A measure based pricing framework for data products

Abstract

It has been widely recognized that data can be viewed as a kind of assets. But accounting for data assets and pricing data transactions are still difficult due to the lack of reasonable measurements of datasets or data products. Literatures of data pricing mainly focus on traditional pricing models including models basing on contents of data, demand of market, data quality, etc.. However, due to the particularity of data, the above models may not coincide with the measure theory and thus suffer from some problems. For example, they do not consider how to price datasets sharing common contents; whether we should pay for a repeat purchase; and how to define peak-valley tariff formally for usage-based pricing. To tackle the above problems, in this paper, we formally define measure spaces for datasets and data products. Specifically, we introduce the measures on discrete, continuous and product data spaces respectivaly. Further we introduce the integral and propose a measure based pricing framework for data products. Our work is parallel to existing pricing models. We fouce on how to measure data, and pricing data is a natural extension by integrating the unit price function under the measure. In contrast, existing models focus on determining total prices directly by considering lots of factors like contents of data, demand of markets, etc. By doing analyses on several real-world applications and cases, we prove the effectiveness and generality of our proposal.

Keywords

Data pricing data measure measure theory data assets

1. Introduction

In the era of big data, data are widely viewed as a vital element, production material and fundamental resource, and thus has drawn great attention from researchers. The importance of viewing data as an asset has been widely recognized. However, it is still difficult to determine the prices of data assets and data transactions.

We note that pricing songs, electronic books, etc., is relatively easy and may have been accepted by majority. The reason why it is easy lies in the fact that they have counterparts in the real world. For instance, a physical CD may take us $10, while iTunes charges $0.99 per song online. We price songs in a similar way that we price CDs in the real world, i.e., piece by piece, despite their unit prices may depend on different aspects. But data that have physical equivalents are mainly from digitalization processes and they only account for a small portion of data in the world. For “native-born” data in cyberspace, it is still controversial how to price them since we could not find their physical equivalents, and thus it is unclear how to “enumerate” them like songs or books.

The absence of reasonable measurements of data is the main obstacle that impedes pricing. Since a dataset (or a data asset) means differently to different buyers, it would result in price discrimination and opaque markets [24]. Sellers and buyers may hardly agree on prices if we do not have a universal and widely accepted way to determine the size of data. For example, a dataset of detailed climate records containing sensor locations, temperature, humidity, rainfall etc., seems less valuable for a researcher who merely studies relations between temperature and rainfall than those who study weather forecast. The former may have a misunderstanding that this dataset is “small” since he only needs temperature and rainfall data. But the essence of dataset, i.e., its content, is a constant. What really differ are their psychological prices reflected by their needs. It is necessary to separate measuring from pricing so that we can eliminate these misunderstandings and conflicts by setting user-independent measurements of data. Considering that the cyberspace is overwhelmed by native-born data, e.g., system logs and sensor records, and data trading and distribution inevitably involve these data, it is crucial to find a way to measure them. In this paper, we study data measuring with the help of measure theory and further study data pricing by introducing integrals. We explictly measure datasets and data assets with mathematical tools, so their sizes can be undisputed determined, and then prices of datasets and data assets can be computed given unit prices. By separating measuring from pricing, more widely accepted prices can be made, and more transparent markets can be establied. Meanwhile, we still have flexibility to manipulate prices by determining unit prices.

Though there have been many data pricing models, which we will discuss in Section 2, we find that most of them mainly treat a dataset as a whole without coinciding with the requirements of measures. For example, how much should we pay for two datasets that share a common part? This is an usual situation when we would like to buy two digital collections, i.e., music albums, with common items, i.e., songs. Moreover, some data suppliers like tradestation.com promise that they will keep datasets updated. If some users have different versions of the same dataset, how should the supplier determine personalized prices of an update? Formally speaking, assume that A and B are two datasets with $C = A \cap B \neq \emptyset$ , and the prices are $v_{1}$ and $v_{2}$ respectively. What would be the cost if we buy A and B simultaneously? Could it be $v_{1} + v_{2}$ ? We may not agree with the price, since we pay twice for the intersection of A and B. If we want to exclude the influence of C, we need to set the price of $A ∖ B$ or $A ∖ C$ again. Similarly, what is the price of two copies of A, considering that the marginal cost of datasets is almost zero, and $A \cup A = A$ ? What if A represents data services like Pay-Per-Click advertisements rather than datasets?

Fig. 1.

AWS Spot Instance. Source: https://aws.amazon.com/blogs/compute/new-amazon-ec2-spot-pricing/.

On the other hand, for usage-based data products, the unit price may vary like peak-valley tariff, and it is complicated to model. For instance, in AWS Spot Instance, the price is determined by the bidding prices from different users, as shown in Fig. 1. Moreover, an user’s instance would be stopped at some time if his bidding price are lower than the market price at that time. He needs to raise his bidding price to re-obtain the right of use of the instance. From this example we can see that an effective and flexible framework is needed for pricing.

These problems are due to the special properties of data products, e.g., near zero marginal costs. Most existing pricing models do not meet the requirements of measures, like countable additivity, and thus have flaws. So, prior to making data pricing models, we need to research measures for datasets and data products. Not until measures for data are made, could we study pricing and accounting problems, which then facilitates the data distribution and data trading.

