SlowGuess/NeurIPS_2023_OCR huggingface dataset:
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 to estimate them. Suppose a firm uses the maximum likelihood estimator to find the parameters and chooses the optimal action  $s_{\mathrm{fin}}$  based on this estimate. In this case, the result of optimization  $\mathbf{E}_X(c(s_{\mathrm{fin}}, X))$  will approximately have a generalized chi-square distribution (Jones, 1983), resulting in the following approximation
$$
\mathbf {E} _ {D, \boldsymbol {X}} \left(c \left(\boldsymbol {s} _ {\text {f i n}}, \boldsymbol {X}\right)\right) \approx a + \frac {b}{n}. \tag {8}
$$
We refer to Appendix A.4 for further details. This result implies the same dependence of the expected marginal costs on the total number of samples  $n$  as Equation (7) with  $\beta = 1$ .
Stochastic optimization A similar dependence arises when the company uses stochastic optimization to directly optimize the expected cost  $c(s) = \mathbf{E}_X(c(s, X))$ . If the company uses an algorithm with provable generalization guarantees (e.g., SGD with a single pass over the data, Bubeck 2015) and the function  $c(s)$  is strongly convex, the outcome  $s_{\mathrm{fin}}$  will satisfy
$$
\mathbf {E} _ {D} \left(c \left(\boldsymbol {s} _ {\text {f i n}}\right) - c \left(\boldsymbol {s} ^ {*}\right)\right) = \mathrm {O} \left(\frac {1}{n}\right),
$$
where  $s^*$  is the optimal action, resulting in the same dependence as in the previous paragraph.
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 a firm trains a classifier used in the production process. In this case, the cost  $c$  depends on the classifier accuracy  $a$ , and the cost overhead will satisfy
$$
c (a) - c \left(a ^ {*}\right) \approx - c ^ {\prime} \left(a ^ {*}\right) \left(a ^ {*} - a\right),
$$
where  $a^*$  is the optimal accuracy achievable by a chosen family of classifiers. If the family of classifiers has a finite VC dimension  $d$ , a well-known statistical bound (Shalev-Shwartz & Ben-David, 2014) gives
$$
a ^ {*} - a = \operatorname {O} \left(\sqrt {\frac {d}{n}}\right),
$$
motivating the same dependence as Equation (7) with  $\beta = 1 / 2$
5 Data sharing between two firms
Having introduced example market and data impact models, we can specify a 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 at 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 we 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 a part of their dataset for the collaborative learning. While, previously, the competitor gets access to all data of the firm  $n_{-i}^{\mathrm{share}} = n_{-i} + n_i$ , now they will get access to only data of people who agree with data sharing  $n_{-i}^{\mathrm{share}} = n_{-i} + n_{i,\mathrm{consent}}$ .
For Cournot competition, Lemma 4.3 and Equation (7) give the following collaboration criterion
$$
\forall i \quad \Pi_ {\text {s h a r e}} ^ {e} > \Pi_ {i, \text {i n d}} ^ {e} \Longleftrightarrow (2 - \gamma) (n _ {i} ^ {- \beta} - (n _ {1} + n _ {2}) ^ {- \beta}) > \gamma (n _ {- i} ^ {- \beta} - n _ {i} ^ {- \beta}),
$$
where  $n_{-i}$  is the size of the data of the player that is not  $i$ ,  $\Pi_{i,\mathrm{share}}^e$  is the expected profit in collaboration, and  $\Pi_{i,\mathrm{ind}}^e$  is the expected profit when no collaboration occurs.
Similarly, in the case of Bernstrand competition, Lemma 4.4 gives
$$
\forall i \quad \Pi_ {\text {s h a r e}} ^ {e} > \Pi_ {i, \text {i n d}} ^ {e} \iff (2 - \gamma - \gamma^ {2}) (n _ {i} ^ {- \beta} - (n _ {1} + n _ {2}) ^ {- \beta}) > \gamma (n _ {- i} ^ {- \beta} - n _ {i} ^ {- \beta}).
$$
The theorem below describes the properties of these criteria (see proof in Appendix A.5).
