SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Thus, the firms use the same quantities as in Section 4.1.2. In the case of non-collaboration, the expected profits are
$$
\Pi_ {i, \mathrm {i n d}} ^ {e} = \left(\frac {2 - \gamma + \gamma c _ {- i} ^ {e} - 2 c _ {i} ^ {e}}{4 - \gamma^ {2}}\right) ^ {2}.
$$
Similarly, in the case of collaboration,
$$
\Pi_ {\mathrm {s h a r e d}} ^ {e} = \left(\frac {(2 - \gamma) (1 - c _ {\mathrm {s h a r e d}} ^ {e})}{4 - \gamma^ {2}}\right) ^ {2}.
$$
The collaboration criterion is the same as in Section 4.1.2.
B.2 Oil drilling
Assume that an oil company manager needs to optimize the costs of oil rig construction. Construction of an oil rig happens in two steps. First, geologists look at a new place for a 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 more profitable to repeat the geological expedition.
The noisy signal of geologists predicts the presence of oil in the following manner
$$
\Pr (\text {s u c c e s s} \mid X) = \Pr (X + \epsilon \geq 0 \mid X), \epsilon \sim \mathrm {N} (0, 1).
$$
Also, the manager knows that
$$
X \sim \mathrm {N} (0, \sigma^ {2}).
$$
However, the manager does not know  $\sigma^2$  and needs to estimate it from the previous observations. The new expedition costs  $a$ , and drilling a new rig costs  $b$ .
The natural strategy is to repeat expeditions until the geologist find  $X \geq r$  and then try to build a rig. Let  $p$  be the probability of finding a good spot and  $q$  be the probability of success in a good spot
$$
\begin{array}{l} p := \Pr (X \geq r) = \Phi \left(\frac {- r}{\sigma}\right), \\ q := \operatorname * {P r} (X + \epsilon \geq 0 \mid X \geq r) = \frac {1}{p} \int_ {r} ^ {\infty} \int_ {- x} ^ {\infty} \frac {\exp \left(\frac {- x ^ {2}}{2 \sigma^ {2}}\right)}{\sqrt {2 \pi \sigma^ {2}}} \frac {\exp \left(\frac {- y ^ {2}}{2}\right)}{\sqrt {2 \pi}} \mathrm {d} y \mathrm {d} x = \frac {M (r , \sigma^ {2})}{p}, \\ \end{array}
$$
where we denote the last integral by  $M$ . So, the expected cost is
$$
f (r) = \frac {\frac {a}{p} + b}{q} = \frac {a + \Phi \left(\frac {- r}{\sigma}\right) b}{M (r , \sigma^ {2})}.
$$
Assume that the company estimates  $\widehat{\sigma}^2$  of  $\sigma^2$  and chooses  $r$  according to that estimate. The loss will approximately be
$$
\begin{array}{l} f (r) - f \left(r ^ {*}\right) \approx f ^ {\prime} \left(r ^ {*}\right) \left(r - r ^ {*}\right) + \frac {f ^ {\prime \prime} \left(r ^ {*}\right)}{2} \left(r - r ^ {*}\right) ^ {2} = \frac {f ^ {\prime \prime} \left(r ^ {*}\right)}{2} \left(r - r ^ {*}\right) ^ {2} \\ \approx \frac {f ^ {\prime \prime} (r ^ {*})}{2} r ^ {\prime} (\sigma^ {2}) ^ {2} (\widehat {\sigma} ^ {2} - \sigma^ {2}) ^ {2}. \\ \end{array}
$$
Now, if the company uses MLE to estimate  $\sigma^2$ , then the costs will asymptotically look like
$$
f (r) \approx A + \frac {B}{n} \chi^ {2} (1),
$$
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  $(\gamma = 1)$  and Cournot competition if not told otherwise.
