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For venture capitals, it is a long process from an entry to its exit. In this paper, the activity of venture investment will be divided into multistages. And, according to the effort level entrepreneurs will choose, the venture capitalists will provide an equity structure at the very beginning. As a benchmark for comparison, we will establish two game models on multistage investment under perfect rationality: a cooperative game model and a noncooperative one. Further, as a cause of pervasive psychological preference behavior, reciprocity motivation will influence the behavior of the decision-makers. Given this situation, Rabin’s reciprocity motivation theory will be applied to the multistage game model of the venture investment, and multistage behavior game model will be established as well, based on the reciprocity motivation. By looking into the theoretical derivations and simulation studies, we find that if venture capitalists and entrepreneurs both have reciprocity preferences, their utility would have been Pareto improvement compared with those under perfect rationality.

Venture capitals have been playing a crucial role in new unlisted SMEs for a long time, especially in new high-tech enterprises. For example, in America, venture capital institutions invest 70% of their capitals into high-tech industries dominated by IT. Many successful high-tech enterprises, such as Microsoft and Netscape, all once received support during their development by venture capitals. According to our data, in China, 29.27% of venture capital institutions focus on high-growth fields, and 16.75% of them focus on high-tech areas. Venture capital is risky simultaneously, even though it is vital to the economic development. Therefore, how to maintain and increase the value of venture capital has been a significant study field for scholars. Scholars target designing a certain kind of financial contracts to motivate venture entrepreneurs (later denoted as EN) to pay more effort that will improve the venture projects’ success probability. However, several problems arise. Firstly, the efforts and actions of EN are implicit; thus there is no contracts to motivate all the efforts of EN, so how to make up for this defect? The actions that financial contracts cannot include can be resolved through mutual reciprocal affection. Such research is inspired by practice. For example, CEO quietly prepared for an employee a birthday party (the cost of which is not included in the wage contract), thus leading to the staff’s willing to pay more efforts to the CEO. Employees’ paying more effort will thus naturally improve the output and, therefore, both the CEO and employees will benefit from it. This reciprocal behavior, in behavioral economics, is called the theory of reciprocal fairness preference. There are many other examples in which reciprocity theory has been applied to improve the performance of enterprises examples; such cases are outstanding in South Korea, Japan, and China. There also exists such a situation in a venture capital: venture capitalist (later denoted as VC) paid the fixed income to EN, and EN will increase efforts to repay VC; therefore, the mutual reciprocal of both VC and EN improves the utility of both sides. Secondly, in general VC prefers multistage investment to venture projects. This is because VC will continue investment if the venture project is of good quality, and will withdraw from the project if the project fails. Thirdly, VC and EN’s reciprocity need to be extended into the long term. Therefore, in this paper we will study the venture investment’s decision-making from two perspectives: multistages and the reciprocal fairness preference as well.

Maintaining and increasing the value of venture capital was an important task in the management of venture capital, VCs hoped to choose venture projects which had good prospects in the future and also hoped to select the qualified enterprise and the manager operated the venture capital to make it appreciated. However, at the beginning of venture capital, the VCs usually did not clearly know the ENs’ management ability, the effort level, and the return and risk condition of the project. As a result, they often made decisions according to the principle of maximizing their own benefit, such as capital abuse and over investment, which result in moral hazard problem [

According to current literatures on venture capital, staged financing could mitigate moral hazard [

Early researches only considered the efforts of EN and established the principal-agent model called a unilateral moral hazard model to motivate EN to pay more efforts. However, the success of the project required both EN’s expertise and the need for VC’s rich experience in marketing. Therefore, both sides needed to pay more efforts. As with the previous analysis, EN and VC might hide their efforts, which might lead to the double moral hazard, and some scholars had established a double moral hazard model. Based on Ramy and Arieh’s research, Zhang and Wei [

From above, we might see that the efforts of VC and EN could be divided into the single moral hazard and the double moral hazard in the management of venture capital.

Then, these research literatures on multistage venture capital were all involved with a hypothesis: the VC and the EN were both perfectly rational. In recent years, many researchers had been challenging this hypothesis of traditional economics. They argued that not all the behaviors could be explained by utility maximization of neoclassical economics; that is, decision-makers were bounded rationality. The behavior of decision-makers with bounded rationality could be presented in many aspects, and one of them was reciprocity motivation. Ultimatum game, dictator game, gift-exchange game, trust game, and the empirical study of researchers all clearly showed that participants’ fair preference was very compelling. Since fair preference motivation had not been subordinated to theoretical frame work of mainstream economics, many new theoretical models were based on it.

