Appendix
Proof of Lemma 1
The change in W(g) by adding link ik can be decomposed into the change in consumer surplus \([Q^{2}(g+ik)-Q^{2}(g)]/2\), and the change in the aggregate profit. Since each additional link always leads to the same amount of increase in Q, consumer surplus is convex in links. Thus we only need to show that the aggregate profit is also convex in links.
Let u be any firm other than i, j, and k. Due to increasing returns, it can be readily seen that \(\pi _{i}(g+ij+ik)-\pi _{i}(g+ik)>\pi _{i}(g+ik)-\pi _{i}(g)\). For the same reason, \(\pi _{u}(g+ij+ik)-\pi _{u}(g+ik)>\pi _{u}(g+ik)-\pi _{u}(g)\). For firm j, we have
$$\begin{aligned}&[\pi _{j}(g+ij+ik)-\pi _{j}(g+ik)]-[\pi _{j}(g+ik)-\pi _{j}(g)] \\&\quad =\frac{(n-1)\gamma }{n+1}[q_{j}(g+ij+ik)+q_{j}(g+ik)]+\frac{2\gamma }{n+1} [q_{j}(g+ik)+q_{j}(g)]. \end{aligned}$$
And for firm k,
$$\begin{aligned}&[\pi _{k}(g+ij+ik)-\pi _{k}(g+ik)]-[\pi _{k}(g+ik)-\pi _{k}(g)] \\&\quad =-\frac{2\gamma }{n+1}[q_{k}(g+ij+ik)+q_{k}(g+ik)]-\frac{(n-1)\gamma }{n+1} [q_{k}(g+ik)+q_{k}(g)]. \end{aligned}$$
The summation of the above two terms is positive if \(q_{j}(g+ik)\ge q_{k}(g+ik)\), since we have \(q_{k}(g)<q_{k}(g+ik)\), \(q_{j}(g+ij+ik)>q_{j}(g+ik)\), \(q_{j}(g)>q_{j}(g+ik)\), and \(q_{k}(g+ij+ik)<q_{k}(g+ik)\). Therefore, when \(q_{j}(g+ik)\ge q_{k}(g+ik)\), which is equivalent to \(c_{j}(g+ik)\le c_{k}(g+ik)\), the aggregate profit is also convex in links. \(\square\)
Proof of Proposition 1
Part (i). Suppose there are a pair of firms i and j, \(i<j\), but \(\eta _{i}(g)<\eta _{j}(g)\). Derive network \(g^{\prime }\) from g by just switching the network positions of i and j. Thus, \(\eta _{i}(g^{\prime })=\eta _{j}(g)>\eta _{i}(g)=\eta _{j}(g^{\prime })\). We show that \(g^{\prime }\) is more efficient than g. Since the total number of links is the same under g and \(g^{\prime }\), \(Q(g^{\prime })=Q(g)\). Thus both the consumer surplus and the total link formation cost are the same under g and \(g^{\prime }\). Now consider firms’ aggregate profit. For any firm k other than i and j, \(q_{k}(g^{\prime })=q_{k}(g)\), thus \(\pi _{k}(g^{\prime })=\pi _{k}(g)\). Now it is sufficient to show that \(\pi _{i}(g^{\prime })+\pi _{j}(g^{\prime })>\pi _{i}(g)+\pi _{j}(g)\). To see this, note that \(q_{i}(g^{\prime })+q_{j}(g^{\prime })=q_{i}(g)+q_{j}(g)\), and \(q_{i}(g^{\prime })-q_{i}(g)>0\). Thus \(q_{i}(g^{\prime })+q_{i}(g)>q_{j}(g^{\prime })+q_{j}(g)\) is sufficient. But this is the case as firm i has a lower initial cost than firm j.
