When organizations, particularly commercial firms, consider starting an innovation project, the new cognitive concepts that inform the imagined new business are still tentative and incomplete. Accordingly, beliefs in whether the project will become a success or failure are not yet consolidated. In the team responsible for pursuing the innovation, typically consisting of developers, engineers and scientifically trained staff, marketing specialist, and managers with entrepreneurial function, not everyone is necessarily convinced of a successful outcome. Opinions on this question are formed, pro and con, in an ongoing interaction between the involved team members processing newly gained information. Proponents and opponents of the project may develop second thoughts and change their respective opinion, depending on what argument or evidence is presented in their exchange. The discussed epistemic bounds notwithstanding, the credibility of the exchanged arguments is not entirely disconnected, of course, from already existing knowledge about the properties of the innovative project. However, the uncertainty about whether this knowledge is sufficiently representative of the unknown true feasibility and costs, not to forget social costs, of the innovation leave room for interpretative differences.

It is for these reasons that the teams responsible for an innovation project in an organization often open up the innovation process. They may mainly wish to consult outside expertise in the form of inbound knowledge flows as explained in Section 2. Such external opinions and advice are supposed to help improving, and sometimes result in modifications of, the understanding of the technical feasibility and the benefits and costs of the intended innovation. However, even when the consulting of external expertise is mainly organized in the form of inbound knowledge flows, some information about the innovation is likely to be disclosed in the exchange with the consulted outside actors. The more specific the information requested is, the more conclusive it may be for guessing what is going to be developed. Hence, an outbound information flow can only be prevented completely in a closed innovation process.

In effect, the consulting of external expertise adds weight to, and sometimes changes the weights of, the arguments exchanged within the innovating team. At the end of the day, opening up the innovation processes comes down to strengthening the not yet consolidated beliefs in either success or failure of the project within the innovating team. The question is whether, in comparison to closed innovation processes, open innovation activities also make a difference with respect to the likelihood of an earlier discovery of severe negative externalities of the considered innovation.

The conditions under which this can be expected to be the case can be explored by means of a stylized model of the opinion formation process within the team which is responsible for the innovation project. The interactions underlying the opinion formation can result in a typical feature of a self-organizing process called phase transition in self-organization theory: An implicit bias in the assessment of the project can accumulate and eventually drive the process beyond a tipping point. The pro-innovation beliefs with which the project was started then turn into the conviction that the innovation is to be abandoned because of lacking success prospects or negative externalities. What has to be discussed here is the following. In case of severe negative externalities, is the tipping point more likely to be reached – implying a shorter expected waiting time for a transition – in open rather than in closed innovation processes?

For expository convenience let us assume a bi-modal framework in which the members of the team responsible in the organization for the innovation project either believe in the success of the innovation or in its failure. Hence, if the share of team members doubting the success – for brevity let us call them *opponents* – at time *t* is denoted by *F(t)*, the share of team members believing in the success, the *proponents*, is 1 – *F(t).* It can be assumed that an innovation project will only be started if (1 − *F*(0)) ≫ *F*(0). As a result of the exchange of arguments within the team, these beliefs can change so that opponent become proponents and vice versa.

In a probabilistic representation of the belief changes, switching from being proponent to becoming opponent can occur with probability *p(t)* and the reverse switching from being opponent to becoming proponent with probability *q(t).* A characteristic feature of opinion formation processes is the frequency-dependency of the switching probabilities, more specifically a nonlinear conformism effect: The more opponents there are in the team, i.e. the more members do not believe in a success of the innovation, the more than proportionately greater is the probability for proponents to become opponents, too. This implies a functional relationship *p(t) = φ (F(t)), φ’* > 0*, φ”* > 0. The conformism effect also works in the opposite case. Hence, the functional relationship for *q(t) = ϕ (1-F(t)), ϕ’* > 0*, ϕ”* > 0. For the sake of the present argument it is sufficient to choose quadratic specifications for these functions with parameters 0 < *α* ≤ 1 and 0 < *β* ≤ 1 so that

$$ p(t)=\alpha F{(t)}^2 $$

(1)

and

$$ q(t)=\beta {\left(1-F(t)\right)}^2. $$

(2)

As has been said, due to the epistemic bound there is an irreducible *ex ante* uncertainty about the implications of the projected innovation. However, sooner or later these implications will gradually be revealed while the organization continues to pursue the innovation. The revealed information is likely to feed back on the interactive opinion formation process of the innovating team. In the model, the ongoing revelation process can therefore be assumed to result in a bias, however small, in the switching probabilities. Accordingly, there will be innovations for which the pieces of information revealed over time cast doubts on their success, especially when they give rise to fears that the innovation might develop severe negative externalities for which the innovating organization may at least in part be made liable. In the model this bias is represented by a ratio *α/β > 1* for this kind of innovations and in the opposite case by *α/β < 1*.