To tackle the above problems, we formally define the domains, measurable spaces and measures for data products. We consider data products that are accompanied with discrete and continuous variables, or discrete/continuous data products for short. We propose measure spaces that are suitable for datasets or data products, on which we further introduce integrals to provide a flexible way to price data.

We need to highlight that our work is parallel to existing pricing models. Most of the existing models focus on how to determine prices by considering lots of factors like contents of data, demand of market and data quality, but neglect the basic requirements of measure theory. In this work, we study a more fundamental problem – how to measure a dataset or a data asset. Once we figured out how to measure data and introduced measure spaces, we can naturally price them with integrals. Moreover, our framework can incorporate previous pricing models seamlessly by using classic pricing models to determine the unit prices, which will act as a part of integrands. Thus the framework we proposed decouples data measuring and unit pricing, and is flexible enough to adapt to various domains.

The contribution of this paper is two-fold:

We propose a pricing framework basing on the measure theory. We demonstrate the measure spaces for discrete and continuous data products. Some common measures are also illustrated.

We study lots of examples and cases from real-world applications, which can be formulated under our proposed framework. This proves the effectiveness and generality of our proposal.

The organization of this paper is as follows. In Section 2, we review and analyze the related work. In Section 3, we introduce the basic concepts and notations of the measure theory, and then define the measure space for discrete data products, i.e., those can be organized as discrete and finite sets. We also analyze several real-world applications and show how to formulate them under our proposal. In Section 4, we define the measure space for continuous data products, i.e., those are accompanied with a continuous variable like usage or duration and do show some examples. In Section 5, to cover more complicated applications, we study the product measures basing on the previous two sections. Examples prove the flexibility and effectiveness of our pricing framework. In Section 6, we review several data trading platforms and show how our proposal can facilitate them. At last, we conclude this work in Section 7.

2. Related work

Currently, data markets are still in their infancy. The economic principles guiding the pricing of data, data products and the data services have not been largely explored [5]. Although a standardized pricing model would facilitate transparent transactions and improve efficiency in the data market [12], existing data-pricing mechanisms lack a data measuring model [27].

Literatures of data pricing mainly build upon traditional pricing models. For example, Moody and Walsh [17] introduced a number of laws to define information as an asset and modified the historical cost method for valuing information. We briefly classify literatures into the following aspects.

Contents of data: Jain and Kannan [13] discussed the most commonly used pricing schemes for network products on the online servers, including connect-time-based pricing, search-based pricing, and subscription-fee pricing, provided a unified framework and presented an approach to analyze the optimal conditions. Sundararajan [25] described two schemes of usage-based pricing and unlimited-usage fixed-fee pricing, analyzed the optimal pricing for network goods, provided a approach of determining optimal the combination of the two schemes, and discussed the implications of the results for bundling and vertical differentiation of network goods. Balasubramanian, Bhattacharya and Krishnan [4] analyzed two pricing mechanisms for network products, namely the unrestricted usage rights selling and pay-per-use mechanisms. They discussed the influence of the two factors on seller’s profits in various market environments. Niu et al. [20] proposed a contextual dynamic pricing mechanism with the reserve price constraint, which used the properties of ellipsoid for efficient online optimization, and can support both linear and non-linear market value models with uncertainty. Chawla et al. [6] studied the problem of revenue maximization in the context of query-based pricing, proposed three types of pricing models, namely uniform bundle pricing, additive or item pricing, and XOS or fractionally subadditive pricings. Mao et al. [16] proposed an IoT data market model from an information design perspective based on unique properties of IoT data, and in turn presented pricing mechanisms that were claimed to be able to achieve revenue maximization.

Demand of market: Bakos and Brynjolfsson [3] studied the bundling model applicable to information goods, analyzed the strategy of bundling a large number of information goods, and found that bundling different information goods may achieve higher profits and greater efficiency than selling these goods separately. Altinkemer and Jaisingh [2] studied the pricing strategy of bundling information goods, and discussed the bundling sales strategy model of enterprises producing only information goods and those producing both physical goods and information goods. Balazinska, Howe and Suciu [5] observed that simple pricing models are more attractive to buyers. They discussed existing pricing models in cloud-based data markets and presented fine-grained data pricing model in which the granularity of the structure is associated with the price. Rao and Keong [22] proposed a method of valuing information based on Shannon’s information value, and used the Sharpe ratio to compare the values of the information with and without the risk. Oh et al. [21] proposed a model based on both willingness-to-sell (WTS) of data providers and willingness-to-buy (WTB) of consumers, while also taking personal data quality as well as different types of personal data into consideration. In addition, they proved that the model has an optimal point can be reached to maximize profit through the application of the convex optimization technique, and in turn they proposed a corresponding multivariate gradient ascent (MGA) algorithm.

Data quality: Heckman et al. [12] conducted a survey to examine the importance of various attributes of data. They then trained a machine learning regression model to fit the prices of datasets using these attributes. Yu and Zhang [27] proposed a bi-level mathematical programming model that takes into account both the versioning strategies and data quality.

Data Privacy: Li et al. [14] proposed a theoretical framework for assigning prices to noisy query answers based on key principles from differential privacy and query pricing in data markets. Gkatzelis, Aperjis and Huberman [10] noticed that due to diverse privacy attitudes of individuals, one needs to set the price high enough to lower the sampling bias, where almost all the individuals choose to participate in the market. So they discussed how to bundle buyer demand to minimize the expected payment to a risk averse seller. Wagner and Eckhoff [26] considered the diversity and complexity of privacy metrics in the literature. They proposed a categorization based on aspects of privacy, expected inputs, and types of data, so as to make a landscape of various privacy metrics. Naghizadeh and Sinha [19] addressed data monetization problem by using a contract-theoretic approach and proposing an adversarial contract design framework, in which different prices are offered for bundles of queries with varying privacy levels.