Fairness preference was mainly from two aspects. First, players concerned about whether the final outcome was fairness or not. Fehr and Schmidt [

Fehr and Falk [

How did reciprocity motivation affect the behavior of the decision-makers? Based on the sequential reciprocity game model of Dufwenberg and Kirchsteiger [

Recently, the amount of literatures about venture capital under bounded rationality was small. Fairchild [

This paper focuses on designing behavior capital contracts based on reciprocity motivation through introducing reciprocity motivation theory into the design of multistage venture capital contract. Our starting point is mainly based on the following three aspects. First, multistage venture capital is pervasive in practice. Instead of providing all the investment upfront, the VC invests in stages to control risks and makes refinancing decisions according to the project condition. There are some research results currently, but further study is still needed. Second, in practice, the VC and the EN are bounded rationality, and reciprocity motivation is also a common psychological preference. If the EN inputs more effort, the VC may offer a higher share in return. Of course, this reciprocity is bilateral. Third, majority of research results study the design of capital structure contracts under perfectly rationality and derive corresponding utility or returns. Thus, if we can find a capital structure contract under reciprocity motivation in which the utility or returns of both sides are Pareto improvement of perfect rationality, our research perspective will have more practical and theoretical significance.

Our main ideas are as follows. If the VC cooperates with the EN, the EN’s effort will achieve the optimal. We call the effort in this case the first-best. Therefore, the first step is to calculate the first-best solution of optimal effort during the cooperation. Then secondly, we calculate the optimal decision of both sides under perfect rationality in the assumption that the two parties make their decisions independently. We call the solution here the second-best. The first-best solution is the optimal upper limit of both sides and the second-best solution is the lower limit of both sides. The third step is to construct a multistage game model on venture capital based on reciprocity motivation through introducing Rabin’s [

We have two goals. The first is to find the optimal effort level between the first-best solution and the second-best solution and to confirm that the capital structure contract is also between the first-best and the second-best solution. The second goal is to confirm that, under reciprocity motivation, the utility or returns of both sides are Pareto improvement compared with that under perfect rationality.

This paper proceeds as follows. In the next section, we put forward the assumptions, notations, and basic descriptions of the model. As a basis for comparison, we consider the multistage decision-making model and solutions under perfect rationality in Section

Consider an innovative entrepreneur (EN) who relies on a venture capitalist (VC) for investment. The VC provides a capital contract for the EN. If the EN accepts, the contract will be executed. If not, the EN’s reservation utility will be zero. We assume that both the EN and the VC are risk neutral. The VC stages the investment into

The time line.

We assume that both the EN and the VC are risk neutral. That is, they have equivalent risk tolerance facing possible profit. The EN’s decision goal is to maximize the utility in revenue; the VC’s decision goal is to maximize the utility of capital gains.

VC invests capital to the venture company. It is a long-term process and the venture capital is divided into

If the VC and the EN reach the investment agreement, the VC will provide the external capital

It is a long-term process after the VC inputs capital to the venture enterprise. The effort of the EN is different in different periods. Naturally, venture investment should be carried out in stages. So we divide the long-term partnership into

The probability of success is

For simplicity, we assume that the return in every period is equal; that is,

Success after

After the VC invests the capital, at the beginning of the project, the VC gives fixed income

For simplicity, there is no consideration for the time value of the return. We can also assume that venture capitalist’s discount factor is

Let

The EN chooses effort level in each stage,

Since the VC is risk neutral, VC’s expected utility equals the expected return:

The VC chooses appropriate equity contract (

The EN and the VC realize cooperation through one side completely controlling project; that is, the owner of the project and the provider of the investment are the same decision-maker. Consequently, adding (

The decision-maker chooses effort level in each stage,

We derive the optimal solution

Under noncooperation, the VC and the EN make independent decisions to maximize their own utility or return. The VC’s goal is the maximization of the future return. Thus, the VC designs a capital structure contract to maximize the total return at the end of stage

However, the EN also cares about the future utility maximization while focusing on maximizing the current utility [

In stage

In this game, the VC provides capital structure contract firstly, and then the EN chooses his own effort level. Therefore, this game is Stackelberg game. It can be described as follows (I):

In model (I), the first-order condition about

By introducing the first-order condition to model (I), the VC’s problem becomes

Put formula (

From formula (

In (

Therefore, the EN’s optimal effort level in each stage is

Put (

So there is Proposition

If the VC cooperates with the EN, the EN’s optimal effort level is

When studying the decision-making in venture capital, the traditional view assumes that the VC and the EN are pure self-interest. That is, they only pursue their individual maximum benefits rather than care about the fairness of the welfare allocation or behavioral motivation. Yet, a series of game experiments (like ultimatum game, trust game, and gift-exchange game) in recent years indicate that fairness preferences also exist in addition to self-interest preferences. The theory believes that they will also focus on the fairness of the welfare allocation or behavioral motivation when they pursue personal interests. As well as the self-interest preferences, fairness preferences can influence the decision-making of the participants in venture capital. For instance, people may sacrifice part of their own interests to preserve the fairness in the revenue allocating, to revenge hostile behaviors or to reward kindness.

