Research Diversity and Invention

Abstract

This paper explains that when there is great uncertainty about which elements of knowledge must be combined to make an invention, the likelihood of invention increases markedly—by many orders of magnitude—when there are numerous diverse research organizations, rather than just a few. The paper examines the possibility that competition (antitrust) policy toward mergers would be improved if enforcement efforts placed more emphasis on protecting the diversity that is provided by numerous research rivals in a market.

Introduction

This paper explains why having numerous diverse research organizations greatly increases the likelihood of invention. An open question is discussed: Does the invention-increasing effect of research diversity justify adjusting competition (antitrust) merger policy by encouraging greater stringency in merger approvals so as to protect the diversity that numerous research rivals in a market would provide?

The importance of diversity for successful research is a theme that is found in the literature about invention and innovation. That literature defines an invention as a new combination of essential elements of knowledge—envisaged together in their working configuration for a new product or service—and defines an innovation as the commercialization of an invention.

The key reason for the beneficial effects of diversity on invention is that an invention combines existing ideas in new ways, and the ideas that are combined in a successful invention are often diverse, coming from very different sources of knowledge and experience. Because diversity within an organization expands the set of ideas that the organization has available for invention, diversity fosters the inventions that become innovations.1 This paper adds an explanation for why a diverse collection of research organizations further stimulates invention.

In terms of the set of knowledge elements from which elements combined in an invention are chosen, Sect. 2 defines uncertainty for an invention and establishes that, given such uncertainty, numerous diverse research organizations make invention more likely.2 Section 3 discusses the possibility that the pace of invention could be increased if competition (antitrust) merger policy placed more emphasis on preserving numerous diverse research organizations.

Section 4 concludes by emphasizing that the proposed change in merger policy is efficiency-based and is important because of its potential for increasing the pace of discovery of new ideas and increasing socially beneficial economic growth.

Uncertainty and Invention with Diverse Research Organizations

Inventions combine existing elements of knowledge in new ways, and the process of discovering an invention is uncertain because of the large number of different elements of knowledge in the set from which must be chosen those that are needed for the right combination of elements for the invention.3 A research organization—wherever situated in the range from large to as small as an individual researcher—confronts the uncertainty of finding the requisite combination of elements from the set of knowledge elements.

A research organization as a practical matter can work with just a subset of the complete set of knowledge elements.4 Successful invention requires that the diversity of ideas in a research organization’s knowledge set must be limited in practice so that the organization can be effectively managed.5 Consequently, there is the possibility that the subset with which it works will not include one or more of the knowledge elements that are needed for the invention.

The type of uncertainty that is the focus of this paper implies that the organization does not know whether the knowledge set with which it works lacks the needed elements. As will be detailed subsequently, the organization can add knowledge elements—for example, additional knowledge elements could be acquired in the market—but although with the larger collection of knowledge elements there is an increased chance that the requisite knowledge elements for the invention are included, whether they are included is unknown.

The uncertainty in the process of invention—the possibility that a research organization is working with a set of knowledge elements that does not include the full set of requisite elements for the invention that is sought—implies that when the uncertainty is great, numerous diverse research organizations will increase the likelihood that some of them will be working with a set of knowledge elements from which the right combination for the invention could be found. To develop that point, we formulate the uncertainty about whether a research organization is working with a set of knowledge elements containing the requisite ones.

An invention combines t elements of knowledge from among the s knowledge elements that a research team has acquired with study and experience from the universe of n knowledge elements. The number n is much larger than s, which in turn is much larger than t.6

Let $p\left(f\right)$ denote the proportion of all possible combinations of s elements chosen from the universe of elements that is taken by cases for which f of the s elements that are available to the research team coincide with f elements in the set of t elements that together define the invention.7 For a research team to be able to discover the essential combination of knowledge elements necessary for an invention, the t essential elements for the invention must be among the s elements that are available to the research team. In that case, $f=t$; and as was shown in Scott (2016, p. 420):

$$P_t\equiv p\left(f,=,t\right)=\prod _{i=0}^{t-1}\left(\frac{s-i}{n-i}\right).$$ 1

If we think of a research organization abstractly as the set of s knowledge elements with which it works, successful invention requires that the organization has the requisite t elements of knowledge among the s elements; yet the ways that the organization could have the essential t elements among its set of s constitute a minuscule proportion of the potential knowledge sets of s elements with which the research organization could have chosen to work.8

For an example, suppose that there is a universe of a million knowledge elements (n = 1,000,000), an invention that combines three elements (t = 3), and a research organization’s knowledge set with a thousand elements (s = 1,000): Then the proportion of all possible combinations of knowledge elements that include the cases where the organization’s set of knowledge elements include the necessary t elements for the invention is 0.9970 × 10^− 9 or about one billionth.

