Sufficient Conditions that Lead to Synergistic Innovations

Abstract

This chapter, which is mainly based on (Forrest et al., 2019, Proceedings of the 2019 Annual Conference of Decision Sciences Institute (pp. 2061–2080)), investigates issues related to the following questions: (1) how can producer-side synergies be created by employing the strategy of economies of scope? And (2) how can demand-side synergies be developed by making use of simultaneous consumer utilities and multi-sided markets? Because of the different approach taken, we are able to describe how resources interact with each other and how simultaneous consumer utilities, two-sided markets and consumers’ willingness to pay react to each other. We first look at how such economic entities and basic concepts as business firm, resource, innovation, diversification, synergy, segment of consumers, etc., can be respectively modeled as interacting systems. We then address the previous two questions by establishing a series of 8 propositions by examining how relevant systems exert forces on each other so that some general conclusions follow. Because of the systemic certainty our discussions offer, this chapter is expected to provide practically useful guidance for managers, entrepreneurs, and retailers to create values for consumers and capture values for their companies.

Appendix: Proof of Theorem 8.1

Given a system S₀ = (M₀, R₀), another system S_n = (M_n, R_n) is said to be an nth-level object system of system S₀, provided that there are systems.

$$ {S}_i=\left({M}_i,{R}_i\right),\mathrm{for}\ i=1,2,\dots, \mathrm{n}\hbox{--} 1, $$

such that S_i is an object in M_i − 1, 1 ≤ i ≤ n. In this case, each element in M_n is referred to as an nth-level object of system S₀. When the system S₀ has at least one nth-level object system, for n = 1, 2, …, then the system is referred to as a multileveled (or multilevel) system.

A chain of object systems of S₀ is a sequence {S_i = (M_i, R_i) : i < α}, for some ordinal number α, of different-level object systems of S₀, such that for each pair i, j < α satisfying i< j, there exists an integer n = n(i, j), a function of i and j, such that S_j is an nth-level object system of S_i, Fig. 8.6, where each oval area stands for a system with its objects indicated by dots and relations by enclosed regions. Then the following result (Lin, 1999, p. 193) is true:

Fig. 8.6

A chain of object system

$$ \left(\ast \right)\ \mathrm{Each}\ \mathrm{chain}\ \mathrm{of}\ \mathrm{object}\ \mathrm{systems}\ \mathrm{of}\ S\ \mathrm{must}\ \mathrm{be}\ \mathrm{finite}. $$

Now, we define the set M(S) of all fundamental objects as follows: First, let us rewrite the system S = (M, R) as S = (M₀, R₀) for the sake of notational convenience. Define

$$ {}^{\ast }{M}_0=\left\{x\in {M}_0:x\ \mathrm{is}\ \mathrm{not}\ \mathrm{a}\ \mathrm{system}\right\} $$

and

$$ {\overset{\sim }{M}}_0=\bigcup \left\{{M}_x:x=\left({M}_x,{R}_x\right)\in {M}_0-{}^{\ast }{M}_0\right\} $$

Assume that for a natural number n, two sequences {^∗M_i : i = 0, 1, 2, …, n} and {${\overset{\sim }{M}}_i$: i = 0, 1, 2, …, n} have been defined, satisfying that

$$ {}^{\ast }{M}_i=\left\{x\in {\overset{\sim }{M}}_{i-1}:x\ \mathrm{is}\ \mathrm{not}\ \mathrm{a}\ \mathrm{system}\right\} $$

and

$$ {\overset{\sim }{M}}_i=\bigcup \left\{{M}_x:x=\left({M}_x,{R}_x\right)\in {\overset{\sim }{M}}_{i-1}-{}^{\ast }{M}_i\right\}. $$

Then the following two sets can be defined for the index number i + 1,

$$ {}^{\ast }{M}_{i+1}=\left\{x\in {\overset{\sim }{M}}_i:x\ \mathrm{is}\ \mathrm{not}\ \mathrm{a}\ \mathrm{system}\right\}\ \mathrm{a}\mathrm{nd}\ {\overset{\sim }{M}}_{i+1}=\bigcup \left\{{M}_x:x=\left({M}_x,{R}_x\right)\in {\overset{\sim }{M}}_i-{}^{\ast }{M}_{i+1}\right\}. $$

From mathematical induction, it follows that a sequence {^∗M_i : i ∈ ω} is defined. Now, by letting M(S) =  ⋃ {^∗M_i : i ∈ ω}, result (*) above guarantees that M(S) consists of all fundamental objects in system S.

As for the conclusion that |M| < |M(S)|, it follows from the assumptions that (1) each object system of S has a finite object set, (2) each object that appears in any relation in the system S or an object system of S only appears once in that relation, and (3) S has at least one chain of object systems of more than one level and each object system on the chain contains more than one object. Here, conditions (1) and (2) are necessary for producing the resultant inequality by limiting our counting cardinalities |M| and |M(S)| within the realm of natural numbers. End of the proof.