Assumptions

The study adopts real options combining game theory, it evaluates theoretical models to figure out the threshold for earnings per share (EPS) based on following the geometric Brownian motion (GBM) involved in a Poisson jump-process. Assuming that there are only two VC firms in the newly created market, when they are interested in investing in the start-up company, the investment scale is equal, which means the same investment input. The different competitive and investment strategies of the two VC firms will be distinguished by the condition of a leader’s dominating strategies (duopoly), an entry-deterred game (specific monopoly), and simultaneous investment based on the market condition and competitor strategy of the project. Based on the investment of two VC firms, the start-up company can gain more additional returns to obtain extra and added values. Different strategies verify the additional returns to obtain extra. The two VC firms have different competitive advantages; although their investment inputs are the same, contributions and sharing are different.

The project value of the start-up company is affected by the external market environment and its operating condition. Assuming that the VC firm investing in the start-up company follows the stochastic EPS, P , the value variation growth over time is described by the jump-diffusion process which includes the GBM (continuous process) P and Poisson process (discrete process) _G

P as follows: D

G D

dPdP dP PdtPdW Pdq (1)

Among the above factors,  : drift over time,  : volatility over time, dW is the increment of a standard Wiener process W of zero mean and unit standard deviation dt , and

：the deterministic amplitude specifying the jump size (fall) in the jump-process 0   .  1

The jump-process follows the Poisson process with an arrival rate of  and then dq and dW are independent (so that E dW( dq) ) : 0

1, with prob.

0, with prob. 1 dq dt

dt





   (2)

The stochastic P follows the jump-diffusion process because the start-up company is also affected by the competition from other firms. When the start-up company applies for new product development, P will decrease if competitors also apply for new patents.

Then, the different competitive and investment strategies of the two VC firms will be distinguished by the condition of a leader’s dominating strategies (duopoly), an entry-deterred game (specific monopoly), and simultaneous investment based on the market condition and competitor strategy of the project. Assume the start-up company and the two VC firms negotiate the additional returns to obtain extra and D x x( ,_{1 2}) belongs to the two VC firms. That is the investments of the two VC firms’ different competition and investment strategies reflect the additional returns to obtain extra investment behavior function as D x x( ,_{1 2}). D x x₁( ,_{1 2}) belongs to VC firm 1. D x x₂( ,_{1 2}) belongs to VC firm 2. Moreover D x x( ,_{1 2})=D x x₁( ,_{1 2})+D x x₂( ,_{1 2}), among which:

1

2

x : VC firm 1 project decision.

(0: if VC firm 1 has not invested; 1: if VC firm 1 has invested).

x : VC firm 2 project decision.

(0: if VC firm 2 has not invested; 1: if VC firm 2 has invested).







The study examines the market acceptance of the products of new ventures against the background of the market uncertainties of the two VC firms. It estimates the responses of the opponents under a competitive landscape in order to determine whether investments are viable.

Table 1 illustrates possible scenarios. The study assumes that VC firm 1 is the leading competitor.

Table1. The function of investment behavior VC firm’s strategy Explanation

(0, 0)

In order to conduct a further study on the effect generated by the additional returns to obtain extra D x x( ,_{1 2}) behavior of both parties under a duopoly market, the correlation of variables x ₁ and x is assumed as follows: ₂

1 2 1 2

( , ) ( )( )

D x x  x h x  (3) k

It is assumed that the additional returns to obtain extra function are a hyperbolic function (Kalashnikov et al., 2009; De Giovanni et al., 2008; Huang and Hsu, 2008). Where h k,  and R

1

k  . k and h separately represent the technology, finance, market, and business h know-how in VC firm 1 and VC firm 2. The study assume VC firm 1 as the market leader. Its business know-how has a comparative advantage, which is hard for VC firm 2 to compete. When dealing with the start-up company, VC firm 1 can have a better bargain. According to the assumption in the study, VC firm 1 can have larger shares of the pies in the start-up company. As the two VC firms intend to invest in the start-up company, their investment strategies are a function of mutual speculation and influence.

