Cherry-picking or frog-kissing? A theoretical analysis of how investors select entrepreneurial ventures in thin venture capital markets

Abstract

We propose a formal model that analyzes which entrepreneurial ventures actively seek and subsequently obtain venture capital (VC) financing in thin VC markets. The model shows that in thin VC markets, (1) VC investors will invest in companies in need (frog-kissing) rather than in best performers (cherry-picking), and (2) the best performing ventures will self-select out of the market for VC. These conclusions are in line with the results from the literature, which note that in Europe many entrepreneurial firms do not actively seek VC investment and that VC investors do not appear to possess the same cherry-picking ability that they have in the US.

Appendix

1.1 Proof of Eq. (6)

Equation (6) follows directly from the theory of auctions with endogenous participation (Menezes and Monteiro 2000). We provide here a proof adapted to our particular framework. If a venture enters into the market for VC, it enters a competition in which, at time t = 2, it will only gain if it gets financed. The selection process of VC, which has been discussed in Sect. 3.3, is known by the entrepreneurs at time t = 1. Let I _max be the maximum impact that VC has on all other N − 1 ventures that potentially compete for VC. The cumulative distribution of I _max is the following: $H\left(i \right) = \Pr \left[{I_{\max} < i} \right] = G\left(i \right)^{N - 1}$. The probability density function of I _max is thus the following: $h\left( i \right) = \left( {N - 1} \right)G\left( i \right)^{N - 2} g\left( i \right)$. The entrepreneur knows that $\tilde{I}$ is the threshold for participation in the market for VC for all players. For $I_{j} > \tilde{I}$, the expected gain $V\left( {I_{j} } \right)$ from being in the market for VC for the entrepreneur is the following:

$$V\left({I_{j}} \right) = \underbrace {{\mathop \int \nolimits_{o}^{{\tilde{I}}} I_{j} h\left(i \right){\text{d}}i}}_{{{\text{Expected}}\,{\text{gain}}\,{\text{if}}\,I_{\max} < \tilde{I}}} \,+\, \underbrace {{\mathop \int \nolimits_{{\tilde{I}}}^{{I_{j}}} \left({I_{j} - i} \right)h\left(i \right){\text{d}}i}}_{{{\text{Expected}}\,{\text{gain}}\,{\text{if}}\,I_{\max} > \tilde{I}}}$$

(8)

The first term of Eq. (8) derives from the fact that if $I_{\hbox{max} } < \tilde{I}$, no other firm will be in the market for VC and entrepreneurs will internalize all the value I _j . The second term in (8) derives from the fact that if $I_{\hbox{max} } > \tilde{I}$, the entrepreneur will have to share some of the value creation with the VC because of competition in the auction. If I _max > I _j , the venture will not be financed by VC and will not receive any benefit from being in the market for VC. Both terms of V(I _j ) are strictly increasing in I _j . The equilibrium level of $\tilde{I}$ is thus unique and such that $V\left( {\tilde{I}} \right) = s$. Integrating the first term of (8) (the other being null) leads to the following:

$$V\left({\tilde{I}} \right) = \mathop \int \nolimits_{0}^{{\tilde{I}}} \tilde{I}h\left(i \right){\text{d}}i = \tilde{I}\left({N - 1} \right)\mathop \int \nolimits_{0}^{{\tilde{I}}} G\left(i \right)^{N - 1} g\left(i \right){\text{d}}i = \tilde{I}G\left({\tilde{I}} \right)^{N - 1}$$

(9)

By equating Eq. (9) to s, we obtain the following:

$$\tilde{I} = \frac{s}{{G\left({\tilde{I}} \right)^{N - 1}}}.$$

Which proves Eq. (6). $\blacksquare$

1.2 Proof of Proposition 1

For this proof, we use the same notation as in the previous Appendix sub-section. If a venture j enters into the market for VC, it enters a competition in which, at time t = 2, it will gain from IVC if I _j  > I _max or, else, will gain from GVC with a probability π. Let $\tilde{I}_{\text{GVC}}$ be the threshold impact to be on the market for VC. For $I_{j} > \tilde{I}_{\text{GVC}}$, the expected gain from being in the market for VC for the entrepreneur is the following:

$$V_{\text{GVC}} \left({I_{j}} \right) = \underbrace {{\mathop \int \nolimits_{o}^{{\tilde{I}_{\text{GVC}}}} I_{j} h\left(i \right){\text{d}}i + \mathop \int \nolimits_{{\tilde{I}_{\text{GVC}}}}^{{I_{j}}} \left({I_{j} - i} \right)h\left(i \right){\text{d}}i}}_{{{\text{Expected}}\,{\text{gain}}\,{\text{from}}\,{\text{IVC}}}} + \,\underbrace {{\pi \mathop \int \nolimits_{{I_{j}}}^{+ \infty} \varphi I_{j} h\left(i \right){\text{d}}i,}}_{{{\text{Expected}}\,{\text{gain}}\,{\text{from}}\,{\text{GV}}}}$$

(10)

where the first two terms in (10) are similar to those in Eq. (8) and the last term represents the expected benefit from GVC. It is easy to prove that the three terms are strictly increasing with I _j, which means that the equilibrium level of $\tilde{I}_{\text{GVC}}$ is thus unique and such that $V_{\text{GVC}} \left( {\tilde{I}_{\text{GVC}} } \right) = s$. Integrating the first and last term of $V_{\text{GVC}} \left( {\tilde{I}_{\text{GVC}} } \right)$ (the second being null) leads to the following:

$$V_{\text{GVC}} \left({\tilde{I}_{\text{GVC}}} \right) = \mathop \int \nolimits_{o}^{{\tilde{I}_{\text{GVC}}}} \tilde{I}_{\text{GVC}} h\left(i \right){\text{d}}i + \pi \mathop \int \nolimits_{{\tilde{I}_{\text{GVC}}}}^{+ \infty} \varphi \tilde{I}_{\text{GVC}} h\left(i \right){\text{d}}i = \tilde{I}_{\text{GVC}} \left({G\left({\tilde{I}_{\text{GVC}}} \right)^{N - 1} + \pi \varphi \left({1 - G\left({\tilde{I}_{\text{GVC}}} \right)^{N - 1}} \right)} \right).$$

(11)

The equilibrium level of $\tilde{I}_{\text{GVC}}$ is thus obtained by equating (11) to s. It is easy to prove that $\pi \varphi = 0 \to \tilde{I}_{\text{GVC}} = \tilde{I}$ and that $\tilde{I}_{\text{GVC}}$ is decreasing in $\pi \varphi$. These two results, combined, prove that: $\tilde{I}_{\text{GVC}} < \tilde{I}$.

Finally, in presence of a GVC, Eq. (7) becomes as follows:

$$\varPi_{\text{VC}}^{{{\text{With\,GVC}}}} = \underbrace {{E\left[{I_{1} - I_{2}} \right]}}_{{{\text{Profit\,if}}\,n > 1}}\underbrace {{\left({1 - G\left({\tilde{I}_{\text{GVC}}} \right)^{N} - G\left({\tilde{I}_{\text{GVC}}} \right)^{N - 1} \left({1 - G\left({\tilde{I}_{\text{GVC}}} \right)^{N}} \right)} \right)}}_{{{\text{Probability}}\,{\text{that}}\,n > 1}} > \varPi_{\text{VC}}^{{{\text{Without\,GVC}}}},$$

(12)

where the first term of (12) is the same as in Eq. 7 (because it only depends on N and on the distributional properties of I), and the second term is greater than in Eq. 7 because $\tilde{I}_{\text{GVC}} < \tilde{I}$.$\blacksquare$