The Single-Level Hierarchical Logit Model

Random Utility Theory

3.3 Some Random Utility Models

3.3.2 The Single-Level Hierarchical Logit Model

case described: namely, that the decision-maker perceives the alternatives as com- pletely distinct, and therefore that their random residuals are independent. A more realistic choice model can be obtained by introducing a covariance between the random residuals of alternativesB andC, as shown in the following sections. In general, as shown below, a multinomial logit model has the property that any vari- ation in the choice probability of one alternative (resulting from a change in its attributes) leads to proportional variations in the choice probabilities of all other alternatives.

In applications, the multinomial logit model should be used with choice alterna- tives that are sufficiently distinct for the assumption of independent random residu- als to be plausible.

3.3 Some Random Utility Models 101 Fig. 3.5 Choice tree of a

single-level hierarchical logit model

These assumptions imply that the decision-maker perceives alternatives belonging to the same group as similar; the similarity is captured by the covariance among the overall random residuals of these alternatives. In a mode choice situation, for example, the available modes can be divided into two groups: public modes (bus and train) and private modes (car and motorbike). Assumption (3.3.8) implies that the decision-maker perceives the modes belonging to the same group to be similar inasmuch as they share a number of attributes (flexibility, privacy, etc.).

The utility structure and the choice mechanism corresponding to a single-level hierarchical logit model can be represented by a choice tree, as shown in Fig.3.5.

In the choice tree, “elementary” choice alternatives (e.g., transport modes) corre- spond to nodes with no exit links (“leaves” of the tree), whereas the root nodeo has no entering links. The intermediate nodesk, one for each group, represent com- pound alternatives: groups of elementary alternatives. The random residualsηk and τ_{j/ k} are associated with the branches that correspond to groups and to elementary alternatives, respectively.

The choice tree can be viewed as the representation of a hypothetical choice process. Starting from the root node, the decision-maker first chooses groupkfrom the available groups (represented by nodes linked to the root); she then chooses elementary alternativejfrom those belonging to groupk(represented by the leaves connected to the nodek). The expression for the overall choice probability of an alternativej,p[j], is obtained as the product of the conditional probabilityp[j/ k]

of choosing elementary alternativej within groupk(lower level), multiplied by the probabilityp[k]of choosing groupk(upper level):

p[j] =p[j/ k] ·p[k] (3.3.9) The name of the model is derived, in fact, from this probability structure.

To specify the probabilities in (3.3.9), further assumptions on the distribution of random residuals must be introduced. For the single-level hierarchical logit model, it is assumed that the random residuals relative to the alternatives available at each de- cision node (the root and the intermediate nodes) are identically and independently distributed Gumbel random variables.

Considering first the lower-level nodes (elementary alternatives), the residuals τ_{j/ k} are assumed to be i.i.d. Gumbel variables with zero mean and the same pa- rameterθ for all groupskand all alternatives j. In the choice among alternatives belonging to groupk, the perceived utility associated with alternativej, U_{j/ k}, can

be expressed as

U_{j/ k}=V_j+τ_{j/ k} ∀j∈I_k, ∀k E[τ_{j/ k}] =0 ∀j∈I_k, ∀k

Var[τj/ k] =π²θ²/6 ∀j ∈Ik, ∀k

(3.3.10)

Under these assumptions, the conditional choice probability of the elementary alternativej can be expressed as

p[j/ k] =Pr[U_{j/ k}> U_{i/ k}] =Pr[V_j−V_i> τ_{i/ k}−τ_{j/ k}] ∀i∈I_k, i=j (3.3.11) and, given the assumptions on the distribution of the residuals τ_{j/ k}, probability (3.3.11) can be expressed as a multinomial logit model:

p[j/ k] = exp(Vj/θ )

i∈Ikexp(Vi/θ ) (3.3.12)

At the upper level, the choice is made among groups of alternatives, with each groupkbeing considered as a compound alternative. Groupkwill be chosen if any one of the elementary alternatives belonging to it is chosen. Because the perceived utilities of elementary alternatives are random, the probabilityp[k] that group k is chosen is the same as the probability that one of its elementary alternatives has the maximum perceived utility among all elementary alternatives in the choice set.

Equivalently, probabilityp[k]can be obtained by assigning to groupkan inclusive perceived utilityU_k^∗equal to the utility of its most attractive alternative, that is, the maximum utility of all the elementary alternatives belonging to the group

U_k^∗=max

j∈Ik

{Uj} =max

j∈Ik

{Vj+τ_{j/ k}} +η_k (3.3.13) which is again a random variable. The probability that groupk is chosen is then the probability that its inclusive perceived utilityU_k^∗is greatest among the different groups.