Part (ii). Suppose link kl has a higher ordering than link ij, and \(ij\in g\) but \(kl\notin g\). Derive network \(g^{\prime }\) from g as follows: sever link ij but add link kl. We show that \(g^{\prime }\) is more efficient than g. By construction, the total number of links is the same under \(g^{\prime }\) and g. Thus the consumer surplus remains the same, and the profit of each firm u other than i, j, k, and l does not change either. Therefore, it is enough to show that \(\sum _{s\in \{i,j,k,l\}}[\pi _{s}(g^{\prime })-\pi _{s}(g)]>0\). Since link kl has a higher ordering than link ij, by part (i) it implies that \(\eta _{i}(g)\le \eta _{k}(g)\) and \(\eta _{j}(g)\le \eta _{l}(g)\). It means that \(c_{i}(g)\le c_{k}(g)\) and \(c_{j}(g)\le c_{l}(g)\), and at least one of the inequalities is strict. In terms of quantities, \(q_{i}(g)\le q_{k}(g)\) and \(q_{j}(g)\le q_{l}(g)\), and at least one of the inequalities is strict. Thus we have
$$\begin{aligned}&\sum _{s\in \{i,j,k,l\}}[\pi _{s}(g^{\prime })-\pi _{s}(g)] \\&\quad \propto \{[q_{k}(g^{\prime })+q_{k}(g)]-[q_{i}(g^{\prime })+q_{i}(g)]+[q_{l}(g^{\prime })+q_{l}(g)]-[q_{j}(g^{\prime })+q_{j}(g)]\}>0. \end{aligned}$$
Part (iii). Fix \(\overline{\eta }(g)\) at any \(\overline{\eta }\), and consider any network \(g^{\prime }\) that has the same average degree \(\overline{\eta }\). It is obvious that \(Q(g^{\prime })=Q(g)\), \(\overline{q} (g^{\prime })=\overline{q}(g)\), \(\overline{c}(g^{\prime })=\overline{c}(g)\), and the total link formation cost is the same under \(g^{\prime }\) and g. Thus, for g to be efficient, g has to maximize the aggregate profit: \(g\in \arg \max _{g^{\prime }}\sum _{i}q_{i}^{2}(g^{\prime })\). The aggregate profit can also be written as \(\sum _{i}q_{i}^{2}(g^{\prime })=n\overline{q} ^{2}+V_{q}(g^{\prime })\). Thus an efficient g should maximizes the variance of quantities \(V_{q}(g^{\prime })\). But
$$\begin{aligned} V_{q}(g^{\prime })= & {} \sum _{i}(q_{i}(g^{\prime })-\overline{q})^{2}=\sum _{i}\left( \frac{\overline{c}-nc_{i}(g)+\sum _{j\ne i}c_{j}(g)}{n+1}\right) ^{2} \\= & {} \sum _{i}(c_{i}(g^{\prime })-\overline{c})^{2}=V_{c}(g^{\prime }). \end{aligned}$$
Thus an efficient g should maximize the variance of final costs \(V_{c}(g^{\prime })\).
Part (iv). We first show that among regular networks \(g^{r}\), \(g^{r}\) with \(0<r<n\) cannot be efficient. In such a \(g^{r}\), there must be a firm k with which firm 1 is not linked. By part (ii), the monotonicity of link ordering, for \(g^{r}\) to be efficient it is necessary that link \(1n\notin g^{r}\). Since firm n also has degree r, there must be a \(j>1\) such that link \(jn\in g^{r}\). But \(1n\notin g^{r}\) and \(jn\in g\) violate the monotonicity of link ordering. Therefore, \(g^{r}\) cannot be efficient.
We next consider irregular networks g that have a two-point degree distribution \(\{h_{1}(g),h_{m}(g)\}\), with firms in \(h_{1}(g)\) being isolated. We show that an efficient g must be a dominant group network. Let \(|h_{m}(g)|=k\). By part (i), the monotonicity in degrees, in an efficient g it must be the case that firms \(1,2,..,k\in h_{m}(g)\). Suppose the degree for firms in \(h_{m}(g)\) is not \(k-1\). Then there is a firm \(j\le k\) such that link \(1j\notin g\). Since firm j and firm 1 have the same degree, there must be a firm l—\(1<l\le k\)—such that link \(lj\in g\). But \(1j\notin g\) and \(lj\in g\) violate the monotonicity of link ordering. Thus, g cannot be efficient. This shows that all firms in \(h_{m}(g)\) must have degree \(k-1\), or that they are all linked with each other. That is, an efficient g must be a dominant group network.
