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Disaggregate Hierarchical Decision Huff Model Incorporating Consumer Kaiyu Choices Among Shopping Sites Read chapter Read first chapter

2023 | OriginalPaper | Chapter

Basics of Kaiyu Markov Models: Reproducibility Theorems—A Validation of the Infinite Kaiyu Representation

Authors : Saburo Saito, Kenichi Ishibashi, Kosuke Yamashiro, Masakuni Iwami

Published in: Recent Advances in Modeling and Forecasting Kaiyu

Publisher: Springer Nature Singapore

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Abstract

The Kaiyu Markov model is formulated as an absorbing stationary Markov chain model in which consumers’ Kaiyu (shop-around) behaviors are expressed as an infinite process of state transitions in a stationary Markov chain. This chapter gives the basics of mathematical formulations of the Kaiyu Markov model. It proves reproducibility theorems that provide validity for representing consumers’ Kaiyu behaviors as infinite stationary state transitions.

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previous chapter Kaiyu Markov Model and Evaluation of Retail Spatial Structures
next chapter Kaiyu Markov Model with Covariates to Forecast the Change of Consumer Kaiyu Behaviors Caused by a Large-Scale City Center Retail Redevelopment
Appendix
Available only for authorised users
Footnotes
1
Mathematics assumed here is high school mathematics level.
 
2
More precisely, xA is the conditional Kaiyu effect given that the consumer stays at shop A in the first step, which equals the sum of the conditional probabilities that the consumer visits shop A after the second step. Similarly, yA is the conditional Kaiyu effect given that the consumer stays at shop B in the first step, which equals the sum of the conditional probabilities that the consumer visits after the second step.
 
3
The following paragraphs can be skipped for the first reading without missing essential understandings of the Kaiyu Markov model. Readers are asked to return to these paragraphs after reading later sections.
 
4
More formally, the assertion is derived as follows.
\( {\displaystyle \begin{array}{l}P\left({X}_3^{Au}=1|{X}_1=A\right)=\sum \limits_{X_2=A,B}P\left({X}_3^{Au}=1,{X}_2|{X}_1=A\right)\kern0.5em \left(\because \left\{{X}_2=A\ \mathrm{or}\ B\right\}=\varOmega \right)\\ {}\kern0.75em =\sum \limits_{X_2=A,B}P\left({X}_3^{Au}=1|{X}_2,{X}_1=A\right)P\left({X}_2|{X}_1=A\right)\ \left(\because \mathrm{chain}\ \mathrm{rule}\right)\\ {}\kern0.75em =\sum \limits_{X_2=A,B}P\left({X}_3^{Au}=1|{X}_2\right)P\left({X}_2|{X}_1=A\right)\ \left(\because \mathrm{Markov}\ \mathrm{property}\right)\\ {}\kern0.75em =\underset{q}{\underbrace{P\left({X}_3^{Au}=1|{X}_2=B\right)}}\underset{p}{\underbrace{P\left({X}_2=B|{X}_1=A\right)}}\ \left(\because P\left({X}_2=A|{X}_1=A\right)=0\right)\\ {}\kern0.75em = qp\ \left(\because \mathrm{from}\ \mathrm{the}\ \mathrm{decision}\ \mathrm{tree}\right)\end{array}} \)
 
5
As for the Law of Iterated Expectation (LIE), see the appendix A of this chapter. The second equality of Eq. (13) uses the Law of Ierated Expectation stated in Sect. A.2 of Appendix A as Eq. (140).
 
6
For details about Markov chain taken up in this example, refer to Sects. 2–4 of this chapter.
 
7
Also, refer to the Law of Iterated Expectations in the expanded form with the 0–1 random variable stated in Sect. A.2 of Appendix A as Eq. (142), which implies the right-hand side of Eq. (14).
 
