SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Nikita Tsoy
INSAIT, Sofia University
nikita.tsoy@insait.ai
nikola.konstantinov@insait.ai
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.
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., 2022). 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 to enhance data access and machine learning models is collaboration via data sharing (Rieke et al., 2020; Durrant et al., 2022). Data-sharing schemes can bring benefits in many industries, such as agriculture, finance, and healthcare (Durrant et al., 2022; FedAI; Rieke et al., 2020). However, at least two barriers exist to such collaborations. The first is privacy and the associated regulatory concerns, which can be 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 at all, free-ride (Blum et al., 2021; Karimireddy et al., 2022), or attack the shared model (Blanchard et al., 2017).
Such conflicting incentives arise naturally between market competitors. On the one 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 act 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 to capture
more revenue<sup>1</sup> or engage in a price war to win a market.<sup>2</sup> 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 is 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.
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., 2021). Some notable lines of work concern incentives under data heterogeneity (Donahue & Kleinberg, 2021; Werner et al., 2022); compensation for computation, energy, and other costs related to training (Yu et al., 2020; Tu et al., 2022; Liu et al., 2022); fair distribution of the benefits resulting from collaboration (Lyu et al., 2020a,b; Blum et al., 2021); and free-riding (Richardson et al., 2020; Karimireddy et al., 2022). We refer to Zeng et al. (2021); Yang et al. (2020); Zhan et al. (2021) for a detailed overview.
A 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 on 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 as 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 an 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 et 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.
<sup>1</sup>https://www.nytimes.com/2021/01/17/technology/google-facebook-ad-deal-antitrust.html
$^{2}$ https://www.reuters.com/article/us-uber-results-breakingviews-idUSKCN1UY2X5
3 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 a taxi market dominated by a few firms. Each firm collects data (e.g., demand for taxis and traffic situation) to 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 in a 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  $m$  firms,  $F_{1},\ldots ,F_{m}$ , each producing  $q_{i}\in \mathbb{R}_{+}$  units of good  $G_{i}$  and offering them to consumers at price  $p_i\in \mathbb{R}_+$ , where  $\mathbb{R}_+ = [0,\infty)$ . In our example,  $G_{1},\dots,G_{m}$  represent taxi services from different companies, with consumers being city travelers,  $q_{i}$  are the total number of kilometers serviced by company  $i$ , and  $p_i$  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 to standard market models, our framework allows product utilities and costs to depend on the qualities of the machine learning models of the firms  $\pmb{v} = (v_{1},\dots,v_{m})$ . 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  $u^{j}(\pmb{g}^{j}, \pmb{q}^{j}, \pmb{v})$  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  $\pmb{q}^j = (q_1^j, \dots, q_m^j)$  of products  $G_1, \dots, G_m$ . The second is the machine learning models' qualities  $\pmb{v}$ , as these may impact the utility of the corresponding products. The last one is consumed quantities  $\pmb{g}^j = (g_1^j, \dots, g_k^j)$  of goods outside the considered market (e.g., consumed food in our taxi example). Assuming that each consumer  $j$  can only spend a certain budget  $B^j$  on all goods, one obtains the following optimization problem
$$
\max  _ {\boldsymbol {g} ^ {j}, \boldsymbol {q} ^ {j}} u ^ {j} \left(\boldsymbol {g} ^ {j}, \boldsymbol {q} ^ {j}, \boldsymbol {v}\right) \text {s . t .} \sum_ {l = 1} ^ {k} \tilde {p} _ {l} g _ {l} ^ {j} + \sum_ {i = 1} ^ {m} p _ {i} q _ {i} ^ {j} \leq B ^ {j}, \tag {1}
$$
where  $\tilde{\pmb{p}}$  are the outside products' prices (which we consider fixed). The solution to this problem  $q_{i}^{j,*}(\pmb {p},\pmb {v})$  determines the aggregate demanded quantity of goods
$$
q _ {i} (\boldsymbol {p}, \boldsymbol {v}) := \sum_ {j} q _ {i} ^ {j, *} (\boldsymbol {p}, \boldsymbol {v}). \tag {2}
$$
The functions  $q_{i}(\pmb{p}, \pmb{v})$  (known as the demand equations) link the prices  $p_{i}$  and the demanded quantities  $q_{i}$  and constitute the first  $m$  restrictions in our setting.
