Skip to main content

Collaborative Networks in Oligopoly with Asymmetric Firms

Abstract

This paper studies collaborative R&D networks among ex ante asymmetric firms. In the first stage firms form bilateral collaborative links, which reduce firms’ marginal costs. In the second stage firms engage in Cournot competition. We characterize the structure of stable networks and study how firms’ initial sizes (costs) affect their network positions. One of the main results is that in stable networks firms’ initial sizes and their number of links are positively but not perfectly correlated. We also characterize efficient networks, which always amplify the initial asymmetry among firms.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. For instance, both in the IT industries (Hagedoorn and Schakenraad 1992) and in the biotech and pharmaceutical industries (Powell et al. 2005) some firms have many collaborative partners, while others have very few.

  2. Hagedoorn and Schakenraad (1992) identified seven leading collaborating firms in the IT industries in the 1980s: All are big firms. They also found that in each subfield about half of the top ten leading collaborating firms are the top ten firms in terms of size. In the other direction the ratio is about the same: in microelectronics about half of the leading suppliers are also the leading collaborators, while in telecommunications the ratio is 6 out of 10. A similar pattern is found in the biotech and pharmaceutical industries (Powell et al. 2005).

  3. The empty network and the complete network (each pair of firms is linked) are two special cases of dominant group networks.

  4. Goyal and Joshi (2006) extend GJ by studying a more abstract second-stage game. Dawid and Hellmann (2014) extend GJ to dynamic and stochastic settings.

  5. Zirulia (2005) and Zikos (2008) extend GM to settings with ex ante heterogeneous firms. However, both models consider only some special cases, with the number of firms being either three or four.

  6. Billand et al. (2015) show that, in the framework of Westbrock (2010), if an additional condition with regard to players’ payoff functions is satisfied, then efficient networks must be NSG.

  7. In a period, a firm either changes quantity only, changes link formation only, changes both, or changes neither, according to an exogenous probability distribution.

  8. The gain to firm i from linking to firm j is fixed; it does not depend on how many additional links firm j has.

  9. A sufficient condition is \(\gamma _{1}-(n-1)\gamma >0\).

  10. It is possible that some firm might produce zero if its final cost is relatively too high. Incorporating this possibility would complicate the analysis without adding much additional insight.

  11. In other aspects, Westbrock (2010) is more general than the current paper. For instance, in his model firms’ products could be imperfect substitutes, and he also considers price competition in the second stage game.

  12. For instance, we cannot compare links 14 and 23.

  13. More intuitively, adding a link between two firms with lower initial costs means that the cost reduction can be applied to larger quantities, which is more efficient than adding a link between two firms with higher initial costs.

  14. For instance, \(\gamma _{1}=5\), \(\gamma _{2}=5.1\), \(\gamma _{3}=5.2\), \(\gamma _{4}=7\), and \(\gamma =0.5\). Then the variance of final costs under the dominant group network \(V_{c}(g^{3})\) is higher than that under the star network \(V_{c}(g^{s})\).

  15. This property no longer holds if link productivity is link-specific (see Sect. 6 for details).

  16. To be more precise, \(c_{i_{H}}(g^{k}-i_{H}i)\le c_{j_{L}}(g^{k})=\gamma _{j_{L}}\).

  17. The second effect is illustrated by the following example. Suppose \(n=6\), \(\alpha =25\), \(\gamma =0.6\), and \((\gamma _{1},\gamma _{2},\gamma _{3},\gamma _{4},\gamma _{5},\gamma _{6})=(5,6,7,7.5,8,8.5)\). When \(f=1.96\) the dominant group network with firms 1, 2, and 4 in the dominant group is stable. However, when \(f=1.9\) it becomes unstable, as firm 3 now has an incentive to form links with firms in the dominant group.

  18. In an earlier version, we also consider strongly stable networks. In particular, in deviation we allow each individual firm to form multiple additional links at the same time [different from the strong stability concepts that are proposed in Jackson and van den Nouweland (2005) or Dutta and Mutuswami (1997)]. Relative to pairwise stable networks, although adopting strong stability refines the set of stable networks, qualitative results remain the same.

  19. For irregular and unconnected networks, we can show that a network that is SAT must have only one component, and it must be either an interlinked star or a complete network.

  20. In an earlier version, we also prove that the following two properties holds for networks that are SAT: (1) firms with the same degree are linked to the same set of firms, and (2) each firm in \(h_{1}(g)\) must receive net positive transfers from each central firm.

  21. In this example, \(\gamma _{1}=\gamma _{2}=5\), \(\gamma _{3}=\gamma _{4}=\gamma _{5}=\gamma _{6}=7.5\), \(\gamma =0.5\), \(\alpha =25\), and \(f=1\). The network with firms 1 and 2 in the center is stable because each of them has a lower initial cost; for each firm, the overall joint surplus that is created by five links is big enough to cover the total cost of link formation. In the unstable network with firm 3 in the center, it has an incentive to sever all of its existing links. The reason is that the overall joint surplus that is created by firm 3’s five links is not big enough to cover the total cost of link formation 10f.

