Sequential funding the venture project or not? A prospect consensus process with probabilistic hesitant fuzzy preference information

Abstract

Sequential investment problem is a widespread issue in venture capital (VC) process and it is an important key for the success of VC as well. Not only the basic information of projects but also the sunk cost, the network of venture capitalists (VCs), etc., should be considered in the sequential decision-making. In this paper, we study the sequential investment from building a decision-making index system firstly. Then, because of complex and uncertain decision-making circumstance, it is hard for the VCs to give their evaluation information for each alternative as crisp numbers, but it may be easy for them to provide their preferences over each pair of alternatives. Motivated by which, the probabilistic hesitant fuzzy preferences (P-HFPs) are adopted by the VCs in their evaluation process. Moreover, considering the bounded rational psychological characteristic of the VCs in decision-making, the prospect theory has been introduced. Therefore, a prospect consensus with P-HFPs has been constructed due to the grouping of sequential decision-making. It has been applied to a practical case, and an expected consensus under rational framework has been used to analyze the case. The superiority of prospect consensus with P-HFPs has been correspondingly demonstrated in this paper through the comparative analyses of the expected consensus with P-HFPs, the prospect consensus with HFPs and the expected consensus with HFPs.

Introduction

Venture capital (VC), as a special kind of money for entrepreneurs who are lack of capital to realize their desired business dream, has been more and more popular with the maturation of financial system. Moreover, VC is not only a domestic but also an internationalized activity. Hagleitner has addressed the VC activity between entrepreneurial firms and global corporations under the enlarged strategic context [1]. Because of the changeable investment environment, VC is a complex and uncertain process. Hence, staged financing becomes the fundamental characteristic of VC, and it is a general way for the VCs to avoid heavy loss and to disperse risk. However, for staged financing, a special target is set in each stage through the protocol reached before the initial investment. Then, the VCs should re-judge the prospect of the project according to the new information which generates with the progress of the project in each stage, such as the new technology, the change of management team and competitive advantage, etc. Moreover, such dynamic process makes the sequential decision-making to be more complex than the initial one. Not only the dynamic risk but also the static risk should be considered in sequential decision-making [2]. Thus, sequential investment decision-making of VCs differs from the initial one. Generally speaking, if it is a difficult thing for the VCs to decide which project will be funded in the initial stage, then it is even more difficult for them to decide whether the supported project is worthy to be sequentially funded. Since they have already spent a lot of money to the project, then abandoning the project means that nothing will be taken back, and such sunk cost¹ is very torturous for the VCs.

A large number of researches have focused on how to find a promising project with the help of decision criteria [3], [4], [5], [6] and of decision-making models [7], [8], [9], [10], [11] in initial investment. However, little attention has been paid to the decision-making of sequential investment for the VCs. Mcmullen [12] applied the sequential decision-making approach to the entrepreneurial action, but he focused on the entrepreneurial judgment with empathic accuracy. Guler [13] just tested the influence of politics and institution on the sequential decision-making with the investment data between 1989 and 2004. The result showed that the more the VC firms took part in the investment rounds, the less probability for them to terminate such investment even if the negative new information arrives. Furthermore, the intra-organizational politics from the co-investors in syndicate investment or from the limited partner indeed influenced the decision-making of continued or terminated investment. While Bergemann et al. [14] presented the dynamic theoretical model to find an optimal investment strategy with the arrival of new information as the progress of the project. The data of VC investment during 1987 to 2002 in the US were used to test the validity of the proposed dynamic model. They concluded that the information derived from the development of the project indeed affected the investment decision-making more than the information held before the project was launched. If the VCs think that the prospect of the project is uncertain and the probability of success is low, then, they intend to distribute the failure risk through more investment rounds and lower initial investment amount. Furthermore, the investment rounds and investment amount will change with the progress of the project and with the arrival of new information. However, all the researches above just discussed single factor that affected the decision-making of sequential investment from the perspective of empirical analysis. There is a lack of systematic index systems to depict the sequential decision-making criteria comprehensively. So, we will establish such an index system in this paper. Also, none of the above researches disclose how to make a reasonable sequential decision-making for the VCs under highly uncertain environment. Therefore, a mathematical model was built to explore the optimal sequential investment decision-making based on concave or convex return function [15]. In addition, sequential investment decision-making has been studied from the perspective of Bayesian learning [16], but it only considered a two-period model which is incompatible with reality of multistage investment in the VC field. Friedl [17] investigated the multistage sequential decision-making with the help of geometric Brownian motion and the Bellman equation [18]. Those mathematical models heavily rely on the mathematical reasoning and ignore the effect of new situation faced by the project on the decision-making of VCs in each investment stage. Whatever, the sequential decision-making is made by VCs, and not only the price of project but also other factors such as the new market condition, the sunk cost, etc., are considered by them. Along with the progress of the project, the new situation of the project will appear. The VCs feel painful if they do not invest the project continuously which means that the prior capital will be the sunk cost, and the sequential decision-making for them is complicated and difficult. Hence, developing an effective sequential decision-making model which reflects the new situation faced by the project is essential for the VCs to relatively improve their quality of decision-making. In this paper, we study the sequential decision-making of VCs with the help of multi-criteria decision-making (MCDM) method which fully investigates all aspects of the project in the decision-making process and includes all the new information faced by the project in sequential decision-making. It overcomes the disadvantages of pure mathematical reasoning method and increases the flexibility of sequential decision-making with the dynamic new information about the project.

