Strategic Data Sharing between Competitors
Authors: Nikita Tsoy
Nikola Konstantinov
Venue: NeurIPS 2023 poster (2023)
Keywords: Incentives
collaborative learning
federated learning
game theory
competition
oligopolistic markets
strategic behavior
Nash equilibrium

Abstract: Collaborative learning techniques have significantly advanced in recent years, enabling private model training across multiple organizations. Despite this opportunity, firms face a dilemma when considering data sharing with competitors—while collaboration can improve a company’s machine learning model, it may also benefit competitors and hence reduce profits. In this work, we introduce a general framework for analyzing this data-sharing trade-off. The framework consists of three components, representing the firms’ production decisions, the effect of additional data on model quality, and the data-sharing negotiation process, respectively. We then study an instantiation of the framework, based on a conventional market model from economic theory, to identify key factors that affect collaboration incentives. Our findings indicate a profound impact of market conditions on the data-sharing incentives. In particular, we find that reduced competition, in terms of the similarities between the firms’ products, and harder learning tasks foster collaboration.

Strategic Data Sharing between Competitors
Nikita Tsoy
Nikola Konstantinoy
INSAIT, Sofia University Sofia, Bulgaria niki ki www.stossiiiii....m
INSAIT, Sofia University Sofia, Bulgaria
nikola. wowstantinovviissst..e.
Abstract
Collaborative learning techniques have significantly advanced in recent years, enabling private model training across multiple organizations. Despite this opportu- nity, firms face dilemma when considering data sharing with competitors- while collaboration improve company's machine learning model, it may also benefit competitors and hence reduce profits. In this work, we introduce a general frame- work for analyzing this data-sharing trade-off. The framework consists of three components, representing the firms' production decisions, the effect of additional data on model quality, and the data-sharing negotiation process, respectively. We then study an instantiation of the framework, based on conventional market model from economic theory, identify key factors that affect collaboration incentives. Our findings indicate profound impact of market conditions on the data-sharing in- centives. In particular, we find that reduced competition, in terms of the similarities between the firms' products, and harder learning tasks foster collaboration.
Introduction
Machine learning has become integral to numerous business functions in the past decade, such as service operations optimization, the creation of new products, consumer service analytics, and risk analysis (Chui et al.] (022). Despite the transformative power of machine learning, its efficacy hinges significantly on the quality and quantity of the training data, making data a key asset corporations can use to increase their profit.
One way enhance data access and machine learning models is collaboration via data sharing (Rieke iet al.] 22200] Durrant et al.) 2022). Data-sharing schemes can bring benefits in many industries, such as agriculture, finance, and healthcare (Durrant et al. 2002 FedAI Rieke al. 2020). However, least two barriers exist to such collaborations. The first is privacy and the associated regulatory concerns, which can partially addressed by new collaborative learning techniques, such as federated learning (Kairouz et al. 2021). The second is proper incentives for collaboration. If the entities have no such incentives, they may not collaborate all, free-ride (Blum et 2021 Karimireddy et al., 2022). or attack the shared model (Blanchard et al. 2017).
Such conflicting incentives arise naturally between market competitors. On the hand, the competitors are an appealing source of data, since they operate on the same market and have data of a similar type. On the other hand, the collaboration could also strengthen the competitors' machine learning models, potentially leading to downstream losses for the company. This concern is especially important for big firms that can influence prices and strategically by considering how their decisions affect competitors' responses. Strategic actions might increase profits through various channels and produce complex downstream effects. For example, firms might collude capture
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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more revenud or engage in price war to win a market Thus, data sharing between big firms might have a complicated downstream effect on their profits.
Our contributions Despite the significance of this data-sharing trade-off, particularly for large, tech-savvy companies, there is limited research on how market competition affects collaborative learning incentives. Our work aims to fill this gap by proposing a framework to investigate data- sharing incentives. The framework modular and consists of three components: market model, data impact, and collaboration scheme. These components represent the firms' production decisions, the effect of additional data on model quality and the data-sharing negotiation process, respectively.
To investigate the key factors that influence data-sharing decisions, we instantiate our framework using a conventional market model from economic theory and a data impact model grounded in learning theory. We examine three potential collaboration schemes: binary collaboration between two firms, partial collaboration between two firms, and binary collaboration among multiple firms. Our findings consistently indicate a profound impact of market conditions on the collaboration incentives. In the case of two firms, we theoretically demonstrate that collaboration becomes more appealing as market competition, measured by the similarities of the firms' products, decreases or the learning task complexity increases. For multiple firms, we conduct simulation studies that reveal similar trends.
Related work
Data sharing incentives in machine learning Incentives for collaboration in machine learning constitute an important research topic, especially in the context of new learning paradigms, such as federated learning (Kairouz et al. (0021). Some notable lines of work concern incentives under data
heterogeneity (Donahue & Kleinberg 202T Werner et al. (022); compensation for computation,
energy, and other costs related to traiing Yu et al. [2200] Tu et al., 2022 Liu et al.
2022); fair
distribution of the benefits resulting from collaboration (Lyu et al.) 2020a Blum et al.
2021); ; and
free-riding (Richardson et al. 2020 Karimireddy et al. 2022). We refer to Zeng et al. (2021); Yang
al. (2020); Zhan et al. (202) for a detailed overview.
fundamental difference between our work and the research described above is that we explicitly consider the downstream market incentives of the firms and their effects data-sharing decisions. These incentives are orthogonal to those previously studied in the literature. For instance, the data- collection game of Karimireddy et al. (2022) can be included in our framework an additional stage of the game after our coalition formation stage. In addition, our framework is not specific to federated learning, as it can be applied regardless of the learning procedure used on the shared dataset.
To our awareness, Wu & Yu (2022) is the only work considering data sharing between market competitors. However, the authors do not explain the mechanisms behind the effects of ML models' quality on the market. Therefore, their framework can not predict whether a potential coalition will be beneficial for the company but only indicate the benefits of the coalition post factum. In addition, they use the marketing model of Rust & Zahorik (1993) instead of the classic economic models (Tirole 1988) to model competition, potentially constraining the applicability. Finally, their notion of the sustainability of coalitions does not arise from standard concepts in game theory. In contrast, we base our analysis on the standard Nash equilibrium concept (Nash 1951)
Market competition and information sharing Competitive behavior is a well-studied topic in the economic literature (Tirole 1988). We refer to (Shapiro (1989) for extensive review of the theory of competition. From a more applied perspective, Reiss & Wolak (2007) discuss how to test the behavior of firms empirically, while Berry al. (2019) overview the modern empirical studies of industrial organization. A rich economic literature studies the incentives of competing firms to share information (Raith 1996, Bergemann & Bonatti 2019). For example, demand (Vives) 1984) or costs (Fried) 1984) might be unobservable, and firms might share information about them to improve their decisions. In contrast to these studies, in our model, firms share information to improve their products or optimize their costs, not to reduce uncertainty. Thus, the mechanisms and conclusions of these earlier works and our work differ.
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Data-sharing problem
In this section, we provide a general formulation of the data-sharing problem, consisting of three components: market model, data impact, and collaboration scheme. The market model outlines consumers' and firms' consumption and production decisions. Data impact explains how additional data affects machine learning model quality. Lastly, the collaboration scheme details the data-sharing negotiation process among the competitors. Given a specific application, these three components can be derived/modeled by market research or sales teams, operation management and data scientists, and mediators, respectively.
We begin with an illustrative example that will help us to clarify the abstract concepts used in our framework and then introduce the three framework components in their full generality.
3.1 Running example
Consider a city with taxi market dominated by few firms. Each firm collects data (e.g.. demand for taxis and traffic situation) train a machine learning model for optimizing driver scheduling and other internal processes, which is crucial for improving scheduling quality, reducing costs, or enhancing services. Each company can use only its own data or take part collaborative training procedure, like federated learning, with its competitors. Collaboration can substantially improve its machine learning model, but it may also strengthen the models of its competitors. Thus, the company must carefully evaluate the impact of data-sharing on its downstream profits.
3.2 Market model
The market model encompasses consumer actions (demand factors) and firm production actions (supply factors). We consider a market with firms, Fi...., Fmm each producing qi I4 units of good G, and offering them to consumers at price p; E R4, where IR, = [0, oo). In our example, G Gm represent taxi services from different companies, with consumers being city travelers, d are the total number kilometers serviced by company i, and are the prices per kilometer.
In this setting, prices and quantities are 2m unknown variables. Correspondingly, we need 2m constraints to describe the consumers' and firms' market decisions, which we derive from the consumers' and the firms' rationality. In contrast standard market models, our framework allows product utilities and costs to depend on the qualities of the machine learning models of the firms
(e),- .... Um.. This dependence models the impact of the learned models on the services and production pipelines of the firms.
Consumers' behavior Each consumer j optimizes their utility (g'g' (2)"))"2)) v) by purchasing goods given the market prices. We assume that consumers cannot influence prices, as there are many consumers and they do not cooperate.
The consumer's utility depends on three factors. The first one is the consumed quantities q² .gf.....'o .%,) of products GI...., Gm. The second is the machine learning models' qualities v, as these may impact the utility of the corresponding products. The last one is consumed quantities g' = (oI......h ....g)) of goods outside the considered market (c.g., consumed food in our taxi example). Assuming that each consumer can only spend certain budget Bi on all goods, one obtains the following optimization problem
k max u' (g', v) s.t. Σ pioi +
m
Σ pid < B',
g'
where P are the outside products' prices (which we consider fixed). The solution to this problem q. (p, v) determines the aggregate demanded quantity of goods
(p. v) = Σ g. (p.v).
The functions q (p, v) (known as the demand equations) link the prices pa and the demanded quantities qi and constitute the first restrictions in our setting.
3

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Firms' behavior Firms maximize their expected profit,, the difference between revenue and cost,
II = Ep(p. - v.).
(3)
Here Cg U;) is the cost of producing q; units of good for the firm F,, and the expectation taken over the randomness in the models' quality, influenced by the dataset and the training procedure. Note that the model quality often observed only after testing the model in production (at test time). Therefore, we assume that the firms reason in expectation instead. In our running example, C; depends on driver wages, gasoline prices, and scheduling quality.
The firms may act by either deciding on their produced quantities, resulting in what is known as the Cournot competition model (Cournot [838]; on their prices, resulting in the Bertrand competition model (Bertrand] 1883). As the demand equations (2) may interrelate prices and quantities for various products, firms strategically consider their competitors' actions, resulting in a Nash equilibrium. The equilibrium conditions provide another m constraints, enabling us to solve the market model entirely,
3.3 The impact of data on the market
Each company Fü possesses a dataset D; that can be used to train a machine learning model (e.g.. trip data in the taxi example) and may opt participate in a data-sharing agreement with some other firms. Denote the final dataset the company acquired by D The company then uses the dataset to train a machine learning model. We postulate two natural ways that the model quality o; can impact company. The first one is by reducing the company's 's production costs Cg(g,, Ut), for example, by minimizing time in traffic jams. The second by increasing the utility of its products (p v), for example, by minimizing the waiting time for taxi arrival.
3.4 Collaboration scheme
Following classic economic logic, we posit that firms will share data if this decision increases their expected profits 175- Since firms can not evaluate the gains from unknown data, we assume that they know about the dataset characteristics of their competitors (e.g., their number of samples and distributional information). Although expected profit maximization determines individual data- sharing incentives, forming a coalition necessitates mutual agreement. Therefore, the precise game- theoretic formulation of the data-sharing problem depends on the negotiation process details, such as the number of participants and full or partial data sharing among firms.
Consider our taxi example. If only two firms present, it is natural that they will agree sharing their full datasets with each other if and only if they both expect that this will lead to increased profits. Partial data-sharing will complicate the process, leading to an intricate bargaining process between the firms. Finally, if many companies are present, the data-sharing decisions become even more complicated, firm needs to reason not only about the data data-sharing benefits but also about the data-sharing decisions of other firms.
Legal and training costs considerations Other considerations can also enter into our framework through the collaboration scheme. In particular, if firms have legal requirements on data usage, they should impose suitable constraints on possible data-sharing agreements. If the collaborative learning procedure involves training large models, the firms should negotiate how to split training costs. For example, they might split the costs equally among all coalition members or proportionally to the sizes of their datasets.
4 Example market and data impact models
In this section, we instantiate the general framework using a conventional market model from economic theory and natural data impact model justified by learning theory. These models allow us to reason quantitatively about the data-sharing problem in the following sections, leading to the identification of several key factors in the data-sharing trade-off. To focus solely the data-sharing off we make several simplifying assumptions: the firms' data homogeneous, training costs negligible, and legal constraints are not present or are automatically fulfilled.