Theorem 5.2. In the case of  $\gamma \leq 0$ , it is profitable for the firms to collaborate. In the case of  $\gamma > 0$ , there exists a value  $x_{t}(\gamma, \beta)$ , where  $t \in \{\text{Bertrand}, \text{Cournot}\}$  is the type of competition, such that, for the firm  $i$ , it is profitable to collaborate only with a competitor with enough data:
$$
\varPi_ {s h a r e} ^ {e} > \varPi_ {i, i n d} ^ {e} \iff \frac {n _ {- i}}{n _ {1} + n _ {2}} > x _ {t} (\gamma , \beta).
$$
The function  $x_{t}$  has the following properties:
1.  $x_{t}(\gamma ,\beta)$  is increasing in  $\gamma$
2.  $x_{\text{Bertrand}} \geq x_{\text{Cournot}}$ .
3.  $x_{t}(\gamma ,\beta)$  is increasing in  $\beta$
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 ( $\gamma$  is bigger), making the market less competitive. Similarly,  $x$  becomes smaller in the Cournot setting since it is known to be less competitive than the Bertrand one (Shapiro, 1989). Finally, when the learning task is harder ( $\beta$  is smaller),  $x$  decreases, making collaboration more likely. Intuitively, it happens because the decrease in cost (7) from a single additional data point is higher for smaller  $\beta$ .
In the supplementary material, we explore several extensions for the two firms case. In Appendix C.1 and Appendix C.2, we look at different cost and utility functions, respectively. In Appendix C.3, we consider the case where  $a$ ,  $b$  and/or  $\beta$  might differ among firms and derive an analog of Theorem 5.2. In Appendix C.4, we consider the data-sharing problem with two companies with heterogeneous data in the context of mean estimation. In Appendix E, 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 in the case of Cournot competition. In this setup, each firm  $F_{i}$  chooses to share a fraction  $\lambda_{i} \in [0,1]$  of data with its competitor and expects its costs to be
$$
c _ {i} ^ {e} = c ^ {e} (n _ {i} + \lambda_ {- i} n _ {- i}) = a + b (n _ {i} + \lambda_ {- i} n _ {- i}) ^ {- \beta},
$$
where  $\lambda_{-i}$  is the fraction of data shared by the company that is not  $i$ .
Remark 6.1. Notice that we can incorporate some legal requirements into this scheme by constraining the action spaces of participants. For example, one can implement sharing consent constraints, arising from copyright or other data ownership constraint, 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  $\lambda \in [0,\lambda_{\mathrm{consent}}]$ .
We will use the Nash bargaining solution (Binmore et al., 1986) to describe the bargaining process between the firms. Namely, the firms will try to compromise between themselves by maximizing the following Nash product
$$
\max  _ {\lambda_ {i} \in [ 0, 1 ]} \left(\Pi_ {1} ^ {e} \left(\lambda_ {1}, \lambda_ {2}\right) - \Pi_ {1} ^ {e} (0, 0)\right) \left(\Pi_ {2} ^ {e} \left(\lambda_ {1}, \lambda_ {2}\right) - \Pi_ {2} ^ {e} (0, 0)\right) \text {s . t .} \forall i \Pi_ {i} ^ {e} \left(\lambda_ {1}, \lambda_ {2}\right) - \Pi_ {i} ^ {e} (0, 0) \geq 0. \tag {9}
$$
The Nash bargaining solution is a 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., 1986). 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  $n_1 \geq n_2$ . Additionally, assume  $(1 - a) \gg b$  (in particular,  $b < 5(1 - a)$ ). The solution  $(\lambda_1^*, \lambda_2^*)$  to the problem (9) is
$$
\lambda_ {1} ^ {*} = \tilde {\lambda} _ {1} + O \left(\frac {b}{1 - a}\right), \lambda_ {2} ^ {*} = 1, w h e r e
$$
$$
\tilde{\lambda}_{1} = \left\{ \begin{array}{ll}\frac{n_{2}}{n_{1}}\Bigl(\Bigl(1 - \frac{4 + \gamma^{2}}{4\gamma}\Bigl(\frac{n_{2}}{n_{1}}\Bigr)^{\beta}\Bigl(1 - \Bigl(\frac{n_{1}}{n_{1} + n_{2}}\Bigr)^{\beta}\Bigr)\Bigr)^{-1 / \beta} - 1\Bigr), & \frac{4 + \gamma^{2}}{n_{1}^{\beta}}\leq \frac{4\gamma}{n_{2}^{\beta}} +\frac{(2 - \gamma)^{2}}{(n_{1} + n_{2})^{\beta}},\\ 1 & \text{otherwise}. \end{array} \right.