Since a research organization is restricted in the size of the knowledge set that can be managed effectively within an organization, in practice the number of knowledge elements with which its researchers work (s elements for the organization) will be far less than the number of elements n in the universe of knowledge elements. The implication is that a given organization may not have within its knowledge set the requisite t elements for an invention. P_t is extraordinarily small—close to zero—in the case of great uncertainty, defined as the case when n is much larger than s.

For a given research organization, we expect P_t to be small because adding to an organization’s knowledge set is costly, whether for the reasons in the management literature that focus on the difficulty of managing a large number of diverse points of view, or whether because of the direct costs of the knowledge acquisition itself. The knowledge set would not be expected to be increased beyond the point where the marginal benefits equal the marginal costs: where the value of increasing P_t has fallen to the additional cost for the increase in P_t.

The increasing costs that are faced by the research organization as it adds new elements of knowledge to the knowledge set with which it works are necessary for understanding the practical limitation on an organization’s knowledge set because as a research organization expands the number of elements of knowledge in its knowledge set, P_t increases at an increasing rate, as will be shown in the following paragraphs. Each knowledge element that is added to the set essentially doubles the number of possible combinations that could be considered and evaluated.9

If a research organization expands its knowledge set—the organization increases the number of knowledge elements in the organization’s set by v elements, so as to have s + v elements—then the proportion $P_{t,v}$ of the sample space for which the research organization’s now expanded set of knowledge elements will overlap completely with the t elements of the essential idea is:

$$P_{t,v}=\left(\prod _{i=0}^{v-1},\frac{s+1+i}{s-t+1+i}\right)P_t.$$ 2

Thus, the change in the proportion for each unit increase in the number of elements in the research organization’s knowledge set is:

$$P_{t,v}-P_{t,v-1}=\left(\frac{t}{s-t+v}\right)P_{t,v-1}>0.$$ 3

Equation (3) shows that the proportion—of all possible combinations of knowledge elements that are taken by the cases where the organization’s set of knowledge elements include the necessary t elements for the invention—is increasing as the number of elements in the organization’s knowledge set increases.

For successive unit increases in the number of elements that are in the organization’s knowledge set, the Appendix derives Eq. (4), which shows that the successive changes in the proportion are increasing:10

$${{(P}_{t,v+1}-P_{t,v})-(P}_{t,v}-P_{t,v-1})=P_t\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left[\left(\frac{t(t-1)}{\left(s,-,t,+,v\right)(s-t+v+1)}\right)\right]>0.$$ 4

Because the expansion of a research organization’s knowledge set increases P_t at an increasing rate, increasing cost with knowledge acquisition (as was discussed in note 5) is key to understanding why we expect P_t to be a very small proportion for a research organization.

Many diverse research organizations—with their diversity defined by the differences in their knowledge sets—will make it more likely that there will be research organizations with knowledge sets that include the requisite knowledge elements. If there were z independent research organizations and for each organization α was the probability of not having the requisite t elements within its knowledge set of s elements, the probability that at least one organization has the requisite t elements is $\left(1,-,{\left(\alpha \right)}^z\right)$, which approaches 1.0 as z becomes larger.11

Observe that great uncertainty—n is much larger than s—is necessary for the efficacy of numerous diverse research organizations. Suppose that the set of knowledge elements from which the research organization selects s elements to study (in the hope that within those s elements the requisite t elements can be found) has been successfully reduced. And suppose the reduction is sufficiently great, from n to a number $\widehat{n}$, so that s, although less than $\widehat{n},$ is large relative to $\widehat{n}$. In that case, having numerous diverse research organizations is of little value and can even reduce (as will be argued below) the chance that organizations are working with knowledge sets with the requisite elements for the invention sought:

Define the proportion $\widehat{P_t}$ as in Eq. (1) but with n replaced with $\widehat{n}$, and assuming the set of knowledge elements (from which the research organization selects s with which to work) is successfully reduced to $\widehat{n}$ (i.e., the essential t elements of knowledge are among the $\widehat{n}$ elements). $\widehat{P_t}$ is the proportion—of the potential knowledge sets of s elements chosen from among the $\widehat{n}$ and with which the research organization could have chosen to work—that include all t essential elements. With s close to $\widehat{n}$, $\widehat{P_t}$ has become close to 1 for an individual research organization, and the gain from having multiple research organizations is small at best.