In the MCDM method, evaluating the alternative projects is an indispensable step. Generally, it is hard for the VCs to give accurate evaluation information for each alternative project under the complex and uncertain situation of sequential decision-making. However, it is easy for them to express their tendency of preferences (or comparison information) between those alternative projects. Therefore, preference information is a suitable way used by the VCs to depict their evaluation information among alternative projects. Moreover, under the uncertain VC circumstance, giving only one preference information seems to be difficult for the VCs to express their real thoughts. For example, when an investor was asked to evaluate the appearance of Mobike and Ofo, he/she may be hesitant to say that “Mobile is a little beautiful or beautiful than Ofo” or that “Mobile is at least beautiful than Ofo”. In those situations, the fuzzy preference has its superiority in expressing the investor's preferences under uncertainty. Especially, the probabilistic hesitant fuzzy preference (P-HFP) with the probability of each hesitant fuzzy element, which includes more preference information, will perfectly portray the VCs’ idea in the real decision-making situations. Hence, different from the existing researches, the P-HFP has been chosen as the basic information unit to explore the sequential decision-making of VCs for the first time in this paper.

As is widely known, the decision-making of the VCs under uncertainty depends on gains and losses rather than the final wealth, and it embodies different risk attitudes for gains and losses [19]. Moreover, prospect theory (PT) [20] has its superiority in depicting such risk attitudes (risk seeking for losses and risk aversion for gains) during the uncertain decision-making process. According to the PT, the decision-making is influenced by the loss aversion and the non-linear probability weighting. Many researches have been conducted to show the nonlinear weight in the decision-making process through questionnaire [21], [22], [23] and to demonstrate the usefulness of behavioral decision-making revealed by the PT [24], [25]. Also, the PT has been introduced to the MCDM problem to portray the behavioral decision-making [26], [27]. Furthermore, the PT combined with fuzzy information has been adopted to solve the MCDM problem [28], [29], [30], [31], [32], [33]. Considering this point, the different risk attitudes for gains or losses unconsciously lie in the VCs’ preferences as well. When we study the decision-making on the basis of preferences, we should integrate those risk attitudes into the decision-making model. Based on the analysis above, using the PT to develop a MCDM method with fuzzy preference information is reasonable and feasible for decision-making of VCs. In addition, because the social division of labor has gradually become deeper and the VCs differ significantly in their ability to terminate failing projects [34], group decision-making has been more and more popular in order to avoid the limitation of individuals. However, consensus is an important issue in group decision-making. Thus, taking into account all the above phenomenon, we try our best to construct a MCDM method named prospect consensus with P-HFPs to study the sequential decision-making of the VCs.

The main contributions of this paper are as follows: Firstly, a decision-making index system of sequential investment is established, because there is no complete sequential decision-making index system in the existing researches. Secondly, the P-HFPs are put forward to study the sequential decision-making problem and depict the evaluation information of the VCs. Thirdly, due to the universality of group decision-making in the VC field, for the first time, a MCDM method has been introduced to study the sequential decision-making and a consensus procedure that integrates the PT and P-HFPs has been constructed for the VCs to make suitable decisions, which is named as prospect consensus. Most importantly, the PT has been firstly used to explore the consensus problem. Fourthly, relative to the afore-mentioned prospect consensus for the VCs, the expected consensus with P-HFPs or with the hesitant fuzzy preferences (HFPs) and the prospect consensus with HFPs have been proposed too. Although the prospect consensus with P-HFPs are confirmed to be more efficient for the VCs, they can also play a useful role for the decision-making in other application fields.

The rest of this paper is organized as follows: The factors that affect the sequential decision-making of the VCs are presented in Section 2. Then, Section 3 includes some basic concepts and algorithms. The core of this paper lies in Section 4 which constructs the main procedure of the proposed prospect consensus with P-HFPs. In Section 5, the proposed method in Section 4 is used to analyze a practical case, and the detailed comparative analyses are conducted in this section. Finally, this paper ends in Section 6 with some conclusions.