Endogenous formation of entrepreneurial networks

Abstract

The purpose of this paper is to study the incentives of entrepreneurial actors to collaborate with others in order to achieve superior performance. We consider an environment composed of n entrepeneurial actors who are identical with regard to the type of innovation and their costs for innovative activities. Innovation takes place in a contest where actors invest in knowledge in order to increase their chances of winning. Prior to these investments, actors have the opportunity to collaborate with others and to form an entrepreneurial network. The benefits of such a cooperation arise from the commitment to share knowledge. For a given network structure, we first completely characterize the actors’ investments in knowledge and their resulting payoffs. In particular, we show that bigger entrepeneurial networks are not always more beneficial than smaller ones. Using these results, we then analyze the optimal endogenous formation of entrepeneurial networks in the open and the exclusive membership game. Our results indicate that the size of entrepreneurial networks is rather limited.

Appendix

Proof of Proposition 1

• 1. Consider first effective investments. Equation (2) implies that $\tilde {x}_{i}^{\ast } = \tilde {x}_{j}^{\ast }\frac {\alpha _{j}}{\alpha _{i}}< \tilde {x}_{j}^{\ast },$since α_j < α_i for m_j < m_i. Hence,

  $$ \begin{array}{@{}rcl@{}} \tilde{x}_{i}^{\ast }&=&m_{i}x_{i}^{\ast }+\beta m_{j}x_{j}^{\ast }+\sum\limits_{\begin{array}{lll} r=1 \\ r\neq i,j \end{array}}^{k}\beta m_{r}x_{r}^{\ast }<\tilde{x}_{j}^{\ast }\\ &=&m_{j}x_{j}^{\ast }+\beta m_{i}x_{i}^{\ast }+\sum\limits_{\begin{array}{lllll} r=1 \\ r\neq i,j \end{array}}^{k}\beta m_{r}x_{r}^{\ast }, \end{array} $$

  that is, $\left (1-\beta \right ) m_{i}x_{i}^{\ast }<\left (1-\beta \right ) m_{j}x_{j}^{\ast }$. Since m_j < m_i, $x_{i}^{\ast }<\frac {m_{j}}{m_{i}} x_{j}^{\ast }<x_{j}^{\ast }$.

• 2. Using Eq. (2), the probability of being successful in the contest is given by

  $$ p_{i}=\frac{\tilde{x}_{i}^{\ast }}{\tilde{X}^{\ast }}=\frac{\tilde{x}_{i}^{\ast }}{\tilde{x}_{i}^{\ast }\sum\limits_{j=1}^{n}\frac{\alpha_{i}}{\alpha_{j}}}=\frac{1}{\alpha_{i}}\left( \sum\limits_{r=1}^{k}\frac{m_{r}}{\alpha_{r}} \right)^{-1}. $$

  Then, m_i > m_j implies α_i > α_j and the proposition follows.

• 3. Equations (2) and (3) imply that

  $$ \begin{array}{@{}rcl@{}} \tilde{X}^{\ast } &=&\sum\limits_{j=1}^{n}\tilde{x}_{j}^{\ast }=\tilde{x}_{i}^{\ast }\cdot \sum\limits_{j=1}^{n}\frac{\alpha_{i}}{\alpha_{j}}\\ &=&\frac{V}{\alpha_{i}} \cdot \frac{\left( \sum\limits_{j=1}^{n}\frac{1}{\alpha_{j}}\right) -1}{\left( \sum\limits_{j=1}^{n}\frac{1}{\alpha_{j}}\right)^{2}}\cdot \alpha_{i}\sum\limits_{j=1}^{n}\frac{1}{\alpha_{j}} \\ &=&V\cdot \frac{\left( \sum\limits_{j=1}^{n}\frac{1}{\alpha_{j}}\right) -1}{\left( \sum\limits_{j=1}^{n}\frac{1}{\alpha_{j}}\right) }=V\cdot \frac{\left( \sum\limits_{r=1}^{k}\frac{m_{r}}{\alpha_{r}}\right) -1}{\left( \sum\limits_{r=1}^{k} \frac{m_{r}}{\alpha_{r}}\right) }. \end{array} $$