The additional returns to obtain extra of the two VC firms is expressed by Eqn. (3). Thus, it is assumed that D x x ,_i( ,_{1 2}) i1, 2 are the additional returns to obtain extra for VC firm 1 and VC firm 2 upon different investment strategies respectively. Meanwhile, the five investment stages are “Seed Stage”, “Startup Stage”, “Expansion Stage”, “Mezzanine Stage”, and “Turnaround Stage”. This study assumes that two VC firms implement the investment at the Startup Stage. For this stage, the start-up company has completed product development and needs further funds to initiate commercial manufacturing and sales. Logically, the start-up company has passed the seed stage. It owns the potential investment value before the EPS meets threshold, that is, VC firms implement the investment. The investment strategy, ^D

 

^{0, 0} , reflects the additional returns to obtain extra of the value function of the potential response investment ^D

 

^{0, 0   . h k}

When the two VC firms evaluate the benefits of investing in a start-up company, the four expected additional returns to obtain extra investment scenarios by different opponents’ reactions and their investment strategies are shown in Table 2.

Table 2. Share of the additional returns to obtain extra matrix for VC firm 1and VC firm 2 upon different competition strategy

VC firm 2

Note: 1. Distributing the additional returns to obtain extra in each cell for



^{VC firm1,VC firm 2}



^.

2. Eqn. (3) is derived by the two VC firms’ additional returns to obtain extra sharing under different strategies:

(1)D x x( ,_{1 2})(x₁h x)( ₂k)x x_{1 2}hx₂kx₁hkD x x₁( ,_{1 2})D x x₂( ,_{1 2}).

(2)D x x1( ,1 2)Z_{x x}1,2D x x( ,1 2),D x x₁( ,_{1 2})is the additional returns to obtain extra for VC firm1.

(3)D x x2( ,1 2) 



1 Z_{x x}1,2



D x x( ,1 2),D x x₂( ,_{1 2})is the additional returns to obtain extra for VC firm 2.

3. 0Z_{x x}1,2  , 1

1,2

Zx x is the distribution ratio for VC firm 1 under D x x( ,_{1 2}).



1Z_{x x}1,2



is the distribution ratio for VC firm 2.

The four expected additional returns to obtain extra scenarios from the various opponents’

reactions and their investment strategies are as follows: (1)



D1(0, 0),D2(0, 0)



represents the condition that both VC firms are taking a waiting strategy. The expected additional returns to obtain extra are D(0, 0)  . VC firm 1 shares h k D₁(0, 0)Z_0,0  and VC firm 2 shares h k

2(0, 0) (1 0,0)

D  Z   , h k 0Z_0,0  ; (2) 1



D1(1, 0),D2(1, 0)



: when VC firm 1 invests first while VC firm 2 adopts the waiting strategy, the expected additional returns to obtain extra are

(1, 0)

D    . VC firm 1 shares h k k D₁(1, 0)Z_1,0  (h k k) and VC firm 2 shares

2(1, 0) (1 1,0) ( )

D  Z   h k k , 0Z_1,0 ; (3) 1



D1(0,1),D2(0,1)



: VC firm 1 delays investments by forgoing investment opportunities, while VC firm 2 remains interested in making investments. The expected additional returns to obtain extra are D(0,1)   . VC firm 1 h k h shares D₁(0,1)Z_0,1  (h k h) and VC firm 2 shares D₂(0,1) (1 Z_0,1) (  h k h) ,

0Z0,1  ; (4) 1



D1(1,1),D2(1,1)



: VC firm 1 and VC firm 2 adopt a cooperation strategy by

investing at the same time. Here, the expected additional returns to obtain extra are (1,1) 1

D      . VC firm 1 shares h k h k D₁(1,1)Z_1,1    (1 h k k h) and VC firm 2 shares D₂(1,1) (1 Z_1,1) (1     , h k k h) 0Z_1,1 . 1

Therefore, the four expected response investment additional returns to obtain extra are (1,1) (1, 0) (0,1) (0, 0)

D D D D . Since VC firm 1 and VC firm 2 implement the investments simultaneously, they adopt the cooperation strategy with synthetic effects viewing the market optimistically. The expected investment additional returns to obtain extra D(1,1) are the highest.

The second is D(1, 0). VC firm 1 dominates the investment opportunity and sets a high threshold, which then makes VC firm 2 give up the investment opportunity. The third is D(0,1). VC firm 2 views the market optimistically, while VC firm 1 views it pessimistically. VC firm 2 implements the investment and then becomes specific monopoly as illustrated ^D

 

^{1, 0 D(0,1)}. The last is

(0, 0)

D . Neither of the two VC firms enter the market although they are interested in investing and are the start-up company’s consultants. The market will reflect that and increase the returns.

The start-up company will pay the consultation fees to the two VC firms. The descending of ranking is D(1,1)D(1, 0)D(0,1)D(0, 0). Below, the study attempts to conduct the market entry threshold under different market environments formed by different VC competition strategies.

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