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4.1 Market model
In order make quantitative statements about the firms' actions and profits within the general model of Section 3.2 one needs to make further parametric assumptions on utilities u² and cost functions C, and specity the competition game. To this end, we use specific utility and cost model standard in the theoretical industrial organization literature (Tirole) 1988; Carlton & Perloff 1990). Despite its simplicity, this model effectively captures the basic factors governing market equilibrium for many problems and is often used to obtain qualitative insights about them.
4.1.1 Demand
We assume that, in the aggregate, the behavior of consumers (T can be described by representative consumer with quasi-lineat quadratic utility (QQUM, 1979)
Σ3 q. + 2 Eizj 999 + - lq q GaGq tgst. g+p B. (4)
max u(g, q)
Σ qi
9.9
Here, L (l,..., 1)², G (1 (1-7)I+ y and g is the quantity from a single outside good.
Additionally, Y E (- mm 1 is measure of substinutability between each pair of goods: higher corresponds more similar goods. our running example, 7 describes the difference in service of two taxi companies, such as the difference in the cars' quality or the location coverage. We refer to tooChoné& & Linnemer (2019) for a detailed discussion on the QQUM model and the plausibility of assuming an aggregate consumer behavior.
In this case, the exact form of the demand equations
is well-known.
Lemma 4.1 (Amir et al. 2017). Assume that G (1 p) > 0 and p G G -lupp) p) S B. The solution the consumer problem is
- 7 - Dj 7(m - 2)p; + +
Pt = 1 9:
Σ it
9
q:
(1 71 + 7mm 1))
(5)
The technical conditions G G (L- and G p) S B ensure that the consumers want
to buy at least bit of every product and do not spend all of their money in the considered market, respectively. For completeness, we prove Lemma 4.1 4.1 Appendix A.1
4.1.2 Supply
We assume that the cost functions
3 are linear in the quantities
II
E(pag: cq)))
(6)
where the parameter e depends on the quality of the machine learning model. We denote of Ep; (ci(c Remark 4.2. Here we assume that the quality of the machine learning model only affects the production costs, and not the utilities. However, this market model, is equivalent to assuming that the machine learning model affects the consumers' utility via increasing the effective quantities of the produced goods. Specifically, assume that each product has an internal quality wy and it increases the effective amount of good in the consumer's 's utility (4), resulting in utility u(g, -........ Wm This model is equivalent to the model above after substituting qı Pt, and & with their effective versions q w.y P. pi/ l/ww, and ē c./w;
We now derive the remaining constraints on the prices and costs. Note that these equilibria depend on the expected costs of and hence the quality of the machine learning models.
Cournot competition Each firm acts by choosing an output level yis which determines the prices (5) and expected profits (6). The next standard lemma describes the Nash equilibrium of this game. Lemma 4.3. Assume that the expected marginal costs are equal to ct, ..,..., cn and companies maximize their profits (6) in the Cournot competition game. If Vi (2 - 7T - c Eees(ch cg), Nash equilibrium quantities and profits are equal to
- 7 (2 + 7(m 2))c + E
0/ =
11: = (9:)
(2 18-7) (2 + (m - 1)m)
5

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Bertrand competition Each firm acts by setting the price p; for their product, which determines the quantities (5) and expected profits (6) The following lemma describes the Nash equilibrium of this game.
Lemma 4.4. Assume that the expected marginal costs are equal to ci...., ... c. and companies maximize their profits (6) in the Bertrand competition game. If vi d1 cf) > d3 Eett -9))
c9),
Nash equilibrium prices and profits are equal to
di + d2c, + d3 Ej
II = (1+ + +2)) - ct)
p
d1
(1-7)(1+ 7m 1))
where di...., d4 depend only and
In both lemmas, the technical conditions Vi (1 (1) Σe - cf) and vi vi (1)c)) d3 Ejtree - -c}) ensure that every firm produces a positive amount of good in the equilibrium. We prove these Lemmas in Sections A.2 and A.3
4.2 Data impact model
In order to reason strategically about the impact of sharing data with others, firms need understand how additional data impacts their costs. Here, we consider the case where all datasets are sampled from the same distribution D. While heterogeneity is an important concern in collaborative learning, we focus on the homogeneous case since heterogeneity is orthogonal to the incentives arising from market equilibrium considerations.
Homogeneity allows us to consider the expected costs in form of = (mf)) where nf = |D; is the number of points the company has access to. Using the examples below, we motivate the following functional form for (n)
(n) b n 1..3E (0.
In this setting, 3 indicates the difficulty of the learning task, higher 3 corresponds to a simpler task (Tsybakov) 2004), a represents the marginal costs of production given a perfect machine learning model, while + b corresponds to the cost of production without machine learning optimizations. Additionally, we assume that < and I-a is small enough, so that the technical requirements of b
Lemmas 4.1 4.3 and 4.44 are satisfied (see Equations (IO) and (IT). Intuitively, this requirement ensures that firms do not exit the market during the competition stage.
In the examples below, we consider a setting where a company needs to perform action s E TR during production that will impact its costs. However, there is uncertainty in the production process coming from random noise X valued in RT. Thus, the cost of production is a random function c(s, X): R² x R R..
Asymptotic normality Assume that the firms use background knowledge about their production processes in the form of a structural causal model of X. However, they do not know the exact parameters of the model and use data estimate them. Suppose firm uses the maximum likelihood estimator to find the parameters and chooses the optimal action 8än based on this estimate. In this case, the result of optimization Ex (e( &im: X)) will approximately have a generalized chi-square distribution (Jones] 1983), resulting in the following approximation
Ep.x (c(Sin: X)) = a
(8)
We refer to Appendix A.4 for further details. This result implies the same dependence of the expected marginal costs the total number of samples Equation
(7) with 3 1.
Stochastic optimization A similar dependence arises when the company uses stochastic optimiza- tion to directly optimize the expected cost c(s) Ex (c(s, X)). If the company uses algorithm with provable generalization guarantees (e.g.. SGD with a single pass over the data, [Bubec] and the function c(s) is strongly convex, the outcome Sün will satisfy
Ep(c(sna) c(s"))
where s" is the optimal action, resulting in the same dependence in the previous paragraph.

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Statistical learning theory
Another justification for the dependency on the number of data points comes from statistical learning theory. The observations from this subsection are inspired by similar arguments in Karimireddy et al. (2022). Assume that firm trains a classifier used in the production process. In this case, the cost depends on the classifier accuracy a, and the cost overhead will satisfy
c(a) - c(a") c'(aa' (a"
(a' - a),
where a" is the optimal accuracy achievable by a chosen family of classifiers. If the family of classifiers has finite VC dimension d, well-known statistical bound (Shalev-Shward & Ben-David 2014 gives
a" - a
motivating the same dependence Equation 7 with 3 1/2.
Data sharing between two firms
Having introduced example market and data impact models, we can specify negotiation scheme and quantitatively analyze the data-sharing problem. Through this, we aim to obtain qualitative insights into how market parameters impact data-sharing decisions. Following conventional economic logic (Tirole 1988 Carlton Perloff 1990). we start with the simplest possible collaboration scheme, in which two companies make a binary decision of whether to share all their data with each other. In the next sections, we proceed to the more complicated situations of partial data sharing and data sharing between many parties.
Full data sharing between two firms According to our framework, both companies compare their expected profits the Nash equilibrium (from Lemma 4.3 or 4.4) for two cases, when they collaborate and when they do not. Then they share data with each other if and only if they both expect an increase in profits from this action.
Remark 5.1. Notice that can incorporate some legal requirements into this scheme by redefining full data-sharing action. For example, if sharing consent constraints, araising from copyright or other data ownership constraint, are present, we could assume that the companies will share only part of their dataset for the collaborative learning. While, previously, the competitor gets access to all data of the firm n share
n + nJ now they will get access to only data of people who agree with data sharing n share R-; + conconeent
For Cournot competition, Lemma and Equation 77 give the following collaboration criterion
Vi S 17
(2 - (n, - (m1 + -2) > T(n - -n,"))
where n is the size of the data of the player that is not s is the expected profit collaboration. and 17, ind is the expected profit when no collaboration occurs.
Similarly, in the case of Berntrand competition, Lemma 4.4 gives
Vi Share
(2 - - (m) (n1 + m2) > (n >
n, ))
The theorem below describes the properties of these criteria (see proof in Appendix A.5
Theorem 5.2. In the case of I 0, it is profitable for the firms to collaborate. In the case of > O, there exists value 21(7,8) B), where E [Bertrand, Cournot) is the type of competition, such that, for the firm i, is profitable to collaborate only with a competitor with enough data:
n
Tharee > II
> xc 3).
n2 + n2
The function Z, has the following properties:
1. xx is increasing in 7-
2. Berrrrand 2 ECoumot-
3. Ze 2e(r,, B) is increasing in B.
7

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Discussion The theorem above indicates that the firms are more likely to collaborate when either the market is less competitive (properties 1 and 2) or the learning task is harder (property 3). Indeed, the threshold x: becomes smaller when the products are less similar (y is bigger), making the market less competitive. Similarly, x becomes smaller in the Cournot setting since is known to be less competitive than the Bertrand one (Shapiro) 1989). Finally, when the learning task is harder (3 is smaller), z decreases, making collaboration more likely. Intuitively, it happens because the decrease in cost from a single additional data point is higher for smaller B.
In the supplementary material, we explore several extensions for the two firms case. In Appendix and Appendix C.2, we look at different cost and utility functions, respectively. In Appendix C.3 we
C.1
consider the case where a, and/or /8 might differ among firms and derive an analog of Theorem 5.2
In Appendix [.4] we consider the data-sharing problem with two companies with heterogeneous data in the context of mean estimation. Appendix
H.
we look at the welfare implications of data sharing.
6 Partial data sharing
In this section, we study a two-companies model in which companies can share any fraction of their data with competitors. For brevity, we only study this model the case of Cournot competition. In this setup, each firm F; chooses to share a fraction ); E [0, 1J of data with its competitor and expects its costs to be
c + mm= a b(n, + im
where A_ is the fraction of data shared by the company that is not i.
Remark 6. Notice that we can incorporate some legal requirements into this scheme by constraining can implement sharing consent constraints,
the action spaces of participants. For example, araising from copyright or other data ownership constraint, by constraining the firms' choices of X. While, previously, the firms were able share all data À [0, 1], now they can share only data of people who agree with data sharing À E [0, Acoosent)
We will use the Nash bargaining solution (Binmore et al. 1986) to describe the bargaining process between the firms. Namely, the firms will try compromise between themselves by maximizing the following Nash product
max (115 (1, (1) 115 0)) s.t. Vi 115( 11, 2 0. (9) X.E
The Nash bargaining solution is commonly used model for two-player bargaining, as it results in the only possible compromise satisfying invariance to the affine transformations of profits. Pareto optimality, independence of irrelevant alternatives, and symmetry (Binmore et al., (986). The following theorem describes the firms' behavior in this setup (see proof in Appendix A.6).
Theorem 6.2. W.l.o.g. assume that 2 n2. Additionally, assume (1 a) > b (in particular, b < 5(1 a)). The solution (x1,5)) to the problem is
b
i; dΙ +0
1 1-a)
Xラ = 1, where
(1-
1t-²
n2
8 (1
)9)
B -1/3
-1).
1+
47
(2->)²
Ri
17
n1
n1 +1
n
(ni (ni+n
otherwise.
B+2
Moreover, Rg is a decreasing function in 7 and 3, and dς 47
4+-²
n2
(1 + o(1)) when
n2/n1 0.
Discussion We can see that the results of Theorem 6.2 are similar those in Section 5 When there is a big difference between the amount of data (n1 >> n22) the big firm does not collaborate with the small firm (Aj = O((2//n nn )8+2)). However, when the firms have similar amounts of data, the large firm will share almost all of it (Xj x 1), and the sharing proportion increases with decrease in competition (smaller 7) and an increase in learning task hardness (smaller (8). In contrast to the previous results, the smaller firm always shares all its data with the competitor (\ī 1).
8