  Define $f\left (\beta \right ) :=\left (\sum \limits _{r=1}^{k}\frac {m_{r}}{\alpha _{r} }\right )$. Then

  $$ \frac{\partial }{\partial \beta }\tilde{X}^{\ast }=V\cdot \frac{\partial }{ \partial \beta }\left( \frac{f\left( \beta \right) -1}{f\left( \beta \right) }\right) =\frac{f^{\prime }\left( \beta \right) }{f\left( \beta \right)^{2}} . $$

  Since α_r = m_r + β (n − m_r) , we have

  $$ f^{\prime }\left( \beta \right) =-\left( \sum\limits_{r=1}^{k}\frac{m_{r}\left( n-m_{r}\right) }{{\alpha_{r}^{2}}}\right) <0. $$

□

Proof of Proposition 2

Define

$$ \begin{array}{@{}rcl@{}} A&=&\left( \begin{array}{cccc} m_{1} & \beta m_{2} & {\ldots} & \beta m_{k} \\ \beta m_{1} & m_{2} & {\ldots} & \beta m_{k} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ \beta m_{1} & \beta m_{2} & {\ldots} & m_{k} \end{array} \right) \ \text{and }\\ B_{i}&=&\left( \begin{array}{ccccc} m_{1} & {\ldots} & \tilde{x}_{1}^{\ast } & {\ldots} & \beta m_{k} \\ \beta m_{1} & {\ldots} & \tilde{x}_{2}^{\ast } & {\ldots} & \beta m_{k} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ \beta m_{1} & {\ldots} & \tilde{x}_{k}^{\ast } & {\ldots} & m_{k} \end{array} \right) \end{array} $$

where B_i is derived from A by substituting the i-row (βm_i,...,m_i,...βm_i)^T by $\left (\tilde {x}_{1}^{\ast },...,\tilde {x}_{i}^{\ast },...\tilde {x}_{k}^{\ast }\right )^{T}$. Induction over r = 1,…k then shows that the determinants of A and B_i are

$$ \begin{array}{@{}rcl@{}} \text{det }A &=&\prod \limits_{r=1}^{k}m_{r}\cdot \left( 1-\beta \right)^{k-1}\cdot \left( 1+\left( k-1\right) \beta \right) \text{ and} \\ \text{det }B_{i} &=&\prod \limits_{r=1,r\neq i}^{k}m_{r}\cdot \left( 1-\beta \right)^{k-2}\\ &&\cdot \left[ \left( 1+\left( k-1\right) \beta \right) \tilde{x} _{i}^{\ast }-\beta \sum\limits_{r=1}^{k}\tilde{x}_{r}^{\ast }\right] . \end{array} $$

Using Cramer’s Rule, the solution of $A\cdot \left (x_{1}^{\ast },x_{2}^{\ast }\right .$ $\left .,..., x_{k}^{\ast }\right )^{T}=\left (\tilde {x}_{1}^{\ast },\tilde {x} _{2}^{\ast },...\tilde {x}_{k}^{\ast }\right )^{T}\$is determined by

$$ \begin{array}{@{}rcl@{}} x_{i}^{\ast }&=&\frac{\text{det }B_{i}}{\text{det }A}=\frac{1}{m_{i}\cdot \left( 1-\beta \right) }\\ &&\cdot \left[ \tilde{x}_{i}^{\ast }-\frac{\beta }{ 1+\left( k-1\right) \cdot \beta }\cdot \sum\limits_{r=1}^{k}\tilde{x} _{r}^{\ast }\right] . \end{array} $$

With $\tilde {x}_{r}^{\ast }=\tilde {x}_{i}^{\ast }\cdot \frac {\alpha _{i}}{ \alpha _{r}}$ the proposition follows. □

Proof of Proposition 3

• 1. Using the definition of $\tilde {p}_{r}$ and γ it follows that

  $$ \begin{array}{@{}rcl@{}} \tilde{p}_{i}-\tilde{p}_{j} &=&-\frac{1}{m_{i}\cdot \left( 1-\beta \right) } \\ &&\cdot \left[ 1-\frac{\beta }{1+\left( k-1\right) \cdot \beta }\cdot \sum \limits_{r=1}^{k}\frac{\alpha_{i}}{\alpha_{r}}\right] \\ &&+\frac{1}{m_{j}\cdot \left( 1-\beta \right) }\\ &&\cdot \left[ 1-\frac{\beta }{ 1+\left( k-1\right) \cdot \beta }\cdot \sum \limits_{r=1}^{k}\frac{\alpha_{j}}{\alpha_{r}}\right] \\ &=&\frac{1}{\left( 1-\beta \right) }\left[ \frac{1}{m_{j}}-\frac{1}{m_{i}} \right] \\ &&+\frac{\gamma }{\left( 1-\beta \right) }\left[ \frac{\alpha_{i}}{ m_{i}}-\frac{\alpha_{j}}{m_{j}}\right] . \end{array} $$