C.1 Different cost functions
In this subsection, we assume that the cost functions of the firms have the form
$$
C _ {i} (q) = c _ {i} ^ {e} q + \frac {k}{2} q ^ {2}.
$$
The profit maximization problem of the firm now has the form
$$
\max  _ {q _ {i}} (1 - q _ {i} - q _ {- i}) q _ {i} - c _ {i} ^ {e} q _ {i} - \frac {k}{2} q _ {i} ^ {2}.
$$
Therefore, the best responses are
$$
q _ {i} ^ {*} (q _ {- i}) = \frac {1 - q _ {- i} - c _ {i} ^ {e}}{2 + k}
$$
and equilibrium quantities are
$$
q _ {i} ^ {*} = \frac {1 + k + c _ {- i} ^ {e} - (2 + k) c _ {i} ^ {e}}{(1 + k) (3 + k)}.
$$
These equations imply the following expected profits
$$
\varPi_ {i} ^ {e} = \frac {2 + k}{2} (q _ {i} ^ {*}) ^ {2}
$$
and result in the following collaboration criterion
$$
\forall i (2 + k) (n _ {i} ^ {- 1} - (n _ {1} + n _ {2}) ^ {- 1}) > n _ {- i} ^ {- 1} - n _ {i} ^ {- 1}.
$$
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 3, we assume that goods produced by the firms can 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
$$
u \left(q _ {0}, q _ {1}, q _ {2}\right) = \left(q _ {1} + q _ {2} - \frac {q _ {1} ^ {2} + 2 q _ {1} q _ {2} + q _ {2} ^ {2}}{2}\right) ^ {\alpha} q _ {0} ^ {1 - \alpha}.
$$
$(p_{1} = p_{2} = p$  because the goods  $G_{1}$  and  $G_{2}$  are perfect substitutes.) This utility prescribes the consumer to spend the share  $\alpha$  of income on goods  $G_{1}$  and  $G_{2}$  ( $B$  is small enough, so that  $q_{1}$  and  $q_{2}$  are much smaller than 1). So, we have the following consumer problem
$$
\max  _ {q _ {1}, q _ {2}} q _ {1} + q _ {2} - \frac {q _ {1} ^ {2} + 2 q _ {1} q _ {2} + q _ {2} ^ {2}}{2} \text {s . t .} p (q _ {1} + q _ {2}) \leq \alpha B.
$$
So, the demand is
$$
p = \frac {\alpha B}{q _ {1} + q _ {2}}.
$$
Expected profits maximization problems are
$$
\max  _ {q _ {i}} \frac {\alpha B q _ {i}}{q _ {1} + q _ {2}} - c _ {i} ^ {e} q _ {i}.
$$
The first order conditions give
$$
\frac {\alpha B q _ {- i} ^ {*}}{(q _ {1} ^ {*} + q _ {2} ^ {*}) ^ {2}} = c _ {i} ^ {e}.
$$
So, equilibrium quantities are
$$
q _ {i} ^ {*} = \frac {\alpha B q _ {1} ^ {*} q _ {2} ^ {*}}{c _ {i} (q _ {1} ^ {*} + q _ {2} ^ {*}) ^ {2}} = \frac {\alpha B c _ {2} ^ {e} c _ {1} ^ {e}}{c _ {i} ^ {e} (c _ {2} ^ {e} + c _ {1} ^ {e}) ^ {2}}.
$$
And equilibrium profits are
$$
\Pi_ {i} ^ {e} = \frac {\alpha B (c _ {- i} ^ {e}) ^ {2}}{(c _ {1} ^ {e} + c _ {2} ^ {e}) ^ {2}} = \frac {\alpha B}{\left(1 + \frac {c _ {i} ^ {e}}{c _ {- i} ^ {e}}\right) ^ {2}}.
$$
This formula implies the following collaboration criterion
$$
\forall i   1 + \frac {c _ {i , \mathrm {i n d}} ^ {e}}{c _ {- i , \mathrm {i n d}} ^ {e}} <   1 + \frac {c _ {\mathrm {s h a r e}} ^ {e}}{c _ {\mathrm {s h a r e}} ^ {e}} = 2.
$$
As can be seen, the collaboration is always not profitable for the firm with more data.
