# Complexity and Application of Tobacco Manufacturer Pricing Game considering Market Segments

Authors: Shubing Guo; Junhai Ma; Xueli Zhan
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper considers price as a strategy using two methods of market segmentation through game theory based on the collected data of several typical cities in Shandong province. Three oligarchs are discussed to analyze the complexity of the market in tobacco system. The effect of different factors on the equilibrium price of Nash equilibrium is discussed. What is more, the variation of parameters drives the system to change to uncertain status, even to bifurcation or chaos. We also discuss the model under normal circumstances, when the three parties jointly take chaos control measures, and its impact on the system. Finally, we clarify chaos control of economic significance. The results indicate that only reasonable pricing strategy is a critical point to achieve maximized profit and maintain orderly operation at the same time, which is a constructive suggestion for tobacco companies’ operational strategy.


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## Body

## 1. Introduction

Based on the AHP method and using 20 factors, the results of Zhang et al. [1] showed that 29.82% of the total area was highly suitable for tobacco production and 17.74% was unsuitable. To solve the problem where the regular IAHP model cannot describe the probability with an interval number, Xiao et al. [2] carried out the probabilistic distribution of the weights calculation in IAHP. By the study of fuzzy judgment matrices, Zhou and Wei [3] discussed a new method for deriving posterior weight with an application into a real example. In order to determine attribute weights in multiple-attribute decision-making, Cui et al. [4] analyzed the limitations of the present approaches to determining weights of attributes. Based on grey incidence, Xu and Lv [5] proposed a method for group decision-making. According to fuzzy comprehensive judgment applied to many provinces in practice, Zhong et al. [6] used grey correlation analysis for the comprehensive evaluation. Al-Bayati and Hussein [7] applied the evaluation process to obtain the valuable feedback as its major conclusions for further study. With imprecise data, Kao [8] used DMUs under the framework of DEA. In order to develop the status of tobacco industry, Li [9] discussed a few measures for developing modern tobacco agriculture. From a new perspective, Li et al. [10] gave two methods and gave an analysis to solve the problems that the structure of BP neural network will become complex when samples’ data is not preprocessed. By using the grey correlation analysis of the BP neural network, Tang [11] gave a method for improving the network’s forecasting accuracy.With the rise of chaos theory, many scholars are interested in building model with the use of complex system method. Ma and Wang [12] established a closed-loop supply chain and found that the system will become more complicated when the retailers exacerbate competition. Considering that players have different adjustment mechanisms and expectations, Ma and Guo [13] investigated a dynamic duopoly Cournot game model and studied the influence of adjustment factors on the system. Based on a new supply chain model with supplier and two retailers, J. Ma and X. Ma [14] found that the bullwhip effect is impacted by different factors.Market segmentation is the procedure which divides into different consumer groups according to certain rule; under this rule the standard variable segments are chosen. After selecting the standard variable, consumers are assigned to different groups with certain method. Thus market segmentation approach is the main research method to solve these two issues, which is named segmentation criteria to determine the choice of distribution methods.For the breakdown of standard variables, it may be single or plural. When there are more variables, the number of market segments will increase exponentially. For example, if there are three values for each variable and the total number of variable is three, then there will be33=27 separations of the market. And when the number of variables is four, there will be 34=81 separations of the market. Therefore, the number of variables determines the choice of market segmentation. Considering the demand of the actual needs and the degree of difficulty of the two studies, we have to make suitable decision of variable’s quantity.When it comes to market segmentation, there are many ways, such as conjoint analysis, mixed regression model, factor analysis, fuzzy cluster analysis, AHP, and decision tree algorithm.Tobacco companies in the competitive market adjust their product price to maximize operation profits. Economic significance of chaos control is to minimize the impact of its price and the price of its own competitors in sales. Thus, the tobacco industry needs to improve sales channels, increasing sales service quality and establishing a good brand image. And then gradually the tobacco industry wins the market through quality and service and also wins more consumers.Consider the situation where three tobacco companies sell their three tobacco products. These three tobacco companies can be seen as a three-oligarch tobacco market. The three tobacco companies’ Nash equilibrium price and the change rate of the price have the influence on the equilibrium state.In practice, more often with real-time changes in the market, the three parties adjust their prices according to the size of the margin of prices. So we need to model the Nash equilibrium of the three parties’ game. Find the Nash equilibrium price for tobacco companies and analyze the complexity of price just like impact on the price of the game and the impact on profits of the game and time series game prices.Finally, through the research, the reasonable pricing strategy is obtained, which is to maintain the image of the enterprise and the profit balance.