Firms' behavior Firms maximize their expected profits, the difference between revenue and cost,
$$
\pi_ {i} ^ {e} = \mathbf {E} _ {\boldsymbol {v}} \left(p _ {i} q _ {i} - C _ {i} \left(q _ {i}, v _ {i}\right)\right). \tag {3}
$$
Here  $C_i(q_i, v_i)$  is the cost of producing  $q_i$  units of good for the firm  $F_i$ , and the expectation is taken over the randomness in the models' quality, influenced by the dataset and the training procedure. Note that the model quality is 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,  $C_i$  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, 1838); or 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  $F_{i}$  possesses a dataset  $D_{i}$  that can be used to train a machine learning model (e.g., trip data in the taxi example) and may opt to participate in a data-sharing agreement with some other firms. Denote the final dataset the company acquired by  $D_{i}^{c}$ . The company then uses the dataset to train a machine learning model. We postulate two natural ways that the model quality  $v_{i}$  can impact a company. The first one is by reducing the company's production costs  $C_{i}(q_{i},v_{i})$ , for example, by minimizing time in traffic jams. The second is by increasing the utility of its products  $q_{i}(\pmb {p},\pmb {v})$ , 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  $\Pi_i^e$ . 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 are present, it is natural that they will agree on 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, as a firm needs to reason not only about the 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 a 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 on the data-sharing trade-off, we make several simplifying assumptions: the firms' data is homogeneous, training costs are negligible, and legal constraints are not present or are automatically fulfilled.
4.1 Market model
In order to 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  $u^{j}$  and cost functions  $C_{i}$  and specify the competition game. To this end, we use a 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 (1) can be described by a representative consumer with quasi-linear quadratic utility (QQUM, Dixit 1979)
$$
\max  _ {g, \boldsymbol {q}} u (g, \boldsymbol {q}) := \sum_ {i = 1} ^ {m} q _ {i} - \left(\frac {\sum_ {i} q _ {i} ^ {2} + 2 \gamma \sum_ {i > j} q _ {i} q _ {j}}{2}\right) + g = \boldsymbol {\iota} ^ {\intercal} \boldsymbol {q} - \frac {\boldsymbol {q} ^ {\intercal} \mathbf {G} \boldsymbol {q}}{2} + g \text {s . t .} g + \boldsymbol {p} ^ {\intercal} \boldsymbol {q} \leq B. \tag {4}
$$
Here,  $\pmb {\iota} = (1,\dots ,1)^{\mathsf{T}}$ $\mathbf{G} = (1 - \gamma)\mathbf{I} + \gamma \pmb{\mu}^{\mathsf{T}}$  and  $g$  is the quantity from a single outside good.
Additionally,  $\gamma \in \left(-\frac{1}{m - 1}, 1\right)$  is a measure of substitutability between each pair of goods: higher  $\gamma$  corresponds to more similar goods. In our running example,  $\gamma$  describes the difference in service of two taxi companies, such as the difference in the cars' quality or the location coverage. We refer to Chone & 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 (2) is well-known.
Lemma 4.1 (Amir et al. 2017). Assume that  $\mathbf{G}^{-1}(\boldsymbol {\iota} - \boldsymbol {p}) > 0$  and  $\pmb{p}^{\mathrm{T}}\mathbf{G}^{-1}(\pmb {\iota} - \pmb {p})\leq B$ . The solution to the consumer problem (4) is
$$
p _ {i} = 1 - q _ {i} - \gamma \sum_ {j \neq i} q _ {j} \Longleftrightarrow q _ {i} = \frac {1 - \gamma - p _ {i} - \gamma (m - 2) p _ {i} + \gamma \sum_ {j \neq i} p _ {j}}{(1 - \gamma) (1 + \gamma (m - 1))}. \tag {5}
$$
The technical conditions  $\mathbf{G}^{-1}(\boldsymbol {\iota} - \boldsymbol {p}) > 0$  and  $\pmb{p}^{\mathrm{T}}\mathbf{G}^{-1}(\boldsymbol {\iota} - \boldsymbol {p})\leq B$  ensure that the consumers want to buy at least a bit of every product and do not spend all of their money in the considered market, respectively. For completeness, we prove Lemma 4.1 in Appendix A.1.