$$
Moreover,  $\tilde{\lambda}_1$  is a decreasing function in  $\gamma$  and  $\beta$ , and  $\tilde{\lambda}_1 = \frac{4 + \gamma^2}{4\gamma}\left(\frac{n_2}{n_1}\right)^{\beta + 2}(1 + o(1))$  when  $n_2 / n_1 \to 0$ .
Discussion We can see that the results of Theorem 6.2 are similar to those in Section 5. When there is a big difference between the amount of data  $(n_{1} \gg n_{2})$ , the big firm does not collaborate with the small firm  $(\lambda_{1}^{*} = \mathrm{O}((n_{2} / n_{1})^{\beta + 2}))$ . However, when the firms have similar amounts of data, the large firm will share almost all of it  $(\lambda_{1}^{*} \approx 1)$ , and the sharing proportion increases with a decrease in competition (smaller  $\gamma$ ) and an increase in learning task hardness (smaller  $\beta$ ). In contrast to the previous results, the smaller firm always shares all its data with the competitor  $(\lambda_{2}^{*} = 1)$ .
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 a 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, a 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 D we also study alternative formulations: a cooperative setting and two non-cooperative settings with only one non-singleton coalition.
Non-cooperative data-sharing game We order the firms in a decreasing data size order so that firm  $F_{1}$  has the largest dataset. In this order, the firms propose forming a coalition with their competitors. First, the largest company makes up to  $2^{m - 1}$  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 a coalition or exhausts all of its offers, the second firm, in the decreasing data size order, begins to 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 all possible states one step before the end of the game. Since the actions of the last player are rational, the algorithm can identify these actions accurately. This way, the algorithm moves one step earlier in the 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  $P = \mathrm{N}(\mu, \sigma^2)$  clipped at 1 form below. Then, we calculate the equilibrium of the resulting data-sharing game and
compute the average size (number of companies) of the resulting coalitions at the equilibrium, as a measure of the extend to which companies collaborate.
We repeat the experiment 10000 times, for fixed values of  $m, \gamma, \beta, \mu, \sigma$  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 a big number of independent runs, which makes our results very precise.
Repeating the experiment for different values of  $\gamma$  and  $\beta$ , we can observe the dependence of this average coalition size on the two parameters of interest. In Figure 1 we plot these dependencies, for different values of  $m \in \{3,4\}$  and  $\sigma \in \{300,600\}$ . When varying one of these parameters, the default values for the other one is  $\gamma = 0.8$  and  $\beta = 0.9$ . We also provide results for other values of  $m$  and  $\sigma$  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  $\gamma$  and  $\beta$ , across all experimental setups.
In this work, we introduced and studied a framework for data-sharing among market competitors. The framework is modular and consists of three components—market 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 characteristics of competition may have a 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 suggest 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 in 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 are 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 (e.g., Gulrajani & Lopez-Paz, 2021). Moreover, in some cases, additional data from a different distribution may damage model performance, yielding orthogonal data-sharing considerations (Donahue & Kleinberg, 2021).
Second, we assume that legal constraints are not present or are automatically fulfilled in all our settings. However, we see the evaluation of AI regulatory frameworks as an important direction for future work. The same applies to training cost considerations.
