On estimating the impact of information spreading in a consumer market modeled by probabilistic cellular automata and ordinary differential equations
Abstract
Individuals spread information about their lives, their career, and their recently purchased items. In addition, considering that the customer’s opinions may influence their friendship circle, neighbors, and work colleagues, they constitute a very important factor to consider in future actions of companies. In this paper, we present an approach based on probabilistic cellular automata and ordinary differential equations for modeling the consumer market, inspired by a biological system of an infectious disease-spreading model. However, instead of spreading a disease, consumers spread their opinion to other individuals, thereby having consequences on sales levels. Using marketing, the economic situation, and individual opinion as input variables of the consumer market model, simulation output (generated demand) is then used in the business game, helping the mediator (teacher) to create different scenarios and giving feedback to the student at simulation runtime, even if many players are involved in the game. Analytical and numerical simulations show that the proposed approach allows the teacher to create virtual markets for the players just by setting the parameters that describe a scenario, increasing the dynamism of the game, saving time, and allowing the mediator to focus on other important subjects, such as the teaching concepts of business strategies and decision support. Moreover, simulations are conducted to demonstrate the applicability of the proposed model in real life problems, such as the electro-domestic market.
1. Introduction
Information spreading has been analyzed for a wide range of areas in order to understand how we acquire, process, and retransmit information in our contact network. It is considered by online social networks a prime problem concerning specialized advertising to each consumer, for instance. In addition, individuals’ opinions have an excellent value for companies, which sometimes spend time and money to collect them for future strategic decisions.1
Rumor can be considered a piece of information circulating from person to person, although not necessarily being a true story.2 Despite that, rumor spreading has some peculiar characteristics close to that of consumer opinion about a product: the consumer receives the information about a product, and forms an opinion based on their criteria and influenced by social contacts, and spreads such information. After a period, he stops spreading his opinion if he does not receive more information or advertising about the product.3,4
Mathematical models of rumor spreading consider deterministic,5 probabilistic,6 and experimental4 approaches. In addition, complex networks have been used to model individuals (nodes) and their interactions (vertices) in a population propagating a rumor.7,8 Moreover, some effort has been made toward examining the similarity between disease and rumor propagation.9,10
The consumer market proposed has some characteristics of epidemic spreading models and information/rumor spreading in a population. In the classical SIR (Susceptible/Infected/Recovered) model,11,12 each individual of a population may be a Susceptible, being able to contract the disease and become an Infected. An Infected individual spreads the disease and he may be cured, obtaining resistance of disease and going to the state “Recovered.” Other state transitions may be considered, according to the model used.13–16
Since the SIR model is used to model diseases, such as mumps and chickenpox, that confer immunity and are difficulty in controlling or even eliminating from the population,17 it is a good approach to the electro-domestic market, where individuals buy few electro-domestics during their lifetime and, after a certain period, stop spreading information about the product. Cell phone markets, for instance, could be modeled by the SIS (Susceptible/Infected/Susceptible) model, where an infected individual becomes susceptible again, due to the death process or healing without immunity (here, individuals are those that buy some units during their lifetime).
Considering information spreading, individuals may be susceptible to receive information (Susceptible) and after that, starts to spread it (Infected). After the information becomes well spread and known by their relatives, it stops being spread (Recovered).18–20 Such approximation has a good fit in real world information propagation and has been modeled by cellular automata,21,22 complex networks,23–25 and ordinary differential equations (ODEs).26 Social online networks are also analyzed in those models.27
In this work, we propose probabilistic cellular automata (PCA) to model a virtual consumer market that has the characteristics of information spreading based on the SIR model, since neighbors of an individual may influence the decision to buy a product. The main idea is to use the proposed model as the core of a module for generating virtual consumer markets in a business game.
The business game has recently been pointed out to be a valuable tool to support teaching and learning in undergraduate courses, such as Economics and Business Administration.28,29 During the business game runtime, students (players) need to set some parameters of a fictitious company, such as the number of products to be manufactured, investment in human resources and marketing, and sales price, among others. Since companies (represented by students) compete for a demand fixed by the teacher (in the role of mediator), only the best-prepared students achieve reliable results.30
The teacher must create different scenarios in order to require from students an adequate challenge so that they can use all the course content. Such scenarios may be economic crisis, high demand peak, expensive labor, and so forth. However, the amount of data generated in the simulations and the number of variables to deal with it grows according to the number of players, making the task of creating different scenarios very difficult. In this sense, there are several computational intelligence approaches proposed in the literature31–37; however, none of them deals with the simulation of the consumer market to diversify the scenarios.