## 2. Market Segmentation Based on Conjoint Analysis

To study simplicity, select a standard variable: product benefits. Market segmentation methods include conjoint and mixed regression model.Conjoint analysis is a statistical method, which is important for product attribution in the minds of consumers as well as the level of product attributes. For tobacco products, they have different properties, such as aroma, taste, smoke, aftertaste intensity, tar, tobacco, and cigarette paper. Different products have different combinations of attribution and attributing levels; for example, the proportion of white sand tar in Hongtashan, Zhonghua, and other brands of cigarettes is higher than the property tar in Zhongnanhai, Pride, and other low-tar brands of cigarettes. Hongtashan product attribute levels and combinations can be expressed as shown in Table1.Table 1
Hongtashan product attributes.
Hongtashan
Attribute
Description
Raw material
Yuxi
Additive
Appropriate
Taste
Supple
Aroma
Plump
Gas
Soft
Aftertaste
CleanDifferent cigarette brands have different combinations of attributes and attribute levels. For different consumers, their preferences for product attributes are different too. The same sense of property has different satisfaction level and the core of this difference is the consumer demand preference differences. Therefore, when segmenting the product, we must focus on the importance of product attributes and attribute levels to improve people’s unit utility.In general, when using the joint analysis of the product segment, it is divided into two steps. First, the experience of the importance of product attributes and attribute levels brings the utility of the preliminary estimation and then using cluster analysis methods depending on the importance of product attributes and attribute levels brings the utility, which will be assigned to different consumers market. It is easy to think that the first step in the estimation process and the ability to recognize the limitations of the amount of data may result in unreliable model parameters and the second step is based on the analyzing result of the first step. So the first step in the estimation process will affect the final result. In this paper, a mixed regression model is proposed to improve the estimating process of the first step.To study the market segmentation mixed regression model, it was firstly introduced by DeSarbo joint analysis and others. In the conjoint analysis, the respondents need to rate the product of a group. DeSarbo scoring data assume that each respondent residual obeys multivariate normal distribution and the variance-covariance matrix estimation. In the evaluation of the relevance of different product respondents, the efficiency of the model decreases with the increase of the estimated parameters, if you assume that the residuals are scoring one dollar data independently distributed, although there will be some decrease in the estimated accuracy. However, you can make the model much more efficient, so the change is worth trying. At this time, although the independent distribution of the residual will sacrifice the accuracy, the model’s efficiency will be improved, which is worth trying.Complete contour method is a more traditional conjoint analysis, in addition to a hybrid approach, self-described method, and that based on the selected method. Compared with the other three methods, complete contour method is simple with normal accuracy, which is widely used. Complete contour method comprises the following steps:(a)
Determine the product attributes and each attribute specific level.(b)
Prepare an investigation to determine the product mix.(c)
Make a preference based on the likelihood of respondents to rate the degree of purchase for each product combination.(d)
Use the least-squares method to estimate the effectiveness of the model coefficients.The basic model is as follows:(1)UG=∑i=1n∑j=1IiαijDij.This is a utility model expressing the sum of the utility of representing a portfolioG. Product attributes are denoted by i, i=1,2,…,n.    The horizontal property is denoted by j, as li is the number of horizontal values of the property i. αij represents utility value attribute, which means the utility of product attributes i get from level j. Dij is a dummy variable; when product attribute i has selected level j, its value is 1; otherwise, it is 0.I
i indicates the importance of each attribute i. The value for this property at all levels brings a little utility, namely, the difference between the maximum and minimum values, expressed as(2)Ii=max⁡αij-min⁡αij.I
i represents utility full-pitch; attribute weight Wi can represent the whole of its utility from the proportion of the sum of all the properties of the effectiveness of the whole distance; namely,(3)Wi=Ii∑i=1nli.We need to have horizontal joint analysis ofn attributes and li data for each property value. Readily available, the level values to be studied are ∑i=1nli. For li level values of each property, we need to get a reference, so for the property there is a need to estimate li-1 coefficients. Therefore, there are C=∑i=1nli-n estimated model coefficients. Assume that each respondent evaluates the needs of M combinations. Data collected from each respondent are M. For each respondent, the linear regression equation can be expressed as(4)SpM=β0M+β1MX1pM+β2MX2pM+β3MX3pM+⋯+βCMXCpM+δpM.ModelSpM represents respondent p scoring the M case of a portfolio. X1pM to XCpM represents a combination of different levels. Corresponding β1M to βCM are utility coefficients under that level. β0M represents intercept model, as δpM are the residuals.In its estimating coefficients, we use random coefficient model, which overcomes unstable shortcomings in the estimation of OLS regression model caused by fewer coefficients.In order to express a clear and simple scoring model, it can also be represented as a matrix form:Sp=θpXp+δp.From the previous formula,θp=β0Mβ1Mβ2MβCM; Xp=β0pMβ1pMβ2pMβCpM-1.The residual vector isδp; covariance matrix is σp2.Therefore, we can useθp, Xp, and σp2 to represent the density function of p respondent’s scoring, referred to as f(Sp∣θk,Xp,σp2).As mentioned above, for simplicity, in the estimation of the model, we assume thatδp followed single normal distribution rather than a multivariate normal distribution. The reasons are already mentioned above. Under this assumption, the density function is expressed as the product of p normal distribution possibilities. Assuming that the whole market is subdivided into K markets, the proportion of each market is wk, where ∑k=1Kwk=1. For one market k, we give the model coefficient θk and its variances σk2; the respondents density function can be expressed as (5)gSp∣Xp,φ=∑k=1KπkfSp∣θk,Xp,σk2.From (5), φ=(w1⋯wk,θ1⋯θk,σ12⋯σk2). In this case, coefficients θp and θp2 at the individual level of the density function model have been replaced by model coefficients θk and σk2 at the segment level, so that the replacement model is a mixed regression model. With the estimated coefficient φ, the probability p individual that belongs to the q market can be calculated based on Bayes’ formula, which is as follows:(6)Pq∣Xp,Sp,φ=πqfSp∣θq,Xp,σq2∑k=1KπkfSp∣θk,Xp,σk2.Using this method, you can assign each respondent to a segment of the market. The principle is to assign the highest probability of the respondents to its home market. Here we will only divide the respondents to a market rather than assigning them to multiple markets.Each respondent hasM data points; thus the total amount of data is P×M. Based on statistical results, in the case of the model parameter estimation, we use the likelihood function:(7)L=∏p=1Pln⁡gSp∣Xp,φ=∏p=1P∑k=1KπkfSp∣θk,Xp,σk2.Multiplicative model is a simple form by taking the logarithm on both sides of the equation.(8)ln⁡L=∑p=1Pln⁡gSp∣Xp,φ=∑p=1Pln⁡∑k=1KπkfSp∣θk,Xp,σk2.The value for market segments is always determined by the Bayesian Information Criterion and the Akaike information criterion. The calculation method for Bayesian Information Criterion is(9)C1=-2ln⁡L+ln⁡P×R.Akaike information criterion is calculated as follows:(10)C2=-2ln⁡L+2R,where R represents the number of estimated model coefficients needed, when C1 and C2 achieve the minimum; corresponding mode value is the number of market segments.


