P. Billand, C. Bravard, and S. Sarangi, Strict Nash networks and partner heterogeneity, International Journal of Game Theory, vol.40, issue.3, pp.515-525, 2011.
URL : https://hal.archives-ouvertes.fr/halshs-00617713

P. Billand, C. Bravard, and S. Sarangi, Modeling resource flow asymmetries using condensation networks, Social Choice and Welfare, vol.41, issue.3, pp.537-549, 2013.
URL : https://hal.archives-ouvertes.fr/halshs-00728194

P. Billand, C. Bravard, J. Durieu, and S. Sarangi, Efficient networks for a class of games with global spillovers, Journal of Mathematical Economics, vol.61, pp.203-210, 2015.
URL : https://hal.archives-ouvertes.fr/halshs-01247683

R. Blundell, R. Griffith, and J. Van-reenen, Market share, market value and innovation in a panel of british manufacturing firms, Review of Economic Studies, vol.66, issue.3, pp.529-554, 1999.

U. Blum, C. Fuhrmeister, P. Marek, and M. Titze, R&D collaborations and the role of proximity, Regional Studies, vol.51, issue.12, pp.1761-1773, 2017.

A. Galeotti, S. Goyal, and J. Kamphorst, Network formation with heteregeneous players, Games and Economic Behavior, vol.54, issue.2, pp.353-372, 2005.

B. Gomes-casseres, J. Hagedoorn, and A. Jaffe, Do alliances promote knowledge flows, Journal of Financial Economics, vol.80, pp.5-33, 2006.

S. Goyal and S. Joshi, Networks of collaboration in oligopoly, Games and Economic Behavior, vol.43, issue.1, pp.57-85, 2003.

S. Goyal and S. Joshi, Unequal connections, International Journal of Game Theory, vol.34, issue.3, pp.319-349, 2006.

S. Goyal, J. Moraga-gonzalez, and . Networks, The RAND Journal of Economics, vol.32, issue.4, pp.686-707, 2001.

S. Goyal, J. L. Moraga-gonzalez, and A. Konovalov, Hybrid R&D, Journal of the European Economic Association, vol.6, issue.6, pp.1309-1338, 2008.

J. Hagedoorn, Inter-firm R&D partnerships: An overview of major trends and patterns since 1960, Research Policy, vol.31, issue.4, pp.477-492, 2002.

M. D. König, S. Battiston, M. Napoletano, and F. Schweitzer, The efficiency and stability of R&D networks, Games and Economic Behavior, vol.75, issue.2, pp.694-713, 2012.

M. D. König, C. J. Tessone, and Y. Zenou, Nestedness in networks: A theoretical model and some applications, Theoretical Economics, vol.9, issue.3, pp.695-752, 2014.

N. V. Mahadev and U. N. Peled, Threshold graphs and related topics, 1995.

D. C. Mowery, J. E. Oxley, and B. S. Silverman, Technological overlap and interfirm cooperation: implications for the resource-based view of the firm, Research Policy, vol.27, issue.5, pp.507-523, 1998.

D. Persitz, Core-periphery R&D collaboration networks. Working paper, 2014.

W. W. Powell, D. R. White, K. W. Koput, and J. Owen-smith, Network dynamics and field evolution: The growth of interorganizational collaboration in the life sciences, American Journal of Economics and Sociology, vol.110, issue.4, pp.1132-1205, 2005.

, ? 1) 2 + 2(n ? 2) + 1)(V i (g) + V j ) ? 0, Moreover, we have 2?

, We show that if i, j ? N t , t ? {1, 2}, and V i (g) ? V j (g), then g(i) ? g(j)

V. If, V j (g), then c i (g) ? c j (g). If j ? g(j), then firm j has formed a link with j in g

, By Proposition 2 since c i (g) ? c j (g) and v j ,j = v j ,i = v, then there is a link between firms j and i in g

, We show that if i, j ? N t , t ? {1, 2}, then we have either g(i) ? g(j) or g(j) ? g(i)

, we have V i (g) ? V j (g), or V j (g) ? V i (g). By 1., in the first case we have g(i) ? g(j), and in the second case we have g(j) ? g(i)

2. {1, Let a binary relation on N t be defined as in (??). Then is the vicinal preorder of g

?. , Consider a firm i ? N such that |g(i)| ? |g(j)|, i.e., V i (g) ? V j (g) for every firm j ? N

, We distinguish between two cases. First, i is the only firm in N 1 who has formed links. Consider a link ij ? g with j ? N 2 and a firm k ? N 1. We build the network g with g = g ? ij + ik. Suppose that v I = v O. It is clear that W (g ) = W (g)

, Since the number of links is the same in g and g , it is sufficient to show that ? > 0 to establish the lemma. We do this in two steps

. First and L. Let-v-m-=-v,

, using the same reasoning as in the proof of Proposition 2, point 2, we can conclude that ? ? 0 for v M = v L implies ? > 0 for v M > v L

, By using similar arguments as in the proof of Corollary 4, we establish that if i, j ? N t , t ? {H, L}, and V i (g) ? V j (g), then g(i) ? g(j). It follows that if sub-network g

, NSG and there is no link between firms in N L and firms in N H. By construction, there exists a firm, say i L ? N L , such that g(i L ) = ?. We now build the network g similar to g except that every firm j L ? g(i L ) replaces its link with i L by a link with i H ? N H, By Lemma, vol.1

N. First, suppose g is such that for firm i H ? N H we have |g(i H )| ? |g(i L )| > 0 for some i L ? N L. By applying the same arguments as in the proof of point 4 of Corollary 3, we obtain a contradiction. Second, if g is such that for each i H ? N H we have |g(i H )| < |g(i L )| for all i L ? N L

. Suppose-v-i,i-=-v-j,j-=-v, By Lemma 4, given in Appendix D, we know that network g that maximizes the total profit contains a link between firms j and j since it contains a link between firms i and i. Moreover, since the assumptions of Property PCS are satisfied, network g that maximizes the total consumer surplus contains a link between firms j and j since it contains a link between firms i and i. It follows that if v i,i = v j

I. Suppose-v-j,j->-v-i, Let v j,j = v j,j ? v i,i > 0. By the previous point we have W (V(g; v j,j ? v j,j , |g|+1) > W (V(g), |g|), and by property PW, we have W (V(g; v j,j , |g|+1) > W