  Since α_im_j − α_jm_i = nβ (m_j − m_i) by definition of α_r = m_r + β (n − m_r), it follows that

  $$ \tilde{p}_{i}=\tilde{p}_{j}+\frac{m_{i}-m_{j}}{m_{i}m_{j}}\cdot \frac{ 1-\gamma n\beta }{\left( 1-\beta \right) }. $$

  (A1)

  Then, m_i > m_j implies $\tilde {p}_{i}>\tilde {p}_{j},$ since

  $$ \begin{array}{@{}rcl@{}} 1 &=&\frac{kn\beta^{2}}{kn\beta^{2}}\geq \frac{n\beta^{2}}{k\beta +\left( 1-\beta \right) }\cdot \frac{k}{n\beta }\\ &=&\frac{n\beta^{2}}{1+\left( k-1\right) \beta }\cdot \sum \limits_{r=1}^{k}\frac{1}{n\beta } \\ &\geq &\frac{n\beta^{2}}{1+\left( k-1\right) \beta }\!\cdot\! \sum \limits_{r=1}^{k}\frac{1}{n\beta +\left( 1-\beta \right) m_{r}} = \gamma n\beta . \end{array} $$

• 2. Expected payoff is ${\Pi }_{r}^{\ast }=\tilde {p}_{r}\tilde {x}_{r}^{\ast }$ by definition. Using Eq. (2), it follows that ${\Pi }_{i}^{\ast }>{\Pi }_{j}^{\ast }\Leftrightarrow \tilde {p}_{i}\alpha _{j}>\tilde {p}_{j}\alpha _{i}$. From Eq. (A1), we can conclude that

  $$ \tilde{p}_{i}\alpha_{j}=\tilde{p}_{j}\alpha_{j}+\frac{m_{i}-m_{j}}{ m_{i}m_{j}}\cdot \frac{1-\gamma n\beta }{\left( 1-\beta \right) }\alpha_{j}. $$

  Since α_j = α_i −(1 − β) (m_i − m_j), it follows that

  $$ \begin{array}{@{}rcl@{}} \tilde{p}_{i}\alpha_{j}\!&>&\!\tilde{p}_{j}\alpha_{i}\Leftrightarrow \left( m_{i}-m_{j}\right)\\ &&\!\times\left[ \frac{\alpha_{j}}{m_{i}m_{j}}\cdot \frac{1-\gamma n\beta }{\left( 1-\beta \right) }-\tilde{p}_{j}\left( 1-\beta \right) \right] \!>\!0. \end{array} $$

  Since m_i > m_j, the proposition follows.

□

Proof of Corollary 1

To see the first part, note that the overall effective investments $\tilde {X}^{^{\ast }}$tend to zero if the external spillover rate tends to one, for

$$ \lim \limits_{\beta \rightarrow 1}\quad \left( \sum\limits_{r=1}^{k}\frac{m_{r}}{ \alpha_{r}}\right) =\left( \sum\limits_{r=1}^{k}\frac{m_{r}}{n}\right) =1. $$

In fact, the actors’ effective investments in this case are similar to a grand network (1;n).

To see the second part of the corollary, note that according to Eq. (3) and Proposition 2, the optimal effective investment $\tilde {x}_{i}^{\ast }$ and the optimal investment $x_{i}^{\ast }$ for β = 0 are given by

$$ \tilde{x}_{i}^{\ast }=\frac{V}{m_{i}}\cdot \frac{k-1}{k^{2}}\quad \text{ and\quad }x_{i}^{\ast }=\frac{V}{{m_{i}^{2}}}\cdot \frac{k-1}{k^{2}}. $$

