Limited Farsightedness in R &D Network Formation

We adopt the horizon-K farsighted set of Herings, Mauleon and Vannetelbosch (2019) to study the R &D networks that will emerge in the long run when firms are neither myopic nor fully farsighted but have some limited degree of farsightedness. We find that a singleton set consisting of a pairwise stable network is a horizon-K farsighted set for any degree of farsightedness \(K\ge 2\). That is, each R &D network consisting of two components of nearly equal size satisfies both horizon-K deterrence of external deviations and horizon-K external stability for \(K\ge 2\). On the contrary, each R &D network consisting of two components with the largest one comprising three-quarters of firms, predicted when all firms are fully farsighted, violates horizon-K deterrence of external deviations. Thus, when firms are homogeneous in their degree of farsightedness, pairwise stable R &D networks consisting of two components of nearly equal size are robust to limited farsightedness.

1. Mauleon and Vannetelbosch [25] provide a comprehensive overview of the solution concepts for solving network formation games.

2. Various approaches to farsightedness can be found in Chwe [6], Xue [32], Herings et al. [10, 11], Dutta et al. [7], Page et al. [27], Page and Wooders [28], Mauleon et al. [26], and Ray and Vohra [30].

3. Herings, Mauleon and Vannetelbosch (2020) define the myopic-farsighted stable set for two-sided matching problems, while Luo et al. [19] investigate the myopic-farsighted stable set in general network formation problems.

4. Petrakis and Tskas [29] investigate the effect of potential entry on the formation and stability of R &D networks when firms are farsighted, while Roketskiy [31] studies collaboration between farsighted firms competing in a tournament and finds that stable networks consist of two asymmetric mutually disconnected complete components.

5. Caulier et al. [4, 5] propose the concept of contractual stability to predict the stable networks when the consent of coalition partners is needed for adding or deleting links.

6. An exception is the open membership game. Yi [33] finds that only the grand coalition is stable, but this result is not always robust when firms are not identical (see Yi and Shin [34]). See Bloch (2005) for a survey on group and network formation in industrial organization.

7. Mauleon et al. [22] show that if firms are myopic (\(\Delta \)-stability) there is no stable association structure for \(n\ge 8\).

8. Throughout the paper, we use the notation \(\subseteq \) for weak inclusion and \(\subset \) for strict inclusion. Finally, \(\#\) will refer to the notion of cardinality.

9. In a minimally connected component, every pair of firms belonging to the component is connected by exactly one path.

10. Firms collaborate in R &D but do not cooperate on R &D effort choices. For a general background on R &D cooperation in oligopoly the reader is directed to Amir [1], Kamien et al. [16] and Katz [17], among others.

11. In Mauleon et al. [20], the reduction in marginal costs depends on the total number of connected firms, but decreases with the distance. In Goyal and Joshi [8], the reduction in marginal costs only depends on the number of direct links. In Goyal and Moraga-Gonzalez [9], firms also benefit imperfectly from public spillovers, i.e. the research done by firms to whom they are not connected.

12. In a network g, a component \(h\in C(g)\) has no redundant links if and only if h is minimally connected. It reflects the idea that firms avoid wasting resources. When a firm deletes a redundant or superfluous link, it remains connected to the same set of firms and so still benefits from the same reduction in marginal costs.

13. Excluding infinitesimally small costs for maintaining redundant links.

14. Petrakis and Tsakas [29] consider a set-up where R &D effort is costly and endogenous but in an environment with only three farsighted firms that could differ in the initial marginal cost and the levels of substitutability between products.

15. P 1-P 6 can also be satisfied by the equilibrium payoff functions in the case of nonlinear inverse demand functions. For instance, a constant-elasticity inverse demand function, \(p(Q)=aQ^{-t}\) with \(0<t<1\), and an iso-elastic inverse demand function, \(p(Q)= a - Q^{\alpha }\) with \(0<\alpha \), do satisfy the properties P 1-P 6 under some conditions. However, it could happen that the pairwise stable networks under the linear demand function are now defeated by some adjacent network.