Finally, consider irregular networks g in which firms have more than three degrees, and denote the degree distribution as \(\{h_{1}(g),h_{2}(g),\ldots ,h_{m}(g)\}\). We show that an efficient g must be an interlinked star. That is, for any firm i with \(\eta _{i}(g)>0\), it must be linked with each firm in \(h_{m}(g)\). Let \(|h_{m}(g)|=k\). By the monotonicity in degrees, in an efficient g it must be the case that firms \(1,2,..,k\in h_{m}(g)\). If \(k=1\), it is obvious that any firm i with \(\eta _{i}(g)>0\) must be linked with firm 1, since otherwise the monotonicity of link ordering will be violated. Now consider the case that \(k>1\). For any firm i with \(\eta _{i}(g)>0\), note that, by the monotonicity of link ordering, we must have link \(1i\in g\). Pick a firm \(j\le k\) other than firm 1 in \(h_{m}(g)\). Suppose link \(ji\notin g\). Since link \(1i\in g\) and firm 1 and firm j have the same degree, there must be another firm l such that link \(jl\in g\) but link \(1l\notin g\). But this violates the monotonicity of link ordering. Therefore, we must have link \(ji\in g\). This shows that each firm in \(h_{m}(g)\) is linked with any firm with positive degrees, or g is an interlinked star. \(\square\)
Proof of Proposition 3
Part (i). It is straightforward to construct such examples.
Part (ii). Suppose a \(g_{nm}^{k}\) with \(i_{H}>j_{L}\) is stable. Now derive another dominant group network \(g^{\prime k}\) from \(g_{nm}^{k}\) by switching the network positions of \(i_{H}\) and \(j_{L}\). We want to show that \(g^{\prime k}\) is stable. In network \(g^{\prime k}\), let \(i_{H}^{\prime }\) be the highest cost firm in the dominant group \(D^{\prime }\), and let \(j_{L}^{\prime }\) be the lowest cost firm among the isolated firms \(I^{\prime }\). Since the network positions of firms \(i_{H}\) and \(j_{L}\) are switched, it must be the case that \(i_{H}^{\prime }<i_{H}\), and \(j_{L}^{\prime }>j_{L}\). Since \(g_{nm}^{k}\) is stable, by the monotonicity property, we have \(Y_{i_{H}^{\prime }}(k)>Y_{i_{H}}(k)\ge f\) and \(X_{j_{L}^{\prime }}(k)<X_{j_{L}}(k)<f\). Therefore, \(g^{\prime k}\) is stable. The converse is not true: if \(g^{\prime k}\) is stable, it does not necessarily imply that \(g_{nm}^{k}\) is stable. This is because \(Y_{i_{H}^{\prime }}(k)>Y_{i_{H}}(k)\) and \(X_{j_{L}^{\prime }}(k)<X_{j_{L}}(k)\), or the stability conditions (7) and (8) become tighter under \(g_{nm}^{k}\) than under \(g^{\prime k}\). \(\square\)
