Multi-task Research and Research Joint Ventures

A1 Proof of Theorem 1

The basic result that a series of parallel arrangements of components is more likely to succeed than a parallel arrangement of series of components has a status akin to the Folk Theorem in game theory and is similarly tricky to pin down to a definite source. Perhaps the simplest version can be found in Ross (1996, Theorem 9.1) which can be adapted to prove Theorem 1 in the paper as follows:

Theorem 1 (after Ross (1996)):

For any, .

The above theorem is both a special case and a trivial extension of Theorem 9.1 in Ross (1996).

For anyS(.) and T-dimensional vectorsp, p’: , whereis the probability that a series of T components with individual probability of successpsucceeds.

Theorem 1 follows from the above by setting and by iterating Theorem 9.1 N times (and not just twice as in Theorem 9.1).

A2 The bridge structure as a general model of information flows

Consider the “bridge” structure depicted earlier in Figure 1and assume that the probabilities of individual task success for firms A and B are respectively and , as in Figure 5. Moreover, for reasons that will become apparent shortly, introduce a “connecting” unit that succeeds with probability .

Figure 5

The bridge structure and information flows.

It is straightforward to compute the overall probability of success of the “bridge” structure, , as:

[19]

Notice that under symmetry () eq. [19] simplifies to:

[20]

Equation [20] shows that the “bridge” structure encompasses both types of RJV organizations (inefficient, with N teams; efficient, with T teams) as special cases:

[21]

[22]

>

This observation is interesting for three reasons: (i) the organization-efficient RJV, by assigning a team to each task, effectively introduces additional information flows across tasks, which explains why it dominates the alternative arrangement (assigning each team to the full series of tasks); (ii) even when inter-task information flows are imperfect (i.e., ), organization-efficient RJVs are more likely to achieve overall success than independent firms; (iii) the superiority of efficient RJVs may hold also in the case (not examined in this paper) when the probabilities of individual success are not statistically independent.

Using eq. [19] we can get an insight into the quantitative superiority of RJVs compared to independent research. Recalling that in a tournament model the profits from independent research are the same as from membership of an inefficient RJV, we can compute the rate of return from forming an efficient RJV as:

[23]

Figure 6 shows that, for the relevant case when success in individual tasks is “difficult” (i.e., ), this rate of return ranges from nearly 100% to 10%:

Figure 6

Rate of return for efficient RJVs.

The formulation of the alternative organizational structures open to an RJV in terms of the “bridge structure” provides a simple way of comparing the results in this paper with the vast literature on RJVs spawned by d’Aspremont and Jacquemin (1988) and Kamien et al. (1992). A distinctive feature of this literature is to regard the formation of an RJV as a means of endogenizing any spillover effect. For single-task research, this effectively amounts to assuming that any RJV member agrees to share its successful innovation with all other members. Under multi-task research there is a richer network of potential information flows, some of which can be formalized by means of our bridge structure. The canonical RJV model, where members share information about completed projects is captured by setting in eq. [19], whereas an organizationally-efficient RJV would share information about completed tasks, i.e., would set . Notice also that the bridge structure can be used to model the case where different research tasks undertaken by different RJV members may not be fully compatible, i.e., success in task i by member A when combined with success in task j by member B leads to the whole project succeeding with probability .