C.3 Asymmetric costs
In this subsection, we assume that firms have asymmetric cost functions
$$
c _ {i} ^ {e} (n) = a _ {i} + \frac {b _ {i}}{n ^ {\beta_ {i}}},
$$
where  $c_{i}^{e}(n)$  is the expected marginal cost of the company if it use  $n$  points for training. Section 4.1.2 gives the following the expected profits
$$
\Pi_ {i} ^ {e} = \left(\frac {2 - \gamma - 2 c _ {i} ^ {e} + \gamma c _ {- i} ^ {e}}{4 - \gamma^ {2}}\right) ^ {2}.
$$
As in Section 4.1.2, the companies will collaborate only if both of them profit from collaboration
$$
\forall i \Pi_ {\text {s h a r e d}} ^ {e} > \Pi_ {i, \text {i n d}} ^ {e} \iff 2 b _ {i} \left(n _ {i} ^ {- \beta_ {i}} - n _ {\text {s h a r e d}} ^ {- \beta_ {i}}\right) > \gamma b _ {- i} \left(n _ {- i} ^ {- \beta_ {- i}} - n _ {\text {s h a r e d}} ^ {- \beta_ {- i}}\right).
$$
Denote  $f_{i}(n_{1}, n_{2}) = 2b_{i}(n_{i}^{-\beta_{i}} - n_{\mathrm{shared}}^{-\beta_{i}}) - \gamma b_{-i}(n_{-i}^{-\beta_{-i}} - n_{\mathrm{shared}}^{-\beta_{-i}})$ . This function can be interpreted as firm  $F_{i}$  reediness to collaborate: the higher this value is the more firm  $F_{i}$  wants to collaborate with its competitor  $F_{-i}$ .
Theorem C.1. Assume the setting described above. The function  $f_{i}(n_{1}, n_{2})$  has the following properties:
1.  $f_{i}(n_{1},n_{2})$  is decreasing in  $\gamma$
2.  $f_{i}(n_{1},n_{2})$  is decreasing in  $\beta_{i}$  and increasing in  $\beta_{-i}$  if  $\forall i\beta_{i}\ln n_{i} > 1$
3.  $f_{i}(n_{1},n_{2})$  is increasing in  $b_{i}$  and decreasing in  $b_{-i}$ .
Proof. The first property is evident.
$$
\frac {\partial f _ {i}}{\partial \gamma} = - b _ {- i} \left(n _ {i} ^ {- \beta_ {- i}} - n _ {\text {s h a r e d}} ^ {- \beta_ {- i}}\right) <   0.
$$
We can prove the last property similarly
$$
\frac {\partial f _ {i}}{\partial b _ {i}} = 2 \left(n _ {i} ^ {- \beta_ {i}} - n _ {\text {s h a r e d}} ^ {- \beta_ {i}}\right) > 0,
$$
$$
\frac {\partial f _ {i}}{\partial b _ {- i}} = - \gamma \left(n _ {i} ^ {- \beta_ {- i}} - n _ {\mathrm {s h a r e d}} ^ {- \beta_ {- i}}\right) <   0.
$$
The first part of the second property follows from the following
$$
\frac {\partial f _ {i}}{\partial \beta_ {i}} = 2 b _ {i} \left(\frac {\ln n _ {i}}{n _ {\text {s h a r e d}} ^ {\beta_ {i}}} - \frac {\ln n _ {i}}{n _ {i} ^ {\beta_ {i}}}\right).
$$
Notice that the function  $g(x) = \frac{\ln x}{x^{\beta_i}}$  is decreasing in  $x$  on  $x > e^{1 / \beta_i}$  because
$$
\frac {\partial g}{\partial x} = \frac {1 - \beta_ {i} \ln x}{x ^ {\beta_ {i} + 1}} <   0.
$$
Therefore,  $\frac{\partial f_i}{\partial\beta_i} < 0$ :  $f_{i}$  is decreasing in  $\beta_{i}$ . 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 ( $\beta_{i}$  is low or  $b_{i}$  is high) or the competitor's model is good ( $\beta_{-i}$  is high or  $b_{-i}$  is low).
C.4 Heterogeneity
In this section, we solve the same model as in Appendix B, but allowing for heterogeneity between firms. Each firm will need to optimize the MSE costs
$$
c _ {i} (s) = a - b + b \mathbf {E} ((s - X _ {i}) ^ {2}) = a + b (s - \mu_ {i}) ^ {2}, X _ {i} \sim \mathrm {N} (\mu_ {i}, 1).
$$
On the contrary to the Appendix B, the means of the noise will be different among firms and will be drawn at the start of the game from the distribution
$$
\mu_ {i} \sim \mathrm {N} (\mu , \sigma^ {2}).
$$
We assume that  $\mu$  is unknown and  $\sigma^2$  is known.