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7 Data sharing between many firms
In this section, we investigate the data-sharing problem for many companies. For brevity, we only consider the case of Cournot competition. The key challenge in the case of many companies is the variety of possible coalitional structures. Moreover, since the companies may have conflicting incentives and data sharing requires mutual agreement, it is not immediately clear how to assess the "plausibility" of particular structure.
We discuss one possible way to solve this problem using non-cooperative game theory (Kóczy 2018). We assume a coalition formation process between the firms, model it as a non-cooperative game, and study its Nash equilibria. The non-cooperative approach often offers a unique sub-game perfect equilibrium, standard solution notion in sequential games (Mas-Colell et al., 1995). However, this equilibrium depends on the assumptions about the bargaining process. For this reason, in Appendix we also study alternative formulations: cooperative setting and two non-cooperative settings with only one non-singleton coalition.
Non-cooperative data-sharing game We order the firms in decreasing data size order that firm Fi has the largest dataset. In this order, the firms propose forming coalition with their competitors. First, the largest company makes up 2 proposals. For any offer, each invited firm accepts or declines it in the decreasing data size order. If anyone disagrees, the coalition is rejected, and the first firm makes another offer. Otherwise, the coalition is formed, and all companies in it leave the game. After the first company forms coalition or exhausts all of its offers, the second firm, in the decreasing data size order, begins propose, etc.
We use the standard backward induction procedure (Mas-Colell et al. 1995) to calculate the sub-game perfect equilibrium of this game. This method solves the game from the end to the beginning. First, the algorithm looks at possible states one step before the end the game. Since the actions last player are rational, the algorithm can identify these actions accurately. This way, the algorithm moves one step earlier
coalition formation process and can now identify the action taken before this state occurred. The procedure iterates until the start of the game is reached.
Experiments We use the procedure described above to empirically test the conclusions of the previous sections. We sample m dataset sizes, one for each firm, from a distribution P = N(H, o²) clipped at form below. Then, we calculate the equilibrium of the resulting data-sharing game and
3, N(1000, 300";
700 3, P: X01000,600")
m1 3, (20000,,00")
01 3, N(1000, 600")
2.4
1.0991
1.092
2.3
1.455
Average coalition size
3.0
1.8
1,4086
1.430
1.4
1,4064
0.50
0.75
0.55
0.50
0.6
0.0
0.5
4.
N[1000, 300")
4. N01000, 600")
4.
N11000, 300P)
4. N(1000 6110006001) 600
2.70
3.1
3.00
1.70
3.0-
2.2
1.22
25-
2.00
30-
1.75
0.35
0.30
01:55
0.5
Similarity between products (7)
Simplicity of learning task (.)
Figure The dependence of the average coalition size on y (left part) and | (right part) in the synthetic experiments. The y-axes report the mean of the average size of the coalitions in the equilibrium partition, where mean taken over 10000 Monte Carlo simulations of the game. See the main text for details.

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compute the average size (number of companies) of the resulting coalitions at the equilibrium, as measure of the extend to which companies collaborate.
We repeat the experiment 10000 times, for fixed values of m, 7: 3. jr, 7 and compute the mean of the average coalition size over these runs. Our simulation solves each instance of the data-sharing game exactly and average it over big number of independent runs, which makes our results very precise.
Repeating the experiment for different values of y and 3, we can observe the dependence of this average coalition size on the two parameters of interest. In Figure I we plot these dependencies, for different values of m {3, 4} and {300, 600). When varying one of these parameters, the default values for the other one is = 0.8 and B = 0.9. We also provide results for other values of m and in Appendix F
As we can see, the conclusion of Theorem 5.2 transfer to this experiment: the average size of the coalitions and thus cooperation incentives, are decreasing in both and /3 across all experimental setups.
8 Conclusion
In this work, we introduced and studied framework for data-sharing among market competitors. The framework is modular and consists of three components sarrokeett--mrkett model, data impact, and collaboration scheme-which can be specified for any particular application. To examine the effects of various parameters, we instantiated the framework using a conventional market model and a natural model of data impact grounded in learning theory. We then studied several data-sharing games within these models.
Our findings indicate that the characteristies of competition may have significant effect on the data-sharing decisions. Specifically, we found that higher product differentiation generally increases the willingness for collaboration, as does the complexity of the learning task. Interestingly, our results suggests that the firms might not want to participate in the federated learning even in the absence of privacy concerns. On the other hand, we also predict that data-sharing collaborations might emerge between competitors even without any external regulations. We hope our study will inspire further in-depth investigations into the nuanced trade-offs in data sharing, allowing competition and collaboration to coexist data-driven environments.
Limitations and future work While our general framework can describe many data-sharing settings, the quantitative results rely on several important assumptions. First, we assume that the firms' data homogeneous. We expect that designing a suitable model in the case of significant heterogeneity is a significant challenge orthogonal to the focus of our work (c.g.. Gug.ajan & Lopez Paz 2021). Moreover, in some cases, additional data from different distribution may damage model performance, yielding orthogonal data-sharing considerations (Donahue & Kleinberg 2021).
Second, we assume that legal constraints are present are automatically fulfilled in all our settings. However, we see the evaluation of Al regulatory frameworks as an important direction for future work. The same applies to training cost considerations.
Finally, we not consider sociological aspects, such as reputational effects (e.g., reputation loss due to a refusal to share data poorly communicated decision to share data). Due to the inherent non-rivalry of data (Karimireddy et al. 2022). sociological modeling, c.g., the theory of collective action, may provide further valuable insights into the data-sharing problem.
We see our contributions as mostly conceptual and do not aim to provide fully realistic model that can directly inform practitioners about the benefits of data-sharing decisions. However, we hope our results will remain qualitatively valid in real-world settings since we used a market model widely adopted in the economic theoretical literature (Choné & Linnemer 2019) and a data impact model motivated by established theoretical frameworks in machine learning (Tsybakov) 2004). Also, we hope that the three-component decomposition of the data-sharing problem will help practitioners leverage their comaation-spciffi knowledge of their market and MI. models to build more accurate models for their specific needs.
10

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Acknowledgments
This research was partially funded by the Ministry Education and Science of Bulgaria (support for INSAIT. part of the Bulgarian National Roadmap for Research Infrastructure). The authors would like to thank Florian Dorner, Mark Vero, Nikola Jovanovic and Sam Motamed for providing helpful feedback on earlier versions of this work.
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Supplementary Material
The supplementary material is structured as follows.
Appendix
contains the proofs of all results in the main text.
Appendix
contains further examples of the data-sharing problem and costs.
Appendix
contains several extensions the setting in Section 5
Appendix contains alternative models of coalition formation between many firms.
Appendix contains the welfare analysis of the data-sharing problem between two firms.
Appendix presents additional simulations for Section
A Proofs
A.1
Proof of Lemma
First, we will solve the same consumption problem without positivity constraints
X piq: S B.
max 9-0
u(g. ........m!.. s.t. g +
Using the KKT conditions for ya we get
0L Ou
0 0q: 0q:
Xpi
where À the Lagrange multiplier for the budget constraint. The first-order condition for g gives
0 1
1=
Therefore, the first-order conditions for 91. 91..... 9m look like
Σ fj. ifi
qi
9j
Ps pi
q6 7
1
Notice that
m EiP:
Σp. P. mm- Σ - 7 - 1) 5) = Σ qi
1 7(m 1)
Therefore,
(1 7mm 1))(1 - p.) - 7mm E.P()
1- (1 + 7m - 2))p; - - Es Pj.
71 + + 1))
7) 7(m 1))
This solution is a global maximizer because u(g. Q ....... (m) is strongly concave in 91. In and linear in g.
Since the auxiliary problem has less constraints than the original problem, solution to the auxiliary problem will be solution to the original problem long all quantities are non-negative. These constraints give the following restrictions on PI,.... Pm and
Vi (1 (1 p.)
(p) Pt).
m B
P:Pi Pi-
The first series inequalities hold when prices are less than one and close enough. In our setting, is achieved when >> and (1 - a) >> b, where and b are parameters from Equation (7) (since Equations (10) and (11) hold). The last equation holds when big enough, for example, when B > m.
14

---

A.2
Proof of Lemma 4.3
The profit maximization tasks of the firms are
max (( c:)a:) E(1 ~ Σ 93 c:)g: o.
Using the first-order conditions, we get
E(1 E jjts 4n)
qi (q-1)
2
Both equilibrium quantities should the best responses to the opponent's expected amounts
1 = E(1 C: - -ー
Therefore,
E(m E, (m- 117 Ei 9.)
Edi o
Notice that
E E
E(...)
E(q,)
q:
2
2
So,
E, cf
Σ 0
2 2+ 7 -)
The best response equation gives
-== 1 1
1-
of
Σ3 93
2 2 7 + 7 - -m))) + E iiss of.
9f
2
(2 7) + (m 1)7)
We get expression for the quantities produced by the firms. However, need to ensure that these quantities are positive. This property is equivalent to the following inequalities
Vi (2 7) c 7
(4 -cy)
(10)
jyti
These inequalities hold, for example, when > and (1 a) >> b, where and are parameters from Equation 77).
Now, notice that the expected profit function a second-degree polynomial in gi
m q + 8q..
Since the maximum value is achieved at 9, (q this polynomial looks like
11 ( प: (q + q: (2 And it gives the following equilibrium expected profits
11. = (g
A.3 Proof of Lemma 4.4
The expected profit maximization tasks of the firms
1 (1 7(m - 2))p: + E Enjes Pj )
max
(p. - c:)
(1 - 71 + 7 1))
Thus, the best responses are
E(1 (1 + 7 2))cc + 7 EjjiPj)
p (p-1) -)=
2(1 + 7 2))
15

---

Notice that
E E
E E(...)
E(p.)
2(1 + 7mm - -2)) 2(1 + 7 2)) = P..
Thus,
p. 1- + (1 7mm 2f + Ej
2(1 + 7(m 2])
If we these equations, get
Σ ps (1 (1->)m+ + (1 + 7(m 2)) E, G + - 77(m-1)) 1)E, 2(1 + 7(m - 2))
ΣΦ; P (1 - (1- + (1+ 71 2)) E
2+ 2+m(m-3)
After substitution of this sum into the best response equation, we have
p d2e; + dg Ejzig
d4
dS = 2 + 72m - 5) - -22 - 3),
d2 = 2 3y(m - 2) + +m²(m² -4m + 4),
d3 + 22(m 2),
d4 4 6-(m - 2) 722 9m + 9).
Here, we again need to ensure that quantities produced by the firms are positive. This property is equivalent the following inequalities
Vp: 4 20 - d1 - c 2 d3 Σ (cf
-c)-
(11)
jft
These inequalities hold, for example, when and from Equation 77.
a) > b where and are parameters
Notice that the expected profit of the firm is second degree polynomial
I1 = aap + 2,p; - Since best response (p-.) a maximizer of this polynomial, we get
= =aa;p - (p-)))) 0
Also, notice that
(1 = E((e) ()):(c....... p-i)) O. II = aa(( p. (p-1)) (p: p. (p-)))
Therefore,
Direct calculations show that
1 + 7 + 2)
a - (1 7(m 1))
These equations imply the following equilibrium profits
(pT.. (1+ 7m 2)) -c)2 of)2 Formally, we assume that the cost of production of one unit of good C' is random variable that depends on random noise X Rm and the decisions of firm R... The firm knows that the noise is distributed according to a density function
(1 -1 + 7mm 1))
A.4 Derivation of Equation
X
f(. 0). 0 ddd
16

