by reviewers, the authors should provide more empirical results on practically how we should choose which metric to use. Secondly, the technical contributions main originate from Bayesian active learning, which has been well established. Although the authors claim that the goal of active learning differs a lot with data evaluation, the concepts and methodologies are similar in some sense.

We kindly suggest the authors expand numerical demonstrations on the necessity of using instrumental value instead of Shapley and/or other notions. Besides, the difference between Bayesian active learning (and also the necessity of using Bayesian approach) should be stated more clearly in the next version.
Decision: Accept (poster)