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Objective and subjective factors: modelling consumer behaviour from individual to population scale Monica-Gabriela Cojocaru Joint work with “The Team� Assoc. Prof. Mathematics Mathematics & Statistics University of Guelph WICI -Waterloo

October 25, 2011

The team - “Networks and Dynamics HQP Lab” 

Faculty: Dr. Henry Thille (Economics- UoG) Dr. Ed Thommes (Physics - UoG)

The team - “Networks and Dynamics HQP Lab� 

Faculty: Dr. Henry Thille (Economics- UoG) Dr. Ed Thommes (Math & Stats - UoG)

Students: completed D. Nelson N. Ringa S. Hawkins K. Dare C. Hogg C. Kuusela

The team - “Networks and Dynamics HQP Lab” 

Faculty: Dr. Henry Thille (Economics- UoG) Dr. Ed Thommes (Math & Stats - UoG)

Students: current D. Vanin C. Cvetkovic E. Wild M. Andrews V. Gheorghiade S. Greenhalgh


Dynamic adoption of new products - population behaviour via individuals interacting - end result of this interaction: adoption of new products - individual behaviour includes subjective factors: personality type, social networks, changes in product preferences -aim: estimate adoption levels of new product variants over time via a generalization of an economic model with producers


Modelling via: ABM, PDE, game theory Numerical illustrations Future work - acknowledgements - references

Differentiated product markets; adoption of new product variants 

We are interested in modelling the adoption of new variants of a product on a market over a finite time interval

We model the level of adoption of a product variant from a small scale perspective (as emergent dynamics via agent-based models);

The original, static, economic model we present next can be found in [Anderson, Palma, Thisse 89 - Rev. Econ. Studies].

Differentiated product markets Characteristics approach m product characteristics, R m is characteristics space. r r r v 1 m 1 m n product variants at z1 = (z1 ,...z1 ),..., zn = (zn ,...zn ) with zi  z j for all i, j  {1,K,n} and i  j. N consumers with preferences distributed according to a r continuous, positive density function f ( z ) in R m , where r  f(z )dz = N. Rm

Characteristics model -contd r r Utility of consumer at z (where z is the preferred product) 2 m r k k purchasing variant i is U i ( z ) = Vi  c  || z  zi || , k=1

where c > 0, Vi =  i  pi is variant's i value, with  i = quality index, pi = price. We define : the market space of variant i as r r r M i = {z  R m ;U i ( z )  U j ( z ), j = 1,...n} and r r the demand for variant i as X i =  f ( z )dz . Mi

2D example 

For example, a hypothetical housing market could be represented by

2D example 

For example, a hypothetical housing market could be represented by

[CTT ]

Agent-based models of differentiated product markets - I 

An ABM model of the population consists of simulating its individuals, each of them having a set of parameters representing their characteristics, e.g. product preferences, personality, etc. (implemented in NetLogo)

In an ABM model, we discretize the consumer density function over the space of product characteristics;

We compute discrete versions of the market spaces and demands for a product variant, keeping in mind that each agent j has a utility function associated with consuming (adopting) a product variant i m r k k U i ( z j ) = Vi   z j  zi k=1


Original economic model : static, no social effects

Dynamics of differentiated product markets 

We introduce time evolution in several ways: - individuals in the population have different personality types; i - individuals interact according to a given social network structure - individuals can change preferences over time, according with their characteristics, while retaining a memory of his/her initial choice; These modify the position of an individuals preference, z j , in the product characteristics space; We simulate each agent j as having a time-dependent utility for consuming the variant i of a product m r k k U i ( z j (t)) = Vi   z j (t)  zi



At each time step, an agent is evaluating the utility of adopting variant i - if the utility is non-negative, then the agent is considered an “adopter”; - if the utility is negative, the agent is a “non-adopter”.

“Force-based� global social influence and personality types 

The time evolution of an agents preference within the characteristics space is given by solving a 1st order ODE:

d (z j (t)) = F j (z j (t)) dt where we need to define

r r F j (z j )

The expression will be given by 3 different components which we define next.