---

However, it does not know the parameters of the distribution 0.
The firm wants minimize its marginal cost average
c(s) = E (C
To achieve it, the firm decides the maximum likelihood estimates 9 of A using its data about the realizations of the noise. After that, the firm chooses an action that minimizes an estimated cost average
8fin = arg min E ((-
The resulting cost c( is random variable because of randomness of the firm's sample However, the asymptotic normality of MLE allows us to reason about cost distribution.
We the delta method to achieve this goal. By the asymptotic normality,
vn(0 - 0) 4 N(O,I(0)))
Further, by the delta method,
(8a - 8²) vn
(0 -). 0).
00
Finally, by the second-order delta method,
n((($an) - c(s*)) 3 VVe(s") ( ss*) + n1(Sin - s* V2c(s")
(Stin s*)
2
V²c((")
=n($ s*)
(5in s*).
2
Thus, the marginal cost will be distributed approximately as generalized chi-squared distribution. The expected marginal cost will be approximately
Ōs T 30
0s
e = Ec((((an) = c(s*)+
Tr vec(s")
1(0)-
I(0)
2n
00
Therefore, one of the natural choices for the dependency expected marginal cost on the number of data points is
c (n)
A.5 Proof of Theorem 5.2
In the case S O, the collaboration is always profitable because the right-hand side of the criterion is less than zero, and the left-hand side of the criterion is not less than zero.
In the case y > O, the criteria for the firm are the following. In the case of Cournot competition, we have
(2 - (m (n1 + n2) 7nn - n1 n8
fCounoe (I, 7) = 2x - 7 x) -(2-7) x) > 0,
where - =n::===ni//nn +n2). The right-hand side of the criterion is increasing in ær fCoumot (0, 7, B) 0, and /Coumee (1, > Thus, there exists a break-even point ToCurnot (7, 3) such that
< O, < #Coumer (7, 3), I Couurnt (I,, 8)
0, 3'Comoot (7, 33,
0,
FCoumet ().
Similarly, break-even point exists in the Bertrand case.
Now, we prove the properties of
The first property follows from the fact that f is decreasing y. The second property follows from the fact that fCourno BBatrandd The proof of the third property is more complicated.
17

---

In the case of Cournot, the proof is the following. Let 2(33 Cournot (7, We want to show that x(3) is increasing B. To achieve it, we will show that fcoumat (2(3), E) From the definition,
7(1 - xx(3)² -2- -) x11=)
x=3)
So,
(Coumo(=(3)) ( <
7(1 -x(3))
7(1 2tB
74
2 (2 - -1 x(3))}
2 2-
7](1 =(3))ote
2
7) (((1=x(3))38 Stc
-
(1 - x) <1
The last inequality follows from the concavity of the function x ØTC, The proof is similar in the Bertrand case.
A.6
Proof of Theorem 6.2
Direct computations show
(A1, X2) (0,
(2c -yc_ 2c (n: + n-idds) + (n + n1di))x
(4-27 (n:) +nc- (n_=)) 2c, (n: + - ( nędi))
(4 - 2)2
Denote
B
(n; >_
11:
gives
b
b
(m4) (m4 >m = +
buy-
(ni n_,
n
So, the profit gain will be equal to
46
2
6(4-27))
6(4 - 27) a) (4 72)2
(A,, 22) - (0, 0) = (2u; ru-s)
1
3
+
+ E(2u; - -mu-i)
n1:
n
where & [1-27)(1-a]
Denote
h;
(ni + n2)
n
4
2y
91
n
n
Then Equation
9 will simplify to
max (2ug [rg u.E(o.h.]
s..Vi 2m:-ru- 20. 0.
Now, we will change variables
2c + 7リー
2u 7u
ui
72
which will give the following problem
max 1 Egi + (11 - Egg + Ev2) s.t. vi 2e; + S (4-22)mu
22
18

---

Notice that the restrictions U; 2 0 are not binding since the point (U1, U2) = (c, e) delivers positive Nash product and satisfies the restrictions. However, one of the restrictions 20; + yo= _ (4 7 h, should be binding because any internal point (U1,)2) could be transformed to the point ((1 + )11 (1 <)uz),, which would increase the objective function and satisfy the restrictions.
First, consider the case 2ot + yuu < (4 // at an optimum (ef, vg), According to the previous
paragraph, we have 205 + = (4 - y22,, which gives vz - vi > (2 + ( - h11) 0. The
derivative of the objective along the direction (2, -7) equals to
(205 - r1 E91 + Evf) (1- Eg2 + (u2) + Eoi (2
{(292 791 -2 + 7ei)).
Since 205 voi 0, 12ge 791 2ng yuil v 10, and g 2[1-a] TØ. the derivative is positive. Thus, the point (ef, u²) can not be optimal because a small shift in the direction (2, -7)
would increase the objective and would not violate the restrictions. Therefore, in the optimum, will have 2 = (4 - //) 702-
If we substitute the last expression into the objective function, the objective will become fourth- degree polynomial of 022 If the stationary point of the objective function vz satisfy the restriction 2uğ + (-- (v.) (4 -4-7" 22 it will be a solution to the original problem. Otherwise, the solution will come form the equation 20l + yoi = (4 - h2 since all other restrictions are not binding. So, the solution will be equal to 12 min (13. v9) (the restriction 20² y01 (12) S (4 - -2) h2 does
not hold only when 222 is big).
Consider the following relaxed problem
max 11 (1 - E9 + (11)(1 - E92 + (12) s.t. 201 YU2 (4 222hh 0,
The stationary point will be the root of the derivative and hence will satisfy the following equation
(Tuğ - 201 (x)())))(11 - Eo + (v))) Ega + Enğ) =
(v))) Eg + Eml (v))) -711 Egg + +vu))) 70ξ 201 (vz)
0i (15)) - EyI + (vi 7(1 E,2 + Evz))
(1 -1-E8 Egi + (v5)) Eg + Evg)
(4_92)8n?
Since -2 9: 54.0 < 0; < 2, and in (w2) 5
we get
8y
(4 - 72) ff - + (1 - 46)2
+ 12
Iru - 201 <
O(E)
8%
So, the stationary point is approximately vy (4-g2) O
27
Therefore, the solution to the original problem is approximately
05 min (4 72)a + O(E), 2h2- - 7h1
( ((4 (4->2)h1
min
2h2
y/1 + O(E).
2ty
2-
Substituting everything back, we get the solution presented the theorem statement. Now, we will prove that λ₁ is non-increasing in 7. First, the criterion function for choosing non-unit solution
(2 10-39 7)
4y
= 4yh2 - (4 + )2)
(n1 + n2)
)
n2
n
increasing in 7 since derivative 4hg solution becomes more probable.
27/11 2 0 is positive. So, when y increases non-unit
Second, the non-unit solution is decreasing in y since the function 4+22 4 + 7is decreasing in ? on (0, 1). (Note thată, 1, when 7 < 0.)
7
Now, we will prove that X₁ is non-increasing in 3.
19

---

First, notice that the non-unit criterion is equivalent to
fzz B) 1 (A) -
z))
2 0,
where x:
R2 # The criterion function / decreasing in æ and have one root sr on the interval
[0. 1]. We will prove that the root is shifting to the right, making more size pairs produce non-unit solution. Notice the criterion can be also expressed
4y 2)
x s
4 4+ - (2 (1 x))
So, prove that x(3) < s(B + t) it sufficient to show that
41 - x(())²
4y(1 x(B))
4 4 - (2 - -)) x(3)) x(()
42 (2 -)) x(())
(2 7) (1 x²)²
4y
4 [4+12 - (2 7)2 - xx) 2))8te)
S1.
4 72
4 4+ 72
4y
The last inequality follows from the concavity of the function ,t.
Now, we will prove that the non-unit solution is decreasing in 3 given that con-unit criterion holds. Notice that the non-unit solution d (8) satisfies
4+
1
1
122
(n2 + (3)n.)
47
(mi + n2)
So, to show that (3) 2 Ai + c), is sufficient to show that
(x 5(1 x²) 5) 8 5(1 21) + 5) 5))tz,
where r 91
and 8
4+2² This inequality is equivalent to 47
((1 - x) 9 -8 8 8(1 - 21)0) 2 S - - 8
(1-2)² 8 0o (1 -- x² and B+c
Denote Uog
We get the following inequality
B
(wo + 600) I up 5 + 606.
Denote
= 80 (u 8 + 8v)a.
Notice that f(u, 1) 0. Additionally notice that
0f 5a - sa(u a- 50 v 2 u 8 8v. dv
we prove that this derivative is negative for all points (up, x). where x E (wo, 1), we will prove the desired result. Notice that the last inequality is the most stringent for the point (uo, vo). At this point, we have
4y
4+
4
(2 7)
Uo
20p +
5e0
0 <
8
9
+
(n1 + 22)
n2
n
The last inequality holds because we consider the non-unit solution.
The last property of the theorem follows from the fact that, for small ratio 22 the non-unit criterion holds and the Taylor formula for this ratio results in the expression presented in the statement.
20

---

B Examples of data-sharing problem
B.1 Taxi rides
In this section, how our framework works using motivating example from Section 3 In this example, two taxi firms use data optimize the driver scheduling algorithm. This optimization allow them to use less driver time; thus, making their expected marginal costs lower.
We assume that the only uncertainty arising in scheduling algorithm is the duration of one kilometer ride ~ N(4 1). The firms do not know the parameter ju. However, each firm F; has n, observations of this random variable S; = {
Their scheduling algorithm require the estimate of ride duration X. If the estimate too low $ < X, the company will need to attract more drivers and pay them more. If the estimate is too big $ > X, drivers will be underutilized. We assume that costs associated with these losses have MSE form C(s, X)
b + b(s 8 x)². This functional form will imply the following costs average
c(s) = (C X)) b+ bE((s - x) = a + b(s p2)
If the firms do collaborate, F; uses average of its own points
1
Si ~ N N(1 - x
1
74
n4
to estimate This will give the following expected marginal costs
b
.iind ,t ((s,,nn)) + b b 8t,ind 11) =
12
the firms collaborate, they uses both samples to calculate average and the expected marginal costs will be
Σxj
N N(1,
1
Sshared
Sahared
T&1 + 112
ni + n2
Chared
Es, c(Sshmre)) = a +
+ 12
To solve the data-sharing problem, we need to describe the competition between firms. For simplicity will only consider the of Cournot competition.
As in Section 4.1.2 the firms are interested in increasing their expected profits
11 Es,,,, (p:4 c:g:).
Thus, the firms use the same quantities as in Section 4.1.2 expected profits are
the case of non-collaboration, the
2 (2-
7ch, 2c
72
Similarly, in the case of collaboration.
- 7) Cchored)
2
I1 Starrd
4 72
The collaboration criterion is the same in Section 4.1.2
B.2 Oil drilling
Assume that an oil company manager needs to optimize the costs oil rig construction. Construction of an oil rig happens in two steps. First, geologists look at new place for rig and produce a noisy signal of oil presence X. Second, the company might try to build a rig. However, if there is no oil, the company will lose money. Thus, sometimes it is profitable to repeat the geological expedition.
21

---

The noisy signal of geologists predicts the presence of oil in the following manner
Pr(success X) Pr(X + + +620 X), N(O, 1).
Also, the manager knows that
X ~ 0²)
However, the manager does not know o² and needs to estimate from the previous observations. The new expedition costs a, and drilling new rig costs b.
The natural strategy repeat expeditions until the geologist find X 2 r and then try to build a rig. Let be the probability of finding good spot and be the probability of success in good spot
p p==Pr(X 2 r)
q = Pr(X + +E 2002222 = 1 30
exp
exp
M(r, 0²)
dydr
p r
2
/2T
where denote the last integral by M. So, the expected cost is
2 +
f(r)
M(r,0²)
Assume that the company estimates 2² 0² and chooses r according to that estimate. The loss will approximately be
f(r) - f(1") = (r +*) ('(r")
{( 2 (r =²
2 (r - ---)²=
["(r") (02)2(002 -
2
Now, if the company uses MLE estimate o², then the costs will asymptotically look like
f(r) ~ A+ x1),
where n is the number of observations.
After that the process is the same as in previous subsection.
C Extensions of the results
In this section, we investigate how different changes in our setting affect the results. For simplicity, we only study the case of perfect substitutes (7 and Cournot competition not told otherwise.
C.1 Different cost functions
In this subsection, we assume that the cost functions of the firms have the form
C((g) = c + 2
The profit maximization problem of the firm now has the form
k 2
max(1 lo g-:)g;
C
2 9
86
Therefore, the best responses and equilibrium quantities are
1 1- 9-i of
(q-)))
+
1 1 c_ (2 k)
q =
(1+ kk)
22