  22. The reason that this type of network could be stable is because pairwise stability only checks the marginal incentives of each link one at a time.

  23. More precisely, in a stable network g suppose \(ij\in g\), \(ik\notin g\), and \(k<j\) (a link of a lower ordering is in g but a link of higher ordering is not). Then it must be the case that \(c_{k}(g)\ge c_{j}(g-ij)\). This is because firm i has an incentive to form an additional link ik, which is more productive than an existing link ij. Since g is stable, firm k must have no incentive to form link ik, which implies that it must have a higher cost than firm j under \(g-ij\).

References

  • Billand, P., Bravard, C., Durieu, J., & Sarangi, S. (2015). Efficient networks for a class of games with global spillovers. Journal of Mathematical Economics, 61, 203–210.

    Article  Google Scholar 

  • Bloch, F. (1995). Endogenous structures of association in oligopolies. Rand Journal of Economics, 26(3), 537–556.

    Article  Google Scholar 

  • d’Aspremont, C., & Jacquemin, A. (1988). Cooperative and noncooperative R&D in duopoly with spillovers. American Economic Review, 78(5), 1133–1137.

    Google Scholar 

  • Dawid, H., & Hellmann, T. (2014). The evolution of R&D networks. Journal of Economic Behavior and Organization, 105, 158–172.

    Article  Google Scholar 

  • Deroian, F., & Gannon, F. (2006). Quality-improving alliances in differentiated oligopoly. International Journal of Industrial Organization, 24(3), 629–637.

    Article  Google Scholar 

  • Dutta, B., & Mutuswami, S. (1997). Stable networks. Journal of Economic Theory, 76(2), 322–344.

    Article  Google Scholar 

  • Galeotti, A., Goyal, S., & Kamphorst, J. (2006). Network formation with heterogeneous players. Games and Economic Behavior, 54(2), 353–372.

    Article  Google Scholar 

  • Goyal, S., & Joshi, S. (2003). Networks of collaboration in oligopoly. Games and Economic Behavior, 43(1), 57–85.

    Article  Google Scholar 

  • Goyal, S., & Joshi, S. (2006). Unequal connections. International Journal of Game Theory, 34(3), 319–349.

    Article  Google Scholar 

  • Goyal, S., & Moraga-Gonzales, J. (2001). R&D networks. Rand Journal of Economics, 32(4), 686–707.

    Article  Google Scholar 

  • Hagedoorn, J., & Schakenraad, J. (1992). Leading companies and networks of strategic alliances in information technologies. Research Policy, 21(2), 163–190.

    Article  Google Scholar 

  • Hsieh, C.-S., Konig, M., & Liu, X. (2017). Network formation with local complements and global substitutes: The case of R&D networks. Working paper.

  • Jackson, M., & van den Nouweland, A. (2005). Strongly stable networks. Games and Economic Behavior, 51(2), 420–444.

    Article  Google Scholar 

  • Jackson, M., & Wolinsky, A. (1996). A strategic model of social and economic networks. Journal of Economic Theory, 71(1), 44–74.

    Article  Google Scholar 

  • Kamien, M., Muller, E., & Zhang, I. (1992). Research joint ventures and R&D cartels. American Economic Review, 82(5), 1293–1306.

    Google Scholar 

  • Katz, M. (1986). An analysis of cooperative research and development. Rand Journal of Economics, 17(4), 527–543.

    Article  Google Scholar 

  • Powell, W., Koput, K., White, D., & Owen-Smith, J. (2005). Network dynamics and field evolution: The growth of interorganizational collaboration in the life sciences. American Journal of Sociology, 110(4), 1132–1205.

    Article  Google Scholar 

  • Westbrock, B. (2010). Natural concentration in industrial research collaboration. Rand Journal of Economics, 41(2), 351–371.

    Article  Google Scholar 

  • Yi, S. (1998). Endogenous formation of joint ventures with efficiency gains. Rand Journal of Economics, 29(3), 610–631.

    Article  Google Scholar 

  • Zikos, V. (2008). R&D collaboration networks in mixed oligopoly. Working paper.

  • Zirulia, L. (2005). R&D networks with heterogeneous firms. Working paper.

Download references

Acknowledgements

We would like to thank the Editor, Lawrence J. White, and two anonymous referees for their very helpful comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Huanxing Yang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

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 ij,  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 ijk,  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. \(\square\)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gong, Q., Yang, H. Collaborative Networks in Oligopoly with Asymmetric Firms. Rev Ind Organ 56, 357–380 (2020). https://doi.org/10.1007/s11151-019-09701-w

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11151-019-09701-w

Keywords

  • Networks
  • R&D collaboration
  • Oligopoly
  • Firm asymmetry

JEL Classification

  • C70
  • L13
  • L20