“Force-based” global social influence and personality types 

The time evolution of an agents preference within the characteristics space is given by solving a 1st order ODE:

d (z j (t)) = F j (z j (t)) dt 

Part 1 := “Pull of a product”

The influence of the overall adoption rate of the variant i on consumer j : r r  N i  1 r r Fi, j ( z j ) = G j   z j  zi N


r r ( z j  zi ) r r z j  zi

where G j , 1,  2 are constants considered 1 > 0,  2 < 0.

Personality determines consumers reaction to adoption levels Ni/N: - “imitator” agents are attracted to popular buys  Gj>0 - ”innovator” are repelled by popular buys  Gj<0

Part 2 := “Pull of conservatism”

Consumers retain memory of their original r preference z j,0 , meaning that an agent cannot be taken too far r res r r out of its comfort zone : F j = K j ( z j  z j,0 ) where K j > 0 is larger for more conservative consumers.

Adding social networks 

Part 3 := â&#x20AC;&#x153;Pull of the social networkâ&#x20AC;?: - A social network, built on the premise that individuals want to form social links with the most popular individuals; Adding social network ---> we add a social influence term to the force acting on consumer j, namely:

r r N i' r (z  z ) r Fnet,i, j ( z j ) =  Ls r j ri , z j  zi s=1 where the sum is over the N i' link neighbours of j who adopt product i, and Ls is the strength of the link

[CTT - Kuusela]

Putting these together, we solve at each time step the ODE n r r res r d (z j (t)) = [ ( Fi, j + Fnet,i, j ) + F j ]( z j (t)) dt i=1

Note: currently we investigate weighted social links - Ls : Developing strength of social links, by allowing individuals to play a â&#x20AC;&#x153;prisoners dilemmaâ&#x20AC;? social game with memory

Link strengths via prisoners dilemma    

Iterative games : can give insight into an agents behaviour PD: is a game where 2 players have pure strategies C and D (cooperate and defect) Dilemma: the agents choose to defect, although it is better for both if they cooperate Payoff matrix we considered: Pl2 C D C [ 3 0] Pl1 D [ 5 1]

where CC payoff is for mutual cooperation; DD if for mutual defection, CD is the sucker (S) payoff and DC is the temptation (T) payoff  A game is a PD if values in the payoff matrix satisfy: S < DD < CC < T and 2CC < (S+T)



One shot PD does not exhibit CC; but the iterated PD does; An agents fitness is the sum of all payoffs divided by the number of total plays; it shows how far from CC the agents behaviour is We use an iterated PD to deduce the strength of links in our network structure; we consider that if a pair of agents are adjacent neighbours in the network, then they have had some interactions before and have each compounded a â&#x20AC;&#x153;scoreâ&#x20AC;? = the influence one agent has over the other We represent these scores as weights on unidirectional links, where each pair of agents (i, j) is joined by a pair of unidirectional links The scores are obtained by simulating iterated PD on the population of agents prior to the start of the consumer model simulations;


The agents are divided into two groups: shopkeepers and customers ----> in our consumer model they become: ---> popular agents and regular agents Shopkeepers: interact with many customers, for a smaller number of iterations; they retain a memory of their interactions; Customers: interact with much smaller numbers of other customers; they do not retain a memory of their interactions (they reset their internal state machines after every interaction). Bottom line: shopkeepers display different behaviour than regular customers; in essence, they tend to adapt to customers and exploit them, I.e. tend to play AllD with some TFT.

Competition among firms with ABM   


Firms produce the products; the costs of production depends on the characteristics of product. We consider i {1,2,3} products with k {x,y} characteristics: x = “environmental friendliness” and y = “comfort” cost per unit of production is Ci (x, y) = c0 + c1 (x2 + y2 + xy), with c0 and c1 const. c0 captures any unit production costs that do not vary with characteristic; c1 scales the characteristic dependent portion of cost. This cost function implies that a product with very high levels of “comfort” or very high levels of “environmental friendliness” are relatively expensive to produce.


We wish to allow firms to vary their prices in a reasonable rational manner. An analytic solution to the pricing game among firms is impossible due to the complexity of demands resulting from our modelling of consumer behaviour. Instead we allow firms to vary price based on a comparison between prices and profits in the two most recent periods. pti = price charged by firm i in period t ti = corresponding profit realized. Firms alternate adjusting price and when it is is turn to adjust: i i pti +1 = pti (1+ ), if sign( pti  pt1 ) = sign( ti   t1 ), i i pti +1 = pti (1 ), if sign( pti  pt1 )  sign( ti   t1 ),

pti +1 = pti otherwise.