Proof of Lemma 4
Pat (i). By the definition of SAT, \(ij\in g\) implies that \([\pi _{i}(g)-\pi _{i}(g-ij)]+[\pi _{j}(g)-\pi _{j}(g-ij)]>2f\). Note that
$$\begin{aligned} \pi _{i}(g)-\pi _{i}(g-ij)=\frac{(n-1)\gamma }{(n+1)^{2}}\left[ 2\alpha -2nc_{i}(g-ij)+2\sum _{l\notin i}c_{l}(g-ij)+(n-1)\gamma \right] . \end{aligned}$$
Then,
$$\begin{aligned}&[\pi _{i}(g+ik)-\pi _{i}(g)]+[\pi _{k}(g+ik)-\pi _{k}(g)] \\&\quad =\frac{(n-1)\gamma }{(n+1)^{2}}\left[ 4\alpha -2nc_{i}(g)+2\sum _{l\notin i}c_{l}(g)-2nc_{k}(g)+2\sum _{l\notin k}c_{l}(g)+2(n-1)\gamma \right] \\&\quad>\frac{(n-1)\gamma }{(n+1)^{2}}\left[ 4\alpha -2nc_{i}(g-ij)+2\sum _{l\notin i}c_{l}(g-ij)-2nc_{k}(g-ij)+2\sum _{l\notin k}c_{l}(g-ij)+2(n-1)\gamma \right] \\&\quad \ge \frac{(n-1)\gamma }{(n+1)^{2}}\left[ 4\alpha -2nc_{i}(g-ij)+2\sum _{l\notin i}c_{l}(g-ij)-2nc_{j}(g-ij)+2\sum _{l\notin j}c_{j}(g-ij)+2(n-1)\gamma \right] \\&\quad =[\pi _{i}(g)-\pi _{i}(g-ij)]+[\pi _{j}(g)-\pi _{j}(g-ij)]>2f. \end{aligned}$$
In the above derivation, the first inequality is based on \(c_{i}(g-ij)=c_{i}(g)+\gamma\), \(c_{j}(g-ij)=c_{j}(g)+\gamma\), and \(c_{l}(g-ij)=c_{l}(g)\) for \(l\ne i,j\), while the second inequality uses the condition \(c_{k}(g-ij)\le c_{j}(g-ij)\).
Part (ii). By definition, \(\eta _{j}(g)>\eta _{i}(g)\). Thus there exists at least a firm k such that \(jk\in g\) and \(ik\notin g\). Suppose \(c_{i}(g)\le c_{j}(g)+\gamma\). Then \(c_{i}(g-jk)\le c_{j}(g-jk)\). Now by part (i), firm k and firm i have incentives to form link ik. This contradicts the presumption that g is SAT. \(\square\)
Proof of Proposition 4
Let \(i_{H}\) be the highest-cost firm in \(h_{1}(g)\). Since g is connected, \(\eta _{i_{H}}(g)\ge 1\). We first show that if \(j\notin h_{m}(g)\), then \(i_{H}j\notin g\). Suppose, to the contrary, \(i_{H}j\in g\). By part (ii) of Lemma 4, \(c_{i_{H}}(g)\) is the highest cost firm among all firms under g. Following part (i) of lemma 4, firm j must be linked to all firms. Thus \(j\in h_{m}(g)\), a contradiction. Therefore, \(i_{H}\) can only be linked to firms in \(h_{m}(g)\), and must be connected to some firms in \(h_{m}(g)\). Call one of these firms in \(h_{m}(g)\) firm k. Again by part (i) of Lemma 4, firm k must be linked with all firms, as it has a link with the highest-cost firm—firm \(i_{H}\)—among all firms. Therefore, the degree of firms in \(h_{m}(g)\) is \(n-1\), and each firm in \(h_{m}(g)\) is connected to all firms.