Marketplace Design: Deep et al. [8] demonstrated a framework called QIRANA which functions as a layer between end users (such as buyers and sellers) and the database so that arbitrage-free query-based pricing can be implemented at scale. The framework enables sellers to choose from a list of pricing functions to their requirements as well as to specify price points by valuing different parts of the database separately. Agarwal et al. [1] designed a real-time matching mechanism for a data marketplace in which training data for Machine Learning tasks are bought and sold. Through introducing the notion of “fairness” and the application of Myerson’s payment function and the Multiplicative Weights algorithm, the proposed mechanism is able to price training data as well as match buyers and sellers, while at the same time the marketplace may centrally set prices for all features for each buyer. Chen et al. [7] built a model-based pricing (MBP) framework which prices Machine Learning model instances instead of pricing the data. The framework was reinforced by the application of noise-injection mechanism to satisfy required formal properties such as preventing arbitrage opportunities. Based on the framework, they proposed algorithms which may be applied to different market scenarios helping data sellers to price Machine Learning model instances.

Besides, some scholars [18,23,27] summarized existing online and offline pricing models of data products in the data market. (i) Free models are those that data or data services can be used for free, such as the data of some public storage can be obtained for free; (ii) In Freemium models, consumers can obtain or use limited data products for free, and pay for value-added services. (iii) Pay-per-use models are those that fees are charged based on users’ usage counts, which has been applied to some API calls. (iv) In packaging models, consumers pay a fixed fee for a certain amount of data or services. (v) Flat-fee models involve data consumers paying a pre-determined fee for unlimited use of data or services in a certain period of time. (vi) In two-part-tariff models, consumers pay a fixed price for the basic services, while extra payment is required for outside pre-defined quota. Fruhwirth et al. [9] reviewed various data marketplaces and their business models, summarized their pricing models and price discovery mechanisms. They discovered that pricing models used by existing data marketplaces include usage based pricing, package pricing, flat fee tariff, and freemium, while price discovery mechanisms include fixed prices, seller-set prices as well as prices decided through auction or negotiation.

Liang et al. [15] classified different data market structures, data pricing strategies and data pricing models. The data pricing model consists two general branches that are economic-based pricing model and game theory-based pricing model. The cost model, consumer perceived model, supply and demand model, differential pricing model and dynamic data pricing (smart data pricing) model are the group of economic-based pricing models, while game theory-based pricing models include non-cooperative game, stackelberg game and bargaining game, etc..

Relationship between our proposal and existing studies: As we discussed in Section 1, in contrast to traditional products, data must be properly measured before conducting research on data pricing. Existing researches neglect the measure of data and adopt traditional pricing models, therefore may encounter some problems, such as countable additivity and examples mentioned in the introduction. In this paper, we introduce the measure theory and discuss a more fundamental problem – the measure of data. Our work is parallel to aforementioned works and can seamlessly incorporate them: (1) granularity: most previous studies treat a dataset (or a data asset) as a whole, while our proposal views a dataset as a collection of finite (Section 3) or inifinite (Section 4) data points; (2) minimum pricing unit: most existing studies directly determine the price of a dataset since it is treated as a whole, while our proposal relies on the unit price, i.e., the price of a single data point. (3) pricing basis: existing studies focus on how to determine prices based on a lot of factors, as shown in this section, while our proposal focuses on how to measure the sizes of datasets, and then compute total prices by integrating the unit prices. In our proposal we highly abstract the unit prices and treat them as functions. So in practice we can resort to classic methods introduced above to decide the unit prices, and this makes our proposal a flexible framework that can adapt to various applications and domains. From another perspective, by abstracting unit pricing funtions, our framework does not focus too much on contents of data. This is analogous to the way we price traditional publications, e.g., books, whose prices are mainly based on the number of pages, rather than whether the contents are useful for some readers. We hope this work could lay a theoretical foundation for further data pricing researches.

3. Measures on discrete data spaces

To facilitate data pricing and data distribution, We need to find a concise way to describe the sizes of datasets and data assets. Inspired by counting items in the real world, we find that in cyberspace some data can also be counted piece by piece.

In many applications, data consist of several discrete data points. For example, an image dataset like ImageNet1

¹
http://www.image-net.org

contains tens of thousands of images, and a medical dataset contains records of many patients. Similarly, many data products are also composed of discrete “points”. For example, digital music stores contain musics and songs. These datasets and data products can be viewed as finite discrete sets. In this section, we study the measurable spaces of discrete data and measures on them, and propose our pricing framework basing on the integrals. Then we validate the framework by studying some real-world examples.

3.1. Background

Many data products can be viewed as finite sets of discrete data points. For example:

Digital music: In 2003, all music available on iTunes Store2

²
https://www.apple.com/lae/apple-music/features/

are priced at $0.99 regardless of their author, length or popularity. In 2007, iTunes introduced a differentiated pricing system. While basic pricing for a song remains to be $0.99, songs with high popularity may cost up to $1.29. Basic pricing for an album is set to $9.99; if the combined price of all songs in one album is less than $9.99, the price of the album would be the combined price. In addition, music publishers may set higher prices for their album.