If the organization runs an *open* innovation process, the exchanges within the innovating team are complemented by a dialogue with external actors. The knowledge inflow from outside affects the opinion formation among the members of the innovating team. Hence, according to the motto “more eyes see better than few eyes”, the bias expressed by the size of the ratio *α/β* should be strengthened the more so, the more external actors get involved in the open innovation, i.e. the more outside knowledge is obtained over time. Following this intuition, the external influence can be depicted by a variable

$$ x(t)=1-{e}^{- \kappa n(t)}. $$

(3)

It depends on the (cumulative) number *n(t)* of external actors getting involved in the team’s opinion formation up to time *t* and on a parameter *k* > 0 reflecting the average strength of their influence. Now suppose the organization runs a completely *closed* innovation process. This means that *n(t)* = 0 for all *t.* In that case *x*(*t*) = 0. Put differently, the degree of openness of the innovation process is expressed in the model by the number *n(t).* The more *n(t)* grows over time, the more open the innovation process is and *x(t)* approaches the value +1 more or less rapidly, depending on the parameter *k*.

On the basis of different values for *n(t)* and, consequently *x(t)*, we can track how in the opinion formation process an innovation project – whether open or closed – fares over time. The critical variable is the share of opponents (or, conversely, proponents) of the project in the innovation team. By assumption, when the innovation project is initiated, *F(0)* < ½. If the revealed information is positive, i.e. in case of *α/β* < 0, the share *F(t)* is likely to decrease over time and the organization can be expected to continue pursuing the innovation. In the opposite case, *F(t)* is increasing in the interactive opinion formation process until a strong majority or all of the innovation team become opponents and the project is abandoned.

Assuming that *t* only takes integer values, the development of *F(t)* is in the mean^{Footnote 5} determined by the first order difference equation

$$ \begin{array}{l}\kern3.36em =\kern0.5em 0\kern0.5em for\kern0.28em F(t)<0,\\ {}F\left(t+1\right)=F(t)+\left(1-F(t)\right)p(t)\left(1+\gamma x(t)\right)-F(t)q(t)\left(1-\gamma x(t)\right)\mathrm{f}\mathrm{o}\mathrm{r}\kern0.28em 0\le F(t)\le 1,\\ {}\kern3.48em =\kern0.5em 1\kern0.5em for\kern0.28em F(t)>1.\end{array} $$

(4)

The factor *γ* is determined by a sign function such that \( \gamma =+1\kern0.5em if\frac{\alpha }{\beta }>1 \) and \( \gamma =-1\kern0.5em if\frac{\alpha }{\beta }<1 \). This ensures that the bias in the switching probabilities is strengthened in the right direction. For 0 ≤ *F*(*t*) ≤ 1 the second summand on the r.h.s. of Eq. (4) represents the mean “inflow” into the share of opponents, i.e. the switching from proponent to opponent, occurring at time *t.* The third summand gives the mean “outflow” from the share of opponents, i.e. the reverse switching.^{Footnote 6}

The question raised in the previous section was whether, in comparison to closed innovations, open innovation activities make a difference with respect to the time at which an innovation is stopped in case of severe negative externalities. Is an organization that has pushed the innovation the more likely to abandon the project that gradually turns out to be a failure or even disaster the more it relies on an open innovation process? To answer this question we have to explore the mean trajectory resulting from Eq. (4) for the specification \( \frac{\alpha }{\beta }>1 \) and, hence, *γ* = + 1.^{Footnote 7} For expository convenience let us choose a simple numerical specification *α* = 1 and *β* = ½ satisfying this condition. Inserting Eqs. (1), (2) and (3) into Eq. (4) and rearranging yields

$$ F\left(t+1\right)=\frac{1+x(t)}{2}F(t)+2\;{\left(F(t)\right)}^2-\frac{3+x(t)}{2}{\left(F(t)\right)}^3, $$

(5)

subject to the condition *F*(*t* + 1) = 0 for *F*(*t*) < 0 and *F*(*t* + 1) = 1 for *F*(*t*) > 0.