Consider the extreme case where n has been successfully reduced to $\widehat{n}$ and a research organization can study those elements: s = $\widehat{n}$. $\widehat{P_t}$ = 1, and for this special case there is no uncertainty of the type that was addressed with numerous diverse organizations in the foregoing discussion—although having intra-organizational diversity for the organization pursuing the $\widehat{n}$ ideas can be the essence of considering all those ideas.

However, if there were any significant uncertainty as to whether an organization would recognize among the $\widehat{n}$ elements the right combination of t elements for the invention, numerous organizations could serve usefully (if the cost for additional search were outweighed by the expected gain) for a parallel path strategy to increase the probability of invention. But these would not be diverse organizations but would be identical in the sense that for each its s set would coincide with $\widehat{n}$.

In that foregoing extreme case, if one insisted that all organizations be diverse, then for all but at most one organization, the s set would not exactly overlap with $\widehat{n}$, and so the diversity across organizations could reduce the likelihood that organizations were working with knowledge sets within which the requisite elements to be combined in the invention could be found. In the extreme case, introducing diversity across the organizations in the s elements can reduce the chance that the t elements are included in an organization’s s set, especially so if s is small. Such extreme cases are not the cases of great uncertainty of interest here.

In sum, the case of interest to us is t < < s << n, and P_t is vanishingly small, many orders of magnitude removed from the P_t = 1 case—nine orders of magnitude in our 1 in a billion example. Numerous diverse firms are helpful for the case of interest.12

What kind of inventions would be examples of the case of interest? Those that come readily to mind are the more dramatic inventions. A recent example is the massive effort by many diverse organizations to develop a vaccine to mitigate the Covid-19 pandemic (Mullard, 2020). Archetypal examples of inventions that each played out over many decades of research by numerous diverse research organizations, including individual researchers, are the invention of the electric lightbulb (Johnson, 2014, pp. 205–215) and the invention of the airplane (Meyer, 2015).13

Less dramatic and prominent inventions may not entail the case of great uncertainty for which numerous diverse research organizations make the invention more likely. Yet because we are often first cognizant of inventions during the later stages of their discovery, the number of knowledge elements from which research organizations choose s for study may well have been considerably reduced (from n to $\widehat{n}$) by the time that the discovery process is observed. If so, we might not realize that initially the problem of finding the invention was uncertain in our sense that P_t was vanishingly small and having numerous diverse research organizations would have been helpful.14

Implication for Competition Policy Toward Mergers

Greater Diversity Coincides with a Larger Number of Research Organizations

As discussed in Sect. 3.2 below, competition policy toward mergers has not placed an emphasis on maintaining numerous research organizations in markets. Yet if merger policy did serve to maintain numerous research organizations in markets, the industrial organization literature suggests that the policy would also serve to maintain numerous diverse organizations. The papers typically discuss a combination of invention and innovation activities—industrial R&D expenditures cover a range of activities from research for invention to the development of inventions in pursuit of innovations—and support the view that having more organizations increases the diversity of their research.

Scherer (1970, p. 357) emphasizes the diversity across firms of different sizes, with smaller firms disproportionately active in invention—which is consistent with Sect. 2’s description of increasing costs for the expansion of a research organization’s knowledge set—and then with larger firms’ playing important roles in innovation. Cohen and Klepper (1991) emphasize the many possible different productive approaches to innovation that can result from having a larger number of firms in an industry.

Scott (1991, pp. 136–139) emphasizes that each firm among a rivalrous group may have an incentive to bring different knowledge to its research to find a unique invention that when developed will dominate post-innovation competition among competing substitute innovations. The rivals strive to be different.