The previous analysis also shows that firm \(i_{H}\) is connected (and only connected) to all firms in \(h_{m}(g)\). Thus \(\eta _{i_{H}}(g)=|h_{m}(g)|\). Since all firms in \(h_{1}(g)\) have the same degree and they are all connected to all firms in \(h_{m}(g)\), each firm in \(h_{1}(g)\) must be linked and only linked with all firms in \(h_{m}(g)\). Therefore, g is an interlinked star. \(\square\)
Proof of Proposition 5
Let \(|N_{i_{L}}(g)|=\eta _{1}\). Since firm \(i_{H}\) has a higher initial cost, \(c_{i_{L}}(g^{\prime })<c_{i_{H}}(g)\) and \(c_{i_{H}}(g^{\prime })>c_{i_{L}}(g)\). Note that by switching the network positions of \(i_{H}\) and \(i_{L}\), for any i other than \(i_{H}\) and \(i_{L}\), \(q_{i}(g^{\prime })=q_{i}(g)\), and thus \(\pi _{i}(g^{\prime })=\pi _{i}(g)\). We also keep the transfers involving i under \(g^{\prime }\) the same as those under g: \(t_{i}^{\prime }=t_{i}\), \(t_{i_{L}}^{\prime k}=t_{i_{H}}^{k},\) and \(t_{i_{H}}^{\prime k}=t_{i_{L}}^{k}\). Therefore, for any other firm i, since the transfers do not change and its gross payoff does not change, its incentive does not change either. As a result, we need to worry only about firm \(i_{H}\) and firm \(i_{L}\).
Now consider the marginal joint surplus that is created for each link that is involved with either \(i_{H}\) or \(i_{L}\) under \(g^{\prime }\). By the properties of increasing returns and monotonicity, each link that is involved with \(i_{L}\) under \(g^{\prime }\) creates more joint surplus than the corresponding link involved with \(i_{H}\) under g: \([\pi _{i_{L}}(g^{\prime })-\pi _{i_{L}}(g^{\prime }-i_{L}i)]+[\pi _{i}(g^{\prime })-\pi _{i}(g^{\prime }-i_{L}i)]>[\pi _{i_{H}}(g)-\pi _{i_{H}}(g-i_{H}i)]+[\pi _{i}(g)-\pi _{i}(g-i_{H}i)]>2f\). Thus, these links will be kept under \(g^{\prime }\). However, for \(i\in N_{i_{L}}(g)\), the link \(i_{H}i\) under \(g^{\prime }\) becomes less productive than the corresponding link \(i_{L}i\) under g—again due to the properties of increasing returns and monotonicity. Nevertheless, recall that in \(h_{l}(g)\) there is another firm k whose initial cost is higher than \(i_{H}\). Given that g is SAT, for any \(s\in N_{k}(g)\) we have \([\pi _{s}(g)-\pi _{s}(g-sk)]+[\pi _{k}(g)-\pi _{k}(g-sk)]>2f\). By property (i) of Proposition 4, \(N_{i_{H}}(g^{\prime })=N_{i_{L}(g)}=N_{k}(g)\). Since \(c_{i_{H}}(g^{\prime })<c_{k}(g)\), by the properties of increasing returns and monotonicity, for any \(s\in N_{k}(g)=N_{i_{H}}(g^{\prime })\), \([\pi _{s}(g^{\prime })-\pi _{s}(g^{\prime }-i_{H}s)]+[\pi _{i_{H}}(g^{\prime })-\pi _{i_{H}}(g^{\prime }-i_{H}s)]>[\pi _{s}(g)-\pi _{s}(g-sk)]+[\pi _{k}(g)-\pi _{k}(g-sk)]>2f\). Therefore, any link \(i_{H}s\) under \(g^{\prime }\) will still be kept.
Now we check the incentives of firm \(i_{H}\) and firm \(i_{L}\) to add additional links under \(g^{\prime }\). We do not need to worry about firm \(i_{L}\), as it is already linked with all firms under \(g^{\prime }\). Consider firm \(i_{H}\). Since \(c_{i_{H}}(g^{\prime })>c_{i_{L}}(g)\), by the monotonicity property firm \(i_{H}\) has a weaker incentive to add an additional link under \(g^{\prime }\) than does firm \(i_{L}\) under g. Given that firm \(i_{L}\) has no incentive to add any additional link under g, firm \(i_{H}\) should also have no incentive to add an additional link under \(g^{\prime }\).
