Statistical Structure — Discrete Choice Model

Discrete choice models (DCM) describe, explain, and predict choices between two or more discrete alternatives, such as selecting a store or choosing a product. They also assume that consumers are utility maximizers, choosing alternatives that offer higher utility.

For details of the prominent types of discrete choice models refer Wikipedia. The content that follows summarizes the multinomial logit model, one that is often used in marketing models.

Choice Probabilities

DCM models specify the probability that a consumer n chooses a particular alternative i, with the probability expressed as a function G of observed variables that relate to the alternatives:

$$ P_{ni}≡Pr(Consumer \;n \;chooses \;alternative \;i)=G(x_{ni},x_{nj \;j≠i},s_n,β) $$

xni is a vector of attributes of alternative i faced by consumer n,

xnj j≠i is a vector of attributes of the other alternatives (other than i) faced by consumer n,

sn is a vector of characteristics of consumer n, and

β is a set of parameters giving the effects of variables on probabilities, which are estimated statistically.

For a pricing research studies the alternatives are products, and the attributes of the product (xni) are brand and price. These attributes and characteristics of consumer (sn), such as annual income, age, and gender, can be used to calculate choice probabilities.

The properties of the probability model are as follows:

Pni is between 0 and 1

$$ ∀n: \sum_{j=1}^J P_{nj}=1,\;where \;J \;is \;the \;total \;number \;of \;alternatives. $$ $$ (Expected \;fraction \;of \;people \;choosing \;i) = 1/N\sum_{n=1}^N P_ni $$

Where N is the number of people making the choice.

The functional form, G, varies depending on the choice of model.

Utility

The utility that the consumer derives from choosing an alternative is composed number of factors, some are observed, others are not. In a linear form, this is expressed as:

$$ U_{ni} = βz_{ni} + ε_{ni}$$

Uni is the utility that consumer n derives from choosing alternative i.

zni is a vector of observed variables relating to alternative i for consumer n that depends on attributes of the alternative, xni, interacted perhaps with attributes of the consumer, sn, such that it can be expressed as for some numerical function z,

β is a corresponding vector of coefficients of the observed variables, and

εni captures the impact of all unobserved factors that affect the consumer’s choice.

The choice of the consumer is designated by dummy variables, yni, for each alternative:

$$ y_{ni}=\left\{\begin{array}{c} 1 \;\;\;U_{ni}>U_{nj} \;∀ \;j≠i \\0 \;\;\; otherwise \;\;\;\;\;\;\;\;\;\;\end{array}\right. $$

The choice probability is then

$$ P_{ni}=Pr(y_{ni}=1)=Pr\left(⋂_{j≠i} U_{ni}>U_{nj}\right)= Pr\left(⋂_{j≠i} βz_{ni}+ε_{ni}>βz_{nj}+ε_{nj}\right)$$ $$ =Pr\left(⋂_{j≠i} U_{ni}-U_{nj}>0\right)$$

The function G, for the multinomial logit model:

$$P_{ni} = \frac {e^{βz_{ni}}}{\sum_{j=1}^J e^{βz_{nj}}} $$

Here the utility for each alternative depends on attributes of that alternative, interacted perhaps with attributes of the consumer.

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