Describing the contrasting situation in which rivals do not strive to be different, Nelson (1981, p. 107) discusses “a syndrome regarding R&D allocation similar to that described by Hotelling in the case of location decisions” and observes that competitive R&D rivals will not necessarily pursue different opportunities. The rivals strive to take different approaches in Scott (1991) because the dominant version of the innovation is unknown in advance of development and commercialization of the underlying invention, and so the conditions that would cause R&D rivals to cluster their efforts around similar development efforts and a similar version of the innovation are not present.15

Thus, given uncertainty, we expect that independent research entities provide unique perspectives. The evidence that firms in the same industry pursue different R&D strategies is strong. Regarding company-specific research policies, interpreting the results in Scott (1984), Griliches (1984, p. 7) observed, “… there appear to be significant differences in company R&D policy above and beyond what would have been predicted just from their differential location within the industrial spectrum.”

There is also evidence (Scott, 1991, pp. 139–145) that research rivalry and the diversity of the rivals are synonymous. Given the competitors in an area of research, distinct R&D strategies are observed for firms in the systems orientation of their portfolios of patents, in the purposive diversification of their R&D across industries, and in the intensity of their R&D. Moreover, the firm effects in R&D intensity are more significant in markets with more competitive structure—which is consistent with the hypothesis that with more firms competing, firms have more incentive to strive for a distinct innovation so as to avoid the erosion of rents in the post-innovation market.

Taken together, the several types of evidence that different firms in the same industries pursue different research strategies support the inferences that firms’ capabilities differ fundamentally and that firms would choose different strategies even if their capabilities were the same.16

The Need for New Competition (Antitrust) Policy Regarding Diversity

Has competition policy been adequate to protect the desirable diversity that numerous research organizations provide to the process of invention in an area of research? The discussion to follow focuses on the antitrust law of the United States; but although the institutional details differ, the essential problem discussed is present in competition policy throughout the world, and it results from the nexus of statute law, enforcement, and interpretation.17

The statute law that protects competition uses language that is broad enough to protect the desirable diversity that is provided by large numbers of firms competing in a given market; and any inadequacy would result because of the historical evolution of enforcement and interpretation of the statutes.18 When considering mergers, coordinated behavior among firms, or dominant firm behavior that would lessen the number of independent research entities in an area of research, both dynamic efficiency—which relates to new products and processes from innovation—and static allocative efficiency—which is a nexus of price, cost, and output given existing products and processes—are considered desirable attributes of performance.

In an insightful analysis based on well-informed understanding of both the law and the economics, Baker (2007) addresses the question of whether competition policy can be used successfully to promote innovation. His analysis develops and supports well the view that “[o]n the whole, … antitrust rules and enforcement today are appropriately focused to promote innovation” (Baker, 2007, p. 576).

Baker’s view is supported with his careful review of the law and the economics to show that “… the modern economic learning about the connection between competition and innovation helps clarify the types of firm conduct and industry settings where antitrust interventions are most likely to foster innovation. Measured against this standard, contemporary competition policy holds up well” (Baker, 2007, p. 576).19

Baker’s reasoning and his conclusion are compelling. Just one amendment is suggested here because of the importance of diversity for invention: Competition policy toward mergers could potentially be improved by addressing the need for large numbers of diverse research organizations in an area of research to increase the probability that research organizations will have the requisite elements of knowledge for a sought-after invention.20

Baker (2007, p. 592) observes “. . antitrust promotes innovation by challenging horizontal mergers that reduce the number of likely innovators when there are few, absent countervailing innovation efficiencies.” Thus, the enforcement agencies’ challenges to horizontal mergers occur when the relevant markets have only a few likely innovators. Consolidation of likely innovators prior to leaving only a few occurs without a challenge from competition policy. The industrial R&D of the firms that are consolidated to leave a few likely innovators entails a mixture of research and invention and development and innovation; the benefit of numerous firms investing in R&D can be lost.

There are arguably two reasons why competition policy toward mergers does not protect the diversity that would be provided by numerous innovative competitors in a market: One is the consensus view that for good innovation performance a moderate amount of seller concentration is ideal—not so much structural competition that investment in research is discouraged because firms expect insufficient returns in post-innovation markets, and not so much concentration that the profitable firms that are shielded from preemption by innovative rivals lack the incentive to invest in research.21

A second reason is that at least a moderate amount of seller concentration may be needed to achieve scale economies that promote efficiency and may do so to a sufficient extent that the efficiency gains increase total economic surplus by more than what is lost by any increase in market power.22

Thus, performance gains from economies of scale in markets with fewer sellers or from economies of scope across the markets of diversified and vertically integrated firms are recognized, while harm to innovative performance from a loss of independent research units is not anticipated given the consensus that a moderate amount of competition—equivalent in practice to a moderate amount of concentration of resources among sellers—is ideal for innovation.23

The consensus about innovation performance’s being best in moderately concentrated markets is strong, yet it is open to question even without the importance of diversity in research efforts.