## 3. Consumption Region Division with Factor Analysis

Table2 shows people’s income information and nonfood expenditure statistics in different regions of Shandong province.Table 2
Consumption data of 26 regions in Shandong province.
Clothing
Household equipment and services
Healthcare
Transportation and communication
Education, culture, and entertainment
Living
Miscellaneous goods and services
Jinan
924.81
536.37
793.09
1448.33
1181.16
985.72
310.2
Qingdao
1041.7
656.94
804.75
988.28
1435.28
951.56
286.07
Zibo
900.05
536.36
783.44
1048.06
1108.46
1013.74
261.81
Zaozhuang
863.73
440.16
323.91
590.55
945.03
458.01
163.9
Dongying
1509.12
625.28
770.32
1095.2
1842.6
751.32
410.97
Yantai
1458.49
272.42
933.11
882.52
999.45
790.02
338
Weifang
1016.72
538.1
590.06
917.65
947.75
972.66
212.75
Jining
820.83
826.42
417.45
619.61
791.7
616.06
162.04
Tyan
798.03
443.74
538.66
623.12
1348.09
727.06
248.7
Weihai
1369.93
609.62
682.48
1080.86
1261.65
1018.55
542.62
Rizhao
982.9
571.58
454.98
1195.46
825.89
579.91
277.91
Laiwu
863.38
713.7
467.06
978.89
1009.62
714.66
183.19
Linyi
980.49
511.48
378.77
1176.03
1070.76
854.93
240.47
Dezhou
889.33
380.85
436.79
887.1
1160.2
594.6
224.04
Binzhou
778.1
430.06
392.1
946.61
1272.55
672.48
191.03
Heze
804.84
336.32
367.33
516.18
633.27
436.34
183.51
Wendeng
742.33
314.85
377.92
952.96
446.54
769.48
200.35
Zhucheng
819.68
333.71
302.22
945.79
785.61
479.52
134.95
Qingzhou
776.72
333.94
627.8
865.29
806.67
786.54
237.13
Weishan
929.52
694.9
351.51
598.42
1274.28
410.42
299.99
Linqing
539.62
337.68
372.83
433.12
479.88
639.22
149.62
Feixian
629.31
402.05
199.15
450.1
757.74
371.59
61.07
Lijin
795.41
327.8
611.97
760.38
1439.42
640.55
224.19
Wucheng
615.4
426.88
385.36
382.57
747.51
524.08
94.81
Donga
516.72
255.33
594.04
358.94
539.33
468.61
125.64
Juye
603.64
405.99
349.08
625.14
622.33
393.37
196.59In order to study the consumption level of different regions in Shandong Province, we analyze the consumption data of 26 cities or regions in Shandong Province in 2005.Part of the original data is shown in Table2.Firstly make principal component analysis of the data. Original indices and their corresponding attributes are as follows:X1: clothingX2: household equipment and servicesX3: healthcareX4: transport and communicationsX5: education and cultural and recreational servicesX6: livingX7: miscellaneous goods and servicesUsing SPSS data analysis software for analysis, prior to principal component analysis, due to the different units of data, direct dimension transformation may have a negative impact on the analysis results. And data obtained is shown in Table3.Table 3
Total variance explained.
Component
Initial eigenvalues
Extraction sums of squared loadings
Rotation sums of squared loadings
Total
% of variance
Cumulative%
Total
% of variance
Cumulative%
Total
% of variance
Cumulative%
1
4.206
60.085
60.085
4.206
60.085
60.085
2.518
35.971
35.971
2
.984
14.055
74.140
.984
14.055
74.140
2.200
31.429
67.400
3
.681
9.724
83.864
.681
9.724
83.864
1.152
16.463
83.864
4
.446
6.376
90.240
5
.375
5.359
95.599
6
.176
2.517
98.115
7
.132
1.885
100.000
Extraction method: principal component analysis.Not difficult to see from the table, the cumulative contribution rate of the first component is 60.085%, while the contribution of the first three components was 83.864%. It can be seen that the first three factors explain more than 70% of the variance. The three principal components extracted are set toY1, Y2, and Y3.Easy to see for the first principal component,Y1, X1, X5, and X7 have more contribution; for the second principal component, Y2, X4, and X6 have more contribution; for the third main component, X2 has the highest contribution rate.According to the ingredient matrix after rotation, we can conclude that the expression of the three main components is as follows:(11)Y1=0.827X1+0.208X2+0.614X3+0.274X4+0.810X5+0.273X6+0.780X7Y2=0.361X1+0.137X2+0.622X3+0.796X4+0.131X5+0.903X6+0.446X7Y3=0.156X1+0.929X2-0.218X3+0.282X4+0.332X5+0.089X6+0.140X7.The main ingredient formula has been set with the variables for each region obtaining the value of the main component into each region,Y1, Y2, and Y3. The value of the main component into each region is shown in Table 4.Table 4
The value of the main component into each region.
Y
1
Y
2
Y
3
Jinan
3227.98
3236.698
1401.396
Qingdao
3408.522
2928.147
1477.314
Zibo
3002.92
2897.338
1258.334
Zaozhuang
2184.902
1654.14
1017.032
Dongying
4169.335
3084.489
1693.371
Yantai
3366.445
2841.781
975.5045
Weifang
2765.644
2635.576
1219.652
Jining
2212.662
1894.687
1319.879
Tyan
2738.167
2123.986
1142.111
Weihai
3698.176
3189.965
1521.553
Sunshine
2582.717
2423.515
1286.979
Laiwu
2573.238
2338.466
1396.388
Linyi
2760.328
2615.265
1342.437
Texas
2602.786
2139.871
1116.959
Binzhou
2596.418
2196.353
1211.457
Heze
1877.739
1534.8
778.2524
Wendeng
1900.588
2147.431
839.4313
Zhucheng
2064.511
1878.562
961.1083
Qingzhou
2387.455
2327.088
909.5647
Weishan
2671.25
1797.075
1384.279
Linqing
1544.009
1524.539
675.9059
Feixian
1612.519
1226.46
848.3813
Lijin
2825.749
2184.929
1075.911
Wucheng
1761.671
1438.315
824.54
Donga
1606.313
1426.568
527.8837
Juye
1734.097
1512.691
840.6687We use the variance contribution rate as weight, and the score model can be expressed as(12)S=0.60085Y1+0.14055Y2+0.09724Y3.Data analysis software SPSS statistical functions and so forth can be used to calculate the level of consumption in different regions and contribute to market segmentation.The sales strategy of the product is closely related to the local consumption level. It is necessary to have a proper change strategy in areas where consumption levels are low. What is more, companies need to seriously consider issues. Not difficult to see from the above analysis, different levels of consumption require different sales focus. For example, the consumption level is arranged in the top three in Dongying, Weihai, and Qingdao. For Dongying City, consumption levels are somewhat used to dress the cultural and recreational projects; for Qingdao, the consumer inclines more to education. Tobacco products should belong to miscellaneous goods and services, so for tobacco companies, when considering the level of consumption of the country and developing marketing strategies, they should not only consider the total level of consumption of the country but also take into consideration the level of consumption for miscellaneous goods.Product marketing not only needs to consider the local spending power, but also needs to consider the following factors.(a) Socioeconomic Factors. As a result of the rapid economic development, people's living standards continue to improve. The consumption level in the region with low consumption levels has high room for improvement. It is necessary to take into account the future business opportunities in the development of sales strategy. Companies should stand high and see far ahead with sufficient strategic vision for the job.(b) Product Quality. Should we provide low quality goods in areas with low consumption levels? The answer of course is no. Low-quality products can indeed reduce costs and drive down selling prices. In fact, consumers find that getting poor quality products can cause irreversible damage to the image of the business.(c) Consumption Habits. The level of consumption is based on the characteristics of the study area to examine consumer groups and product positioning, but the issue of personal spending habits cannot be ignored. It is difficult for us to meet all the individual needs of consumers, but we cannot view consumers as a static group with specific consumer preferences. With the improvement of living standards, consumers must have higher requirements for the products; for example, the health risks of tobacco products may become increasingly valued. Tobacco companies must respect the habits of consumers in marketing.The level of consumption in the country to some extent can help make the right business regional marketing strategy with not too hasty making the business decision, which should refine consumer groups, pay attention to product quality, and have a broader strategic vision.