About 10,000 students of business administration course from one of the largest private universities of São Paulo, Brazil, use the game considered in this work every semester. Considering this situation, and the importance of information management by students and teachers involved in business administration learning,38 tools to speed up this learning the process are needed.
In the considered business game, groups of students represent companies, which have two production lines of vacuum cleaners: “Luxury” and “Standard.”39 For each iteration of simulation, they must decide about some variable values to reach their goal in the game. Some of these variables directly influence the quantity of products sold, while others affect the operating costs and financial results of the simulated company. The main decision variables handled by the players in the game are: as follows
1.
investments in processes concerning environmental management;
2.
investments in improvements of manufacturing processes;
3.
investments in technology and product development;
4.
investments in advertising;
5.
sale price of the Luxury product;
6.
sale price of the Standard product;
7.
hiring of employees for the Luxury production line;
8.
hiring of employees for the Standard production line;
9.
dismissal of employees for the Luxury production line;
10.
dismissal of employees for the Standard production line;
11.
purchase of raw materials for the production of Luxury products; and
12.
purchase of raw materials for the production of Standard products.
At the end of each iteration, demand is then created by consumer market simulation, which is shared by the companies according to their decisions results. Flexibility of the consumer market allows the teacher to propose distinct levels of challenges and difficulties for players during the simulations. At the end of game, the final score of each player (a grade ranging from 0 to 10) is calculated considering the measurable goals and the results obtained by the player. Thus, if a player reaches 100% or more of the established goals, it will get full marks.
This paper is organized as follows: in Section 2, the contact network, PCA, and ODE models numerical and analytical simulations are presented. Section 3 contains simulations of scenarios close to the business game requirement and, in Section 4, the model is tested by using real data from the Brazilian appliance market. Finally, Section 5 presents the conclusions and some possibilities for future work.
2. Consumer market in cellular automata
2.1 PCA and the contact network
Individuals live in a square matrix formed by
cells with periodic boundary conditions (the top edge of the matrix contacts the bottom edge, and the right-hand edge contacts the left-hand edge – the toroidal surface). A network model, used by Schimit and Monteiro,40 where each cell of the PCA lattice corresponds to an individual and an edge between two cells represents a social contact, configures the neighborhood of each cell. Each cell has social contact with a neighborhood defined as the square matrix of size
centered on such a cell, where r is the maximum radius in which a connection can be made. Each cell makes m connections with other cells pertaining to its neighborhood matrix. The cells with a Moore radius equal to i form the layer i. The probability
of creating a connection between a cell and any cell pertaining to the layer i of its neighborhood matrix is given by
, where
. Consider the case
, where we have
,
, and
, that is, the connection probability between a cell and any of the eight surrounding cells composing layer 1 is 50%, between a cell and any of the 16 cells composing layer 2 is 33%, and so on. Two or more connections between the same two cells are allowed. Such a random network has connections mainly locally connected and presents a “high” clustering coefficient and “small” average shortest path length, like a human social contacts network.40,41 Here, PCA is modeled using a lattice of n = 200, N = 40,000 cells (individuals). Each individual connects to
neighbors inside a radius
.
2.2 Consumer market in PCA
Individuals must decide if they will buy certain product. In this stage, his status is Consumer. He may buy a product due to being influenced by neighbors, marketing, and external situation (good and bad economic situation). From the moment he buys the product, he becomes a Buyer. In this state, he may influence his neighbors about their decision to buy or not. If a Buyer has not been satisfied about the product, he is now a Regretful and does not positively influence his neighbors. All Regretful individuals may die and, in his/her place, a Consumer is born, keeping the individuals number N constant. A state transition diagram is shown in Figure 1.
Pb is the probability of a Consumer buying the product and becoming a Buyer and is given by the following:
where v is the buyer neighbors of each consumer and C is the total connections of each individual, making v/C the influence of the neighborhood. Moreover, k(t) is a function of external factors and m(t) is a function of marketing operations in the population. Constants α, β, and γ represent the influence of neighborhood, external factors, and marketing operations, respectively. Finally, Pr is the constant probability of a Buyer regretting the purchase, Pn is the constant probability of a Buyer become a Consumer (either dying or having to buy the product again for some reason), and Pc is the probability of a buyer individual returning to be a consumer due to the necessity of buying another unit of the same product. Cells states are simultaneously updated at the end of each iteration.