When firms do not collaborate, the expected costs are the same as in Appendix B. When firms collaborate, MLE for the means are
$$
s _ {i, \mathrm {s h a r e d}} = \frac {2 \sigma^ {2} n _ {1} n _ {2} Y _ {i} + n _ {i} Y _ {i} + n _ {- i} Y _ {- i}}{2 \sigma^ {2} n _ {1} n _ {2} + n _ {1} + n _ {2}},
$$
where  $n_i$  is the number of data points of firm  $F_i$ ,  $Y_i$  is the average of noise. Then expected marginal costs will be
$$
c _ {i, \text {s h a r e d}} ^ {e} = a + b \mathbf {E} _ {S _ {1}, S _ {2}} \left(\left(s _ {i, \text {s h a r e d}} - \mu_ {i}\right) ^ {2}\right).
$$
By substitution,
$$
\begin{array}{l} \mathbf {E} \left(\left(s _ {i, \text {s h a r e d}} - \mu_ {i}\right) ^ {2}\right) = \\ \mathbf {E} \bigg (\frac {n _ {- i} ^ {2} (\mu_ {1} - \mu_ {2}) ^ {2}}{(2 \sigma^ {2} n _ {1} n _ {2} + n _ {1} + n _ {2}) ^ {2}} + \frac {(2 \sigma^ {2} n _ {- i} + 1) ^ {2} n _ {i}}{(2 \sigma^ {2} n _ {1} n _ {2} + n _ {1} + n _ {2}) ^ {2}} + \frac {n _ {- i}}{(2 \sigma^ {2} n _ {1} n _ {2} + n _ {1} + n _ {2}) ^ {2}} \bigg) \\ = \frac {2 \sigma^ {2} n _ {- i} + 1}{2 \sigma^ {2} n _ {1} n _ {2} + n _ {1} + n _ {2}}. \\ \end{array}
$$
Therefore,  $c_{i,\mathrm{shared}}^e$  is increasing in  $\sigma^2$ : the more different the firms are, the less collaboration gives. Two corner cases give
$$
\sigma^ {2} = 0 \Rightarrow c _ {i, \text {s h a r e d}} ^ {e} = a + \frac {b}{n _ {1} + n _ {2}},
$$
$$
\sigma^ {2} \rightarrow \infty \Rightarrow c _ {i, \mathrm {s h a r e d}} ^ {e} \rightarrow a + \frac {b}{n _ {i}}.
$$
However, notice that collaboration criteria in the Cournot case does not depend on  $\sigma^2$  and hence does not change
$$
\begin{array}{l} \Pi_ {i, \mathrm {s h a r e}} ^ {e} - \Pi_ {i, \mathrm {i n d}} ^ {e} \geq 0 \iff 2 - \gamma - 2 c _ {i, \mathrm {s h a r e}} ^ {e} + \gamma c _ {- i, \mathrm {s h a r e}} ^ {e} \geq 2 - \gamma - 2 c _ {i, \mathrm {i n d}} ^ {e} + \gamma c _ {- i, \mathrm {i n d}} ^ {e} \\ \Longleftrightarrow - \frac {2 (2 \sigma^ {2} n _ {- i} + 1)}{2 \sigma^ {2} n _ {1} n _ {2} + n _ {1} + n _ {2}} + \frac {\gamma (2 \sigma^ {2} n _ {i} + 1)}{2 \sigma^ {2} n _ {1} n _ {2} + n _ {1} + n _ {2}} \geq - \frac {2}{n _ {i}} + \frac {\gamma}{n _ {- i}} \Longleftrightarrow 0 \geq - 2 \frac {n _ {- i}}{n _ {i}} + \gamma \frac {n _ {i}}{n _ {- i}}. \\ \end{array}
$$
D Other models of coalition formation
In this section, we present models of coalition formation different from the model presented in Section 7.
D.1 Cooperative data sharing
We use an solution concept motivated by the  $\alpha$ -core (Von Neumann & Morgenstern, 1944), which is standard in cooperative game theory. The  $\alpha$ -core consists of partitions that are incentive-compatible, 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  $\alpha$ -core, tailored to our setup. This deviation is necessary due to two reasons, non-transferable utility and the presence of externalities (dependencies of utilities on the
actions of all participants), which arise in our setting because all firms compete with each other on the market.