## 4. Tobacco Manufacturer Pricing Considering Market Segments

Three tobacco companies sell three tobacco products. These three tobacco companies can be seen as representing tobacco market with three oligarchs. The competition of three tobacco companies makes the market price eventually enter the Nash equilibrium.


### 4.1. Tobacco Nash Equilibrium Price

Composition market demand function is(13)q1=a1-b1p1+c1p2+d1p3,q2=a2-b2p2+c2p3+d2p1,q3=a3-b3p3+c3p1+d3p2.All model coefficients are greater than zero. In addition to the fact that the demand for a fixed price is not affected, along with tobacco company’s own price increasing, demand decreases with the increase in the price of other brands; otherwise the demand increases. In addition, the impact of own price changes on demand is greater than the impact of competing prices changes. That is,b1>c1 and b1>d1. Assume that the sum of all costs for per tobacco product is w; then three tobacco companies’ profit function can be expressed as(14)π1=a1-b1p1+c1p2+d1p3p1-w,π2=a2-b2p2+c2p3+d2p1p2-w,π3=a3-b3p3+c3p1+d3p2p3-w.According to Nash equilibrium, after the three parties in the game, the price reached a steady state, so that the three parties are reluctant to change the price again. So the state corresponding to the price for the marginal profit is zero state. This state may be a function of the price of the derivative profits obtained. The equations to calculate the equilibrium price are therefore(15)a1-2b1p1+c1p2+d1p3+b1w=0,a2+d2p1-2b2p2+c2p3+b2w=0,a3+c3p1+d3p2-2b3p3+b3w=0.Solutions of equations can obtain tripartite equilibrium price.In addition, in order to make the best price existing, the need to make profit function of the second derivative is less than zero; namely,(16)∂2π1∂p12=-2b1<0,∂2π2∂p22=-2b2<0,∂2π3∂p32=-2b3<0.For example, we give a set of values for the model simulation. According to the analysis above, the definition, and scope of parameters, makea1=200, a2=280, a3=200, b1=70, b2=90, b3=60, c1=20, c2=10, c3=40, d1=20, d2=10, d3=40, and w=30. Numerical simulation is applied to three different tobacco companies. It can be seen that the system has the following characteristics: sales are mainly affected by their own prices and are less affected by the price of competitors. We have studied the specific trends in the following text.With the use of simulation software for data analysis, we can draw three Nash equilibrium prices:p1=23.67, p2=19.6, and p3=31.09; their profits are π1=2804.82, π2=9738.56, and π3=71.286. So we give Table 5 to compare the three tobacco companies.Table 5
The comparison of the three tobacco companies.
Number
Brand image
Affected by own price
Affected by substitute
Equilibrium price
Equilibrium profit
1
Middle
Middle
Middle
Middle
Middle
2
Good
High
Low
Low
High
3
Middle
Low
High
High
LowIt can be seen that the parameters of their own caused a greater impact and the parameters of the competitors caused a smaller impact. Although the price is low, profits are highest, because the price increase brought about by the increase in sales will increase the balance of profits.


