Random Utility Theory
3.3 Some Random Utility Models
3.3.3 The Multilevel Hierarchical Logit Model *
3.3 Some Random Utility Models 107 whereYk′ is the logsum variable of groupkobtained with the alternative specific systematic utilitiesVj/ k.
Fig. 3.7 Choice tree of multilevel hierarchical logit models
In this notation, single nodes are indicated with lowercase letters (o, i, j, l, r, s), groups of nodes with capital letters (A, I ), and nodes related in some structural way to particular other nodes as lowercase letter functions of those other nodes (a(r), p(r, s)).
At each choice node, whether intermediate or initial, it is assumed that a con- ditional choice is made among all the available alternatives. These alternatives are represented by nodesr, and may be either elementary alternatives (leaves of the tree) or compound alternatives (intermediate nodes). For any such alternative, the node that represents the choice situation directly involving it isa(r), and the full set of alternatives in the choice situation isIa(r).
To model the conditional choice, a perceived utilityUr/a(r) is assigned to each node (alternative)r. This is a random variable that, as usual, is decomposed into the sum of its mean,Vr, and a random residual,εr/a(r), with the following properties.
– Ifr is a leaf of the tree,Vr is the expected value of its perceived utilityUr/a(r). Ifris an intermediate node,Vr is the expected value of the maximum perceived utility (EMPU or inclusive value) of the alternatives, whether elementary or not, belonging toIr;
– The random residualsεr/a(r)of all nodesr that are descendants ofa(r)are as- sumed to be i.i.d. Gumbel variables with zero mean and parameterθa(r). There- fore, the variance Var[εr/a(r)] =π2θa(r)2 /6 is associated with the conditional choice made at nodea(r) from all the elementary alternatives directly or indi- rectly reached froma(r).
From the above assumptions, it follows that
Ur/a(r)=Vr+εr/a(r) ∀r∈Ia(r) E[εr/a(r)] =0
Var[εr/a(r)] =π2θa(r)2 6
(3.3.31)
From the results on the expected value of the maximum of Gumbel variables re- ferred to in Sect.3.3.1, the systematic utility assigned to any node can be determined
3.3 Some Random Utility Models 109 recursively by starting from the choice tree leaves as
Vr=
E[Ur/a(r)] ifr∈I
θrln
h∈Irexp(Vh/θr)=θrYr ifr /∈I (3.3.32) Under the above hypotheses, the conditional probability of choosing alternative rat the choice nodea(r)is expressed by a multinomial logit model:
p r/a(r)
= exp(Vr/θa(r))
r′∈Ia(r)exp(Vr′/θa(r)) (3.3.33) and also, from (3.3.32):
p r/a(r)
=exp(Vr/θa(r))
exp(Ya(r)) (3.3.34)
If the alternativer is a compound alternative (i.e.,ris an intermediate node) in (3.3.32), the numerator of (3.3.33) becomes:
exp(Vr/θa(r))=exp θr
θa(r)
Yr
=exp(δrYr)
whereδr is the ratio of coefficientsθr andθa(r). It is analogous to the coefficientδ introduced in the previous section (see (3.3.18)) and, as such, must be in the inter- val[0,1]. In this case, expressions (3.3.33) and (3.3.34) can be reformulated as
p r/a(r)
= exp(δrYr)
r′exp(Vr′/θa(r))= exp(δrYr)
exp(Ya(r)) (3.3.35) Finally, the absolute (unconditional) probability of choosing the elementary al- ternativej∈I can be obtained from the definition of conditional probability and from the assumptions made on the tree choice mechanism:
p[j] =p j/a(j )
·p a(j )/a
a(j )
· · · j∈I or
p[j] =p j/a(j )
r∈Aj
p r/a(r)
j ∈I (3.3.36)
Replacing expressions (3.3.34) and (3.3.35) in (3.3.36) yields:
p[j] =exp(Vj/θa(j )) exp(Ya(j )) ·
r∈Aj
exp(δrYr)
exp(Ya(r)) j∈I (3.3.37) and also
p[j] =exp(Vj/θa(j )) exp(Yo) ·
r∈Aj
exp(δrYr) exp(Yr)
=exp(Vj/θa(j )) exp(Yo) ·
r∈Aj
exp
(δr−1)Yr
j∈I (3.3.38)
Absolute choice probabilities p[j] can therefore be computed recursively through the following steps.