---

These equations imply the following expected profits
+ & m
(g))
2
and result in the following collaboration criterion
Vi (2 + (),," (m1 + m2)
n
As we can see, the added non-linearity in the cost does not qualitatively change the conclusions of Theorem 5.2
C.2
Multiplicative substitution
In the Section assume that goods produced by the firms be additively substituted by the outside goods (i.e., the utility function function is quasilinear). However, in general, the firms might operate in different markets with different substitution patterns.
To investigate, this question we assume the following form of utility function
-(1 (1 + 02 of + 2q1
45
v(go 91,92)
2
(P1 = P2 p because the goods G1 and G2 are perfect substitutes.) This utility prescribes the consumer to spend the share of income goods G1 and G2 (B is small enough, so that Q1 and Oa are much smaller than 1). So, we have the following consumer problem
oi 29192
o
max Q1 + 02 Q1-92
s.t. p(g: + (2) S aB.
2
So, the demand is
aB
Q1 + Q2 p
Expected profits maximization problems are
aBq
max
Egi
ạ1 + 92
The first order conditions give
aBq (9) + qz)² = c.
So, equilibrium quantities
aBqq 12
aBesef
q c,( + 93)
c) + ct)2
And equilibrium profits are
aB(c_,)2
aB
II
(cf + cg)²
This formula imply the following collaboration criterion
Vi 1
,ind
1
Shaae 2.
Lind
Sharee
As can be seen, the collaboration is always not profitable for the firm with more data.
23

---

C.3 Asymmetric costs
In this subsection, we assume that firms have asymmetric cost functions
b
(n) = a, + n
where of (n) is the expected marginal cost of the company if it use points for training. Section 4.1.2 gives the following the expected profits
2 - + yc 2 4 7-22
II
As in Section 4.1.2 the companies will collaborate only if both of them profit from collaboration
Vi Shared
I1
2b,(n, shhared)) >> {b-((n 8-4
sharrd
8-
Denote (n1, n2) (n; & nshared) (n n sharrd ). This function can be interpreted as firm Fi reediness to collaborate: the higher this value is the more firm Fi wants to collaborate with its competitor F-i.
Theorem C.1. Assume the setting described above. The function f((11, n2) has the following properties:
1. (2) is decreasing in
2. (1(1))2)) is decreasing in 33 and increasing in B_4 if Vi 9. In > 1.
3. f.(m1,n) n2) is increasing in b; and decreasing in b-
Proof. The first property is evident
Qf
b
-8-1 11, shaared < 0.
B_1
θ7
We can prove the last property similarly
0f: 20
2(m1
n nsharedd > 0,
0b.
0f: 3-1 Ti Th
ab_ (11
B_r
< 0.
The first part of the second property follows from the following
0ʃ,
In shared
Iu
2b;
03,
n 9,
Notice that the function g(x)
In z
is decreasing in on 2 >> 4/9, because
Og
3 Inx
3.+1 < 0.
dr
0/. Therefore, 23, < 0: fa is decreasing in 3.. The second part can be proved similarly.
The first property is similar to the first property in Theorem 5.2 The firms with more similar goods have less incentives to collaborate.
The last two properties show that the firm wants to collaborate with its competitor when the firm's machine learning model is bad (3, low or high) or the competitor's 's model is good (8-; is high or b_; is low).
24

---

C.4 Heterogeneity
In this section, we solve the same model as in Appendix B but allowing for heterogeneity between fimms. Each firm will need to optimize the MSE costs
(a(s) = a b + bE((s x,)²) + b(s pa)?, X; ~ N(44, 1).
the contrary to the Appendix B. means the noise will be different among firms and will be drawn at the start of the game from the distribution
Ms N(,,
We assume that f is unknown and o² is known.
When firms do not collaborate, the expected costs are the same as in Appendix collaborate, MLE for the means are
B When firms
20 2 + n;Yi n
Si,shared
20 2 12 + m1 112
where n, is the number of data points of firm F;, Yi is the average of noise. Then expected marginal costs will be
G.shared = a a 6 Es,.s ((Sishhrrd )
By substitution,
E( (Si.sheddd
2)
n² ,2, 1611-10 -22)²
(202n_; + 11)n, + 2202 + nı + n2)² + (202n + n1 + 112)
Th
E
(2022112 + ni + n2)2
20 +
20 712
Therefore, G.saardd is increasing in 2 the more different the firms are, the less collaboration gives. Two corner cases give
0² 0
C .hhared
Mi1 712
à² Gishared a +
n2
However, notice that collaboration criteria in the Cournot case does not depend on o² and hence does not change
II shaar
11 2 += 2 71 2c, share + i,_hre 2 2 7
ind + 7c_
2(2021 n +1+ 1)
7(202 n; +
2 7
2 -2 n-i
n2
2
20 + n1 + 20 + n1 + +12
74 nh
D Other models of coalition formation
In this section, we present models of coalition formation different form the model presented in Section 7
D.1 Cooperative data sharing
We use an solution concept motivated by the a-core (Von Neumann & Morgenstern 1944), which is standard in cooperative game theory. The a-core consists of partitions that are incentive-come compptilee ensuring that no subset of firms wants to deviate. Incentive compatibility indicates that these partitions more stable and likely to occur.
First, we introduce a modification of a-core, tailored to our setup. This deviation necessary due to two reasons, con-transferable utility and the presence of externalities (dependencies of utilitics on the
25

---

actions of all participants), which arise in our setting because all firms compete with each other on the market.
Let N = [Fi...., Pm]] J be the set all firms. A partition N a set of subsets {Si,.... S&. S S N, such that V S; SS 0 and U,S, = N. We denote the set of all partitions of N as P(N).
Definition D.I (a-Core). A partition P' belongs to the a-core of the game if and only if
As N: S,Q P(N\S) (I) (
where (P) expected profit of the firm F if market coalition structure is
Intuitively, partition P belongs the a-core if no of companies wants deviate from this partition. Here we say that subset of companies wants to deviate, if any company in this subset will increase its profits by joining this new coalition, regardless of how the remaining companies split into groups.
In the next theorem we identify one coalition structure that always belongs to the a-core of the data-sharing game, demonstrating that the not-emptiness the core.
Theorem D.2. W.l.o.g. assume that nr 2 n2 2 ... 2 Rms where n, is the number of data points of firm Fg. Consider partitions P. [A, N A.), where A, {Fi, F2,..., Fi). Let 1" arg max, ITF, (P:). Then Pe- belongs to a-core.
Before we start to prove this theorem, we formulate the following lemma. Lemma D.3. Let Q C N and E Q. Then
117(11,, (ON)) (2 (2- (2 + (m QQ)c 7(m 191c
(2 - 7)2 + (m 1)7)22
where 6x aa b/nxx. expected marginal cost of coalition X, and nx = Erex np, the number of data points of coalition X.
Proof. The results Section 4.1.2 give the following expected profits
(2 7 (2 + 7 2) + Ejtt G)2
17
(2 (2- (m 1)7)
In our case,
SCq. F
6
levio
F F
By substituting of into the profit equation for Q, we get
(2 72-7- (2 7m 2))c + r(Q) 1)cg + 7 10))
IT9(1 N Q))
(2 72 (m (m-1)m1 1)
(2 12-7- (2 + 7(m - 1Q1) + + - 1Q)c
(2 - (2- + (m 1))²
Corollary D.4. Let Q CN F Q, and F' Q. Then
IIA(Q, N \ 11) 11 (1Q', \ ((1) +
+ 7m - 1- 1-12
7m- (Q')) >
2 + 7 1- -1-1Q)
(mm (m--11)
niQ
8
n x
3
niQ
3
n
Now, we ready prove Theorem D.2
26

---

Proof. We prove the theorem by contradiction. Suppose that there exists a coalition Q for which deviation is profitable.
First, we consider the case A,- 0 Q + 0. Let E A n nQ Notice that
(P,)) I7 (P:-) 2 M7. (PQ)) 2 (IQ, (Q)).
The first equality follows from Lemma D3 The first inequality follow from the definition of 1". The
last inequality follows from Corollary D.4 and observation n Ajoi 2 ng. So, we get contradiction. Thus, A₁- O Q 0. Let F E Q. In this case,
(P,) 2 (IQ, \ Q)).
This inequality follows from Corollary D.4 the fact that n,, 2 no. and observation JN \ A4- [Q]. This contradiction proves the theorem.
In the partition described in the theorem, the firms form two coalitions: one contains the i" companies with the largest datasets and the other one contains all other firms and leads largest profits for the company with the largest amount of data. However, it may not be the only one in the core since the a-core is one of the most permissive cores the cooperative game literature: a set of companies will deviate only if it is better-off irrespective of the actions of others, sometimes resulting non-cconomically plausible partitions.
The following example illustrates this flaw. Assume that there are three firms in the market and nı n2 >> ni.. In this case, the a-core consists of two equilibria: {{Fi, F2), {Fa}) and ((F)), (F2), [Fs)). but the first equilibrium is less economically plausible. Indeed, for the first firm, the second equilibrium is more profitable than the first. However, this firm will not deviate because of being afraid that the second firm might collaborate with the third. Nevertheless, this fear ungrounded since the second firm expects less profit from the collaboration with the third, than from staying alone.
D.2 Universal data-sharing treaty
The first model consider following situation. Imagine that companies might sign some treaty that will their data accessible for use for other companies. To make companies comply to the outcome, the decision is made by consensus. In terms of coalition formation game, each company either agree or disagree to participate grand coalition. If any company disagrees, the grand coalition is not formed and companies act like singletons.
W.l.o.g., assume that n1 2 n22 2 ... 2 P'm. The grand coalition will be Nash equilibrium in this game only if all firms will be better off in the grand coalition
vF (TF,, F Fm))) > (TF()(()), (FF(( (Fa)....,
Since grand coalition profit is the same for all firms and singleton profit is biggest for the firm with the most amount of data, this system of inequalities equivalent to the inequality for the first firm i (f(F1, F2,... ... Fm)1)) 17 (((F)) ... {Fm))).
By substitution, this inequality takes the form
mm 1
+ 1
Y
8
n
where n = E, nz-
D.3 Data-sharing treaty
The second model is similar to the first, but here the members of the treaty share data only between themselves. Thus, not everybody needs to agree to build coalition and resulting structure will be one coalition with several members and singleton coalitions. Formally, Each firm has two actions: Y and N. All firms that answer Y form a coalition between themselves and all firms that answer N form singleton coalitions. We interested in Nash equilibria of this game.
assume the following game.
W.l.o.g., assume that n 2 n2 2 ... 2 nm- The following lemma is useful for the description of all equilibria of this game
27

---

Lemma D.5. Let S be a set of firm that answer Y in equilibrium and i" minfi Fj E S}. Then Fj
Proof. To show this it is sufficient to show that single firm F always want to join coalition S that has more data than the firm. This is equivalent to the following inequality
U ( G & (U(F)))
> u s))
2(S + ---m---
7|S|
m
m+1-2
(ns + Rr)
where ng = Eoes nG. Denote y = "r The last inequality is equivalent to
(m + 1 - 1 (1 + y)
y
y
7/5|
y
(1
y)
Since 7 1 and |S| < m, have m + 1 7|S|. Furthermore, 1 because 2 y > Therefore, the inequality above holds.
10 (1+0)
y²
y
(1+2)
Lemma D.5 greatly constraints possible equilibria of the game. Indeed, the equilibrium set of companies that say may only have the form S; = {F; j 2 i). check whether S appears the equilibrium, we need to check the following system of inequalities
Vj < i 11 (1S.) G & S() Z M3, (FM) ([G) ( (()))))) Vj N 11 (1S.) G S.)) S MI, (LS, ((G) (CG)))HHHHHHCCCCCCCCC & s
These inequalities are closest for the firms around threshold i. Thus, this system equivalent to two inequalities
(1S. G & s
(IS)) G & 8,)) z ((S. s
E
Welfare analysis
In this section, we consider how collaboration affects the cumulative utility of firms and consumers. We start with the definition of welfare.
Definition E.1. The sum of consumer surplus and firms surplus W is called welfare. In our term this quantity is equal to the sum of consumers utility and firms profits
W W 11+
The definition of welfare in our case is simpler than in general case because consumers have quasilinear preferences.
Welfare an important quantity in policy analysis. Welfare reflects the cumulative benefits of all market participants. Thus, if redistribution between people is possible (e.g.. through government transfers), increase in welfare means increase in utility for all people given the right redistribution scheme.
Now, we want to evaluate how data-sharing in duopoly case affects the expected welfare. This will help us to understand in which cases the firms should be incentivized to collaborate with each other. By direct substitution, we get
(qi) + (o))² + 22999 B ciqs
W vi 26
We show that collaboration increases welfare in the Cournot case if the marginal costs of the firms are close enough: lef - cgl < 7- To accomplish that, we will first demonstrate that aw des < ch= and then show that W decreases along the line of ca These properties will show
28