2.3 Consumer market in ODEs
Consider that individuals are homogeneously distributed over space. Then, the PCA model can be represented by the following ODE model:
(1)
where a is the buying rate constant and b is the regret rate constant. When a Buyer individual returns to be a Consumer due to the necessity for buying another unit of the same product, such transition is expressed by constant rate c. Moreover, when a Regret individual either loses the disappointment about the product or dies, it is expressed by the constant rate e (which is called the death rate constant in epidemiological models). Because dC(t)/dt+dB(t)/dt+dR(t)/dt = 0, the total number of individuals remains constant, then: C(t) + B(t) + R(t) = N. Stationary solutions (C*, B*, R*) (where C*, B*, and R* are constants satisfying dC(t)/dt = 0, dB(t)/dt = 0 and dR(t)/dt = 0 for any instant t), are as follows:
(2)
and
(3)
where
(4)
Stability analysis of Equations (2)40 reveals that the buyers-free stationary state given by Equation (2) is asymptotically stable if R0<1 and unstable if R0>1; the state in which buyers remain in the population given by Equation (3) is stable if R0>1 and unstable if R0< 1. In epidemiological modeling, R0 is a bifurcation parameter called the basic reproduction number, that is, the number of new infected individuals from the influence of a single infected individual.17 Here, it reveals if satisfied buyers will remain in the population, helping to spread information about the product and keeping sales active.
Because the ODE model is a mean-field approximation for the PCA model, parameters a, b, c, and e can be estimated from PCA simulations by the following expressions obtained from Equation (1):
(5)
where ΔB(t)C→B/Δt is the increase per time step of buyer individuals due to the buying process; ΔR(t)B→R/Δt is the increase per time step of regret individuals due to the regret process; ΔC(t)B→C/Δt is the increase per time step of consumer individuals due to the necessity for buying another unit of the same product; and ΔC(t)R→C/Δt is the increase per time step of consumer individuals due to regret individuals that either lose the disappointment about the product or die. Assume that the probability of state transition at each iteration can be calculated using the relative frequency of its occurrence, then
ΔB(t)C→B/[ΔtC(t)];
ΔR(t)B→R/[ΔtB(t)];
ΔC(t)B→C/[ΔtB(t)];
ΔC(t)R→C/[ΔtR(t)]. Consequently:
(6)
where Sv is the number of consumer individuals with v connections to buyer neighbors. The expression for c comes from the fact that a buyer individual is firstly tested to establish if it will need to buy another unit of the same product (with probability Pr); if not, then it is tested to establish if the individual loses the disappointment about the product or dies.
Therefore, product sales parameters b, c, and e of linear terms in Equation (1) are related to the probabilities of regret, loss of disappointment, and death. The value of a is related to average buying probability
, and the term aC(t)B(t) gives the sales rate of the product.
In order to obtain similar evolutions of C(t), B(t), and R(t) in the ODE and PCA approaches, parameters a, b, c, and e can be estimated from simulations with the PCA model, then used in Equations (1), which are numerically solved. Average values of these four parameters are calculated by taking into account the last 20 time steps of PCA simulation (when the system has already reached its permanent regime).
Figure 2 shows the dynamical behavior for PCA (Figure 2(a)) with parameters: for n = 200, α = 0.83, β = 0, γ = 0, Pr = 60%, Pc = 30%, Pn = 10% with initial conditions C(0) = 99.5%, B(0) = 0.5%, and R(0) = 0%; and for the ODE (Figure 2(b)) with parameters: a = 0.716, b = 0.666, c = 0.127, e = 0.098 with initial conditions C(0) = 0.995, B(0) = 0.005, and R(0) = 0%. By using those parameters, R0 = 0.984, confirming that for values R0 < 1, product sales are not successful.
Figure 3 shows the dynamical behavior for PCA (Figure 3(a)) with parameters: n = 200, α = 0.5, β = 1, γ = 1, Pr = 5%, Pc = 10%, Pn = 10% with initial conditions C(0) = 99.5%, B(0) = 0.5%, and R(0) = 0%; and for ODEs (Figure 3(b)) with parameters: a = 1.312, b = 0.050, c = 0.0.095, e = 0.101 with initial conditions C(0) = 0.995, B(0) = 0.005, and R(0) = 0%. By using those parameters, R0 = 9.042, confirming that for values R0 > 1, product sales are successful, with about 60% of the population pertaining to buyer state in the permanent regime. It is worth noting that a different parameter set has been tested, and the results were dynamically the same, that is, the system output is always limited and small variations in input parameters culminate in a small variation in results.