8
Formally, it is derived as follows. Let \( {X}_{\left[4,\infty \right)}^{Au} \) denote the set of random variables \( {X}_4^{Au},\dots \) from t = 4 to infinity.
\( {\displaystyle \begin{array}{l}E\left({RE}_{(b)}^A|{X}_3^{Au}=1,{X}_1=A\right)=\sum \limits_{X_{\left[4,\infty \right)}^{Au}}{RE}_{(b)}^A\left({X}_{\left[4,\infty \right)}^{Au}\right)P\left({X}_{\left[4,\infty \right)}^{Au}|{X}_3^{Au}=1,{X}_1=A\right)\\ {}=\sum \limits_{X_{\left[4,\infty \right)}^{Au}}{RE}_{(b)}^A\left({X}_{\left[4,\infty \right)}^{Au}\right)P\left({X}_{\left[4,\infty \right)}^{Au}|{X}_3^{Au}=1\right)\kern0.5em \left(\because \mathrm{Markov}\ \mathrm{property}\right)\\ {}=E\left({RE}_{(b)}^A|{X}_3^{Au}=1\right)\end{array}} \)
 
9
The precise definition requires measure theory. A rough definition of a real-valued random variable which is sufficient for the following discussion is as follows. A real-valued random variable X is a mapping from a sample space Ω to real space R. In the case when it takes discrete values, for any value r it takes, its inverse image X−1(r) is given some probability, i.e., belongs to the family F of the set of events over Ω. In the case when it takes continuous values, for any intervals (−∞, r], r ∈ R, its inverse imageX−1((−∞, r]) is given some probability. i.e., belongs to the family F of the set of events over Ω.
 
10
More precisely, the event {X = i} should be expressed as {ω ∈ Ω| X(ω) = i} or in another expression, {X−1(i) ∈ Ω| i ∈ N}, which is the inverse image of the mapping of X : Ω → {1, 2, …, 6}. It should be noted that the random variable X is defined as a mapping from the sample space Ω to real space R. In this die case, Ω is composed of {ω1, ω2, …, ω6} that correspond to the number of the die roll.
 
11
Here we employ the stronger sufficient condition. The restriction to nonnegative matrices and the assumption that all rows satisfy the condition (55) may be weaken.
 
12
The formal definition of group is the following. Consider the set G and a binary operation ∘ defined on G, i.e., ∘ : G × G → G. The set G is said group if it satisfies the following three conditions: (1) (associative) (a ∘ b) ∘ c = a ∘ (b ∘ c) for ∀ a, b, c ∈ G, (2) (the existence of a unity element) ∃e ∈ G a ∘ e = e ∘ a = a  for ∀ a ∈ G, (3) (the existence of an inverse element) For ∀ a ∈ G, ∃ b ∈ G a ∘ b = b ∘ a = e The element b is usually denoted as a−1.
 
13
See Saito [17, 18], and Saito et al. [21, 22].
 
14
Saito and Ishibashi [20] in this volume discusses the relationship between this theorem and the supply-driven Input-Output analysis. The discussion is quoted here. “The line of argument in the proof is parallel to that of Input-Output analysis with supply-driven I–O coefficients. It is interesting to note that while the stability of demand-driven I–O coefficients is guaranteed by the well-known non-substitution theorem, it has been argued that the supply-driven I–O coefficient approach does not have a sound theoretical foundation for their stability. In terms of Input-Output terminology, the observed Kaiyu effect can be interpreted as the intermediate input total for each sector. Thus the theorem might imply that even if supply-driven I–O coefficients were not stable or fluctuated, its forecast of the total output and intermediate input total may be robust or stable. These points should be explored further.”
 
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19.
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Saito, S., & Yamashiro, K. (eds). (2019). Advances in Kaiyu studies: From shop-around movements through behavioral marketing to town equity research. Springer Singapore.
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Metadata
Title
Basics of Kaiyu Markov Models: Reproducibility Theorems—A Validation of the Infinite Kaiyu Representation
Authors
Saburo Saito
Kenichi Ishibashi
Kosuke Yamashiro
Masakuni Iwami
Copyright Year
2023
Publisher
Springer Nature Singapore
Book
Recent Advances in Modeling and Forecasting Kaiyu

Print ISBN: 978-981-9912-40-7

Electronic ISBN: 978-981-9912-41-4

Copyright Year: 2023

DOI
https://doi.org/10.1007/978-981-99-1241-4_4