The perceived utilities Uj =Vj +τj/ k of the various alternatives j in group kare, by assumption (3.3.8), independently distributed Gumbel variables with the same scale parameterθ. As stated earlier, the maximum of a set of such random variables is also distributed as a Gumbel variable with parameterθand with mean equal to:

V_k^∗=E[U_k^∗] =E maxj∈Ik

{V_j+τ_{j/ k}}

=θln

j∈Ik

exp(Vj/θ )=θ Y_k (3.3.14)

whereV_k^∗ is the Expected Maximum Perceived Utility (EMPU) or inclusive sys- tematic utility andYk is the corresponding logsum variable. In the expression for the inclusive perceived utility (3.3.13), the r.v. max(Vj+τ_{j/ k})can be replaced by

3.3 Some Random Utility Models 103 its expected value plus a deviationτ_k^∗12from this value, which is another zero-mean Gumbel variable with parameterθ. Then:

U_k^∗=θ Y_k+τ_k^∗+η_k=θ Y_k+ε_k^∗ (3.3.15) Thus, the perceived utility of groupkhas a mean valueθ Y_k and a deviationε_k^∗, which is the sum of the two zero-mean random variablesτ_k^∗andη_k. The basic as- sumption of the hierarchical logit model is that at each choice level the random residuals of the available alternatives are i.i.d. Gumbel variables; that is, it is as- sumed that theε_k^∗are i.i.d. Gumbel variables with zero mean and parameterθo, with ηk distributed in a way that makes this so:

E ε_k^∗

=0 ∀k Var

ε_k^∗

=π²θ_o²/6 ∀k (3.3.16)

In accordance with this assumption, the choice probability of group k is ex- pressed by a multinomial logit model. In fact:

p[k] =Pr

U_k^∗> U_h^∗

=Pr

θ Y_k−θ Y_h> ε_h^∗−ε^∗_k

∀h=k and, given the results of the previous section:

p[k] = exp(θ Yk/θ_o)

hexp(θ Yh/θ_o)= exp(δYk)

hexp(δYh) (3.3.17) whereδis the ratio of parametersθandθ_oassociated with the two choice levels:

δ=θ/θo (3.3.18)

Replacing expressions (3.3.12) and (3.3.17) in (3.3.9), the choice probability of the generic elementary alternativej is obtained:

p[j] =p[j/ k] ·p[k] = exp(Vj/θ )

i∈Ikexp(Vi/θ )· exp(δYk)

hexp(δYh) (3.3.19) Variances and covariances of the random residualsε_j of the elementary alter- natives’ overall perceived utility (3.3.8) can also be derived. The variance of ε_j coincides with that of the random residualε_k^∗because the two variables are the sum of the same variable(η_k)and another independent Gumbel variable (τ_k^∗ andτ_{j/ k}, respectively) with zero mean and the same parameterθ. Therefore:

Var[ε_j] =Var ε_k^∗

=π²θ_o²/6 ∀j (3.3.20)

12From the Gumbel variable’s property of stability with respect to maximization, the r.v.τ_k^∗ is distributed like the variableτ_{j/ k}associated with each alternativejbelonging to groupk, that is, as a Gumbel variable with zero mean and parameterθ.

The variance of the random residualε_jis identical for all elementary alternatives.

There is also a positive covariance between the random residuals of any pair of alternativesiandj belonging to the same group. In fact:

Cov[ε_i, ε_j] =E

(η_k+τ_{i/ k})·(η_k+τ_{j/ k})

=E η²_k

+E[ηkτj/ k] +E[ηkτi/ k] +E[τi/ kτj/ k] ∀i, j∈Ik

Because all the variablesη_k, τ_{i/ k}, and τ_{j/ k} have zero mean and are mutually independent, the first term is equal to the variance ofη_k and the others are zero, because they are the covariances of independent random variables:

Cov(εi, ε_j)Var(ηk) ∀i, j∈I_k (3.3.21) However, if two elementary alternativesiandj belong to different groups, all the terms are zero and so also is the covariance betweenεi andεj.

The variance ofηk can be expressed as a function of the two parametersθ and θ_o:

Var[η_k] =Var[ε_j] −Var[τ_{j/ k}] =π²(θ_o²−θ²)

6 ∀k (3.3.22)

From the previous results, the structure of the random residual variance–

covariance matrix can be determined. The elements of the main diagonal are all equal to the variance of the residualsε_j, expressed by (3.3.20). The covariance be- tween each pair of alternatives belonging to the same group is the same and equal to the value given by (3.3.21) and (3.3.22), whereas the covariance between alter- natives belonging to different groups is zero. Therefore, if the alternatives of each group are ordered sequentially, the resulting variance–covariance matrix has a block diagonal structure. Figure3.6shows a choice tree and the corresponding variance–

covariance matrix.

It is also possible to express the coefficient of correlation between the perceived utilities of two alternativesiandjas a function of the basic model parameters:

ρij=

⎧

⎨

⎩

Cov[εiε_j]

Var[εi]^1/2Var[εj]^1/2 =^θo2−θ2

θ_o² =1−δ² ifi, j∈I_k

0 otherwise (3.3.23)

The parametersθ, θ_o, andδplay a major role in the structure of the hierarchical logit model and in determining the choice probabilities.