– Calculateδr =θr/θa(r)for each noder.
– Recursively calculate valuesYr, with expression (3.3.32).
– Calculate probabilitiesp[j], j∈I, with expression (3.3.38).
Given: θr r /∈I withθr=0 ifr∈I Ir r /∈I withIr= ∅ifr∈I Vj ∀j∈I
The model described can be demonstrated with the choice tree in Fig.3.8. The leaves of the tree (AI,CD,CP,BS,ST,FT) represent the elementary choice alterna- tives that, in this example, are the transport modes available for an intercity trip: air (AI), car driver (CD), car passenger (CP), bus (BS), slow train (ST), and fast train (FT). The intermediate nodes represent groups of alternatives, or compound alter- natives. NodeCRrepresents the car, combining the two alternatives of car driver and car passenger, nodeLT the public land transport modes (bus, slow train, and fast train), and nodeRW combines the railway alternatives. Finally, the respective values of parametersθandδare assigned to each intermediate node and to the root.
Following expression (3.3.36), the choice probability of fast train (FT) can be written as
p[FT] =p[FT/RW].p[RW/LT].p[LT/o]
where
p[FT/RW] = exp(VFT/θRW)
[exp(VST/θRW)+exp(VFT/θRW)]=exp(VFT/θRW) exp(YRW) with
YRW=ln
exp(VST/θRW)+exp(VFT/θRW) p[RW/LT] = exp(θRWYRW/θLT)
exp(θRWYRW/θLT)+exp(VBS/θLT)
= exp(δRWYRW)
exp(δRWYRW)+exp(VBS/θLT)=exp(δRWYRW) exp(YLT) with
YLT =ln
exp(δRWYRW)+exp(VBS/θLT)
(3.3.39)
3.3 Some Random Utility Models 111
= AI CD CP BS ST FT
π2 6
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
AI CD CP BS ST FT
θo2
θo2 θo2−θCR2 θo2−θCR2 θo2
θo2 θo2−θLT2 θo2−θLT2 θo2−θLT2 θo2 θo2−θRW2 θo2−θLT2 θo2−θRW2 θo2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Fig. 3.8 Choice tree and variance–covariance matrix for a multilevel hierarchical logit model
p[LT/o] = exp(θLTYLT/θo)
exp(θLTYLT/θo)+exp(θCRYCR/θo)+exp(VAI/θo)
= exp(δLTYLT)
[exp(δLTYLT)+exp(δCRYCR)+exp(VAI/θo)]=exp(δLTYLT) exp(Yo) with
YCR=ln
exp(VCD/θCR)+exp(VCP/θCR) Yo=ln
exp(δLTYLT)+exp(δCRYCR)+exp(VAI/θo)
The absolute choice probability can be written in the form (3.3.38) as follows.
P[FW] =exp(VFT/θRW) exp(Yo) ·exp
(δLT−1)YLT
·exp
(δRW−1)YRW This choice probability can be thought of as resulting from a choice process in which the decision-maker first chooses the compound alternative “public land trans- port” from the available alternatives, which in this case are air, the compound alter- native “car” and the compound alternative “public land transport”. Subsequently, she chooses the group “train” from the alternatives available within the land trans- port group (bus and train), and finally fast train from the two elementary alternatives (fast and slow train) that make up the train group.
Returning to the general model, it is possible to express the variances and covari- ances of the random residuals as functions of the parametersθr. Rigorous demon- stration of these results involves the use of GEV models described in Sect.3.3.5.
The same results can be obtained in a less rigorous way using the total variance decomposition method described for the single-level hierarchical logit model in the previous section. It is assumed that the total variance of each of the elementary alternativesj is identical and equal to:
Var[εj] =π2θo2/6 (3.3.40) The overall random residual of each elementary alternativeεjis decomposed into the sum of independent zero-mean random variablesτa(r),r associated with each link of the choice tree. The total variance of an elementary alternative is equal to the sum of the variances corresponding to the links of the (single) branch connecting the root to the leaf that represents the alternative. Furthermore, it is assumed that the random residual variance of each elementary alternativej that can be reached from an intermediate noder, and that is associated to the conditional choice represented by noder itself, is equal toπ2θr2/6. It follows that, for all these alternatives, the sum of the contributions of the variances associated with the links that connectrto j must be identical and equal toπ2θr2/6:
Var[εj/r] =π2θr2/6=Var[τa(j ),j] +Var[τa(a(j )),a(j )] + · · · +Var[τr,f (r,j )] wheref (r, j )is the only descendant ofrthat is on the path fromrtoj.
