Choice Set Modeling *

Random Utility Theory

3.5 Choice Set Modeling *

car availability in mode choice models. Attributes with this interpretation can be easily identified in a number of the random utility models described in the next chapter. The implicit approach is undoubtedly simpler from the application point of view, although there is a noticeable lack of consistency because “utility” attributes are mixed with “availability” attributes.

In the explicit approach, the choice probability of an alternativej for decision- makeriis usually expressed through a two-stage choice model:

pⁱ[j] =

Iⁱ∈Gⁱ

pⁱ[j, Iⁱ] =

Iⁱ∈Gⁱ

pⁱ[j/Iⁱ]pⁱ[Iⁱ] (3.5.1)

where

Iⁱ is the generic choice set of decision-makeri

Gⁱ is the set made up of all possible nonempty choice sets for decision-maker i(nonempty subsets of the set of all the possible alternatives)

pⁱ[j, Iⁱ] is the joint probability that decision-makeriwill choose alternativej and thatIⁱ is his choice set

pⁱ[j/Iⁱ] is the probability that decision-maker i will choose alternative j, her choice set beingIⁱ

pⁱ[Iⁱ] is the probability thatIⁱ is the choice set of individuali

The choice probability conditional on setIⁱ, pⁱ[j/Iⁱ], can be represented with one of the random utility models described in Sect.3.3.

An example of an explicit choice set generation model can be obtained, starting from the general model (3.5.1), by assuming that the probabilities that each single alternative belongs to the choice set are independent of each other:

Pr[j∈Iⁱ/ h∈Iⁱ] =Pr[j ∈Iⁱ] ∀j, h (3.5.2) In this case, the probabilityp[Iⁱ]can be expressed as

p[Iⁱ] = ^h∈Iip[h∈Iⁱ] · _{k /∈I}i[1−p[k∈Iⁱ]]

1−p[Iⁱ≡ ∅] (3.5.3)

where the first product is extended to all the alternatives included in Iⁱ and the second to all those not included inIⁱ. The denominator of expression (3.5.3) nor- malizes the probabilitiesp[Iⁱ]to take into account the fact that an empty choice set (Iⁱ≡ ∅)is usually excluded, under the assumption that the decision-maker’s choice set includes at least one alternative; the probability that the choice set is empty is given by

p[Iⁱ≡ ∅] =

j

1−p[j∈Iⁱ]

(3.5.4)

3.5 Choice Set Modeling 141 Replacing expressions (3.5.3) and (3.5.4) in (3.5.1), the choice probability of the generic alternative is:

pⁱ[j] =

Iⁱ∈Gⁱ{ _h∈Iipⁱ[h∈Iⁱ] · _{k /∈I}i[1−pⁱ[k∈Iⁱ]] ·pⁱ[j/Iⁱ]}

1− _j[1−pⁱ[j∈Iⁱ]] (3.5.5) Specification of model (3.5.5) requires a model to represent the probability p[j ∈Iⁱ]that generic alternativej belongs to the choice set. Various authors have proposed a binomial logit model²¹:

p[j∈Iⁱ] = 1 1+exp(

kγ_kY_kjⁱ ) (3.5.6) where theY_kare “availability/perception” variables mentioned above and theγ_kare their coefficients.

The explicit approach, although very interesting and consistent from a theoret- ical point of view, poses some computational problems. The number of all possi- ble choice sets (i.e., the cardinality ofGⁱ) grows exponentially with the number of alternatives. This complicates the calculation of choice probabilities (3.5.1), and therefore the joint calibration of the parametersβ_k in the systematic utility andγ_k in the choice set model.