### 4.2. Complexity of Price Game of Three Oligarchic Tobacco Companies

In practice, more often with real-time changes in the market, the three parties adjust their prices according to the size of the margins of prices. When the profit margin is greater than zero, the price increase is conducive to gaining more income, and when the profit margin is less than zero, we need to lower prices in order to obtain greater profits. Therefore, the price adjustment model can be written as(17)pit+1=pit+gi×pit×∂πit∂pit,wheregi represents the speed of price adjustment of player i (i=1,2,3); then the tripartite price adjustment model is(18)p1t+1=p1t+g1p1ta1-2b1p1+c1p2+d1p3+b1w,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w.We can first obtain the above model Jacobin matrix in order to get a stable domain model:(19)J=j1,c1p1g1,d1g1p1d2p2g2,j5,c2g2p2c3g3p3,d3g3p3,j9.From (19),(20)j1=1-2b1g1p1+g1a1-b1p1+c1p2+d1p3-b1p1-w,j5=1-2b2g2p2+g2a2-d2p1-b2p2+c2p3-b2p2-w,j9=1-2b3g3p3+g3a3-c3p1+d3p2-b3p3-b3p3-w.The characteristic equation of (19) is (21)λE-J=A0+A1λ+A2λ2+A3λ3=Fλ=0,where(22)A0=j1j6j8+j2j4j9+j3j5j7-j1j5j9-j3j4j8-j2j6j7,A1=j1j5+j1j9+j5j9-j3j7-j2j4-j6j8,A2=-j1-j5-j9,A3=1.The Routh-Hurwitz stability criterion proposes that a system whose characteristic roots are both in the unit circle is asymptotically stable, and this equals to satisfying the following conditions:(23)F1=A0+A1+A2+A3>0,F-1=A0-A1+A2-A3<0,A02-A3<0,A3-A022-A1-A2A02>0.Large-scale price adjustment will cause great harm to the system, so that the price balance is no longer stable, endangering the stability of the entire system.In the three vendors’ characteristics prescribed value, make(24)a1=3.6,a2=4.2,a3=3.6,b1=3.2,b2=4.2,b3=2.6,c1=0.4,c2=0.2,c3=0.6,d1=0.5,d2=0.3,d3=0.7,w=0.15.Examine the impact of price adjustment factor on the system.(i) Impact on the Price of the Game. We fixed the adjustment speed of tobacco companies 2 and 3 as 0.1 and studied the impact of the change of the adjustment speed of company 1 on the system.The size of the index indicates the average convergence or divergence of the system phase space orbit. The index is calculated as(25)λ=limn→∞1n∑n=0n-1ln⁡dfxn,udx.The maximum exponent means the mean of the difference after multiple iterations when difference between the initial values is very small. Iterative equation is as follows:(26)xn-yn≈∏n=0n-1dfxn,udxxnx0-y0.Knowing the index and the number of iterationsn, we can know the difference between the original minor differences through n iterations. Index may be positive or negative, when the index is positive, it indicates that the difference is a proliferation of state in this direction; when the index is negative, it indicates that the difference within the shrinkage is in that direction. If the system phase space trajectory is away from the equilibrium point, the system is out of stability. Whenever there is a positive index, the system will present chaotic state. Thus, to determine whether there is a system chaotic state, you need to check whether there is a positive index. According to the model and numerical simulation of this article, we can obtain the index change map paper systematically. See Figure 1.Figure 1
The index change map.Not difficult to see from the figure, wheng1<0.4, the maximum is always less than zero, so the system is stable. When g1=0.41, the maximum for the first time is zero, because there are no indexes greater than zero, so that the maximum of system does not enter a chaotic state but enters the fold bifurcation state. When g1>0.53, the maximum number begins to appear positive; then the system enters a chaotic state.With the increase of retail price adjustment factor 1, the parties of the dynamic evolution of the equilibrium price are shown in Figure2.Figure 2
The impact of price adjustment speed of evolution of retailer’s equilibrium price.Under a normal price adjustment system, game player keeps stable in the game after a certain period into a steady state. In this condition, the profit side of the game has reached the maximum, which cannot change unilaterally by the price to obtain greater profits. So that is a Nash equilibrium. But if either one of price adjustment coefficients is too large, it will break the steady state, so that the chaotic system enters a state of disorder. As is shown in Figure2, when the price adjustment factor is greater than 0.41 g1, the system began to enter doubling bifurcation state; when g1 is greater than 0.53, the system enters a chaotic state. In this state, the market competition is chaotic disorder and vicious competition.(ii) The Impact on Profits of the Game. For the company, the price changes directly affect the acquisition of profits. Therefore, the price of the chaotic state of the game will inevitably lead to chaos game profits. We fixed the adjustment speed of tobacco companies 2 and 3 as 0.1 and analyzed the changes in adjustment speed of tobacco company 1. Please see Figure 3.Figure 3
The impact of price adjustment speed of evolution of a retailer’s balance profit.Through two bifurcation diagrams seen in Figure3, regardless of the nature of the tobacco industry, the impact of the price adjustment factor on the equilibrium price is similar to the acceleration of price adjustment’s speed. Tobacco company’s prices gradually appear to be chaotic. The rapid rate of price adjustments by either party will lead to significant tripartite profits to chaos. This stability of the market is very negative.(iii) Time Series Game Prices. Game price time series reflects the increase in the game with the times and the changes in the equilibrium price. At steady state, the price of the game is always in a stable state, as shown in Figure 4.Figure 4
The price varies with the steady-state period of the game.When the system is in a chaotic state, the price increases with the emergence of the game cycle volatility, as shown in Figure5.Figure 5
The price varies with the chaotic state of the game cycle.Chaos is a so violent price fluctuations phenomenon, which has certain impact on the stability of the market.When one of tobacco’s price adjustments is too fast, the equilibrium prices go into chaos and the nonlinear feedback control is introduced as follows:(27)uk,pn=k∏i=1mpn-pi.After introducing the chaotic control coefficient, the power system of the tobacco company game is shown as follows:(28)p1t+1=p1t+g1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w.According to (28), we can get(29)p1t+1=p1t+g11+k1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w+k2p2t-p2t+1,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w+k3p3t-p3t+1.The impact of coefficient changes on the stability of the system under such conditions is shown in Figure6.Figure 6
Chaos unilateral control factor on system stability.It can be seen that the system is stable when the value is close to 0.47, which means that the tobacco companies can make more rapid price adjustments.If all the tobacco companies on the market work together to stabilize the market, chaos control factor of the system becomes(30)p1t+1=p1t+g1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w+k2p2t-p2t+1,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w+k3p3t-p3t+1.According to (30), we can get(31)p1t+1=p1t+g11+k1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g21+k2p2ta2+d2p1-2b2p2+c2p3+b2w+k2p2t-p2t+1,p3t+1=p3t+g31+k3p3ta3+c3p1+d3p2-2b3p3+b3w+k3p3t-p3t+1.In order to facilitate analysis and comparison, chaos control factor drives to all tobacco companies. Observing chaotic control factor on system stability at this time, see Figure 7.Figure 7
Chaos tripartite control factor on system stability.Under normal circumstances, when the three parties jointly take chaos control measures, the system will be easier to control to enter a stable state. In the diagram, the key point is to make the system’s stablek value smaller. In this case, it seems that the three parties take effect to unilateral chaos control when there is not much difference among them. This is mainly due to the price adjustment factor 1 retailers sensitively affected by too high corresponding unilateral system. And the chaotic state of preselected g1 coefficient is relatively large.Chaos control of economic significance actually reduced by addingk value for each of original price determination coefficients b, c, and so on. So its economic significance is actually to minimize the impact on the sales price.Tobacco companies in the competitive market adjust product price to maximize profits. But the price adjustment speed should not be too fast; otherwise, it will cause the chaotic phenomena in tobacco companies’ equilibrium price, which further causes the chaos phenomenon in profits and huge price fluctuations. Therefore, it is necessary to introduce a method of chaos control. Economic significance of chaos control is to minimize the impact of its price and that of competitors in sales. Thus, the tobacco industry needs to improve sales channels and service quality and establish a good brand image. That helps to gradually win the market through quality and service and finally win more consumers.