Finally, for firm \(i_{H}\) and firm \(i_{L}\) under \(g^{\prime }\), we need to check requirement (3): no incentive to sever all of its current links. We first construct the transfers between these two firms. Specifically, let
$$\begin{aligned} t_{i_{L}}^{\prime i_{H}}-t_{i_{H}}^{^{\prime }i_{L}}=t_{i_{H}}^{i_{L}}-t_{i_{L}}^{i_{H}}+[\pi _{i_{L}}(g)-\pi _{i_{L}}(g_{-i_{L}})]-[\pi _{i_{H}}(g^{\prime })-\pi _{i_{H}}(g_{-i_{H}}^{\prime })]. \end{aligned}$$
Due to the properties of increasing returns and monotonicity, \([\pi _{i_{L}}(g)-\pi _{i_{L}}(g_{-i_{L}})]>[\pi _{i_{H}}(g^{\prime })-\pi _{i_{H}}(g_{-i_{H}}^{\prime })]\). Thus the net transfer from \(i_{L}\) to \(i_{H}\) under \(g^{\prime }\) is bigger than the net transfer from \(i_{H}\) to \(i_{L}\) under g. By this construction, under \(g^{\prime }\) firm \(i_{H}\)’s incentive is the same as that of firm \(i_{L}\) under g:
$$\begin{aligned}&[\pi _{i_{H}}(g^{\prime })-\pi _{i_{H}}(g_{-i_{H}}^{\prime })]-\eta _{1}f+\sum _{j\in N_{i_{H}}(g^{\prime }),j\ne i_{L}}(t_{j}^{\prime i_{H}}-t_{i_{H}}^{\prime j})+(t_{i_{L}}^{\prime i_{H}}-t_{i_{H}}^{^{\prime }i_{L}}) \\&\quad =[\pi _{i_{L}}(g)-\pi _{i_{L}}(g_{-i_{L}})]-\eta _{1}f+\sum _{j\in N_{i_{L}}(g)}(t_{j}^{i_{L}}-t_{i_{L}}^{j})\ge 0. \end{aligned}$$
Thus firm \(i_{H}\) has no incentive to sever all of its current links under \(g^{\prime }\). Now consider firm \(i_{L}\):
$$\begin{aligned}&[\pi _{i_{L}}(g^{\prime })-\pi _{i_{L}}(g_{-i_{L}}^{\prime })]-(n-1)f+\sum _{j\ne i_{L},j\ne i_{H}}(t_{j}^{\prime i_{L}}-t_{i_{L}}^{\prime j})+(t_{i_{H}}^{^{\prime }i_{L}}-t_{i_{H}}^{\prime i_{L}}) \\&\quad =[\pi _{i_{H}}(g)-\pi _{i_{H}}(g_{-i})]-(n-1)f+\sum _{j}(t_{j}^{i_{H}}-t_{i_{H}}^{j})+\{[\pi _{i_{L}}(g^{\prime })-\pi _{i_{L}}(g_{-i_{L}}^{\prime })]-[\pi _{i_{H}}(g)-\pi _{i_{H}}(g_{-i_{H}})]\} \\&\qquad -\,\{[\pi _{i_{L}}(g)-\pi _{i_{L}}(g_{-i_{L}})]-[\pi _{i_{H}}(g^{\prime })-\pi _{i_{H}}(g_{-i_{H}}^{\prime })]\} \\&\quad \ge \{[\pi _{i_{L}}(g^{\prime })-\pi _{i_{L}}(g_{-i_{L}}^{\prime })]-[\pi _{i_{H}}(g)-\pi _{i_{H}}(g_{-i_{H}})]\}-\{[\pi _{i_{L}}(g)-\pi _{i_{L}}(g_{-i_{L}})]-[\pi _{i_{H}}(g^{\prime })-\pi _{i_{H}}(g_{-i_{H}}^{\prime })]\} \\&\quad >0. \end{aligned}$$
In the above derivation, the first inequality uses the fact that firm \(i_{H}\) has no incentive to sever all of its links under g, while the last inequality is implied by the monotonicity property. Thus firm \(i_{L}\) has no incentive to sever all of its links under \(g^{\prime }\).