The fairness function in venture capital is based on the psychological game framework of Geanakoplos et al. Rabin [

Hence, Rabin [

More generally, it provided a crude reference point against which to measure how generous player

This function captures how much more than or less than player

Furthermore, Rabin [

This function was to represent player

Thus, the reciprocal fairness model [

As was shown above,

Because these preferences form a psychological game, we could use the concept of psychological Nash equilibrium defined by GPS. This was simply the analog of Nash equilibrium for psychological games, imposing the additional condition that all higher-order beliefs match actual behavior. Rabin [

If the pair of strategies

According to Rabin’s reciprocity fairness preferences theory, we can construct the utility functions of the participants in venture capital based on the fairness preferences.

Based on Section

For simplicity, remark

Substitute the formulas above into (

And then

Continue to substitute the expression into (

Finally, we put formulas (

In Section

In model (III), the first-order condition about

There is

Put formulas (

It is complicated to work out the first-order condition of

Does we focus on whether there exists an appropriate pair (

Given too many parameters in formula (

Now the two inequalities

For convenience, we remark

The objective of the research is to find a contract structure

That is

We use simulation algorithm to solve the optimization problem. The algorithm is as follows.

Let

Loop over the value of

Store the maximum value of

Search in matrix

We use Matlab editing program, we can work out that the optimal venture capital structure is

Thus, we can conclude Proposition

In multistage venture capital, if the VC and the EN both have reciprocity motivation, there necessarily exists the optimal capital structure contract

Proposition

In multistage venture capital, if the VC and the EN both have reciprocity motivation, compared with the standard game, VC changes the payment structure; that is,

Proposition

Owing to

The parameter values and the optimal numerical solutions are brought into

Obviously,

First, Propositions

The moral hazard in the venture capital has been a widely concerned problem all over the world. The multistage model in venture capital that scholars considered could mitigate the moral hazard problem. Speaking of the models, some scholars researched how to design an incentive contract using the principal-agent model to ensure the venture capital’s appreciation and safety. While some scholars considered the exit of the VC in the multistage problem based on complete information and explained the condition in which the VC will continue or exit, their underlying hypotheses are all perfect rationality.

However, it is impractical for some decision-makers to adopt this perfect rationality hypothesis. In contrast, decision-makers commonly adopt bounded rationality. Therefore, we can say that to study the multistage venture capital model under bounded rationality is of crucial importance. Reciprocity motivation is one of many forms of bounded rationality for certain, and it is also a common one in practice.

In this paper, we have incorporated the reciprocity motivation into the multistage model to study the optimal capital structure contract. As a basis for comparison, at first we offered two game models: a cooperative model and a noncooperative one, both under perfect rationality. Then we introduced the reciprocity motivation into the multistage model and built the multistage behavioral game model based on reciprocity motivation. Our study clearly showed that the utility of the VC and the EN under reciprocity motivation is Pareto improvement compared with that under perfect rationality. This is where we exhibited the innovation point of this paper.

However, our study has limitations in the four following aspects. Firstly, our model has been founded under information symmetry; then how to combine our study of the reciprocity motivation with the principal-agent model? Secondly, we have mainly focused on the designing of the optimal capital structure contract but have not given the exit conditions of the VC. Thirdly, the model is based on the risk neutral assumption, so how will the model change if the VC and EN have fairness preferences? Fourthly, we only considered the EN’s effort, so what are the results for the EN and VC’s efforts? These are valuable challenges in front of us.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are particularly grateful to the associate editor and reviewers for thoughtful, valuable discussions and suggestions. The authors acknowledge the financial support by Humanities and Social Science Project of Ministry of Education of China (14XJCZH001), by Soft Science Research Project of Sichuan Province (2014ZR0027), and by the Fundamental Research Funds for the Central Universities (JBK130401). They also thank their students Litian Xu, Yiqing Zhao, Qinyu Chen, Mengxing Du, Xiao Xu, Zexi Zhang, Feng Yu, and Huan Zhou; Jiaci Wang contributed to this paper.