Initial setup  

Consumers are normally distributed over the two characteristics x and y. Consumer characteristics are drawn from two distributions; hence we assume independence of the consumers preferences over the two characteristics. All consumers draw from the same distribution for the y characteristic, but for the x characteristic there are two distributions. Some consumers have their x characteristic drawn from a distribution with a higher mean that the others, allowing us to control how many â&#x20AC;&#x153;environmentally sensitiveâ&#x20AC;? consumers there are in the model.


Recall - static characteristics model m product characteristics, n product variants N consumers with preferences distributed according to r r m a density function f ( z ) in R , where  f( z )dz = N. Rm

m r k 2 k Utility w.r.t. i is U i ( z ) = Vi   z  zi , Vi =  i  pi , i=1

pi = price. The market space of i and the demand for it are r r r r r m M i = {z  R ;U i ( z )  U j ( z ), j = 1,...n} and X i =  f ( z )dz . Mi

[Nelson - CTT]

PDE model of differentiated product markets - II 

Conservation psychology studies people's motivations for acting in an environmentally conscious way. The most important findings: Social processes influential. Strong normative pressure. Eco-products associated with technology and status.

We use these findings to build a time-dependent model of ecoproduct demand. We take n=2: an eco-product and its standard version and m=1: a single characteristic which measures a product's `eco-level'

We allow consumer preferences and product prices to change in time, i.e.:

f = f (z,t) and X i (t) =

M i ( p(t ))

r r f ( z ,t)dz .

Social interaction between two types of consumers drives this change. Early Adopters = adopt a product because of their own beliefs. Go With the Crowds = adopt a product out of a desire to conform.

PDE model - contd 

We developed an equation to describe how the consumer distribution f(z,t) evolves, given by ft + (fv)z =  fzz , (*) where v=v(z,t), the velocity function, incorporates social processes in the evolution of consumer preferences. To help visualize the role of v, we can think of f as the concentration of a substance in a solution .Thus v describes the movement of the fluid carrying a substance in solution = the fluids velocity if we assume it moves through a passage of const. crosssectional area.

 fzz accounts for influences on consumer preferences that have not 

been explicitly modeled, through a diffusion process derived from a random walk at the individual level.  adjusts the relative strength of the velocity function and the diffusion process.

Diffusion coefficient effect on preference density

Velocity of preference changes  1.

The expressions we chose for this model are as follows: For two products:

v Innov (z,t) = z2

z(z  z2 )(z  L) and 2 2 ((z  z2 ) + L)

v Im it (z,t) = (X1 + 1  ) 2.

z(z  z1 )(z  L) z(z  z2 )(z  L) + (  X + 1  ) 2 ((z  z1 ) 2 + L) 2 ((z  z2 ) 2 + L) 2

For k-products - direct generalization, where Xi is the demand for product i = 1,...,k

Velocity functions for Inn and Imit

L=10, z2=8

Velocity functions for Inn and Imit

L=10, =1, z1 =2, z2 =8, X1 =X2 =0.5 respectively, X1=0.3<X2=0.7

Demand and population distribution simulations  

We solved the model numerically (Matlab) We started each simulation with a normal distribution of consumers over the base product at z=4 and let the model evolve for a few time periods in order to generate initial conditions for the two product model The simulations have initial distributions as follows:

1 f Innov (z,0) = sInnov e 0.4  1 e f Im it (z,0) = sIm it 2

(zz1 )2 0.4

(zz1 )2 2


Demand for the variant as a function of its price

Demand for variant as a function of distance from base product

Preference density changes

Preference density changes with the increase in Innovators fraction

Producers game and policy implications   

We assume a noncooperative game between n firms, each producing one product. Prices become a function of time p=p(t); Each firm has a profit function, given by revenue minus cost i(p(t))=pi(t)Xi(p(t)) - Ci(p(t)) for i in {1,…,n} which they try to maximize through their choice of pi(t). Given a time interval [0,T], we divide it into subintervals 0 = t0 < t1< ... < ts=T, where tk denotes a time where firms may “adjust” the price of their product.