Hence,

$$ p_{i}^{\ast }=\frac{\tilde{x}_{i}^{\ast }}{\tilde{X}^{\ast }}=\frac{\frac{V}{ m_{i}}\cdot \frac{k-1}{k^{2}}}{V\cdot \frac{k-1}{k}}=\frac{1}{m_{i}k}, $$

and expected payoff in equilibrium is

$$ {\Pi}_{i}^{\ast }=V\cdot p_{i}^{\ast }-x_{i}^{\ast }=V\cdot \frac{km_{i}-k+1}{ k^{2}{m_{i}^{2}}}. $$

Note that $x_{i}^{\ast }$ and $p_{i}^{\ast }$ are decreasing in the size m_i of the network and in the number k of competing networks. Moreover, according to Proposition 3, a bigger network N_i is more profitable than a smaller network N_j iff

$$ m_{i}<\frac{m_{j}\cdot \left( k-1\right) }{1+k\cdot \left( m_{j}-1\right) }. $$

□

Proof of Proposition 4

Suppose the grand network {N} has been formed in equilibrium leading to a network structure (1;n). As we already know, an entrepreneurial actor’s payoff in equilibrium is given by

$$ {\Pi}_{i}^{\ast }\left( 1;n\right) =\frac{1}{n}. $$

To show that (1;n) is not an equilibrium network structure, consider an entrepreneurial actor i’s payoff if it unilaterally decides not to participate in the grand network. Using the characterization of equilibrium payoffs, actor i’s payoff is

$$ {\Pi}_{i}^{\ast }\left( 2;n-1,1\right) =\frac{n^{3}\beta +2n^{2}\beta^{2}-3n^{2}\beta +n^{2}-4n\beta^{2}+6n\beta -2n+3\beta^{2}-4\beta +1}{ \left( 2n+2\beta -2n\beta +n^{2}\beta -2\right)^{2}}. $$

Then, ${\Pi }_{i}^{\ast }\left (2;n-1,1\right ) >{\Pi }_{i}^{\ast }\left (1;n\right )$ if

$$ \begin{array}{@{}rcl@{}} &&-\left( 1-\beta \right) \left( n-1\right) \left( \beta \left( 7n+n^{3}-5n^{2}-4\right) \right.\\ &&\left. +\left( n-1\right) \left( n-4\right) \vphantom{n^{3}}\right) <0 \end{array} $$

For n = 3, the LHS of this inequality is positive and equal to 2 (1 − β) (β + 2). For n > 3, the LHS is negative since (7n + n³ − 5n² − 4) and increasing in n. Hence, the grand coalition is only an equilibrium structure for n = 3. If n > 3, a single actor profits from a deviation.

Consider now partition {1,...,1} in which all actors form degenerated networks and the network structure is (n;1,...,1). The equilibrium payoff of an entrepreneurial actor i is then given by

$$ {\Pi}_{i}^{\ast }\left( n;1,...,1\right) =\frac{\beta n^{2}-\beta +1}{ n^{2}\left( n\beta -\beta +1\right) }. $$

Suppose that actor i unilaterally decides to form a network with some other actor $j\in \mathbb {N}$. Using the characterization of equilibrium payoffs for m_i = 2, m_j = 1 for j≠i, actor i’s payoff is

$$ \begin{array}{@{}rcl@{}} &&{\Pi}_{i}^{\ast }\left( n-1;2,1...,1\right) \\ &=&\frac{\left( n\beta -\beta +1\right) \left( n\beta -\beta +2\right) }{2\left( 2n+2\beta -2n\beta +n^{2}\beta -2\right)^{2}}\frac{A}{B}. \end{array} $$

$$ \begin{array}{@{}rcl@{}} && \text{with } A:=\left( \begin{array}{c} 2n^{4}\beta^{3}-10n^{3}\beta^{3}+8n^{3}\beta^{2}+21n^{2}\beta^{3}-30n^{2}\beta^{2}+9n^{2}\beta -23n\beta^{3} \\ +42n\beta^{2}-21n\beta +2n+10\beta^{3}-24\beta^{2}+14\beta \end{array} \right)\\ &&\text{and } B:=\left( n^{3}\beta^{3}-4n^{2}\beta^{3}+4n^{2}\beta^{2}+5n\beta^{3}-11n\beta^{2}+5n\beta -2\beta^{3}+7\beta^{2}-7\beta +2\right). \end{array} $$

Tedious calculations then show that

$$ {\Pi}_{i}^{\ast }\left( n-1;2,1...,1\right) >{\Pi}_{i}^{\ast }\left( n;1...,1\right) $$

is always satisfied. □

Proof of Corollary 2

We prove the corollary in three steps: In Step 1, we ask under which circumstances a single actor has an incentive to participate in a network of size m_i. As we will see, this sets an upper bound on the maximal size of a network to be more profitable than a non-cooperating actor. In Step 2, we argue that in equilibrium there can only exist networks whose sizes differ by one member. In Step 3, we then determine the equilibrium network structure in the open membership game.