Therefore, \(g^{\prime }\) is SAT. \(\square\)
Proof of Lemma 5
Part (i). By (11), we can compute the difference in gross profits
$$\begin{aligned}&[\pi _{i}(g+ij+ik)-\pi _{i}(g+ij)]-[\pi _{i}(g+ij)-\pi _{i}(g)] \\&\quad =\frac{(n-1)\gamma _{ik}}{n+1}[q_{i}(g+ij+ik)+q_{i}(g+ij)]-\frac{ (n-1)\gamma _{ij}}{n+1}[q_{i}(g+ij)+q_{i}(g)]>0. \end{aligned}$$
The inequality follows because \(q_{i}(\cdot )\) is increasing in additional links and \(\gamma _{ik}>\gamma _{ij}\) since \(k<j\).
Part (ii). Again by (11), we have
$$\begin{aligned}&[\pi _{i}(g+ij)-\pi _{i}(g)]-[\pi _{j}(g+ij)-\pi _{j}(g)] \\&\quad =\frac{(n-1)\gamma _{ij}}{n+1} \{[q_{i}(g+ij)+q_{i}(g)]-[q_{j}(g+ij)+q_{j}(g)]\}>0, \end{aligned}$$
where the inequality follows \(c_{i}(g)<c_{j}(g)\).
Part (iii). Suppose \(ik\notin g\). We will show that g is not stable. Since \(ij\in g\), \([\pi _{i}(g)-\pi _{i}(g-ij)]>f\). Because \(k<j\), by part (i), \(\pi _{i}(g+ik)-\pi _{i}(g)>\pi _{i}(g)-\pi _{i}(g-ij)>f\). Thus firm i has an incentive to form link ik. By a similar logic, \(kl\in g\) and \(i<l\) imply that \(\pi _{k}(g+ik)-\pi _{k}(g)>\pi _{k}(g)-\pi _{k}(g-kl)>f\). Thus firm k also has an incentive to form link ik. Therefore, g is not stable: a contradiction. \(\square\)
Proof of Proposition 6
Part (i). Suppose \(12\notin g\). Since both \(\eta _{1}(g)>0\) and \(\eta _{2}(g)>0\), firm 1 must be linked to some firm \(i>2\) and firm 2 must be linked to some firm \(j>2\). Applying part (iii) of Lemma 5, we must have \(12\in g\): a contradiction.
Part (ii). Suppose \(1k\in g\), but there is an i, \(2\le i<k\), such that \(1i\notin g\). Since \(\eta _{i}(g)>0\), there is an \(j>1\) such that \(ij\in g\). Now applying part (iii) of Lemma 5, we must have \(1i\in g\): a contradiction.
Suppose \(1k\notin g\) for some \(k>2\), and there is some \(s\in N_{k}(g)\) such that \(c_{k}(g-ks)\ge c_{1}(g)\). Because g is stable, \(\pi _{k}(g)-\pi _{k}(g-ks)>f\). By parts (i) and (ii) of Lemma 5, \(c_{k}(g-ks)\ge c_{1}(g)\) implies that \(\pi _{1}(g+1k)-\pi _{1}(g)>\pi _{k}(g)-\pi _{k}(g-ks)>f\). Therefore, \(1k\in g\): a contradiction.
Part (iii). If \(1n\in g\), then by part (ii) all firms are linked to firm 1. Thus g is an interlinked star and firm 1 is in the center. If there is an \(s\in N_{n}(g)\) such that \(c_{1}(g)\le c_{n}(g-ns)\), then by part (ii) \(1n\in g\), and the result follows. If \(\gamma _{1}-\gamma _{12}\le \gamma _{n}-\sum _{3\le j<n}\gamma _{jn}\), then the second condition is always satisfied, and the result follows. \(\square\)