Figure 4 shows the sales rate (that is,
) in function of parameters α, β, and γ; for α data, β and γ are fixed β = γ = 0.05 and α = 0, 0.1, 0.2, …, 2.0, considering an average of 10 simulations for each value. The same is valid for β and γ data. Note that when sales rates have low values, it is better to invest in word of mouth advertising, since it returns betters results. When the sales rate increases, the external situation (and traditional marketing, since it has the same influence in Pr) variation leads to higher sales rates. The professor has to understand such variation in order to compose scenarios for the players.
When α and β vary for α = 0, 0.05, 0.1,…, 2.0, and β = 0, 0.05, 0.1,…, 1.0 (γ = 0.1), the sales rates are as presented in Figure 5, where, for each pair of parameters, the average of 10 simulations is considered. Note that, again, it is possible to see that the better external scenario is more efficient than word of mouth advertising, since for equals variations of α and β, the sales rate is higher for the β variation.
Figure 6 presents sales rates for the variation of probabilities Pr and Pc (Pr = 0, 0.05, 0.1, …, 2.0 and Pc = 0, 0.05, 0.1, …, 1.0), where, for each pair of parameters, the average of 10 simulations is considered. Note that investment to increase the probability of the buyer needing to buy another product (Pc, which increases
) is more efficient than reducing the probability of the buyer regretting the purchase (Pr). Although those probabilities are pre-fixed in the business game, it is possible to let the professor manage them, if product quality is considered as a variable of the game.
3. Results
3.1 Scenarios for the business game
The module proposed to the game for generating virtual consumer markets by the teacher is illustrated in Figure 7. By using the PCA model, parameters of the business game and teacher decisions will vary in order to analyze the influence over the consumer market. The teacher will set variables α, β, γ, m(t), and k(t), and here we will simulate different market scenarios.
Figure 8 shows the dynamical behavior for PCA with parameters: n = 200, α = 0.05, β = 0.1, γ = 0.1, Pr = 5%, Pc = 10%, Pn = 10% with initial conditions C(0) = 99.5%, B(0) = 0.5%, R(0) = 0%, k(0) = 0.6, m(0) = 0.3. Between instants t = 0 and t =100, k(t) = 0.6, then at instant t = 101, k(t) drops to k(t) = 0.3, expressing the worst economic scenario. At instant t = 201, m(t) increased from m(t) = 0.3 to m(t) = 0.6, expressing higher levels of investment in advertising (IAD) from students. At instant t = 301, α increased from α = 0.05 to α = 0.1, in an attempt to change the model, giving higher priority to neighborhood contact on consumer decisions. At instant t = 400, the economic scenario is better, increasing k(t) from k(t) = 0.3 to k(t) = 0.8.
By calculating R0 at different instants, we have R0(100) = 2.045, R0(200) = 1.732, R0(300) = 2.055, R0(400) = 2.247, and R0(500) = 2.709. Note that parameter R0 is directly related to units sold, reflecting the economic general situation at each instant.
Figure 9 shows the dynamical behavior for PCA with parameters: n = 200, α = 0.1, β = 0.1, γ = 0.1, Pr = 5%, Pc = 10%, Pn = 10% with initial conditions C(0) = 99.5%, B(0) = 0.5%, R(0) = 0%, k(t) = 0.6–0.0016t for 0 ⩽t ⩽ 250 and k(t) =−0.2+0.0016t for 250 < t ⩽ 500; in this case, k(t) starts at k(0) = 0.6, linearly decreases to k(250) = 0.2 and, from that, linearly increases to k(500) = 0.6. Moreover, m(0) = 0.6, then it changes at instant 200 to m(200) = 0.9; and Pc has a new value at instant 400, Pc = 30%.
4. Using real data
The business game frequently considers the appliances market in order for players to differentiate their products.33 Here, we use appliances market data obtained from the Brazilian public foundation Instituto Brasileiro de Geografia e Estatística (IBGE, Brazilian Institute of Geography and Statistics), considering the period from 2004 to 2013.42 The variable considered was the normalized sales of appliance per month. Parameters are chosen to reproduce this real scenario.
Therefore, the time dependent parameter k(t) has been set accordingly to real external factors to simulate an external influence in the market. Each time step is considered to be one month, and t = 0 is equivalent to January 2004. The function k(t) has been separated in three intervals:
The simulation runs until time step 144, which is December 2014. Note that a small reduction in sales happened during the 2008 stock market crash. Figure 10 shows the real data and the adjusted model data. Real data is presented by a thick dark line and model data by a thin dark line. Other parameters used in this simulation were: n = 200, α = 0.15, β = 0.05, γ = 0.05, Pr = 5%, Pc = 10%, Pn = 10% with initial conditions C(0) = 64%, B(0) = 36%, R(0) = 0%, m(t) = 0.6.