• 1. Consider an network structure (k;m₁,1,m₃ ,...m_k) , k ≥ 2, with m₁ actors in network N₁ and one actor i in network N₂ = {i}. Then, actor i has an incentive to join the network N₁ if

  $$ \begin{array}{@{}rcl@{}} &&{\Pi}_{i}^{\ast }\left( k-1;m_{1},m_{3},...m_{k}\right)\\ &>&{\Pi}_{i}^{\ast }\left( k;m_{1},1,m_{3},...m_{k}\right) ,\text{ that is, if} \\ &&\frac{\left( k-1\right) \left( m_{1}+1\right) -\left( k-1\right) +1}{\left( k-1\right)^{2}\left( m_{1}+1\right)^{2}}\\&>&\frac{1}{k^{2}}. \end{array} $$

  Simple calculation then shows that this inequality is satisfied as long as

  $$ m_{1}<\frac{k}{2\left( k-1\right) }\left( k+\sqrt{\left( k^{2}-4k+8\right) } \right) -1. $$

  If k = 2, this equality reduces to m₁ < 3. Hence, there could only be three actors in the environment, n = 3. Since n > 3 by assumption, k > 2, the maximal size m₁ which satisfies this inequality is m₁ = k − 1, since

  $$ k-1\!<\!\frac{k}{2\left( k - 1\right) }\left( k + \sqrt{\left( k^{2} - 4k + 8\right) } \right) -\!1\!\iff\! 4\!<\!8 $$

  and

  $$ \begin{array}{@{}rcl@{}} k&>&\frac{k}{2\left( k-1\right) }\left( k + \sqrt{\left( k^{2}-4k + 8\right) } \right) - 1\!\iff\! 1\\ &>&k^{2}\left( 3-k\right) \end{array} $$

  for m_i = k > 2. Hence, as long as m₁ ≤ k − 1, a non-cooperating actor always has an incentive to join the network N₁. As a consequence, an equilibrium network structure has to satisfy this property, since otherwise actors in a network with size greater than k − 1 would be better off by not participating in this network.

• 2. Consider an equilibrium network structure (k;m₁, m₂, m₃,...m_k) , k ≥ 2, with m₁ actors in network N₁ and m₂ actors in network N₂. Suppose that m₂ > m₁ ≥ 2. Note that a switch in membership between network N₁ and N₂ does not affect the total number of networks, so k remains unchanged. Since the equilibrium payoff of an entrepreneurial actor is decreasing in the size of network it is a member of,

  $$ \frac{\partial {\Pi}_{i}^{\ast }}{\partial m_{i}}<0, $$

  the members of the larger network N₂ then have an incentive to switch to the smaller network N₁. This is profitable as long as m₁ + 1 < m₂.

• 3. Steps 1 and 2 imply that there can only be equilibrium network structures for which either all networks are of identical size m^∗ > 1, or some networks have size m^∗ > 2 while the others have size m^∗− 1. Moreover, if $k_{1}^{\ast }$ denotes the number of networks of size m^∗ and $k_{2}^{\ast }$ the number of networks of size m^∗− 1. Then,

  $$ \begin{array}{@{}rcl@{}} m^{\ast }k_{1}^{\ast }+\left( m^{\ast }-1\right) k_{2}^{\ast } &=&n\text{ and } \\ k_{1}^{\ast }+k_{2}^{\ast }-1 &\geq &m^{\ast }. \end{array} $$

□

Proof of Corollary 3

Let (k;m₁, m₂, m₃,...m_k) be an equilibrium network structure in the exclusive membership game for β = 0. To prove the corollary we proceed in three steps: In Step 1, we show that if all actors in network N_i have an outstanding invitation for another non-member actor j∉N_i, the equilibrium network structure remains unchanged and, vice versa, if a non-member actor j∉N_i asks for membership in N_i, this is not in the interest of the member actors. We can therefore restrict attention to equilibrium network structures where no actor in network N_i has an incentive to announce a list which excludes one member actor j ∈ N_i. In Step 2, we ask under which circumstances a member of a network of size m_i ≥ 4 has no incentives to do so. As we will see, this sets a lower bound on the maximal size of a network to be more profitable with m_i instead of m_i − 1 actors. In Step 3, we consider the same question for the remaining cases m_i = 2 and m_i = 3. In Step 4, we then determine the equilibrium network structure in the exclusive membership game.