## 4.1. Tobacco Nash Equilibrium Price

Composition market demand function is(13)q1=a1-b1p1+c1p2+d1p3,q2=a2-b2p2+c2p3+d2p1,q3=a3-b3p3+c3p1+d3p2.All model coefficients are greater than zero. In addition to the fact that the demand for a fixed price is not affected, along with tobacco company’s own price increasing, demand decreases with the increase in the price of other brands; otherwise the demand increases. In addition, the impact of own price changes on demand is greater than the impact of competing prices changes. That is,b1>c1 and b1>d1. Assume that the sum of all costs for per tobacco product is w; then three tobacco companies’ profit function can be expressed as(14)π1=a1-b1p1+c1p2+d1p3p1-w,π2=a2-b2p2+c2p3+d2p1p2-w,π3=a3-b3p3+c3p1+d3p2p3-w.According to Nash equilibrium, after the three parties in the game, the price reached a steady state, so that the three parties are reluctant to change the price again. So the state corresponding to the price for the marginal profit is zero state. This state may be a function of the price of the derivative profits obtained. The equations to calculate the equilibrium price are therefore(15)a1-2b1p1+c1p2+d1p3+b1w=0,a2+d2p1-2b2p2+c2p3+b2w=0,a3+c3p1+d3p2-2b3p3+b3w=0.Solutions of equations can obtain tripartite equilibrium price.In addition, in order to make the best price existing, the need to make profit function of the second derivative is less than zero; namely,(16)∂2π1∂p12=-2b1<0,∂2π2∂p22=-2b2<0,∂2π3∂p32=-2b3<0.For example, we give a set of values for the model simulation. According to the analysis above, the definition, and scope of parameters, makea1=200, a2=280, a3=200, b1=70, b2=90, b3=60, c1=20, c2=10, c3=40, d1=20, d2=10, d3=40, and w=30. Numerical simulation is applied to three different tobacco companies. It can be seen that the system has the following characteristics: sales are mainly affected by their own prices and are less affected by the price of competitors. We have studied the specific trends in the following text.With the use of simulation software for data analysis, we can draw three Nash equilibrium prices:p1=23.67, p2=19.6, and p3=31.09; their profits are π1=2804.82, π2=9738.56, and π3=71.286. So we give Table 5 to compare the three tobacco companies.Table 5
The comparison of the three tobacco companies.
Number
Brand image
Affected by own price
Affected by substitute
Equilibrium price
Equilibrium profit
1
Middle
Middle
Middle
Middle
Middle
2
Good
High
Low
Low
High
3
Middle
Low
High
High
LowIt can be seen that the parameters of their own caused a greater impact and the parameters of the competitors caused a smaller impact. Although the price is low, profits are highest, because the price increase brought about by the increase in sales will increase the balance of profits.