16. \(\lceil x \rceil \) is the function that takes as input a real number x and gives as output the lowest integer greater than or equal to x.

17. We use the notational convention that \(\phi _{-1}(g)=\emptyset \) for every \(g\in {\mathcal {G}}\).

18. Since the degree of farsightedness of players is equal to K,  Herings et al. [12] distinguish farsighted improving paths of length less than or equal to \(K-2\) after a deviation from g to \(g+ij\) and farsighted improving paths of length equal to \(K-1\). In the former case, the reasoning capacity of the players is not yet reached, and the threat of ending in \(g^{\prime }\) is only credible if it belongs to the set G. In the latter case, the only way to reach \(g^{\prime }\) from g requires K steps of reasoning or even more; one step in the deviation to \(g+ij\) and at least \(K-1\) additional steps in any farsighted improving path from \(g+ij\) to \(g^{\prime }\). Since this exhausts the reasoning capacity of the players, the threat of ending in \(g^{\prime }\) is credible, irrespective of whether it belongs to G or not.

19. Similarly to Jackson and Watts [15], a set of networks C is a cycle if for any \(g^{\prime }\in C\) and \(g\in C\setminus \{g^{\prime }\}\), \(g^{\prime }\in \phi _{1}^{\infty }(g)\). A cycle C is a closed cycle if \(\phi _{1}^{\infty }(C)=C\). For every pairwise stable network \(g\in P_{1}\), the set \(\{g\} \) is a closed cycle.

20. Herings et al. [11] define a pairwise farsightedly stable set as a set \(G_{\infty }\) of networks satisfying horizon-\(\infty \) deterrence of external deviations and minimality, but with horizon-\(\infty \) external stability replaced by the requirement that for every \(g^{\prime }\in {\mathcal {G}}\setminus G_{\infty }\), \(\phi _{\infty }(g^{\prime })\cap G_{\infty }\ne \emptyset \).

21. \(\lfloor x \rfloor \) is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x.

22. By adding the link \(i_{1}j_{1}\), there are now two paths connecting \(i_{1}\) and \(j_{1}\) in the network, and so there is a superfluous link that does not belong to the end network on the path that connects indirectly \(i_{1}\) and \(j_{1}\).

23. Since the number of links in the smallest component (i.e. \(\# N \setminus {\bar{S}}(g)-1\) links) is greater than or equal to the maximum number of links that belong to the end network and are still missing in the largest component (i.e. 2 links), we only need to repeat one time the process from Step 1 to Step 4, and so, we go directly from Step 4 to Step 6 after the first repetition.

Ana Mauleon and Vincent Vannetelbosch are, respectively, Research Director and Senior Research Associate of the National Fund for Scientific Research (FNRS). Financial support from the Fonds de la Recherche Scientifique—FNRS research grant T.0143.18 is gratefully acknowledged. Jose J. Sempere-Monerris gratefully acknowledges financial support from the Spanish Ministry of Science and Innovation, AEI and FEDER under the project PID2019-107895RB-I00, as well as from Generalitat Valenciana under the project PROMETEO/2019/095.

Authors and Affiliations

1. CEREC, UCLouvain Saint-Louis, Brussels, Belgium

   Ana Mauleon

2. CORE/LIDAM, UCLouvain, Louvain-la-Neuve, Belgium

   Ana Mauleon, Jose J. Sempere-Monerris & Vincent Vannetelbosch

3. Department of Economic Analysis and ERI-CES, University of Valencia, Valencia, Spain

   Jose J. Sempere-Monerris

Corresponding author

Correspondence to Vincent Vannetelbosch.

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This article is part of the topical collection ‘Group Formation and Farsightedness’ edited by Francis Bloch, Ana Mauleon and Vincent Vannetelbosch.

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