• 1. Consider first the case in which in equilibrium all members of a network N_i with size m_i have an outstanding invitation for another actor j ∈ N_j. (k;m₁, m₂, m₃,...m_k) then is an equilibrium if actor j∉N_i has no incentive to join this network. If 1 < m_j < m_i, the number of networks remains constant and ∂π^∗/∂m < 0 ensures that actor j indeed is not willing to switch to the bigger network. If m_j = 1, the number of networks reduces to k − 1 and, according to Step 1 in the proof of Corollary 2, actor j has an incentive to join N_i if m_i ≤ k − 1. However, (k;m₁, m₂, m₃,...m_k) is then still an equilibrium network structure if one actor in N_i drops actor j from her list of announcements. This requires

  $$ \frac{\left( k+1\right) \left( m_{i}+1\right) -\left( k+1\right) +1}{\left( k+1\right)^{2}\left( m_{i}+1\right)^{2}}<\frac{k\left( m_{i}-1\right) +1}{ k^{2}{m_{i}^{2}}}. $$

  Since the term on the left side of this inequality is decreasing in m_i,

  $$ \begin{array}{@{}rcl@{}} &&\frac{\left( k+1\right) \left( m_{i}+1\right) -\left( k+1\right) +1}{\left( k+1\right)^{2}\left( m_{i}+1\right)^{2}}\\ &<&\frac{\left( k+1\right) m_{i}-\left( k+1\right) +1}{\left( k+1\right)^{2}{m_{i}^{2}}}. \end{array} $$

  But

  $$ \frac{\left( k+1\right) m_{i}-\left( k+1\right) +1}{\left( k+1\right)^{2}{m_{i}^{2}}}<\frac{k\left( m_{i}-1\right) +1}{k^{2}{m_{i}^{2}}} $$

  since − k² (m₁ − 1) − k (m₁ + 1) − 1 < 0.

• 2. Consider now a network structure (k;m₁, m₂, m₃,...m_k) , k ≥ 2, with m₁ ≥ 4 actors in network N₁. Then an entrepreneurial actor in network N₁ has an incentive to exclude one member of her network if

  $$ \begin{array}{@{}rcl@{}} &&{\Pi}_{i}^{\ast }\left( k+1;1;m_{1}-1,m_{2},m_{3},...m_{k}\right) \\ &>&{\Pi}_{i}^{\ast }\left( k;m_{1},m_{2},m_{3},...m_{k}\right) , \end{array} $$

  that is, if

  $$ \frac{\left( k+1\right) \left( m_{1}-1\right) -k}{\left( k+1\right)^{2}\left( m_{1}-1\right)^{2}}\geq \frac{k\left( m_{1}-1\right) +1}{ k^{2}{m_{1}^{2}}}. $$

  Since m₁ ≥ 4 by assumption, the minimal number k of networks which satisfies this inequality is k = m₁ + 1, since for k = m₁:

  $$ \begin{array}{@{}rcl@{}} &&\frac{\left( m_{1}+1\right) \left( m_{1}-1\right) -m}{\left( m_{1}+1\right)^{2}\left( m_{1}-1\right)^{2}}-\frac{m_{1}\left( m_{1}-1\right) +1}{{m_{1}^{4}}} \\ &=&-\frac{\left( m_{1}+1\right) \left( m_{1}-1\right)^{2}+{m_{1}^{3}}}{ {m_{1}^{4}}\left( {m_{1}^{2}}-1\right)^{2}}<0, \end{array} $$

  and for k = m₁ + 1 it holds that

  $$ \begin{array}{@{}rcl@{}} &&\frac{\left( m_{1}+2\right) \left( m_{1}-1\right) -\left( m_{1}+1\right) }{ \left( m_{1}+2\right)^{2}\left( m_{1}-1\right)^{2}}\\ &&-\frac{\left( m_{1}+1\right) \left( m_{1}-1\right) +1}{\left( m_{1}+1\right)^{2}{m_{1}^{2}}} \\ &=&\frac{\left( m_{1}-1\right)^{2}-8}{\left( m_{1}-1\right)^{2}\left( m_{1}+2\right)^{2}\left( m_{1}+1\right)^{2}}>0 \end{array} $$

  for m_i ≥ 4. Hence, as soon as k ≥ m₁ + 1, a member of network N₁ has an incentive to exclude another actor in N₁.²⁹