---

that collaboration increases welfare since after collaboration the costs of the companies equalizes and become smaller.
The proof of the first part follows from the derivations below. W.I.o.g. assume that ny 2 n2, then ci c Direct computations show
ow 0G2
0qi
0q2
0qi dgg
0qg
ci
+
(0i + 792)
(02 + +07) -2-
Oc²
The results of Section
4.1.2 give
9 7 26 70
4- 7-
Substituting the expressions above, have
aw
(12 ( (17 - (12 - 8y 72 +99
0cg
(4 72)
(12 (12- (87 - 70) )cg 7) (12 87 - +2++7)
(4 (4-2
-(12 8y - 72 + )1 c5) + 2 73)
72)2
The restriction 02 S 1 gives
oW
-(12 87 -22 + 7 787 7D))
Oc²
(4-
This inequality equivalent to the following inequality
36 247 37 3 87 73 < (36- 336-372)(199)) + +73 >0. 0.
The last inequality holds since y < 1.
The proof of the second part comes from the computations below. Assume oj = cG C, we want to show that W decreases in We have
W (12 87 72 + )² (24 - 167 - -27² + +-y>)0 + 12 8y - 72 + 73 B
(4 7222
(12 8y + c)2
+ B.
(4 22-2
Clearly, this function is decreasing in c for 1.
F Additional experiments for Section 7
29

---

m
3, P
N(1000,3003
m
3, P
N
(1000,, 300
3.0
1.4994
1.4992
2.5
1.4990
1.4988
2.0
1.4986
1.5
1.4984
0.25
0.50
0.75
1.00
0.6
0.8
1.0
m = 3, P = (1100,, 6002)
m = 3, P = N11000, 6002)
Average coalition size
2.4
1.460
2.2
2.0
1.455
1.8
1.450
1.6
1.4
0.25
0.50
0.75
1.00
0.6
0.8
1.0
m
3, P = (1000,,000)
m = 3, P (0000,9000)
2.0
1.410
1.405
1.8
1.400
1.6
1.395
1.4
1.390
0.25
0.50
0.75
1.00
0.6
0.8
1.0
Similarity between products (v)
Simplicity of learning task (B)
The dependence of the average coalition size on 7 and for in synthetic experiments. The y-axes report the mean of the average size of the coalitions in the equilibrium partition, where mean taken over 10000 Monte Carlo simulations of the game. The firms' dataset sizes are sampled from clipped (at 1) distribution P. There are 3 firms in the market engaging in the Cournot competition. The default values of y and are equal to 0.8 and 0.9, respectively.
Figure 2:
30

---

m
4, P
= 110000,00**
m
4, P = N(1000,300
3.5
1.97
3.0
1.96
2.5
2.0
1.95
0.25
0.50
0.75
1.00
0.6
0.8
1.0
m = 4, P = N(1000, 6002)
m = 4, P N(1000,600'
1.76
Average coalition size
2.75
2.50
1.74
2.25
1.72
2.00
1.70
1.75
0.25
0.50
0.75
1.00
0.6
0.8
m
4, P = N(1000,900'
m = 4, P = N(1000,900*)
1.62
2.2
1.60
2.0
1.58
1.8
1.6
1.56
0.25
0.50
0.75
1.00
0.6
0.8
1.0
Similarity between products (v)
Simplicity of learning task (B)
Figure 3: The dependence of the average coalition size on 7 and for in synthetic experiments. The y-axes report the mean of the average size of the coalitions in the equilibrium partition, where mean taken over 10000 Monte Carlo simulations of the game. The firms' dataset sizes are sampled from clipped (at 1) distribution P. There are 4 firms in the market engaging in the Cournot competition. The default values of y and are equal to 0.8 and 0.9, respectively.
31

---

m = 5, P N(1000,300 3002)
m 5, N(1000,300) 3002)
2.42
4.0
2.40
3.5
2.38
3.0
2.36
2.5
2.34
0.25
0.50
0.75
1.00
0.6
0.8
1.0
m = 5, P N(1000,,
m = 5, P = N(100,,
3.00
Average coalition size
1.950
2.75
1.925
2.50
1.900
2.25
1.875
2.00
1.850
1.75
0.25
0.50
0.75
1.00
0.6
0.8
m = 5, P = N(1000,
5, P = (1000,90002)
m
1.78
2.4
1.76
2.2
1.74
2.0
1.72
1.8
1.70
1.6
1.68
0.25
0.50
0.75
1.00
0.6
0.8
1.0
Similarity between products (7)
Simplicity of learning task (3)
Figure 4: The dependence of the average coalition size on 7 and for in synthetic experiments. The y-axes report the mean of the average size of the coalitions in the equilibrium partition, where mean taken over 10000 Monte Carlo simulations of the game. The firms' dataset sizes are sampled from clipped (at 1) distribution P. There are 5 firms in the market engaging in the Cournot competition. The default values of y and are equal to 0.8 and 0.9, respectively.
32