PPC (Pay Per Click): Keyword search engines such as Google Adsense3

https://www.google.com/adsense

and Yahoo! Publisher Network4

⁴

http://contextualads.yahoo.net

provide PPC service to advertising agencies. After advertising agencies acquired the right to post advertisements through ways such as bidding, they only need to pay when their advertisement receives a click from search engine users.

Subscription services: Many service providers now offer subscription services. Spotify,5

⁵

https://www.spotify.com

for example, provides music streaming service for a monthly subscription fee of $9.99. Apple Music adopted a differentiated subscription scheme – $9.99/month and $99.9/year for personal subscribers, $4.99/month for students, $14.99/month for family subscribers. Netflix,6

⁶

https://www.netflix.com

who provides online video streaming service, operates a three-level subscription system – $8.99/month without HD streaming, $12.99/month with HD feed, and $15.99/month with UHD feed.

3.2. Model

Let us review some definitions and notations of the measure theory [11]. Given a set X, we denote $F$ as the σ-algebra of X, which contains part of subsets of X. $(X, F)$ is called a measurable space. A non-negative set function $μ : F \to [0, \infty]$ satisfying the following properties is a measure:

$μ (\emptyset) = 0$ ;

(Countable Additivity) for any disjoint $A_{i}$ we have $μ (⋃_{i = 1}^{\infty} A_{i}) = \sum_{i = 1}^{\infty} μ (A_{i})$ .

The triple $(X, F, μ)$ is a measure space. The integral of f with respect to μ is $\begin{matrix} F (A) = \int_{A} f d μ . \end{matrix}$

Now, we introduce the domain we use in this section. We denote $X = {x_{1}, \dots, x_{n}}$ as the set of data points, where $x_{i}$ represents a data point, e.g., an image, a music, etc.. A dataset or data product can be treated as a subset of X. Formally, the dataset D satisfy $D \subset X$ . The power set of X, i.e., the set of all subsets of X, is used as its σ-algebra. Given a measure μ, we have the measure space $(X, F, μ)$ . A popular measure is the counting measure $μ (D) = # (D)$ . The measure of D is the number of elements in D.

Given a function $f : X \to R$ , the integral on $D \in F$ is $\begin{matrix} F (D) = \int_{D} f d μ . \end{matrix}$ The function f, i.e., the integrand, corresponds to the unit price function. It can be determined by various pricing models. We will not discuss how to design a price function according to multiple factors in this paper since our work focus on how to measure data, and pricing is a natural application of measure.

3.3. Examples

In this subsection, we prove the effectiveness of our pricing framework by analyzing several real-world cases, and answer the first two questions in Introduction.

Example 3.1.
Given a set of records $X = {x_{1}, \dots, x_{n}}$ . The subsets of X form several datasets. For instance, we have $A = {x_{1}, x_{2}, x_{3}}$ and $B = {x_{3}, x_{4}, x_{5}}$ . We use the counting measure in this example. We can observe that $μ (A \cup B) = μ (A) + μ (B) - μ (A \cap B) = 3 + 3 - 1 = 5$ . This means that, when buying datasets A and B simultaneously, we will not be charged repeatedly by using our proposal.
Example 3.2.
Let X denote the set of files, where $a_{i}$ is the number of bytes of the file $x_{i}$ . The constant function $f = c$ represents the cost per byte. Then the integral $\begin{matrix} \int_{D} f d μ = c \cdot \sum_{x_{i} \in D} a_{i} \end{matrix}$ gives use the total cost of transferring the entire dataset $D \in F$ , where μ is the generalized counting measure.
Example 3.3.
Let X denote the set of all ads in a PPC advertisement system. Clearly, an ad is charged by clicks. Unlike Example 3.1, clicking an ad twice is different from clicking once. So, we solve this problem by using the integral. We use $f : X \to N$ to describe the number of clicks with $f (x_{i}) = n (x_{i}) \times c (x_{i})$ , where $n (x_{i})$ is the number of clicks of the ad $x_{i}$ , and $g (x_{i})$ is the cost per click of $x_{i}$ . Then, the integral $\begin{matrix} \int_{A} f d μ = \sum_{x_{i} \in A} f (x_{i}) = \sum_{x_{i} \in A} n (x_{i}) \cdot c (x_{i}) \end{matrix}$ models the pricing strategy of PPC systems, where μ is the counting measure.
Example 3.4.
In the scenario of subscription services, we denote X as the set of all tiers of subscription and $f : X \to N$ as the number of purchases times the unit prices as Example 3.3. Then the integral $\int_{D} f d μ$ is the total cost of subscriptions D.
Example 3.5.
We denote X as the set of all musics in iTunes and use the counting measure as μ. If we use the constant function $f = 0.99$ , the integral $\int_{D} f d μ = 0.99 \times # (D)$ represents the pricing strategy of the early iTunes. If we partition X into several disjoint subsets, i.e., $X = X_{1} \cup \dots \cup X_{m}$ , and let f be $\begin{matrix} f (x) = \{\begin{matrix} 0.99 & if x \in X_{1}, \\ 1.29 & if x \in X_{2}, \\ \dots & \dots \end{matrix} \end{matrix}$ then the integral $\int_{D} f d μ$ models the pricing strategy since 2007. Unit prices, e.g., $0.99, $1.29, may be determined by examining quality of songs.

