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individuals making choices (the consumers), and p : I × 2 C × Θ −→ [0, 1] the
choice probability function, such that p(i|B,θ) is the probability of alternative
i being selected given that the selection must be made from the choice
set B ⊂ C and that the decision maker has characteristics θ ∈ Θ.
For the case analysed, I is the set of indices for market places (establishments
or firms locations), each of them with attributes Z that may include
the spatial coordinates, the selling price, amenities offered, and other. Θ
may specify demographic oreconomicvariablesoftheconsumers,orany
otheraspectinfluencing tastes.
The distribution of tastes in the population of decision-makers (consumers)
is given by a probability measure µ(.|θ) inthespaceU(I) of utility
functions with arguments in I, depending on their characteristics θ.
The introduction of a supplementary random component in the utility
function leads to the Random Utility Maximisation (RUM) paradigm, extensively
studied again by McFadden (1977), which allows considering a population
of consumers with both known and unmeasured covariates influencing
their decision, and their distribution in a geographical space. The formal
integration of all information elements may be provided by utility functions
of the form
U ≡ W + ε
where W is the deterministic or systematic part of the utility and ε is a
random term, capturing the uncertainty whose sources are the unobserved
attributes of the alternative establishments, the unobserved individual characteristics
(such as psychological factors), measurement errors (for example,
of distances and transportation costs), and other.
MacFadden demonstrates that a PCS is compatible with the RUM hypothesis
(or can be generated from the RUM hypothesis) and a family of
choice sets B ∈ B via the following mapping: p : I × 2 C × Θ −→ [0, 1]
defined by
p(i k |B,θ) =µ {U ∈ U(I) / U(i k )=maxU(i j )}, θ
j
for each B = {i 1 , ..., i n } ∈ C,and θ ∈ Θ.
Finding econometrically feasible PCS consistent with RUM is done then
by generating choice probabilities p from parametric families of probabilities
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