From Figure 10, one can observe that, although a fine model adjustment is needed, it is clear that there is a similar behavior of the curves (real and simulated market). Nevertheless, two points may be highlighted: (i) a good fit between the data with minor variations; and (ii) slow and steady growth with a fall during the 2008 crisis. Thus, this simulation result demonstrates the applicability of the proposed model in real situations.
5. Conclusions
Based on the epidemiological SIR model, we presented an information spreading model that influences individuals on the buying decision of a determined product. By using the buying rate, regret rate, and a second buying rate (a, b, and c, respectively), it is possible to analyze a product’s success in the population, calculating
; if R0> 1, individuals buying the product will always exist in population, and if R0< 1, product sales will be unsuccessful. Numerical simulations were used to show how the players of a business game influence the product success, sales rate, and consumer market. Furthermore, such a model is stressed to assist the teacher, who inputs some conditions to the game in an attempt to create several scenarios. Flexibility of the model was an objective to enable fine-tuning of the consumer market by the teacher without either abruptly breaking its dynamics or diminishing to zero the sales rate. For instance, the teacher may use analytical data to check how parameter changes will influence the buying rate, before inputting the data to the business game.
The probability of buying a product clearly has a saturation point, that is, a point that even if players multiply marketing investments, the difference in information spreading is minimal. This is interesting to the game and it also has real consumer market characteristics, since some studies indicate the existence of marketing saturation and also the negative influence on consumers, which may even feel annoyed.43,44 Other contributions of the paper are as follows: (i) the relation between PCA and ODE models by using mean-field properties for a consumer market; and (ii) a good fit of model data with real data for a market similar to the market handled in the considered business game.
Although modeled by a simple model, some considerations are valid, as shown by the result from Figure 6, which indicates that if the regret probability of a buyer is high, there is no point in trying to increase the sales among individuals that have already bought the product. Regret buyers may indirectly influence the market negatively, since they would never buy the product again, and this disturbs the information spreading about the product.
Future works should try to stress the following point: disassociate the buying process and spreading information, that is, there could be a process to control the buying decision based on another process that could control the spreading model. Furthermore, consumers, buyers, and regrets individuals could spread positive and negative points about the product without the necessity to buy.
Funding
PHTS is partially supported by grant #2015/01032-9 of São Paulo Research Foundation (FAPESP), and by grant #303743/2016-6 of Brazilian National Research Council (CNPq). SA Araújo would like to thank for his CNPq scholarship grant #311971/2015-6.
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Biographies
Pedro Henrique Triguis Schimit received his BSc degree in Electrical Engineering with emphasis on Automation and Control from the University of São Paulo, Brazil (2005) and his PhD degree in Sciences (2010) from the same institution. He is a permanent lecturer in the Informatics and Knowledge Management Graduate Program University at the Nove de Julho University. Currently, he is a research productivity fellow at CNPq–Brazilian National Research Council (level 2) in the Production and Transportation Engineering Committee, Game Theory area. His research areas of interest include computer modeling and simulation, dynamical systems, game theory, graph theory, numerical analysis, and modeling of economic and biological systems.
Daniel Ferreira de Barros Junior received his BSc degree in Computer Science (2003) and his MSc degree in Industrial Engineering (2014) from the Nove de Julho University – UNINOVE, São Paulo, Brazil. Since 2014 he has been teaching at UNINOVE in Computer Science for an undergraduate course. His research interests include computational intelligence and processes optimization.
Sidnei Alves de Araújo received his MSc and PhD degrees in Electrical Engineering, respectively, from Mackenzie University, Brazil (2002), and from the University of São Paulo (USP), Brazil (2009). Currently, his is an associate professor for the Informatics and Knowledge Management Graduate Program of the Nove de Julho University, São Paulo, Brazil, Associate Researcher at the Signal Processing Laboratory at the Polytechnic School of the USP, and a Technological Development Productivity and Innovation Extension Fellow at CNPq–Brazilian National Research Council (level 2). His research areas of interest include image processing and computer vision, computational intelligence, metaheuristics, and machine learning.
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Information spreading in a population modeled by continuous asynchrono...
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Modelling the Impact of Transit Media on Information Spreading in an U...
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