4. Measures on continuous data spaces

Besides finite discrete sets, many datasets or data products are charged according to a continuous variable like usage or duration, which can be represented by continuous and infinite sets. Considering counting data points piece by piece can hardly be applied here, we need to resort to Lebesgue measure to define the sizes of data. So in this section, we study measures on continuous data spaces with the help of the Lebesgue measure and integral on data.

4.1. Background

Many data products are priced with respect to the usage or duration.

Massively multiplayer online (MMO) games: MMO games such as World of Warcraft7

⁷
https://worldofwarcraft.com

and Fantasy Westward Journey8

⁸

https://game.163.com/en/xy2.html

charge players based on their in-game time. Fantasy Westward Journey charges player 0.6 RMB every in-game hour, and World of Warcraft is about $0.50 a day.

Servers (Cloud computing): AWS (Amazon web service)9

⁹

https://aws.amazon.com

offers a pay-as-you-go approach for pricing for a variety of cloud services. With AWS users pay only for the individual services they need, depending on how long users use them. Paperspace,10

¹⁰

https://www.paperspace.com/

FloydHub,11

¹¹

https://www.floydhub.com/

etc. also adopt the pay-as-you-go mode for individual, and charge users for how long they have used.

Online streaming platform: Datacast streaming service12

¹²

https://www.dacast.com/

charges users based on the amount of data usage in addition to a differentiated pricing scheme based on provided streaming bandwidth.

Cloud service: Alibaba Cloud IOT platform13

¹³

https://www.alibabacloud.com/

charges connection between terminals and the platform based on the time duration. A limited free duration of usage is provided every month.

Amazon EC2 Spot Instances:14

¹⁴

https://aws.amazon.com/ec2/spot/

Users are able to bid for unused EC2. The instance would start when the current price of Spot instances is lower than the highest acceptable price set by the user. Spot price is determined by long-term supply-demand of unused EC2 instances. Users pay the Spot price that’s in effect at the beginning of each instance-hour for their running instance, billed to the nearest second.

4.2. Model

In this subsection, we describe the pricing framework for data products that are associated with a continuous variable. Considering that the usage or duration is generally non-negative, we use the half line $R^{+} = [0, \infty)$ as the domain. The Borel set $B$ is the σ-algebra of $R^{+}$ . $(R^{+}, B)$ is the measurable space of continuous data products. $(R^{+}, B, ν)$ is the measure space. Here, we adopt the Lebesgue measure, where $ν ((a, b)) = b - a$ for the interval $(a, b)$ . Moreover, in real-world applications, the set of usage/duration M is often the union of several disjoint intervals, i.e., $M = ⋃_{i = 1}^{m} (a_{i}, b_{i})$ . In this case, we have $ν (M) = \sum_{i = 1}^{m} b_{i} - a_{i}$ . Given a function $g : R^{+} \to R$ , the integral with respect to ν, i.e., the Lebesgue integral, is defined as $\begin{matrix} G (M) = \int_{M} g d ν . \end{matrix}$ Similarly, the function g represents the unit price function and can be determined by various pricing models.

4.3. Examples

Example 4.1.
In applications that are charged by duration, we could use a constant function $g = c$ representing the unit price. The set M describe the usages. Then the integral $\int_{M} g d ν = c \cdot ν (M)$ models the pricing strategy basing on duration.
Example 4.2.
In Alibaba Cloud IOT platform, we use a piece-wise function $\begin{matrix} g (t) = \{\begin{matrix} 0 & if t < t_{1}, \\ c & if t ⩽ t_{1} . \end{matrix} \end{matrix}$ The integral $\int_{M} g d ν$ models the pricing strategy basing on duration with a free trial period.
Example 4.3.
In the market of AWS EC2 Spot instances, we denote the market price at time t as $g (t)$ . As shown in Fig. 2, the total cost can be represented by the Lebesgue integral $\int_{M} g d ν$ , where M indicates the user’s usage.
5. Measures on the product of data spaces

In previous sections, we have discussed measures for discrete and continuous data spaces. However, in many applications, data products may have complicated pricing strategies that involve multiple variables. In this section, we utilize product measurable spaces and product measures to cover these complicated cases.

Fig. 2.

The Lebesgue integral.

5.1. Background

There are many applications that involve multiple discrete or continuous variables for pricing.

Membership subscription: Amazon Prime membership subscription service15

¹⁵
https://www.amazon.com/amazonprime

includes video streaming service. A Prime member may purchase different videos (such as digital films or TV series) on Amazon platform, with various prices. Using Amazon UK as an example, early or lower-resolution series are priced at 1.39 pound per episode or 9.99 pound per season; normal series cost 2.49 pound per episode. In case of Prime exclusive series, Prime members may watch free of charge.

Patreon: Patreon16

¹⁶

https://www.patreon.com/

is a crowdfunding membership platform that provides service for content creators/artists. Membership of certain creator/artist are presented with varied subscription options. This allows the members or patrons to receive different bonus service provided by the creator/artist based on their payment tiers.

Amazon EC2 Spot instances:17

¹⁷

https://aws.amazon.com/ec2/spot/

Users of Amazon EC2 may not only bid for unused instances and pay according to the duration of usage, but also could choose between different configurations of instances with varied performance at differentiated price level.

5.2. Model

In this part, we talk about product measures. Without loss of generality, we only consider the product of two spaces.