An intermediate approach, named Implicit Availability Perception (IAP), ac- counts for the availability and perception of an alternative by modifying its sys- tematic utility in the random utility model. This approach is based on a generaliza- tion of the conventional concepts of availability and choice set membership. Instead of assuming that an alternative is either available or not, the approach considers that an alternative may have intermediate levels of availability and perception to a decision-maker. The decision-maker’s choice set is then viewed as a “fuzzy set”;

it is no longer represented as a set of[0/1]Boolean variables (1 if the alternative is available or perceived, 0 otherwise), but rather as a set of continuous variables μ_I(j )defined on the interval[0,1]. This representation could apply, for example, to an alternative that is theoretically available but not completely perceived as such for a particular journey, due to factors that may be either subjective (lack of infor- mation, time constraints, state of health, etc.) or objective (weather conditions, etc.) Obviously, extreme values ofμ_I(j )are still possible, corresponding respectively to the nonavailability and the complete availability and perception of alternativej.

The model accounts for different levels of availability and perception of an al- ternative by directly introducing an appropriate functional transformation ofμ_I(j ) into the alternative’s utility function:

U_jⁱ =V_jⁱ+lnμⁱ_I(j )+εⁱ_j (3.5.7) where

21In this application, the Binomial Logit model (3.5.6) should be seen as a convenient functional relationship rather than a random utility model since it does not represent any “choice”.

U_jⁱ is the perceived utility of alternativej for decision-makeri V_jⁱ is the systematic utility of alternativejfor decision-makeri εⁱ_j is the random residual of alternativej for decision-makeri

μⁱ_I(j ) is the level of membership of alternativej in the choice setIⁱ of decision- makeri (0≤μ≤1)

In this way, all the alternatives can be considered as theoretically available. If alternativej is not available(μⁱ_I(j )=0), the term lnμⁱ_I(j ) forces its perceived utilityU_jⁱ to minus infinity and the probability of choosing it to zero, regardless of the value ofV_jⁱ. Furthermore, choice probabilities of all the other alternatives are no longer influenced by alternativej. If, on the other hand, an alternativejis definitely available and taken into consideration(μⁱ_I(j )=1), the additional term is equal to zero and the perceived utility has the conventional expression. Intermediate values ofμⁱ_I(j )reduce the utility of the alternative according to its level of availability.

For a generic individuali, the true value of the availability and perception level, and therefore of the term lnμⁱ_I(j ), is unknown to the analyst. It can therefore be modeled as a random variable, which in turn can be expressed as the sum of its mean value,E[lnμⁱ_I(j )], and a random residual,ηⁱ_j, defined by the difference lnμⁱ_I(j )− E[lnμⁱ_I(j )]. Expression (3.5.7) then becomes:

U_jⁱ=V_jⁱ+E

lnμⁱ_I(j )

+η_jⁱ +εⁱ_j (3.5.8) In order to make expression (3.5.8) more tractable, E[lnμⁱ_I(j )] can be ap- proximated by its second-order Taylor series expression around the pointμ¯ⁱ_I(j )= E[μⁱ_I(j )]. Substituting this approximation in (3.5.8) yields:

U_jⁱ∼=V_jⁱ+lnμ¯ⁱ_I(j )−1− ¯μⁱ_I(j )

2μ¯ⁱ_I(j ) +σ_jⁱ withσ_jⁱ=ε_jⁱ +ηⁱ_j (3.5.9) The choice probability of alternativej can therefore be calculated using the ran- dom utility models described in Sect.3.3; it will depend on the systematic utility of each alternative, on the mean availability and perception of each alternative and on the joint distribution of the random variablesσ_jⁱ. For example, if the latter are assumed to be i.i.d. Gumbel(0, θ )variables, a new multinomial logit model is ob- tained:

pⁱ[j] =

exp1 θ ·

V_jⁱ+lnμ¯ⁱ_I(j )−^{1− ¯μiI(j )}

2μ¯ⁱ_I(j )

hexp₁

θ·

V_hⁱ+lnμ¯ⁱ_I(h)−^{1− ¯μiI(h)}

2μ¯ⁱ_I(h)

(3.5.10)

where the sum in the denominator is extended to all the alternatives theoretically available to decision-makeri. From the above expression, it can be deduced that,