Formulation of the Dynamic Game 

Find p*(tk)=(p1*(tk),…,pn*(tk)) in Rn+, an optimal solution of the game: i(pi*,(p^)i*)  i(pi,(p^)i*) , for all pi  pi*, pi  0,

i in {1,…,n}, (p^)i *=(p1*,…,pi-1*,pi+1*,…,pn*).  Also, for t in [tk,tk+1): Xi(p*(tk))=Mi(p*(tk)) f(z,t) dz and Mi(p*(tk))={z in Rm: Ui(z, p*(tk)  Uj(z, p*(tk), j=1,…,n}  We solve the game by the reaction curve method. The intersection of the reaction curves gives a Nash equilibrium solution. where:

Effects of Dynamic Prices  

We compare the results of simulations with dynamic prices to those with fixed prices; We see that allowing the producer of the variant to adjust their price can eliminate lock-in at the base.

Effects of the degree of eco-ness  

We examine the effect of distance between the base and the variant in the characteristics space. Adjusting the distance is a trade-off between speed of adoption and longterm adoption levels

Effects of subsidies for the ecoproducts 

We look at the effect of a subsidy, given for the first 25% of the time interval, on variants with varying distance from the base product.


ABM approach - organic foods - III 

Utility expression = a linear combination of food attributes and consumer attributes, as:

U i = Bsi + (n green  n notg ) f i  ( porg  ps )w i , where B is the perceived health benefit of consuming the organic vs. the standard food, si is the sensitivity to perceived health benefits of consuming the organic food, f i is the "folower tendency" of consumer i, and w i is the sensitivity to the price difference We choose : si and w i  [0,1], and f i statistically distributed in [-1,1]. 

Then a consumer is said to adopt the organic food product if Ui > 0, is said to “not adopt” if Ui < 0, and is said to “stay with previous choice” if Ui =0


This model allows one to analyze the adoption level of an organic food product over a given time window [0,T], while at the same time; It allows the examination of the adoption level across different social networks (grid, PA, random) It allows one to determine values for the perceived benefit parameter B, for fixed price differences and given social network structures; Example: for organic milk on a PA network with

porg = 1.50058 ps and f i normally distributed we find BPA  [1.7,2.3] < Bran  [2,2.5] < Breg  [2.7,3.5] 

This models allows the introduction of time by considering that

porg  ps = ( porg  ps )(t)



Our results could have implications for policymakers and for firms The long-term effect of subsidies depends on the product's characteristics Firms should consider both short-term and longterm goals in choosing what type of product to produce. Estimation of perceived benefits of adopting organic/eco-products Analysis of social networks and heterogeneity of consumers on the markets of eco-products

Future work 

Future directions - ABM: - scaling to real – world data; - allow changes in the social network as a consequence of agents interactions - relation to previous work: diffusion of product adoption in small world networks and epidemic models Adoption of new products - can be done in continuous time, via a 2-dim PDE model using a diffusion-like equation; one can track the evolution of the population density - can be “compared” with its ABM counterpart

Acknowledgements and references 

Selected References:

[1] S.P. Anderson, A. del Palma and J.-F. Thisse, Discrete Choice Theory of Product Differentiation, Cambridge: The MIT Press, 1992. [2] A.-L. Barabasi and R. Albert, Emergence of Scaling in Random Net-works, Science: 286: 509-512, 15 October 1999. [3] M.A. Janssen and W. Jager, Fashions, habits and changing preferences: Simulation of psychological factors affecting market dynamics, Journal of Economic Psychology 22: 745-72, 2001. [4] A. Sengupta, D. V. Greetham and M. Spence, An Evolutionary Model of Brand Competition, Proceedings of the 2007 IEEE Symposium on Artificial Life: 100-107, 2007. [5] U. Wilensky. NetLogo. Center for Connected Learning and Computer-Based Modeling, Northwestern Uni- versity, Evanston, IL. 1999. 

Funding: NSERC, CFI, OCE, City of Guelph

Further information

Further information

Thank you !

Monica Cojocaru WICI Seminar  
Monica Cojocaru WICI Seminar  

Objective and subjective factors: modelling consumer behaviour from individual to population scale