First, we illustrate the product of two spaces with discrete sets. Given two measure spaces $(X_{1}, F_{1}, μ_{1})$ and $(X_{2}, F_{2}, μ_{2})$ , we have the product measurable space $(X_{1} \times X_{2}, F_{1} \times F_{2})$ , where $X_{1} \times X_{2}$ is the Cartesian product of $X_{1}$ and $X_{2}$ , and $F_{1} \times F_{2}$ is the σ-algebra of $X_{1} \times X_{2}$ . Since $X_{i}$ is finite, $F_{1} \times F_{2}$ is also the Cartesian product of $F_{1}$ and $F_{2}$ . The product measure $μ_{1} \times μ_{2}$ is defined as a measure satisfying $\begin{array}{l} (μ_{1} \times μ_{2}) (D_{1} \times D_{2}) \\ = μ_{1} (D_{1}) μ_{2} (D_{2}), \forall D_{1} \in F_{1}, D_{2} \in F_{2} . \end{array}$ If $(X_{1}, F_{1}, μ_{1})$ and $(X_{2}, F_{2}, μ_{2})$ are both σ-finite, the product measure $μ_{1} \times μ_{2}$ can be uniquely defined. Luckily, the counting measure and the generalized counting measure we discussed in Section 3 are σ-finite. Now we have the measure space $(X_{1} \times X_{2}, F_{1} \times F_{2}, μ_{1} \times μ_{2})$ . The integral on $D \in F_{1} \times F_{2}$ is $\begin{matrix} \int_{D} h d μ_{1} \times μ_{2}, \end{matrix}$ where $h : X_{1} \times X_{2} \to R$ .

For the product of two spaces with continuous sets, i.e., $R^{+} \times R^{+}$ , we directly adopt the Lebesgue measure. The measure space is $(R^{+} \times R^{+}, B^{'}, ν^{'})$ , where $B^{'}$ is the σ-algebra, and $ν^{'}$ is the Lebesgue measure. Note that $ν^{'} \neq ν \times ν$ , since the product measure of Lebesgue measures is not complete. That is why we directly adopt the Lebesgue measure on this two-dimensional situation. Again, the integral is the two-dimensional Lebesgue integral $\begin{matrix} \int_{M} h d ν^{'}, \end{matrix}$ where $M \in B^{'}$ and $h : R^{+} \times R^{+} \to R$ .

Similarly, given $(X, F, μ)$ , a measure space for discrete sets, and $(R^{+}, B, ν)$ , a measure space for continuous sets, we can have $(X \times R^{+}, F \times B, μ \times ν)$ . The integral is $\begin{matrix} \int_{T} h d μ \times ν, \end{matrix}$ where $T \in F \times B$ and $h : X \times R^{+} \to R$ . However, $μ \times ν$ is not a complete measure. Consider that $(μ \times ν) (\emptyset \times M) = μ (\emptyset) ν (M) = 0$ $\forall M \in B$ . If we let M be a non-measurable set, e.g., Vitali set, then the measure of $\emptyset \times M$ is not defined. Thus there exists a subset of the null set that is not measurable, which proves that $μ \times ν$ is not complete. We could do completion to tackle this problem. On the other hand, in real-world applications, M is often a union of intervals, and whether the measure is complete is not important.

5.3. Examples

Example 5.1.
In Patreon, patrons could have different bonus services according to their payment tiers. Here, we denote $X_{1}$ as the set of membership tiers, and $X_{2}$ as the set of services provided by content creators. We simply use the counting measure for $μ_{1}$ , $μ_{2}$ . Since the domain is discrete, the function $h : X_{1} \times X_{2} \to R$ can be organized as Table 1.

Table 1
The unit price function h of Example 5.1

Tiers Services

$x_{2} = 1$ $x_{2} = 2$ $x_{2} = 3$ $\dots$

$x_{1} = 1$ ∞ 20 10 $\dots$

$x_{1} = 2$ 100 18 10 $\dots$

$x_{1} = 3$ 75 15 10 $\dots$

Now, the integral $\begin{matrix} \int_{D} h d μ_{1} \times μ_{2} = \sum_{(x_{1}, x_{2}) \in D} h (x_{1}, x_{2}) \end{matrix}$ represents the cost of services under different membership tiers. Note that the value $h (1, 1) = \infty$ means that the Service 1 is not provided to the patrons with Tier 1, which models the membership subscription services.
Example 5.2.
We extend the Example 4.3. We denote X as the set of instances with different configurations. Since X is discrete, we could formulate the $h : X \times R^{+} \to R$ as a set of functions, i.e., $\begin{matrix} h (x, t) = h_{x} (t), \forall x \in X . \end{matrix}$ As shown in Fig. 1, $h_{x} (t)$ illustrates the market prices of the instance x. We use the counting measure as μ and the Lebesgue measure as ν, and we have the integral $\begin{matrix} \int_{T} h d μ \times ν \end{matrix}$ represent the total cost of using multiple Spot instances simultaneously.
6. Case study

Tiers	Services
$x_{1} = 1$	∞	20	10	$\dots$
$x_{1} = 2$	100	18	10	$\dots$
$x_{1} = 3$	75	15	10	$\dots$

In this section, we briefly review some data trading platforms and see how our proposal can facilitate their showcases as well as trading processes.