## 4.2. Complexity of Price Game of Three Oligarchic Tobacco Companies

In practice, more often with real-time changes in the market, the three parties adjust their prices according to the size of the margins of prices. When the profit margin is greater than zero, the price increase is conducive to gaining more income, and when the profit margin is less than zero, we need to lower prices in order to obtain greater profits. Therefore, the price adjustment model can be written as(17)pit+1=pit+gi×pit×∂πit∂pit,wheregi represents the speed of price adjustment of player i (i=1,2,3); then the tripartite price adjustment model is(18)p1t+1=p1t+g1p1ta1-2b1p1+c1p2+d1p3+b1w,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w.We can first obtain the above model Jacobin matrix in order to get a stable domain model:(19)J=j1,c1p1g1,d1g1p1d2p2g2,j5,c2g2p2c3g3p3,d3g3p3,j9.From (19),(20)j1=1-2b1g1p1+g1a1-b1p1+c1p2+d1p3-b1p1-w,j5=1-2b2g2p2+g2a2-d2p1-b2p2+c2p3-b2p2-w,j9=1-2b3g3p3+g3a3-c3p1+d3p2-b3p3-b3p3-w.The characteristic equation of (19) is (21)λE-J=A0+A1λ+A2λ2+A3λ3=Fλ=0,where(22)A0=j1j6j8+j2j4j9+j3j5j7-j1j5j9-j3j4j8-j2j6j7,A1=j1j5+j1j9+j5j9-j3j7-j2j4-j6j8,A2=-j1-j5-j9,A3=1.The Routh-Hurwitz stability criterion proposes that a system whose characteristic roots are both in the unit circle is asymptotically stable, and this equals to satisfying the following conditions:(23)F1=A0+A1+A2+A3>0,F-1=A0-A1+A2-A3<0,A02-A3<0,A3-A022-A1-A2A02>0.Large-scale price adjustment will cause great harm to the system, so that the price balance is no longer stable, endangering the stability of the entire system.In the three vendors’ characteristics prescribed value, make(24)a1=3.6,a2=4.2,a3=3.6,b1=3.2,b2=4.2,b3=2.6,c1=0.4,c2=0.2,c3=0.6,d1=0.5,d2=0.3,d3=0.7,w=0.15.Examine the impact of price adjustment factor on the system.(i) Impact on the Price of the Game. We fixed the adjustment speed of tobacco companies 2 and 3 as 0.1 and studied the impact of the change of the adjustment speed of company 1 on the system.The size of the index indicates the average convergence or divergence of the system phase space orbit. The index is calculated as(25)λ=limn→∞1n∑n=0n-1ln⁡dfxn,udx.The maximum exponent means the mean of the difference after multiple iterations when difference between the initial values is very small. Iterative equation is as follows:(26)xn-yn≈∏n=0n-1dfxn,udxxnx0-y0.Knowing the index and the number of iterationsn, we can know the difference between the original minor differences through n iterations. Index may be positive or negative, when the index is positive, it indicates that the difference is a proliferation of state in this direction; when the index is negative, it indicates that the difference within the shrinkage is in that direction. If the system phase space trajectory is away from the equilibrium point, the system is out of stability. Whenever there is a positive index, the system will present chaotic state. Thus, to determine whether there is a system chaotic state, you need to check whether there is a positive index. According to the model and numerical simulation of this article, we can obtain the index change map paper systematically. See Figure 1.Figure 1
The index change map.Not difficult to see from the figure, wheng1<0.4, the maximum is always less than zero, so the system is stable. When g1=0.41, the maximum for the first time is zero, because there are no indexes greater than zero, so that the maximum of system does not enter a chaotic state but enters the fold bifurcation state. When g1>0.53, the maximum number begins to appear positive; then the system enters a chaotic state.With the increase of retail price adjustment factor 1, the parties of the dynamic evolution of the equilibrium price are shown in Figure2.Figure 2
The impact of price adjustment speed of evolution of retailer’s equilibrium price.Under a normal price adjustment system, game player keeps stable in the game after a certain period into a steady state. In this condition, the profit side of the game has reached the maximum, which cannot change unilaterally by the price to obtain greater profits. So that is a Nash equilibrium. But if either one of price adjustment coefficients is too large, it will break the steady state, so that the chaotic system enters a state of disorder. As is shown in Figure2, when the price adjustment factor is greater than 0.41 g1, the system began to enter doubling bifurcation state; when g1 is greater than 0.53, the system enters a chaotic state. In this state, the market competition is chaotic disorder and vicious competition.(ii) The Impact on Profits of the Game. For the company, the price changes directly affect the acquisition of profits. Therefore, the price of the chaotic state of the game will inevitably lead to chaos game profits. We fixed the adjustment speed of tobacco companies 2 and 3 as 0.1 and analyzed the changes in adjustment speed of tobacco company 1. Please see Figure 3.Figure 3