First, parameterδdefined by (3.3.18) must take on values in the interval[0,1]. It is defined by the ratio between two nonnegative quantities and, because the variance ofε_j(π²θ_o²/6)cannot be less than that of one of its componentsτ_{j/ k}(π²θ²/6), the following must hold.

θ_o≥θ → 0≤δ≤1

As the variance ofτj/ ktends to that ofεj (i.e., asθtends toθo), parameterδtends to one. In this case, the variance ofηk(3.3.22) and the covariance between two alter-

3.3 Some Random Utility Models 105

car motorcycle

walking bus metro

π² 6

⎡

⎢

⎢

⎢

⎣

car motorcycle walking bus metro

θ_o² θ_o²−θ² 0 0 0

θ_o²−θ² θ_o² 0 0 0

0 0 θ_o² 0 0

0 0 0 θ_o² θ_o²−θ²

0 0 0 θ_o²−θ² θ_o²

⎤

⎥

⎥

⎥

⎦

Fig. 3.6 Choice tree and variance–covariance matrix of a single-level hierarchical logit model

natives belonging to the same group (3.3.21) both tend to zero, and the hierarchical logit model (3.3.19) reduces to the multinomial logit model.

This can be seen by substitutingδ=1 in (3.3.19), yielding:

p[j] = exp(Vj/θ )

i∈Ikexp(Vi/θ )· exp[ln

i∈Ikexp(Vi/θ )]

hexp[ln

i∈Ihexp(Vi/θ )]

= exp(Vj/θ )

h

i∈Ikexp(Vi/θ ) (3.3.24)

which is a multinomial logit model with a different expression for the summation in the denominator.

If the variance ofτj/ k tends to zero (i.e.,θtends to zero), parameterδ will also tend to zero. In this case, the two probabilities in the model (3.3.19) will be modified as follows.

– The conditional choice of an elementary alternative within a group degenerates into a deterministic choice of the alternative with maximum systematic utility:

θ→0limp[j/ k] =lim

θ→0

exp(Vj/θ )

i∈Ikexp(Vi/θ )=

1 ifVj=maxi∈Ik(Vi)

0 otherwise (3.3.25)

– The systematic utilities of alternative groups, equal toθ Y_k, assume the value of the maximum systematic utility among the elementary alternatives in each group:

θ→0limθ Y_k=lim

θ→0θln

i∈Ik

exp(Vi/θ )=max

i∈Ik

(V_i)

The choice probability of the group therefore becomes p[k] = exp[max_i∈Ik{V_i}/θ_o]

hexp[max_i∈Ih{V_i}/θ_o] (3.3.26) Thus, if parameterδis zero, the random residuals associated with the conditional utilities of elementary alternatives within a group are zero(Var[τj/ k] =0). In this case, the choice between groups is modeled by comparing, using a probabilistic logit model, the alternatives having maximum systematic utility within each group:

a random residual at the group level still exists, and the maximum utility alternative is deterministically chosen within each group.

Some special cases of the model presented can be analyzed. If a groupkconsists of a single alternativej, then p[j/ k] =1 and the general expression (3.3.19) for this alternative becomes

p[j] = exp(Vj/θo) exp(Vj/θ_o)+

h=kexp(δYh) (3.3.27) In some applications of the single-level hierarchical logit model, and in particular for systems of partial share models covered in the next chapter, the systematic utility V_jof alternativej in groupkis decomposed into two parts: one part,V_k, associated with groupk itself; and a second part,V_{j/ k}, associated with the alternative within the group:

Vj=Vk+Vj/ k (3.3.28)

This decomposition leads to an alternative formulation of the choice probabilities p[j/ k]andp[k]. By replacing (3.3.28) in (3.3.12) and (3.3.17), respectively, it fol- lows that

p[j/ k] = exp(Vj/θ )

i∈Ikexp(Vi/θ )= exp[(Vk+Vj/ k)/θ] exp(Vk/θ )·

i∈Ikexp(Vi/ k/θ )

= exp(Vj/ k/θ )

i∈Ikexp(Vi/ k/θ ) (3.3.29)

and

p[k] = exp(Vk/θ_o+δY_k^′)

hexp(Vh/θ_o+δY_h^′) (3.3.30) because

δY_k=δln

j∈Ik

exp(Vj/θ )=δln

j∈Ik

exp

(V_k+V_{j/ k})/θ

=δln

exp(Vk/θ )·

j∈Ik

exp(Vj/ k/θ )

=δV_k/θ+δln

j∈Ik

exp(Vj/ k/θ )

=V_k/θ_o+δY_k^′

3.3 Some Random Utility Models 107 whereY_k^′ is the logsum variable of groupkobtained with the alternative specific systematic utilitiesVj/ k.