youedata.com and juhe.cn are online datasets/API trading platforms. Each dataset is marked with a total price. Like what we discussed before, mixing measuring and pricing together on the one hand complicates the process of pricing, and results in counterintuitive prices that do not meet the requirements of the measure theory. On the other hand, a consumer may find it hard to figure out how many records are contained and what the unit price is. For example, in the index page, a cheaper dataset with few records attracts more consumers, but it may have a higher unit price than other competing products. This harms the fairness of the platform. By using our framework, the finer granularity of data products can be examined. We set the price as the integral $\int_{D} f d μ$ , so we can separately and precisely examine how many records a data set D contains by utilizing the measure μ, and define the price of unit data points by modeling the unit price function f. The minimum pricing unit in our proposal is data points. We can directly define the unit price of a single data point, and a dataset with a higher unit price may implies its high quality, and consumers then can choose suitable products easier. For the platforms, our framework could enhance the way they showcase different products. A platform could display the unit price along with the total price of a dataset, in which case consumers will gain a clearer insight into the value of datasets. Though in this paper we only discussed the counting measure and the Lebesgue measure, and they do cover a lots of cases, we point out that many other measures are also available in case we face data products with more complicated formats.

Some data suppliers in platforms like tradestation.com, datatang.com and datasl.com may promise that they will keep datasets updated. But how do updates charge? This is a critical question for both consumers and suppliers, but has not been dealt with appropriately in existing platforms. As we mentioned earlier, cares should be taken when we face maybe overlapping data. If some consumers bought different versions of the same dataset, and the data content of each consumer is slightly different from others’. The data supplier will suffer from determining the personalized prices of the update for different consumers. With the help of measure theory adopted in our proposal, we resolve this problem nicely. Since we introduce the measure μ, the size of updated part can be easily computed by $μ (Δ D ∖ D)$ , where D is the user’s current dataset and $Δ D$ is the released update. Charges for updates can be computed by integrals of unit prices, i.e., $\int_{Δ D ∖ D} f d μ$ , with almost no extra efforts since we have defined the measure of data. Consumers will also be satisfied with the personalized prices.

Another popular option for data trading is to negotiate prices between consumers and suppliers, where platforms like factual.com, datasl.com and finndy.com act as intermediaries and only charge management fees. Under our framework, only the unit price f needs to be determined through negotiation, at which time demand of market, data quality and other factors may come into effect. Then the total pricing along with management fees can be computed by the integrals $\int f d μ$ . From this perspective our framework provides a flexible way to satisify various needs of data pricing since preivous studies as we discussed in Section 2 can be seamlessly incorporated into determining f.

Discussion From the examples introduced in previous sections and the aforementioned platforms, we can see that the framework proposed in this paper can be adopted well across various domains. The core idea is to separate the measuring process from pricing data. Measures are also independent of consumers and suppliers, so that everyone can agree with the “volume” of a dataset. Once we make measures for data, we could “enumerate” them just like physical things, and experiences of pricing physical things will help us price data by determining unit prices. Our work is parallel to existing pricing models and we also could take benefits from them. The measuring and pricing framework we proposed in turn makes it possible to solve accounting problems and facilitate data distribution and data trading.

Limitations Due to the high-level abstraction, these are still some gaps to fill before applying our framework. The granularity of data points, or equivalently speaking, the elements of the dataset D, need to be determined. For example, the suitable unit for ImageNet dataset is images, rather than pixels. Moreover, we need to define the measure μ and the unit pricing function f when we encounter with a new scenario, where exploration of data properties, market needs, etc., is still inevitable.

7. Conclusion

In this paper, we review multiple datasets and data products, and formulate the measure spaces for discrete and continuous data. By introducing measure spaces, we could measure the volume of $A \cup B$ reasonably, which solves the problem of measuring joint datasets or data products. For data services like PPC advertisements, the integral could model repeat purchases by using a unit price function. Moreover, the Lebesgue integral is utilized for modeling usage-based pricing products. Finally, we use product measures to formulate more complicated pricing scenarios. The case studies prove the effectiveness and generality of our proposed framework. Our framework decouples data measuring and unit pricing, and thus can adapt to various domains by adopting different ways of unit pricing. We hope our framework could shed new light on data pricing researches and become a new paradigm.

The future work may involves the following aspects:

In addition to the counting measure and the Lebesgue measure used in this paper, we could consider more measures to cover more applications.

We may research pricing models of datasets or data products in specific fields. Based on the existing pricing framework, it is possible to study the pricing models of data assets in certain domains.

Practicing the pricing of data products while taking existing economic principles into consideration is also a potential direction. Accounting is a process of measurement. Now that lots of pricing frameworks concerning data products have been summarized, it is possible to conduct pricing studies on these data products. Standardizing and regulating the pricing of data products are beneficial to the data trading and distribution of them in data market, as well as researching the pricing of other data assets.

References

Agarwal ,

Dahleh and

Sarkar , A marketplace for data: An algorithmic solution, in: ACM Conference on Economics and Computation, 2019, pp. 701–726.

Altinkemer and

Jaisingh , Pricing bundled information goods, in: IEEE International Workshop on Advanced Issues of E-Commerce and Web-Based Information Systems, IEEE, 2002, pp. 89–96.

Bakos and

Brynjolfsson , Bundling information goods: Pricing, profits, and efficiency, Management science 45(12) (1999), 1613–1630. doi:10.1287/mnsc.45.12.1613.