Finally, we do not consider sociological aspects, such as reputational effects (e.g., reputation loss due to a refusal to share data or a poorly communicated decision to share data). Due to the inherent non-rivalry of data (Karimireddy et al., 2022), sociological modeling, e.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 a 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 (Chone & 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 situation-specific knowledge of their market and ML models to build more accurate models for their specific needs.
This research was partially funded by the Ministry of 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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The supplementary material is structured as follows.
- Appendix A contains the proofs of all results in the main text.
- Appendix B contains further examples of the data-sharing problem and costs.
- Appendix C contains several extensions of the setting in Section 5.
- Appendix D contains alternative models of coalition formation between many firms.
- Appendix E contains the welfare analysis of the data-sharing problem between two firms.
- Appendix F presents additional simulations for Section 7.
First, we will solve the same consumption problem without positivity constraints
$$
\max  _ {g, q _ {1}, \dots , q _ {m}} u (g, q _ {1}, \dots , q _ {m}) \text {s . t .} g + \sum_ {i = 1} ^ {m} p _ {i} q _ {i} \leq B.
$$
Using the KKT conditions for  $q_{i}$  we get
$$
0 = \frac {\partial L}{\partial q _ {i}} = \frac {\partial u}{\partial q _ {i}} - \lambda p _ {i},
$$
where  $\lambda$  is the Lagrange multiplier for the budget constraint. The first-order condition for  $g$  gives
$$
0 = 1 - \lambda \Longrightarrow \lambda = 1.
$$
Therefore, the first-order conditions for  $q_{1},\ldots ,q_{m}$  look like
$$
0 = 1 - q _ {i} - \sum_ {j \neq i} q _ {j} - p _ {i} \Rightarrow p _ {i} = 1 - q _ {i} - \gamma \sum_ {j \neq i} q _ {j}.
$$
$$
\sum_ {i} p _ {i} = m - \sum_ {i} q _ {i} - \gamma (m - 1) \sum_ {i} q _ {i} \Rightarrow \sum_ {i} q _ {i} = \frac {m - \sum_ {i} p _ {i}}{1 + \gamma (m - 1)}.
$$
$$
q _ {i} = \frac {(1 + \gamma (m - 1)) (1 - p _ {i}) - \gamma (m - \sum_ {i} p _ {i})}{(1 - \gamma) (1 + \gamma (m - 1))} = \frac {1 - \gamma - (1 + \gamma (m - 2)) p _ {i} - \gamma \sum_ {j \neq i} p _ {j}}{(1 - \gamma) (1 + \gamma (m - 1))}.
$$
This solution is a global maximizer because  $u(g, q_1, \ldots, q_m)$  is strongly concave in  $q_1, \ldots, q_m$  and linear in  $g$ .
Since the auxiliary problem has less constraints than the original problem, a solution to the auxiliary problem will be a solution to the original problem as long as all quantities are non-negative. These constraints give the following restrictions on  $p_1, \ldots, p_m$  and  $B$
$$
\forall i (1 - \gamma) (1 - p _ {i}) \geq \gamma \sum_ {j \neq i} (p _ {j} - p _ {i}),
$$
$$
B \geq \sum_ {i = 1} ^ {m} q _ {i} ^ {*} p _ {i}.
$$
The first series of inequalities hold when prices are less than one and close enough. In our setting, it is achieved when  $1 > a$  and  $(1 - a) \gg b$ , where  $a$  and  $b$  are parameters from Equation (7) (since Equations (10) and (11) hold). The last equation holds when  $B$  is big enough, for example, when  $B > m$ .
A.2 Proof of Lemma 4.3
The profit maximization tasks of the firms are
$$
\max _ {q _ {i}} \mathbf {E} ((p _ {i} - c _ {i}) q _ {i}) = \mathbf {E} (1 - \gamma \sum_ {j \neq i} q _ {j} - c _ {i}) q _ {i} - q _ {i} ^ {2}.
$$
Using the first-order conditions, we get
$$
q _ {i} ^ {*} (q _ {- i}) = \frac {\mathbf {E} (1 - c _ {i} - \gamma \sum_ {j \neq i} q _ {j})}{2}.
$$