In the consensus view, the problem with too much competition is that investors might expect too little return from research given the probability that competitors would market competing substitute innovations—perhaps imitations that use the results of the investments of others, or that a competitor would have a dominant innovation. Yet government subsidies can make research investments profitable even in highly competitive markets (Link and Scott, 2001).

The consensus-view problem with too little competition is that there will be insufficient incentive for research investment by firms that are not anticipating preemption by the innovations of rivals. Yet, theory (Scott and Scott, 2014, Appendix Remark 3 and Remark 4, pp. 52–53) predicts and evidence (Scott and Scott, 2014) finds higher R&D investments in more highly concentrated markets when the research expenditures of the oligopolists are sufficiently strong strategic substitutes.

If an increased number of research organizations is associated with an increased diversity of knowledge sets, and consequently it is more likely that there will be research organizations with the requisite knowledge elements for a sought-after invention and that the invention will be found, then at least one aspect of innovation performance—invention—would not be subject to an inverted-U relationship. Thus, the inverted-U relationship, if it exists, would derive from other aspects of the innovation process—presumably ones that are related to commercialization rather than to invention.24

Conclusion

In practice, competition policy toward mergers does not pay attention to maintaining large numbers of independent research entities that provide diversity for inventive efforts that support innovation in a market. The inattention occurs even though the goal of competition policy is to promote competition—in part because “[c]ompetition often spurs firms to innovate” (U.S. Department of Justice and Federal Trade Commission, 2010, p. 23).

The consensus view that a moderate amount of concentration of research resources promotes innovation has aligned in the application of competition policy with the view that economic performance given existing technologies is improved when markets are sufficiently concentrated to achieve significant economies of scale and when firms’ operations across markets capture available efficiency gains from vertical integration and from economies of scope.25 Attention to the improvement in innovation performance that results from the diversity that is provided by numerous research competitors is simply not a focus of competition policy.26

The focus in this paper on the benefits of diversity in research efforts is a narrow economics argument about the efficiency of the research process; thus, the focus is different than the one that has been emphasized by the Biden administration’s FTC.

The FTC has proposed new approaches to competition policy that include, among other things, actions “to ensure that all members of the public reap the benefits of competition, particularly members of underserved, marginalized and diverse groups … [and] to identify and redress the potentially disparate impact anticompetitive transactions or business practices have on traditionally underserved and marginalized communities, particularly when conduct may seek to deliberately exploit or prey upon the disadvantages of these diverse communities.” (Federal Trade Commission, October 2021, p. 21).27

The FTC’s initiatives are controversial because competition policy—as it has evolved with the enforcement and interpretation of the statute law—has had the narrow goal of promoting competition in markets to increase efficiency and innovation for the benefit of consumers.28 The FTC proposes to use competition policy to address equity among diverse groups—an approach that is distinct from but complementary with the use of competition policy to ensure diverse research efforts.29

Changing competition policy to address a goal of preserving the diversity that is provided by large numbers of competitors in a market has a narrowly economic, efficiency-based justification: improving innovative performance.30 An economy’s dynamic performance is important; when good, it can create productivity gains and growth that allow better living standards for households throughout the distributions of income and wealth.

Thus, an implication of this paper’s findings about how numerous diverse research organizations can increase the likelihood of invention would be that in the calculus of antitrust enforcement decisions about mergers, additional weight should be assigned to the benefits for invention from avoiding the loss of independent research organizations because of mergers. Translating the implication into policy is difficult, and several key issues must be addressed:

1. It is important to keep conceptually separate—at least in principle—the size and structure for the inventing (or R&D) organization from the size and structure of the production organization in which the inventing organization may or may not be housed. As was discussed in note 5, there are economies of scale in inventing; yet the metric for the scale of research output differs from the metric for the scale of the production of goods and services to which an invention may apply, and the differences in the metrics are important when examining tradeoffs: Mergers decrease the number of independent research centers but increase the scale of production and possibly of invention and R&D activities.

2. Seller concentration in the relevant goods/services market need not be the same as—or have the same effects as—the concentration in the underlying research “market” that is especially important for evaluating a merger’s impact on the number of diverse research organizations.