4.1.2 Supply
We assume that the cost functions (3) are linear in the quantities
$$
\pi_ {i} ^ {e} = \mathbf {E} \left(p _ {i} q _ {i} - c _ {i} q _ {i}\right), \tag {6}
$$
where the parameter  $c_{i}$  depends on the quality of the machine learning model. We denote  $c_{i}^{e} = \mathbf{E}_{D^{c}}(c_{i})$ .
Remark 4.2. Here we assume that the quality of the machine learning model only affects the production costs, and not the utilities. However, in this market model, it 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  $w_{i}$  and it increases the effective amount of good in the consumer's utility (4), resulting in utility  $u(g, w_{1}q_{1}, \ldots, w_{m}q_{m})$ . This model is equivalent to the model above after substituting  $q_{i}$ ,  $p_{i}$ , and  $c_{i}$  with their effective versions  $\tilde{q}_{i} = w_{i}q_{i}$ ,  $\tilde{p}_{i} = p_{i} / w_{i}$ , and  $\tilde{c}_{i} = c_{i} / w_{i}$ .
We now derive the remaining constraints on the prices and costs. Note that these equilibria depend on the expected costs  $c_{i}^{e}$  and hence on the quality of the machine learning models.
Cournot competition Each firm acts by choosing an output level  $q_{i}$ , 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  $c_1^e, \ldots, c_m^e$ , and companies maximize their profits (6) in the Cournot competition game. If  $\forall i (2 - \gamma)(1 - c_i^e) > \gamma \sum_{j \neq i} (c_i^e - c_j^e)$ , Nash equilibrium quantities and profits are equal to
$$
q _ {i} ^ {*} = \frac {2 - \gamma - (2 + \gamma (m - 2)) c _ {i} ^ {e} + \gamma \sum_ {j \neq i} c _ {j} ^ {e}}{(2 - \gamma) (2 + (m - 1) \gamma)}, \Pi_ {i} ^ {e} = (q _ {i} ^ {*}) ^ {2}.
$$
Bertrand competition Each firm acts by setting the price  $p_i$  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  $c_1^e, \ldots, c_m^e$ , and companies maximize their profits (6) in the Bertrand competition game. If  $\forall i d_1(1 - c_i^e) > d_3 \sum_{j \neq i} (c_i^e - c_j^e)$ , Nash equilibrium prices and profits are equal to
$$
p _ {i} ^ {*} = \frac {d _ {1} + d _ {2} c _ {i} ^ {e} + d _ {3} \sum_ {j \neq i} c _ {j} ^ {e}}{d _ {4}}, \Pi_ {i} ^ {e} = \frac {(1 + \gamma (m + 2)) (p _ {i} ^ {*} - c _ {i} ^ {e}) ^ {2}}{(1 - \gamma) (1 + \gamma (m - 1))},
$$
where  $d_1, \ldots, d_4$  depend only on  $\gamma$  and  $m$ .
In both lemmas, the technical conditions  $\forall i (2 - \gamma)(1 - c_i^e) > \gamma \sum_{j \neq i} (c_i^e - c_j^e)$  and  $\forall i d_1(1 - c_i^e) > d_3 \sum_{j \neq i} (c_i^e - c_j^e)$  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 to understand how additional data impacts their costs. Here, we consider the case where all datasets are sampled from the same distribution  $\mathcal{D}$ . 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  $c_{i}^{e} = c^{e}(n_{i}^{c})$ , where  $n_i^c = |D_i^c|$  is the number of points the company  $i$  has access to. Using the examples below, we motivate the following functional form for  $c^{e}(n)$
$$
c ^ {e} (n) = a + \frac {b}{n ^ {\beta}}, \beta \in (0, 1 ]. \tag {7}
$$
In this setting,  $\beta$  indicates the difficulty of the learning task, higher  $\beta$  corresponds to a simpler task (Tsybakov, 2004),  $a$  represents the marginal costs of production given a perfect machine learning model, while  $a + b$  corresponds to the cost of production without machine learning optimizations. Additionally, we assume that  $a < 1$  and  $\frac{b}{1 - a}$  is small enough, so that the technical requirements of Lemmas 4.1, 4.3, and 4.4 are satisfied (see Equations (10) and (11)). 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  $s \in \mathbb{R}^n$  during production that will impact its costs. However, there is uncertainty in the production process coming from random noise  $X$  valued in  $\mathbb{R}^m$ . Thus, the cost of production is a random function  $c(\boldsymbol{s},\boldsymbol{X}) \colon \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}_+$ .