The cubic difference Eq. (5) implies a bifurcation by which a tipping point emerges in the share of opponents, i.e. a critical mass or frequency *F*
^{crit}(*t*) of opponents. *F*
^{crit}(*t*) represents an unstable equilibrium point. Once *F*(*t*) > *F*
^{crit}(*t*) the mean process of Eq. (5) will over time be attracted to *F** = 1. This is the all-opponent equilibrium which, by assumption, means that the innovation is abandoned. The question to be discussed can therefore be translated into whether and how the position of the tipping point varies in the interval [0, 1] with the openness of the organization’s innovation activities. The answer can be given by analyzing the extreme cases of, on the one side, an entirely closed and, on the other side, a very open innovation process. In the former case, *n* = 0 and, by Eq. (3), *x(t)* = 0 while in the latter case a growing *n* drives *x(t)* close to 1.

Let us start with an entirely closed innovation regime. For *x(t)* = 0 Eq. (5) has three equilibrium points in the interval [0, 1] which can be found by setting *F(t + 1) = F(t)* and solving: *F** = 1 which is stable, the instable equilibrium *F*
^{crit} = 1/3, and another stable equilibrium in *F** = 0. Hence, there is the possibility that in a closed innovation regime, an innovating team starting with a share *F(0) <* 1/3 is in a self-amplifying way attracted to an all-proponent equilibrium – despite the bias in the revealed information pointing to an impending failure. Only if the innovation project is from the very beginning controversial (i.e. *F(0)* > 1/3) will the revealed information lead the innovating team to quickly refrain from continuing the innovation process.

A very different result obtains in very open innovation regimes. Consider he limiting case *x(t)* = 1. In this case, Eq. (5) has two equilibrium points in the interval [0, 1]: *F** = 1 and *F*
^{crit} = 0. This means that *F(t)* will (in the mean) be attracted to the only stable equilibrium *F** = 1, in which everyone in the innovating team opposes the project, once the instable equilibrium *F*
^{crit} = 0, in which no one opposes, is left. Put differently, in an entirely open innovation process, one single member of the team opposing the innovation suffices to set in motion a development in which an innovation suspected of causing severe negative externalities is soon abandoned. Even if the bias expressed by the ratio \( \frac{\alpha }{\beta }>1 \) is very small *F(t)* will then grow successively as a result of the conformism pressure building up through the external expertise influencing the opinion formation within the innovating team.

Since *F*
^{crit} varies parametrically with *n(t),* a growing openness of the innovation process, i.e. *n(t)* going from 0 to large values and *x(t)* from 0 to 1, implies a parametric shift of the tipping point *F*
^{crit} from the value 1/3 (associated in the chosen numerical specification with *n(t)* = 0) downwards to zero. Consequently, the share of opponents in the innovating team which suffices to drive *F(t)* beyond *F*
^{crit} in the direction of *F** = 1 becomes smaller and smaller and is therefore more likely to be reached.^{Footnote 8}

Thus, under the assumption of this model we obtain a clear result. The openness of the innovation process is crucial for reacting early to signs of a failure or even disaster threatening to follow from the pursuit of the innovation. The more open the innovation activity is, the more likely will an innovation be abandoned when there are hints pointing at a failure. In the interest of keeping social costs caused by negative externalities of innovations down it would therefore be desirable to have innovation processes that are as open as possible. As discussed in section 2, such openness may however conflict with the motivation of the innovating organization to protect the private knowledge on which its innovation is based from being diffused to competitors. To create this private innovative knowledge and to implement it in the form of a commercial business requires substantial investments. If the knowledge would diffuse, the innovation could easily be imitated by competitors and the expected innovation rent be quickly competed away.

There is thus a conflict of interests here. If openness of the innovation process would be made a legal requirement, potential innovators can be expected not to undertake many of the innovation projects they would be willing to start as closed activities. If, in contrast, choosing the degree of openness is left to the strategic discretion of the innovating organization, the choice is in most cases a more or less closed innovation activity. It can then happen that even disastrous innovations continue for long until rising social costs induce the public or the government to intervene and to force the innovating organization to let outside expertise in.