Let  $N = \{F_1, \ldots, F_m\}$  be the set of all firms. A partition of  $N$  is a set of subsets  $\{S_1, \ldots, S_k\} : S_i \subseteq N$ , such that  $\forall i, j, i \neq j : S_i \cap S_j = \emptyset$  and  $\cup_i S_i = N$ . We denote the set of all partitions of  $N$  as  $P(N)$ .
Definition D.1 ( $\alpha$ -Core). A partition  $P$  belongs to the  $\alpha$ -core of the game if and only if
$$
\nexists S \subseteq N: \forall F \in S, Q \in P (N \setminus S) \quad \Pi_ {F} ^ {e} (P) <   \Pi_ {F} ^ {e} (Q \cup \{S \}),
$$
where  $\Pi_F^e (P)$  is expected profit of the firm  $F$  if market coalition structure is  $P$ .
Intuitively, a partition  $P$  belongs to the  $\alpha$ -core if no set of companies wants to deviate from this partition. Here we say that a 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  $\alpha$ -core of the data-sharing game, demonstrating that the not-emptiness of the core.
Theorem D.2. W.l.o.g. assume that  $n_1 \geq n_2 \geq \dots \geq n_m$ , where  $n_i$  is the number of data points of firm  $F_i$ . Consider partitions  $P_i = \{A_i, N \setminus A_i\}$ , where  $A_i = \{F_1, F_2, \ldots, F_i\}$ . Let  $i^* = \arg \max_i \Pi_{F_1}^e(P_i)$ . Then  $P_{i^*}$  belongs to  $\alpha$ -core.
Before we start to prove this theorem, we formulate the following lemma.
Lemma D.3. Let  $Q \subset N$  and  $F \in Q$ . Then
$$
\Pi_ {F} ^ {e} (\{Q, N \setminus Q \}) = \frac {(2 - \gamma - (2 + \gamma (m - 1 - | Q |) c _ {Q} ^ {e} + \gamma (m - | Q |) c _ {N \setminus Q} ^ {e}) ^ {2}}{(2 - \gamma) ^ {2} (2 + (m - 1) \gamma) ^ {2}},
$$
where  $c_{X}^{\epsilon} = a + b / n_{X}^{\beta}$ , expected marginal cost of coalition  $X$ , and  $n_{X} = \sum_{F\in X}n_{F}$ , the number of data points of coalition  $X$ .
Proof. The results of Section 4.1.2 give the following expected profits
$$
\Pi_ {i} ^ {e} = \frac {(2 - \gamma - (2 + \gamma (m - 2)) c _ {i} ^ {e} + \gamma \sum_ {j \neq i} c _ {j} ^ {e}) ^ {2}}{(2 - \gamma) ^ {2} (2 + (m - 1) \gamma) ^ {2}}.
$$
In our case,
$$
c _ {i} = \left\{ \begin{array}{l l} c _ {Q}, & F _ {i} \in Q, \\ c _ {N \setminus Q}, & F _ {i} \notin Q, \end{array} \right.
$$
By substituting  $c_{i}^{e}$  into the profit equation for  $F \in Q$ , we get
$$
\begin{array}{l} \Pi_ {F} ^ {e} (\{Q, N \setminus Q \}) = \frac {(2 - \gamma - (2 + \gamma (m - 2)) c _ {Q} ^ {e} + \gamma (| Q | - 1) c _ {Q} ^ {e} + \gamma (m - | Q |) c _ {N \setminus Q} ^ {e}) ^ {2}}{(2 - \gamma) ^ {2} (2 + (m - 1) \gamma) ^ {2}} \\ = \frac {(2 - \gamma - (2 + \gamma (m - 1 - | Q |) c _ {Q} ^ {e} + \gamma (m - | Q |) c _ {N \setminus Q} ^ {e}) ^ {2}}{(2 - \gamma) ^ {2} (2 + (m - 1) \gamma) ^ {2}}. \\ \end{array}
$$
Corollary D.4. Let  $Q, Q' \subset N$ ,  $F \in Q$ , and  $F' \in Q$ . Then
$$
\begin{array}{l} \Pi_ {F} ^ {e} (\{Q, N \setminus Q \}) \geq \Pi_ {F ^ {\prime}} ^ {e} (\{Q ^ {\prime}, N \setminus Q ^ {\prime} \}) \iff \\ \frac {2 + \gamma (m - 1 - | Q ^ {\prime} |)}{n _ {Q ^ {\prime}} ^ {\beta}} - \frac {\gamma (m - | Q ^ {\prime} |)}{n _ {N \setminus Q ^ {\prime}} ^ {\beta}} \geq \frac {2 + \gamma (m - 1 - | Q |)}{n _ {Q} ^ {\beta}} - \frac {\gamma (m - | Q |)}{n _ {N \setminus Q} ^ {\beta}}. \\ \end{array}
$$
Now, we are ready to prove Theorem D.2.