• 3. Consider finally the case of a network structure (k;m₁, m₂, m₃,...m_k) , with m₁ ∈ {2,3} actors in network N₁. Using the inequality from Step 1, an entrepreneurial actor in network N₁ has an incentive to exclude a member of this network for m₁ = 3, if k ≥ 6 since

  $$ \frac{2\left( k+1\right) -k}{4\left( k+1\right)^{2}}-\frac{2k+1}{9k^{2}}= \frac{k^{3}-2k^{2}-16k-4}{36k^{2}\left( k+1\right)^{2}}>0. $$

  Moreover, for m₁ = 2

  $$ \frac{\left( k+1\right) -k}{\left( k+1\right)^{2}}-\frac{k+1}{4k^{2}}=- \frac{k^{3}-k^{2}+3k+1}{4k^{2}\left( k+1\right)^{2}}<0 $$

  for all k ≥ 2.

• 4. Steps 2 and 3 imply that a network structure (k;m₁, m₂, m₃,...m_k) forms in equilibrium in the exclusive membership game if for all networks k < m_i + 1 if m_i ≥ 4 or k ≤ 5 if m_i = 3. The first condition ensures that each actor in a network N_i with m_i ≥ 4 has no interest to exclude another actor if k < m_i + 1 according to Step 2. The second property follows from Step 3 for the case m₁ = 3.

□

Proof of Proposition 5

Define

$$ \begin{array}{@{}rcl@{}} A&=&\left( \begin{array}{cccc} m_{1} & \beta_{2}m_{2} & {\ldots} & \beta_{k}m_{k} \\ \beta_{1}m_{1} & m_{2} & {\ldots} & \beta_{k}m_{k} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ \beta_{1}m_{1} & \beta_{2}m_{2} & {\ldots} & m_{k} \end{array} \right) \ \text{and }\\ B_{i}&=&\left( \begin{array}{ccccc} m_{1} & {\ldots} & \tilde{x}_{1}^{\ast } & {\ldots} & \beta_{k}m_{k} \\ \beta_{1}m_{1} & {\ldots} & \tilde{x}_{2}^{\ast } & {\ldots} & \beta_{k}m_{k} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ \beta_{1}m_{1} & {\ldots} & \tilde{x}_{k}^{\ast } & {\ldots} & m_{k} \end{array} \right) \end{array} $$

as in the proof of Proposition 2. Induction over r = 1,…k then shows that the determinants of A and B_i are

$$ \begin{array}{@{}rcl@{}} \text{det }A \!&=&\!\prod \limits_{r=1}^{k}m_{r}\cdot \left( \displaystyle\sum \limits_{j=0}^{k}\left( -1\right)^{k-1-j}\left( k-1-j\right) \right.\\ &&\left. P_{k-j}\left( \beta_{1},...,\beta_{k}\right) \vphantom{\displaystyle\sum \limits_{j=0}^{k}}\right) \\ \text{det }B_{i} \!&=&\!\prod \limits_{r=1,r\neq i}^{k}m_{r}\!\cdot \left( \tilde{x }_{i}^{\ast }\left( \displaystyle\sum \limits_{j=0}^{k-1}\left( -1\right)^{k-2-j}\left( k - 2 - j\right)\right.\right.\\ && \left.\left. P_{k-1-j}\left( \beta_{1},...,\beta_{i-1},\beta_{i+1},...,\beta_{k}\right) \vphantom{\sum \limits_{j=0}^{k-1}}\right) \right. \\ &&\left. -\displaystyle\sum \limits_{j=1,j\neq i}^{k}\tilde{x}_{j}^{\ast }\beta_{j}\displaystyle\prod \limits_{r=1,r\neq i,j}^{k}\left( 1-\beta_{r}\right) \right) . \end{array} $$