Reviews:
[{"review_id": "OIO86ICv1l", "reviewer": "Reviewer_Jw59", "summary": "* This paper analyzes the economic consequences of data sharing between competitors.\n* Interaction is modeled as a market:\n  * Market has $m$ firms ($F_1,\\dots,F_m$), where firm $i$ produces $q_i$ units of good $G_i$ at price $p_i\\ge 0$, and quality $v_i$.\n  * In addition, there are “outside goods” $\\{1,\\dots,k\\}$ offered at fixed prices $\\tilde{p}_l$. \n  * Each consumer $j$ optimizes their utility $u^j$ by deciding on consumption of firm goods $q^j\\in\\mathbb{R}^m_+ $ and outside goods $g^j\\in\\mathbb{R}_+^k$ under budget constraint $B^j$ - Leading to Firms maximize their expected utility $\\mathbb{E}_v[p_i q_i - C_i(q_i,v_i)]$.\n  * Different solution concepts are considered: Firms act either by deciding on $q_i$ (Cournot competition), prices $p_i$ (Bertrand competition), or by strategically considering the response of their competitors (Nash equilibrium).\n* In section 4, a concrete market model is instantiated: Consumers make decisions according to a quasi-quadratic utility model characterized by a substitutability parameter $\\gamma$, and the cost associated with each firm is $C_i(q_i,v_i)=c_i q_i$, where $c_i$ depends on the quality of the machine learning model. Machine learning quality is assumed to affect production costs only.\n* For analysis, data is assumed to be homogeneous (data of all firms is sampled independently from the same distribution), and coefficient $c_i$ is assumed to take a concrete power-law parametric form ($c_i = a+b/n^\\beta$). Attention is restricted to competition between two firms.\n* Theorem 5.1 characterizes the equilibria as a function of the $\\gamma$ parameter, and the ratios between the amounts of data collected by the two firms. Theorem 6.1 characterizes the equilibrium in the case of partial data sharing. Finally, a numerical simulation is conducted on a competition setting with more than two firms.\n", "questions": "* In the proposed model, do the firms have the ability to invest part of their budget in independent data collection? (e.g by conducting surveys to improve training set quality, or buying data from an external provider which is not a direct competitor). If the market model does not include this possibility, what would be the consequences of considering it?\n* What are typical values of the constants $a$,$b$,$\\gamma$ in real-world systems, and why?\n* L224: “We assume that $b/(1-a)$ is small enough...” - Which values of $b/(1-a)$ are small enough for the lemmas to hold? Are they realistic in real-world systems?\n* How would the results be different if the cost $c_i$ had a different parametric form? For example, using similar arguments to the ones presented in the paper, one could consider costs that relate to the parametric forms described in Viering et al. 2022 (Table 1).\n", "limitations": "Limitations are not thoroughly discussed. In particular, I feel it would be helpful to provide an indication of where we expect the main assumptions in this paper to hold, and discuss the consequences of making wrong assumptions.\n", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Topic is well motivated.\n* Clean presentation, connects contemporary topics in machine learning to classic economic notions in a creative and interesting way.\n", "weaknesses": "* Parametric assumptions for main theorems are not validated against real-world data. Not clear which parameter regimes are likely in practice.\n* Data homogeneity assumption may be too conservative - Homogeneity means that datasets collected by all firms are assumed to be sampled independently from the same distribution. This is unlikely to be the case - For instance, in the taxi running example, different operators are likely to observe different data distribution, e.g. because they operate in different parts of town.\n* Limitations of the method are not thoroughly discussed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable and constructive feedback. In the following we address the weaknesses and questions raised in your review.\n\n**Weaknesses**\n\n**Parametric assumptions for main theorems are not validated against real-world data. Not clear which parameter regimes are likely in practice.**\n\nThank you for your feedback. We show how one can reason about parameter values in our response in the \"Questions\" section below. As we note in the general response, a systematic analysis and validation of a real-world market model and the most exact data impact model are probably outside our submission's scope. Please refer to the main response for our motivation. \n\n**Data homogeneity assumption may be too conservative**\n\nWe certainly agree that heterogeneity may be an important aspect in some settings. Please refer to our shared response for a discussion of the numerous challenges and orthogonal incentives arising when considering a heterogeneous setting. Additionally, we note that a simple model of heterogeneity is considered in Appendix C.4. Our framework allows even more intricate models of heterogeneity, providing that an appropriate data impact model is formulated.\n\n\n**Questions**\n\n**In the proposed model, do the firms have the ability to invest part of their budget in independent data collection? … If the market model does not include this possibility, what would be the consequences of considering it?**\n\nThank you for the valuable suggestion. One can incorporate this possibility within our framework by adding a data-collection stage of the game before the data-sharing stage. Intuitively, it should incentivize firms with small amounts of data to gather more of it to be accepted into desirable coalitions. We expect the results of Theorem 5.1 qualitatively stay the same, but this mechanism may lower the threshold for collaboration.\n\n**What are typical values of the constants $a, b$, $\\gamma$ in real-world systems, and why?**\n\nWhile our market model is widely used in economic theory, one typically uses domain-specific models to describe real-world systems. As we pointed out in the general response, we hope that our three-component decomposition of the data-sharing problem will help practitioners to leverage their situation-specific knowledge of their market and ML models to formulate an appropriate application-specific model.\n\nThat said, the parameters in our manuscript do have natural interpretations that can inform relevant magnitudes based on economic intuition. In particular, $\\gamma$ should be positive (since we consider competing firms) but not very close to $1$ (since firms usually do not produce identical products). The parameter $a$ should correspond to the costs of the firms with a perfect ML model, and $a + b$ should correspond to the costs of the firms with a very bad model. Since $1$ should correspond to the largest possible price for a product (when there is only an infinitesimally small amount of this good on the market), the ratio $\\frac{1}{a}$ should roughly correspond to the ratio of the highest possible price of the product (like during supply shortage) to the lowest possible price. For example, this multiplier for the natural gas in the EU during 2020–2022 was around 50 (FRED), implying that $a$ was around $\\frac{1}{100}$. As for the ratio of $\\frac{b}{a+b}$, we could use consulting firms' data about the impact of machine learning on costs. In Espel et al. (2020), this ratio was around $\\frac{1}{5}$, yielding values of $b$ around $\\frac{1}{500}$.\n\nFRED. Global price of Natural gas, EU. https://fred.stlouisfed.org/series/PNGASEUUSDM\n\nEspel, P., Herbener, M., Rupprecht, F., Schröpfer, C., and Venus, A. How industrial companies can cut their indirect costs—fast. https://www.mckinsey.com/industries/automotive-and-assembly/our-insights/how-industrial-companies-can-cut-their-indirect-costs-fast, McKinsey, 2020.\n\n**Which values of $b / (1 - a)$ are small enough for the lemmas to hold? Are they realistic in real-world systems?**\n\nWe provided these restrictions in the proofs of our lemmas (Appendices A.3, L490 and A.4, L504). For example, in the Cournot case, the restriction would follow from the following property: $1 - a > m b$. Given our discussion of magnitudes in the previous paragraph, we expect this property to hold.\n\nIntuitively, in our market model, the ratio being small is equivalent to the assumption that firms do not exit the market during the competition stage. While modeling firms' entry into and exit from the market is certainly interesting, we see this mechanism as complementary to our focus on the trade-off described in this work between improving your model versus risking increased competition. \n\n**How would the results be different if the cost had a different parametric form? For example, costs that relate to the parametric forms described in Viering et al. 2022 (Table 1).**\n\nWe do not expect the results to change qualitatively. However, instead of a threshold for the ratio between the number of data points, we will have another disparity function between them (for example, difference), and the notion of task simplicity would be different."}, {"review_id": "Cf9WMPJV91", "reviewer": "Reviewer_4Akt", "summary": "This paper explores the dilemma faced by firms when considering strategic data sharing with their competitors. It introduces a framework to investigate the incentives for data sharing and examines its impact on collaboration and profitability. The author discusses the barriers to data sharing, such as privacy concerns, and proposes a market model, data impact model, and collaboration scheme as components of the framework. The findings suggest that reduced competition and harder learning tasks foster collaboration. An illustrative example of a taxi market is provided to demonstrate the concepts discussed. Overall, this study aims to understand how market competition affects collaborative learning incentives and provides insights into the data-sharing trade-off. \n", "questions": "- The study mentions the use of a data impact model grounded in learning theory, but it doesn't provide detailed information about the model's validity and discussion about alternative formulations. Is there any other forms of data impact model could also be considered?\n\n- I can see there are plenty extensions provided in the supplementary materials, is there a way to get comparable results for an analog of the problem that consider a sequential game(such as Stackelberg) rather than simultaneous game? This set up may happens in some mega tech companies get the drop on some other small firms and most often the big companies take the advantages. \n\n- The big companies tend to have more bargaining power than small size firms, and the cost functions are often asymmetric. Does Theorem 6.1 still holds when bigger company has different cost function as it is discussed in C.3?\n", "limitations": "The authors didn't discuss the limitations of their theoretical results. This study is mainly theoretical and primarily focused on introducing a novel framework on strategic data sharing dilemma. There are still plenty rooms for in-depth investigations into the data sharing studies.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The study introduces a novel and comprehensive framework to analyze data sharing between competitors, considering factors such as machine learning model quality's impact on production cost.\n\n- The study investigates the incentives for data sharing and examines the impact of market conditions, product similarities, the complexity of learning task and firm size on collaboration incentives. Albeit it's simply based on a conventional market model and a natural model of data impact grounded in learning theory, the findings are inspiring and the exposition of the results is clear and easy to follow.\n", "weaknesses": "- Since this research is primarily about economic modeling and analysis, it would be great to see a discussion of the various real-world examples in the context rather than just one taxi market example and a oil-market example in the appendix.\n- This study uses simulation to examine collaboration incentives among multiple firms, but it would be valuable to discuss the limitations of the simulations and potential biases inherent in such studies.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable and constructive feedback. In the following we address the weaknesses and questions raised in your review.\n\n**Weaknesses**\n\n**It would be great to see a discussion of the various real-world examples in the context**\n\nThank you for your valuable suggestion. We will add more examples of industries where such problems arise in the next version of the manuscript. We expect that such problems are particularly relevant for industries such as agriculture, finance, and healthcare (Durrant et al., 2022; FedAI; Rieke et al., 2020), as well as other industries interested in cross-silo FL (Kairouz et al., 2021).\n\nDurrant, A., Markovic, M., Matthews, D., May, D., Enright, J., and Leontidis, G. The role of cross-silo federated learning in facilitating data sharing in the agri-food sector. In: *Computers and Electronics in Agriculture*, 2022.\n\nFedAI. WeBank and Swiss Re signed Cooperation MoU. https://www.fedai.org/news/webank-and-swiss-re-signed-cooperation-mou\n\nRieke, N., Hancox, J., Li, W., Milletari, F., Roth, H. R., Albarqouni, S., Bakas, S., Galtier, M. N., Landman, B. A., and Maier-Hein, K. The future of digital health with federated learning. In: *NPJ digital medicine*, 2020.\n\nKairouz, Peter, et al. \"Advances and open problems in federated learning.\" In: *Foundations and Trends in Machine Learning*, 2021.\n\n**It would be valuable to discuss the limitations of the simulations and potential biases inherent in such studies.**\n\nThank you for pointing this out. We provide a brief discussion here and will update the manuscript's next version accordingly.\n\nFirst, our simulation solves each instance of the data-sharing game exactly, so we do not anticipate any biases coming from here. Moreover, given that we average over many independent runs (10000), we expect our results to be very close to the true expectation.\n\nSecond, we analyze this game to test if the findings from Sections 5 and 6 apply to more complex scenarios involving multiple companies. Therefore, we see Section 7 as further validation of the theoretical findings in the paper rather than a complete characterization of the data-sharing trade-off between multiple firms. Investigating other negotiation schemes between multiple companies will certainly be an interesting direction for future work.\n\n**Questions**\n\n**No detailed information about the data impact model's validity and discussion about alternative formulations. Can other forms of data impact models be considered?**\n\nWe refer to the shared response for a discussion on the practical evaluation of the data impact (and market) model. Additionally, please see our response to Reviewer Jw59 for how we expect our results to transfer in case of other data scaling laws suggested by Viering et al. 2022. We also note that Appendices C.3 and C.4 briefly discuss different data impact models. We will make sure to better highlight these aspects in future versions of the manuscript.\n\nOverall, we consider our parametric family to be quite general, at least in the i.i.d. setting, since it covers multiple model-quality convergence rates grounded in standard statistics and ML theory frameworks. In practical applications, we expect the best parametric form of the data impact model to be sensitive to the real-world context.\n\n**Is there a way to get comparable results for an analog of the problem that consider a sequential game (such as Stackelberg) rather than simultaneous game?**\n\nThank you for the valuable suggestion. There are several ways to include sequential decision-making in our framework. One can assume a sequential competition (for example, according to the Stackelberg model) during the competition phase. Coalition formation could also be made sequential (as in Section 7). If one wants to cover the entry and exit of the firms, they could also iterate our data-sharing game multiple times (similar to super-games in the industrial organization literature). We would be happy to discuss other possibilities if the reviewer provides more details about their setup of interest.\n\n**Does Theorem 6.1 still hold when the bigger company has different cost function as it is discussed in C.3?**\n\nQualitatively the solution to the bargaining problem should not change, but the numerical constants will be different."}, {"review_id": "rWBmChTEAt", "reviewer": "Reviewer_DZoM", "summary": "The authors present a trainable model for optimising the benefits of data sharing based valuation of data, benefits of the shared data sets for the inference model, in a conventional market model.  The authors point out that the model can be used to find an optimal data sharing strategy.\n\nThe model is nicely developed and mathematically sound, presented clearly in an understandable manner. It states the assumptions that are used in the development of the model and provides a good a good summary of the consequences, the effect of the simplicity of the learning task, and similarity of the products build using the shared data sets. The mean coalition size is shown as function of these parameters. ", "questions": "I would like the authors to address the points discussed in the weaknesses part of the review, especially redoing their data value analysis based on cases where not including the data from others would lead to discriminative AI models that are generally not acceptable.\n\nAlso, even the current analysis should contain the estimation how long time would it take to build an own model (and estimate the cost of being late in the market with AI features) compared to the complications of sharing the data. The equivalent extra time for a attaining a similar product alone could be estimated already with the current model in the manuscript.", "limitations": "The libations are discussed in the weakness part - as these are the major weakness, of the otherwise very good paper.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper is written and goes through the relevant steps in creation of the model.  It provides excellent argumentation and conclusions from the assumptions made.", "weaknesses": "The biggest weakness I see in the paper is its assumption that the collaborative partners share the same data distribution and the data is modelled as iid samples of a common distribution.  This is generally the situation when there less sense of sharing data, as one has the capability, in time,  to  the get a representative data set that is enough for an highly accurate inferred model.\n\nHowever, the real problem, is in a case that the data sharing partners work with differently distributed,  where the parties are do not have the capability to reproduce the data that the other parties have. This is for example the case where medical X-rays are taken with different X-ray machines, in diverse set of hospitals and the aim is to build an inference model that works generally with a variable types of machines.  An other example is, again in the medical domain, where the ethnic origin of the subjects has to be taken into account. Then one cannot really build a good model that is not ethnically non-discriminating without schemes of sharing data.\n\nAn other aspect that has not been taken account, is the effects of legislation, for example EZU Data governance act that mandates data sharing for data recorder from what the act calls connected devices. \n\nAlso, the effects of data sharing based on EU GDP  rights for subjects, not companies, to share their personal data with third parties. These require more complicated data strategy consideration that should be discussed to make the paper valuable for real evaluation of companies data sharing sharing strategies ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable and constructive feedback. In the following we address the weaknesses and questions raised in your review.\n\n**Weaknesses**\n\n**The assumption that the collaborative partners share the same data distribution and the data is modeled as iid samples of a common distribution.** \n\nWe certainly agree that heterogeneity can be an important aspect in some collaborative learning settings. Please refer to our shared response for a discussion of the numerous challenges and orthogonal incentives arising when considering a heterogeneous setting. In particular, we would like to highlight that since directly incorporating data from other distributions may actually damage the model performance (due to the additional data being too different), heterogeneity yields additional strategic data-sharing considerations (Donahue & Kleinberg, 2021), which are orthogonal to the trade-off considered in this work between improving your model versus risking increased competition. \n\nAdditionally, we note that we consider a simple model of heterogeneity in Appendix C.4. We believe that our framework allows for considering even more intricate models of heterogeneity, providing that an appropriate data impact model is formulated. We certainly agree with the reviewer that this is an interesting direction for future work.\n\nDonahue, K. and Kleinberg, J. Model-sharing games: Analyzing federated learning under voluntary participation. In: *AAAI Conference on Artificial Intelligence*, 2021.\n\n**One cannot really build a good model that is not ethnically non-discriminating without schemes of sharing data (in the heterogeneous case).**\n\nWe certainly agree that fairness and non-discrimination are important concerns in collaborative learning. In fact, one can explicitly analyze fairness concerns within our framework. To do so, one could consider the impact of fairness constraints on learning outcomes by changing the data impact model and adding some diversity constraints on the action spaces of the firms within their collaboration scheme.\n\nAdditionally, we note that already in the homogeneous case, our framework provides a possibility to improve these measures in practice. In particular, even for homogeneous data, the performance of the ML models on rare population groups can be significantly improved by having access to more data from the same distribution (and therefore observing more of the rare group samples).\n\n**Another aspect that has not been taken account, is the effects of legislation, for example EZU Data governance act that mandates data sharing for data recorder from what the act calls connected devices.**\n\nThank you for the valuable suggestion. One can explicitly incorporate and study the effect of legislation using our framework by constraining the action spaces of the firms in the collaboration scheme (for example, by forcing them to share some parts of data regardless of other data-sharing actions). We see this as an exciting direction for future work, though orthogonal to the current focus of the paper, which analyzes the data-sharing trade-off from the perspective of market incentives only.\n\nAdditionally, we believe that our work is aligned with the values and objectives of the EU Data Governance Act since it shows that data sharing can be beneficial even between competing market participants and without the presence of legal requirements to share data.\n\n**The effects of data sharing based on EU GDP rights for subjects, not companies, to share their personal data with third parties … should be discussed to make the paper valuable for real evaluation of companies data-sharing strategies.**\n\nThank you for the valuable suggestion. We will discuss data ownership aspects in the next version of the manuscript. We expect that this requirement can also be incorporated into our framework. For example, in Section 6, one can implement sharing consent constraints by constraining the firms’ choices of $\\lambda$. While, previously, the firms were able to share all data $\\lambda \\in [0, 1]$, now they can share only data of people who agree with data sharing $\\lambda \\in [0, \\lambda_\\text{consent}]$.\n\n**Questions**\n\n**Even the current analysis should contain the estimation how long time would it take to build an own model (and estimate the cost of being late in the market with AI features) compared to the complications of sharing the data. The equivalent extra time for a attaining a similar product alone could be estimated already with the current model in the manuscript.**\n\nThank you for the valuable suggestion. While training and local data collection costs are certainly interesting to consider, we see these aspects as orthogonal to the focus of our work, which is on the trade-off between receiving access to competitors' data and risking increased market competition. While training can be a bottleneck in some cases (for example, in the context of foundation models), we target the fairly common situation where the lack of sufficient data (and the expensiveness of its collection) are of a much bigger concern.\n\nAdditionally, we are unsure how the reviewer suggests to estimate the training costs using our current framework. In order to estimate the effect of being late with the ML model, one needs to model temporal characteristics of the market, as well as costs for local data generation. Both of these aspects are not currently modeled.\n\nWe will be happy to expand on our answer, if the reviewer's concern was related to different effects or some other misunderstanding occurred."}, {"review_id": "xcWNe3X1nu", "reviewer": "Reviewer_JkGv", "summary": "This paper uses algorithmic game theory to study potential data sharing between competing firms that train machine learning models with similar goals. The authors propose a new framework that can be used to analyze trade-offs faced by data-using agents who make decisions about whether to collaborate (via data sharing) to improve their own ML performance, while potentially losing out on profits because they also improved their competitors ML performance.\n\nThe authors draw on conventional market models from academic economics to make predictions about how different structural factors might impact collaboration decisions. This analysis has implications for the development of regulations and norms pertaining to data sharing.\n", "questions": "Early in the Introduction, free-riding and non-collaboration concerns are mentioned. The authors may want to engage briefly with sociological work on collective action (though perhaps this better suited for future work and out of scope for this paper). I think a big open question for future work along these lines is which types of data are best handled with an “economic-leaning” model vs a “sociological-leaning” model that includes non-monetary incentives facing individual data generating agents.\n\nIn general, the framework also seems to lack any notion of the possibility of commons or public goods. I could imagine adding an additional stage (or several) to the game to account for this. Fully engaging with this topic is almost certainly out of scope given the current space constraints, but perhaps worth a brief mention. \n\nWhile I expect the core audience for this kind of paper (i.e. readers familiar with some of the references already or interested in data collection games) will follow most sections, there’s potential to strengthen the broader impact of the paper just by adding a bit more high-level summarization of each sections.. Sections 3.1 and the end of Section 5 do a nice job of this kind of discussion, but there’s opportunities to emphasize this kind of recap in other sections.\n\nAs a minor comment, it may help the paper to discuss whether data-dependent products lend themselves to Bertrand vs. Cournot competition. It seems Cournot may be preferred, but I didn’t quite understand why. This relates to my main high-level comment about the work, which is that I think readers will want to know how different assumptions here map to different data-dependent technology contexts.\n", "limitations": "The work is reasonable in discussing its own limitations.\n\nIn terms of societal implications, I think the discussion in the paper is commensurate to potential concerns. Overall, the paper is a bit light on discussion of how data sharing would work in practice (not touching on topics like public goods, anti-trust concerns, consumer welfare, etc.) but I think this is reasonable given space constraints and paper scope.\n\nAs noted above, I think some engagement with literature on collective action, commons, and public goods could be insightful, or even a brief mention of why the authors think the collective action / commons perspective on data-dependent technologies is not relevant.\n", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This paper has a number of strengths. The motivation, combination of theoretical arguments with simulation evidence, and reasonable use of assumptions all stood out.\n\nFirst, the paper’s motivation is quite strong. The authors argue that the incentives underlying data sharing between machine learning operating firms is understudied, and that using the lens of theoretical modeling can highlight how market conditions might impact sharing behaviors.\n\nThe work combines theory (drawing on published economics literature) and a simulation experiment. I thought this combination was convincing: the experimental component is likely to help readers understand the implications of the theoretical framework.\n\nThere are a lot of assumptions at play, but they seem fairly plausible on the whole. Thinking in terms of whether this theory-focused paper could guide practical decision-making (by firms) or policy making (by regulators), the current draft devotes enough attention to arguing for plausibility. I do think there’s room to see more discussion of where these assumptions are more or less plausible (see below), though this may be out of scope for a theory paper that’s trying to propose a new framework and ask for some re-thinking.\n", "weaknesses": "In my view, the main threat to the validity and impact of the work is the reliance on assumptions from conventional economics. The authors are very upfront about signposting which prior works are most foundational (e.g. the 1979 work defining representative consumer with quasi-linear quadratic utility, and classical models of competition that focus on quantities vs. prices) but empirical validation could help this line of work have more impact in the long-term. \n\nPut another way, the current draft does a great job of pointing out relevant literature that it builds off, but could do more to explicitly state why certain assumptions fit a specific empirical context. Of course, the authors may not wish to zoom in on a specific context, which is a fair choice for scoping the paper.\n\nThe paper also discusses data as a “key asset” (e.g. in the Introduction) in a very broad sense. None of the claims are situated relative to specific use cases of data and ML. Examples like ad tech and a “production process” are gestured at, and of course the running example of taxi driver scheduling is helpful. On the whole, however, I think it may be hard for readers to assess how contextually dependent some of the claims and results are. Overall, the impact of the work could be improved by clarifying the extent to which these analyses are or are not contextually dependent (even in terms of specific numerical characterizations of data scaling behavior / “data impact model”).\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable and constructive feedback. In the following we address the weaknesses and questions raised in your review.\n\n**Weaknesses**\n\n**In my view, the main threat to the validity and impact of the work is the reliance on assumptions from conventional economics … The current draft … could do more to explicitly state why certain assumptions fit a specific empirical context. Of course, the authors may not wish to zoom in on a specific context, which is a fair choice for scoping the paper.**\n\nThank you for your comments. We certainly agree that designing and validating market and data impact models for specific applications is important for the framework's applicability, and we deem this direction interesting for future work. The current paper indeed stays away from focusing on a specific real-world application: please refer to our general response for a thorough justification on this matter. \n\n**The paper also discusses data as a \"key asset\" (e.g. in the Introduction) in a very broad sense. None of the claims are situated relative to specific use cases of data and ML … Overall, the impact of the work could be improved by clarifying the extent to which these analyses are or are not contextually dependent**\n\nThank you for your valuable suggestion. Indeed, the parts of the framework may vary depending on the context, due to the generality of the framework. We plan to provide further explanation on the framework adaptation by giving more specific examples of relevant industries (also suggested by Reviewer 4Akt) and discussing example numerical characterizations of our models (also suggested by Reviewer Jw59). \n\n**Questions**\n\n**The authors may want to engage briefly with sociological work on collective action (though perhaps this better suited for future work and out of scope for this paper). I think a big open question for future work along these lines is which types of data are best handled with an \"economic-leaning\" model vs a \"sociological-leaning\" model that includes non-monetary incentives facing individual data generating agents.**\n\nThank you for your valuable suggestion. We certainly agree that analyzing the differences between economic and sociological modeling of data sharing is an exciting direction for future work. We believe that our framework can capture other (e.g., sociological) data-sharing approaches by adapting the collaboration scheme. We see this as orthogonal to the focus of the current work, which studies non-cooperative games since they align with the classic economic modeling of firms as rational, profit-driven, and self-interested. However, we will be happy to add a discussion on possible sociological extensions as interesting future work.\n\nIn fact, Appendix E provides some evidence about the relevance of collective action theory toward finding widely beneficial schemes for data sharing. This section analyzes global welfare in the context of the data-sharing problem. The result suggests that the welfare is maximized when data sharing occurs (in the case of full data sharing between two firms).\n\n**In general, the framework also seems to lack any notion of the possibility of commons or public goods. I could imagine adding an additional stage (or several) to the game to account for this. Fully engaging with this topic is almost certainly out of scope given the current space constraints, but perhaps worth a brief mention.**\n\nWe certainly agree that the public goods perspective might be relevant here since data is non-rival. However, data does not seem to be the commons because it belongs to either companies or individuals in most legal systems. At the same time, machine learning models and data do not seem to be public goods since they are excludable. Therefore, we are unsure how to include the discussion of public goods in our framework. That said, one aspect that may be possible to model is the creation of public datasets (e.g., ImageNet).\n\n**There's potential to strengthen the broader impact of the paper just by adding a bit more high-level summarization of each sections.**\n\nThank you for the valuable suggestion. We will seek to provide further summaries and intuition for the next version of the paper.\n\n**It may help the paper to discuss whether data-dependent products lend themselves to Bertrand vs. Cournot competition. It seems Cournot may be preferred, but I didn't quite understand why. I think readers will want to know how different assumptions here map to different data-dependent technology contexts.**\n\nWe focus on the Bertrand and Cournot competition because these models are the most popular in empirical research and are a basis for all other competition models. In some cases, we only provided the analysis for the Cournot model since the derivations were less tedious with the parametrization of the market model adopted in the paper. However, we expect these results to also transfer to the Bertrand model.\n\nRegarding the mapping to different contexts, completely answering this question is beyond speculation on our side, requiring a separate economic study since we are not aware of any current economic work that empirically studies the question of data sharing for machine learning (see also our main response). We hope to discuss how one could adopt our framework to real-world contexts in the next version of our paper, for instance by covering examples of relevant industries and numerical characterizations of our models, as mentioned above. "}]