Balasubramanian ,

Bhattacharya and

V.V.

Krishnan , Pricing information goods: A strategic analysis of the selling and pay-per-use mechanisms, Marketing Science 34(2) (2015), 218–234. doi:10.1287/mksc.2014.0894.

Balazinska ,

Howe and

Suciu , Data markets in the cloud: An opportunity for the database community, Proceedings of the VLDB Endowment 4(12) (2011), 1482–1485. doi:10.14778/3402755.3402801.

Chawla ,

Deep ,

Koutris and

Teng , Revenue maximization for query pricing, 2019, arXiv preprint arXiv:1909.00845.

Chen ,

Koutris and

Kumar , Towards model-based pricing for machine learning in a data marketplace, in: International Conference on Management of Data, 2019, pp. 1535–1552.

Deep ,

Koutris and

Bidasaria , Qirana demonstration: Real time scalable query pricing, Proceedings of the VLDB Endowment 10(12) (2017), 1949–1952. doi:10.14778/3137765.3137816.

Fruhwirth ,

Rachinger and

Prlja , Discovering business models of data marketplaces, in: International Conference on System Sciences, 2020.

10.

Gkatzelis ,

Aperjis and

B.A.

Huberman , Pricing private data, Electronic Markets 25(2) (2015), 109–123. doi:10.1007/s12525-015-0188-8.

11.

P.R.

Halmos , Measure Theory, Vol. 18, Springer, 2013.

12.

J.R.

Heckman,

E.L.

Boehmer,

E.H.

Peters,

Davaloo and

N.G.

Kurup, A pricing model for data markets, in: iConference 2015 Proceedings, 2015.

13.

Jain and

P.K.

Kannan , Pricing of information products on online servers: Issues, models, and analysis, Management Science 48(9) (2002), 1123–1142. doi:10.1287/mnsc.48.9.1123.178.

14.

Li ,

Li Yang ,

Miklau and

Suciu , A theory of pricing private data, ACM Transactions on Database Systems 39(4) (2014), 34. doi:10.1145/2691190.2691191.

15.

Liang ,

Yu ,

An ,

Yang ,

Fu and

Zhao , A survey on big data market: Pricing, trading and protection, IEEE Access 6 (2018), 15132–15154. doi:10.1109/ACCESS.2018.2806881.

16.

Mao ,

Zheng and

Wu , Pricing for revenue maximization in iot data markets: An information design perspective, in: IEEE Conference on Computer Communications, IEEE, 2019, pp. 1837–1845.

17.

D.L.

Moody and

Walsh , Measuring the value of information-an asset valuation approach, in: European Conference on Information Systems, 1999, pp. 496–512.

18.

Muschalle ,

Stahl ,

Löser and

Vossen , Pricing approaches for data markets, in: International Workshop on Business Intelligence for the Real-Time Enterprise, Springer, 2012, pp. 129–144.

19.

Naghizadeh and

Sinha , Adversarial contract design for private data commercialization, in: ACM Conference on Economics and Computation, 2019, pp. 681–699.

20.

Niu ,

Zheng ,

Wu ,

Tang and

Chen , Online pricing with reserve price constraint for personal data markets, in: IEEE International Conference on Data Engineering, IEEE, 2020, pp. 1978–1981.

21.

Oh,

Park,

G.M.

Lee,

Heo and

J.K.

Choi, Personal data trading scheme for data brokers in iot data marketplaces, IEEE Access 7 (2019), 40120–40132. doi:10.1109/ACCESS.2019.2904248.

22.

Rao and

N.W.

Keong , A method to price your information asset in the information market, in: IEEE International Congress on Big Data, IEEE, 2016, pp. 307–314.

23.

Schomm ,

Stahl and

Vossen , Marketplaces for data: An initial survey, ACM SIGMOD Record 42(1) (2013), 15–26. doi:10.1145/2481528.2481532.

24.

Shapiro and

H.R.

Varian , Information Rules: A Strategic Guide to the Network Economy, Harvard Business Press, 1998.

25.

Sundararajan , Nonlinear pricing of information goods, Management Science 50(12) (2004), 1660–1673. doi:10.1287/mnsc.1040.0291.

26.

Wagner and

Eckhoff , Technical privacy metrics: A systematic survey, ACM Computing Surveys 51(3) (2018), 57. doi:10.1145/3168389.

27.

Yu and

Zhang , Data pricing strategy based on data quality, Computers & Industrial Engineering 112 (2017), 1–10. doi:10.1016/j.cie.2017.08.008.

Tiers	Services

	$x_{2} = 1$	$x_{2} = 2$	$x_{2} = 3$	$\dots$
$x_{1} = 1$	∞	20	10	$\dots$
$x_{1} = 2$	100	18	10	$\dots$
$x_{1} = 3$	75	15	10	$\dots$

A measure based pricing framework for data products

Abstract

Keywords

1. Introduction

3. Measures on discrete data spaces

1 http://www.image-net.org

2 https://www.apple.com/lae/apple-music/features/

3.3. Examples

4.1. Background

7 https://worldofwarcraft.com

4.3. Examples

15 https://www.amazon.com/amazonprime

5.3. Examples

7. Conclusion

References

¹
http://www.image-net.org

²
https://www.apple.com/lae/apple-music/features/

⁷
https://worldofwarcraft.com

¹⁵
https://www.amazon.com/amazonprime