Proof. We prove the theorem by contradiction. Suppose that there exists a coalition  $Q$  for which deviation is profitable.
First, we consider the case  $A_{i^*} \cap Q \neq \emptyset$ . Let  $F \in A_{i^*} \cap Q$ . Notice that
$$
\Pi_ {F} ^ {e} \left(P _ {i ^ {*}}\right) = \Pi_ {F _ {1}} ^ {e} \left(P _ {i ^ {*}}\right) \geq \Pi_ {F _ {1}} ^ {e} \left(P _ {| Q |}\right) \geq \Pi_ {F} ^ {e} \left(\{Q, N \backslash Q \}\right).
$$
The first equality follows from Lemma D.3. The first inequality follows from the definition of  $i^*$ . The last inequality follows from Corollary D.4 and observation  $n_{A_{|Q|}} \geq n_Q$ . So, we get a contradiction. Thus,  $A_{i^*} \cap Q = \emptyset$ .
Let  $F\in Q$  .In this case,
$$
\Pi_ {F} ^ {e} \left(P _ {i ^ {*}}\right) \geq \Pi_ {F} ^ {e} (\{Q, N \backslash Q \}).
$$
This inequality follows from Corollary D.4, the fact that  $n_{N\setminus A_{i^{*}}}\geq n_Q$  , and observation  $|N\setminus A_{i^*}|\geq |Q|$  . 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 to largest profits for the company with the largest amount of data. However, it may not be the only one in the core since the  $\alpha$ -core is one of the most permissive cores in 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-economically plausible partitions.
The following example illustrates this flaw. Assume that there are three firms in the market and  $n_1 \gg n_2 \gg n_3$ . In this case, the  $\alpha$ -core consists of two equilibria:  $\{\{F_1, F_2\}, \{F_3\}\}$  and  $\{\{F_1\}, \{F_2\}, \{F_3\}\}$ , 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 is 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 a 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 in grand coalition. If any company disagrees, the grand coalition is not formed and companies act like singletons.
W.l.o.g., assume that  $n_1 \geq n_2 \geq \dots \geq n_m$ . The grand coalition will be Nash equilibrium in this game only if all firms will be better off in the grand coalition
$$
\forall F \Pi_ {F} ^ {e} (\{\{F _ {1}, F _ {2}, \dots , F _ {m} \} \}) > \Pi_ {F} ^ {e} (\{\{F _ {1} \}, \{F _ {2} \}, \dots , \{F _ {m} \} \}).
$$
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 is equivalent to the inequality for the first firm
$$
\left. \Pi_ {F _ {1}} ^ {e} (\{\{F _ {1}, F _ {2}, \dots , F _ {m} \} \}) > \Pi_ {F _ {1}} ^ {e} (\{\{F _ {1} \}, \{F _ {2} \}, \dots , \{F _ {m} \} \}). \right.
$$
By substitution, this inequality takes the form
$$
\frac {\gamma m - m - 1}{n ^ {\beta}} > - \frac {m + 1}{n _ {1} ^ {\beta}} + \sum_ {j = 1} ^ {m} \frac {\gamma}{n _ {j} ^ {\beta}},
$$
where  $n = \sum_{j} n_{j}$ .
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 a coalition and resulting structure will be one coalition with several members and singleton coalitions. Formally, we assume the following game. 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 are interested in Nash equilibria of this game.