The impact of price adjustment speed of evolution of a retailer’s balance profit.Through two bifurcation diagrams seen in Figure3, regardless of the nature of the tobacco industry, the impact of the price adjustment factor on the equilibrium price is similar to the acceleration of price adjustment’s speed. Tobacco company’s prices gradually appear to be chaotic. The rapid rate of price adjustments by either party will lead to significant tripartite profits to chaos. This stability of the market is very negative.(iii) Time Series Game Prices. Game price time series reflects the increase in the game with the times and the changes in the equilibrium price. At steady state, the price of the game is always in a stable state, as shown in Figure 4.Figure 4
The price varies with the steady-state period of the game.When the system is in a chaotic state, the price increases with the emergence of the game cycle volatility, as shown in Figure5.Figure 5
The price varies with the chaotic state of the game cycle.Chaos is a so violent price fluctuations phenomenon, which has certain impact on the stability of the market.When one of tobacco’s price adjustments is too fast, the equilibrium prices go into chaos and the nonlinear feedback control is introduced as follows:(27)uk,pn=k∏i=1mpn-pi.After introducing the chaotic control coefficient, the power system of the tobacco company game is shown as follows:(28)p1t+1=p1t+g1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w.According to (28), we can get(29)p1t+1=p1t+g11+k1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w+k2p2t-p2t+1,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w+k3p3t-p3t+1.The impact of coefficient changes on the stability of the system under such conditions is shown in Figure6.Figure 6
Chaos unilateral control factor on system stability.It can be seen that the system is stable when the value is close to 0.47, which means that the tobacco companies can make more rapid price adjustments.If all the tobacco companies on the market work together to stabilize the market, chaos control factor of the system becomes(30)p1t+1=p1t+g1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g2p2ta2+d2p1-2b2p2+c2p3+b2w+k2p2t-p2t+1,p3t+1=p3t+g3p3ta3+c3p1+d3p2-2b3p3+b3w+k3p3t-p3t+1.According to (30), we can get(31)p1t+1=p1t+g11+k1p1ta1-2b1p1+c1p2+d1p3+b1w+k1p1t-p1t+1,p2t+1=p2t+g21+k2p2ta2+d2p1-2b2p2+c2p3+b2w+k2p2t-p2t+1,p3t+1=p3t+g31+k3p3ta3+c3p1+d3p2-2b3p3+b3w+k3p3t-p3t+1.In order to facilitate analysis and comparison, chaos control factor drives to all tobacco companies. Observing chaotic control factor on system stability at this time, see Figure 7.Figure 7
Chaos tripartite control factor on system stability.Under normal circumstances, when the three parties jointly take chaos control measures, the system will be easier to control to enter a stable state. In the diagram, the key point is to make the system’s stablek value smaller. In this case, it seems that the three parties take effect to unilateral chaos control when there is not much difference among them. This is mainly due to the price adjustment factor 1 retailers sensitively affected by too high corresponding unilateral system. And the chaotic state of preselected g1 coefficient is relatively large.Chaos control of economic significance actually reduced by addingk value for each of original price determination coefficients b, c, and so on. So its economic significance is actually to minimize the impact on the sales price.Tobacco companies in the competitive market adjust product price to maximize profits. But the price adjustment speed should not be too fast; otherwise, it will cause the chaotic phenomena in tobacco companies’ equilibrium price, which further causes the chaos phenomenon in profits and huge price fluctuations. Therefore, it is necessary to introduce a method of chaos control. Economic significance of chaos control is to minimize the impact of its price and that of competitors in sales. Thus, the tobacco industry needs to improve sales channels and service quality and establish a good brand image. That helps to gradually win the market through quality and service and finally win more consumers.


## 5. Conclusion

This paper describes two methods of market segmentation. When considering segmentation, the equilibrium price can be determined based on game theory. By using the theory of complexity, we build a three-oligarch competitive model in tobacco market. Lastly, it discusses the stability of the equilibrium price. By numerical simulation, we calculate the stable region with the change of adjustment speed. We also observe the phenomenon of bifurcation and chaos under different values of parameters. In the practice of management, the decision-maker should avoid the price entering into bifurcation and chaos. The reasonable pricing strategy, which is to maintain the image of the enterprise, obtains the profit balance at last. A constructive suggestion is put forward for the development of the tobacco companies.



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*Source: 6701286-2017-07-31.xml*