Author rebuttal:
We thank all reviewers for their valuable and constructive feedback. Below we provide a general response to several recurring comments. We also offer individual answers for all remaining questions.

**Real-world evaluation of the market and data impact models.**

Multiple reviewers brought up the importance of real-world evaluation of the considered market and data impact models studied in Sections 4-7. While we recognize the significance of this concern, we see the core contributions of our paper in a general framework for the data-sharing trade-off and general qualitative insights into this problem. Our approach is more theoretical in nature and does not aim to provide a fully-realistic model that can directly inform practitioners. We will make sure to further clarify the scope of our work in the next version of the manuscript.

We opted for a theoretical approach for the following reasons.

First, we do not anticipate a one-size-fits-all model to cover all data-sharing scenarios, considering the diversity of relevant industries (e.g., online platforms, finance, agriculture, healthcare) and machine learning tasks (e.g., mean estimation, regression, deep learning). In practice, designing the most appropriate models will likely be an application-dependent problem, and we hope that our three-component decomposition of the data-sharing problem will help practitioners to leverage their situation-specific knowledge of their market and ML models.

Moreover, both economics and machine learning lack a definitive theory to predict parametric forms of market and data impact models for particular situations. In economics, market modeling is more often art than science, and parametric models (e.g., Berry et al., 1995; Nevo, 2001) require the scope of one economic article to explain and justify (since they require us to model counterfactuals). In machine learning, the generalization of deep models, especially for non-i.i.d. data, remains poorly understood.

Third, already with the basic market and data impact models, a complete analytical solution to the problem in Sections 6 and 7 remains elusive (although we provided various qualitative insights about them). With more nuanced market and data impact models, we expect to get only numerical results without rigorous theoretical characterizations. At the same time, we used a market model widely adopted in the theoretical literature (Choné & Linnemer, 2019) and a data impact model motivated by established theoretical frameworks in machine learning (Tsybakov, 2004). Given the wide adoption of these parametric families, we hope that our results will be useful and qualitatively valid in real-world data-sharing settings. 

**The assumption of homogeneous data**

We certainly agree that heterogeneity is an important concern in collaboration learning. In case of significant heterogeneity, our framework will only require appropriate changes to the data impact model. We expect that designing such a model is a significant challenge orthogonal to the focus of our work (see, for example, Gulrajani & Lopez-Paz, 2020). Moreover, in some cases, additional data from a different distribution may damage model performance, yielding additional data-sharing considerations (Donahue & Kleinberg, 2021). Given these concerns, we decided to focus on the homogeneous case and also discuss a simple model of heterogeneous learning in Appendix C.4. We certainly agree the analysis of heterogeneity is an exciting direction for future work.

Berry, S., Levinsohn, J., and Pakes, A. Automobile Prices in Market Equilibrium. In: *Econometrica: Journal of the Econometric Society*, 1995.

Choné, P. and Linnemer, L. The quasilinear quadratic utility model: An overview. *CESifo Working Paper*, 2019.

Donahue, K. and Kleinberg, J. Model-sharing games: Analyzing federated learning under voluntary participation. In: *AAAI Conference on Artificial Intelligence*, 2021.

Gulrajani, I. and Lopez-Paz, D. In Search of Lost Domain Generalization. In: *International Conference on Learning Representations (ICLR)*, 2020.

Nevo, A. Measuring market power in the ready‐to‐eat cereal industry. In: *Econometrica*, 2001.

Tsybakov, A. B. Optimal aggregation of classifiers in statistical learning. In: *The Annals of Statistics*, 2004.

Meta-review: This paper presents a general framework for analyzing data-sharing trade-off and analysis of an instance of this framework. Some interesting insights can be drawn from the analysis. We hope the reviews and feedbacks during the discussions could be helpful for the authors to better prepare the final version of this paper.
Decision: Accept (poster)