3. The tradeoffs must be enumerated. The key tradeoff to confront would be to weigh gains from a larger number of diverse research organizations with the advantages that are provided by fewer firms that may not only realize scale economies in research and production but also appropriate more of the social returns when an invention is developed and commercialized (Cohen and Klepper, 1991).31 The tradeoff also entails the different roles that are played by small and large firms in the process of invention and innovation (Scherer, 1970, p. 357), with the need to maintain a sufficient presence of larger firms so as to ensure the successful innovations that build on the inventions that are generated by smaller firms.32

The point here is that competition policy should increase the weight that is placed on the diversity that is provided by a larger number of innovative rivals because of the increased likelihood that there will be research organizations with the requisite knowledge elements for the invention that precedes further development work and commercialization of an innovation.33 Without the invention, there will be no innovation. But clearly there are the foregoing issues to address if merger policy is to implement an increase in the weight that is placed on the value of numerous diverse research centers.

Moreover, although the statute law that governs competition policy is stated broadly enough to encompass enforcement to protect desirable diversity of numerous research rivals in a market, new legislation would probably be needed to implement the proposed change in merger policy. Using competition policy toward mergers to ensure the diversity of research that is important for invention would probably require new legislation, given the current interpretation of the current statutes and because interpretation of the statute law develops slowly through case law.34

Such new legislation would state that protecting the diversity of research efforts is important for effective competition and thus is an important goal of competition policy. The legislation would add information about the intent of the broadly-worded statute laws—adding language to state explicitly that maintaining the diversity that is provided by numerous competitors in a market is an important part of what is meant by maintaining competition.

The proposed change in antitrust policy toward mergers—to emphasize the importance of maintaining numerous diverse research organizations in markets—could have a significant impact on the pace of invention; this is an impact that would be especially helpful in the context of evidence that ideas are becoming harder to find (Bloom et al., 2020).35

We described above the increasing costs that are faced by a research organization when expanding its own knowledge set, but also showed that a collection of diverse research organizations could increase the likelihood that some of the organizations would have within their knowledge sets the requisite knowledge for invention. Policy that promotes numerous diverse research organizations may therefore unleash the possibilities of combinatorial growth that is described by Weitzman (1998).

Appendix

From Eq. (3),

$${{(P}_{t,v+1}-P_{t,v})-(P}_{t,v}-P_{t,v-1})=P_{t,v}\left(\frac{t}{s-t+v+1}\right)-P_{t,v-1}\left(\frac{t}{s-t+v}\right)$$

From Eq. (2), the right-hand side equals

$$\left(\prod _{i=0}^{v-1},\frac{s+1+i}{s-t+1+i}\right)\left(\frac{t}{s-t+v+1}\right)P_t-\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left(\frac{t}{s-t+v}\right)P_t$$

$$=P_t\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left[\left(\frac{s+v}{s-t+v}\right),\left(\frac{t}{s-t+v+1}\right),-,\left(\frac{t}{s-t+v}\right)\right]$$

$$=P_t\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left[\left(\frac{t(s+v)}{{(s-t+v)}^2+(s-t+v)}\right),-,\left(\frac{t}{s-t+v}\right)\right]$$

$$=P_t\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left[\left(\frac{t\left(s,+,v\right)\left(s,-,t,+,v\right)-t[{\left(s,-,t,+,v\right)}^2+\left(s,-,t,+,v\right)]}{[{\left(s,-,t,+,v\right)}^2+\left(s,-,t,+,v\right)](s-t+v)}\right)\right]$$

$$=P_t\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left[\left(\frac{t\left(s,-,t,+,v\right)[\left(s,+,v\right)-(\left(s,-,t,+,v\right)+1\left)\right]}{[{\left(s,-,t,+,v\right)}^2+\left(s,-,t,+,v\right)](s-t+v)}\right)\right]$$

$$=P_t\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left[\left(\frac{t(s-t+v)(t-1)}{[{\left(s,-,t,+,v\right)}^2+\left(s,-,t,+,v\right)](s-t+v)}\right)\right]$$

$$=P_t\left(\prod _{i=0}^{v-2},\frac{s+1+i}{s-t+1+i}\right)\left[\left(\frac{t(t-1)}{\left(s,-,t,+,v\right)(s-t+v+1)}\right)\right]>0$$

Declarations

Conflict of interest

Not Applicable