Let α_r = β_rn + (1 − β_r) m_r. Since the first-order conditions for a Nash equilibrium imply that $\tilde {x} _{j}^{\ast }\alpha _{j}=\tilde {x}_{i}^{\ast }\alpha _{i}$, the optimal investments according to Cramer’s Rule are given by

$$ \begin{array}{@{}rcl@{}} x_{i}^{\ast } &=&\frac{1}{m_{i}\left( \displaystyle\sum \limits_{j=0}^{k}\left( -1\right)^{k-1-j}\left( k-1-j\right) P_{k-j}\left( \beta_{1},...,\beta_{k}\right) \right) } \\ &&\cdot\left( \left( \displaystyle\sum \limits_{j=0}^{k-1}\left( -1\right)^{k-2-j}\left( k-2-j\right) P_{k-1-j}\right.\right.\\ &&\times\left.\left. \left( \beta_{1},...,\beta_{i-1},\beta_{i+1},...,\beta_{k}\right) \vphantom{\sum \limits_{j=0}^{k-1}}\right) \right. \\ &&\left. -\displaystyle\sum \limits_{j=1,j\neq i}^{k}\frac{\alpha_{i}}{\alpha_{j}} \beta_{j}\displaystyle\prod \limits_{r=1,r\neq i,j}^{k}\left( 1-\beta_{r}\right) \right) \tilde{x}_{i}^{\ast }. \end{array} $$

and the proposition follows. □

Proof of Proposition 6

Similar to the argumentation in Section 4, the individual and overall effective investments are given in equilibrium by

$$ \begin{array}{@{}rcl@{}} \tilde{X}^{\ast }&=&\alpha_{i}\tilde{x}_{i}^{\ast }\cdot \sum\limits_{r=1}^{k}\frac{ m_{r}}{\alpha_{r}}\text{ and }\tilde{x}_{i}^{\ast }\\ &=&\frac{V}{\alpha_{i}} \cdot \frac{\left( \sum\limits_{r=1}^{k}\frac{m_{r}}{\alpha_{r}}\right) -1}{\left( \sum\limits_{r=1}^{k}\frac{m_{r}}{\alpha_{r}}\right)^{2}} \end{array} $$

where α_r = β_rn + (1 − β_r) m_r. Using the result of Proposition 5, the equilibrium payoff of an actor in network i reads as

$$ {\Pi}_{i}=\frac{V}{\alpha_{i}\sum\limits_{r=1}^{k}\frac{m_{r}}{\alpha_{r}}}\left( 1-\frac{\left( \sum\limits_{r=1}^{k}\frac{m_{r}}{\alpha_{r}}\right) -1}{\left( \sum\limits_{r=1}^{k}\frac{m_{r}}{\alpha_{r}}\right) \left( \displaystyle\sum\limits_{j=0}^{k}\left( -1\right)^{k-1-j}\left( k-1-j\right) P_{k-j}\left( \beta_{1},...,\beta_{k}\right) \right) }{\Psi}_{i}\right) $$

with

$$ \begin{array}{@{}rcl@{}} {\Psi}_{i}&=&\displaystyle\sum \limits_{j=0}^{k-1}\left( -1\right)^{k-2-j}\left( k-2-j\right) P_{k-1-j}\\ &&\times\left( \beta_{1},...,\beta_{i-1},\beta_{i+1},...,\beta_{k}\right)\\ &&-\displaystyle\sum \limits_{j=1,j\neq i}^{k}\frac{\alpha_{i}}{\alpha_{j}}\beta_{j}\displaystyle\prod \limits_{r=1,r\neq i,j}^{k}\left( 1-\beta_{r}\right) . \end{array} $$

To show that $\left (\beta _{1}^{\ast },...,\beta _{k}^{\ast }\right ) =\left (0,....,0\right )$ is an equilibrium in spillover rates, consider the marginal payoff of an actor in network i, given all other networks choose β_r = 0. Since Ψ_i = 0 for (β₁,..,β_i,..,β_k) = (0,..,β_i,..,0), it follows that

$$ \left. \frac{\partial }{\partial \beta_{i}}{\Pi}_{i}\right\vert_{\left( 0,..,\beta_{i},..,0\right) }=\frac{\partial }{\partial \beta_{i}}\left( \frac{V}{m_{i}+\alpha_{i}\left( k-1\right) }\right) <0, $$

and $\beta _{i}^{\ast }=0$ is a best response for network i. □