W.l.o.g., assume that  $n_1 \geq n_2 \geq \dots \geq n_m$ . The following lemma is useful for the description of all equilibria of this game
Lemma D.5. Let  $S$  be a set of firm that answer  $Y$  in equilibrium and  $i^{*} = \min \{i\mid F_{i}\in S\}$ . Then  $\forall j\geq i^{*}F_{j}\in S$ .
Proof. To show this it is sufficient to show that a single firm  $F$  always want to join a coalition  $S$  that has more data than the firm. This is equivalent to the following inequality
$$
\begin{array}{l} \left. \Pi_ {F} ^ {e} (\{S \cup \{F \} \} \cup \{\{G \} \mid G \notin S \cup \{F \} \}) > \Pi_ {F} ^ {e} (\{S \} \cup \{\{G \} \mid G \notin S \}) \right. \\ \Longleftrightarrow \frac {\gamma (| S | + 1) - m - 1}{(n _ {S} + n _ {F}) ^ {\beta}} > \frac {\gamma | S |}{n _ {S} ^ {\beta}} - \frac {m + 1 - \gamma}{n _ {F} ^ {\beta}}, \\ \end{array}
$$
where  $n_S = \sum_{G \in S} n_G$ . Denote  $y := \frac{n_F}{n_S}$ . The last inequality is equivalent to
$$
(m + 1 - \gamma) \left(1 - \frac {y ^ {\beta}}{(1 + y) ^ {\beta}}\right) > \gamma | S | \left(y ^ {\beta} - \frac {y ^ {\beta}}{(1 + y) ^ {\beta}}\right).
$$
Since  $\gamma < 1$  and  $|S| < m$ , we have  $m + 1 - \gamma > \gamma |S|$ . Furthermore,  $1 - \frac{y^{\beta}}{(1 + y)^{\beta}} \geq y^{\beta} - \frac{y^{\beta}}{(1 + y)^{\beta}} > 0$  because  $1 \geq y > 0$ . Therefore, the inequality above holds.
Lemma D.5 greatly constraints possible equilibria of the game. Indeed, the equilibrium set of companies that say  $\mathrm{Y}$  may only have the form  $S_{i} = \{F_{j} \mid j \geq i\}$ . To check whether  $S_{i}$  appears in the equilibrium, we need to check the following system of inequalities
$$
\begin{array}{l} \forall j <   i \Pi_ {F _ {j}} ^ {e} (\{S _ {i} \} \cup \{\{G \} \mid G \notin S _ {i} \}) \geq \Pi_ {F _ {j}} ^ {e} (\{S _ {i} \cup \{F _ {j} \} \} \cup \{\{G \} \mid G \notin S _ {i} \cup \{F _ {j} \} \}), \\ \forall j \geq i \Pi_ {F _ {j}} ^ {e} (\{S _ {i} \} \cup \{\{G \} \mid G \notin S _ {i} \}) \leq \Pi_ {F _ {j}} ^ {e} (\{S _ {i} \cup \{F _ {j} \} \} \cup \{\{G \} \mid G \notin S _ {i} \cup \{F _ {j} \} \}). \\ \end{array}
$$
These inequalities are closest for the firms around threshold  $i$ . Thus, this system is equivalent to two inequalities
$$
\begin{array}{l} \Pi_ {F _ {i - 1}} ^ {e} \left(\left\{S _ {i} \right\} \cup \left\{\left\{G \right\} \mid G \notin S _ {i} \right\}\right) \geq \Pi_ {F _ {i - i}} ^ {e} \left(\left\{S _ {i} \cup \left\{F _ {i - 1} \right\} \right\} \cup \left\{\left\{G \right\} \mid G \notin S _ {i} \cup \left\{F _ {i - 1} \right\} \right\}\right), \\ \Pi_ {F _ {i}} ^ {e} \left(\left\{S _ {i} \right\} \cup \left\{\left\{G \right\} \mid G \notin S _ {i} \right\}\right) \leq \Pi_ {F _ {i}} ^ {e} \left(\left\{S _ {i} \cup \left\{F _ {i} \right\} \right\} \cup \left\{\left\{G \right\} \mid G \notin S _ {i} \cup \left\{F _ {i} \right\} \right\}\right). \\ \end{array}
$$
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
$$
W := u + \sum_ {i = 1} ^ {m} \Pi_ {i}.
$$
The definition of welfare in our case is simpler than in general case because consumers have quasilinear preferences.