In Fig.3.8, for example, the variances of the elementary alternativesBS,ST, and FT, corresponding to the conditional choice between public land transport modes represented by intermediate nodeLT, are all equal toπ2θLT2 /6. This variance will correspond to the fraction of variance associated to the link (LT,BS) and to the sum of the variances associated with links (LT,RW) and (RW,ST) or to the links (LT, RW) and (RW,FT).
The random residual variance of the elementary alternatives relative to the con- ditional choice represented by nodea(r), the predecessor ofr, is in turn the sum of the variance corresponding torand the nonnegative term Var[τa(r),r], associated with link(a(r), r); this variance will therefore not be less than that associated with r, or:
θa(r)≥θr (3.3.41)
The variance contribution associated with each link(a(r), r)of the graph can be expressed as
Var[τa(r),r] =π2 6
θa(r)2 −θr2
(3.3.42) Inequality (3.3.41) can be generalized, assigning zero variance andθj =0 to the leaves of the graph, thus yielding:
θj≤θa(j )≤ · · · ≤θo (3.3.43)
3.3 Some Random Utility Models 113 From the preceding expression and the definition of the coefficientsδr =θr/θa(r), it follows that these coefficients must belong to the interval[0,1].
Continuing with the example in Fig.3.8, the variance of alternatives ST and FTinvolved in the conditional choice between railway services (nodeRW) will be π2θRW2 /6, whereas the variance of alternatives involved in the choice between public land transport modes (nodeLT) will be π2θLT2 /6, withθLT ≥θRW. The variance contribution assigned to link (LT,RW) will therefore beπ2(θLT2 −θRW2 )/6.
The variance decomposition model described here allows one to derive the co- variances between the perceived utilities of any two elementary alternativesiandj. This covariance will correspond to the sum of the variances of the random residuals τa(r),r(which are independent with zero mean) associated with the links common to the two branches connecting the root to leavesiandj. Because of the tree structure, these branches can have in common only links from the root to the node where they separate, which is their last node in common. By repeatedly applying (3.3.42), the covariance ofεiandεjis found to be:
Cov[εi;εj] =π2(θo2−θp(i,j )2 )
6 ∀i, j∈I (3.3.44) wherep(i, j )is the first common ancestor of elementary nodesiandj.
If two alternatives have the root node as their first common ancestor, that is, if they do not belong to any intermediate compound alternative, their covariance is zero. The correlation coefficient between two elementary alternatives can be de- duced from expression (3.3.40) and (3.3.44) as follows.
ρ[i, j] = Cov[εi;εj]
[Var[εi] ·Var[εj]]1/2 =θo2−θp(i,j )2
θo2 =1−θp(i,j )2
θo2 (3.3.45) For the tree in Fig.3.8, the covariance between alternativesSTandFT is given byπ2(θo2−θRW2 )/6, the sum of the variances relative to links(o,LT)and (LT,RW).
The covariance betweenSTandBSwill beπ2(θo2−θLT2 )/6 which, as stated before, is less than or equal to the covariance betweenFTandST.
In the literature, the parameterθois sometimes taken to be equal to one because, as shown in Chap. 8 on travel-demand estimation, only the parametersδr can be statistically estimated. Because all the parametersθr but one can be obtained from the coefficientsδr, specifying one of theθrs immediately allows the others to be determined. Settingθo=1 leads to a simple expression for the other parameters. In this case, the covariance and the correlation coefficient between any two elementary alternatives become, respectively,
Cov[εi, εj] =π2(1−θp(i,j )2 ) 6 ρ[εi, εj] =1−θp(i,j )2
In conclusion, the structure of the choice tree is also the structure of the covari- ances between the perceived utilities of the elementary alternatives. Two alternatives
that have no nodes in common along the branches connecting them to the rootoare independent. On the other hand, the covariance between elementary alternativesi andj belonging to the same group (their branches meet at an intermediate node) increases with greater “distance” of their first common ancestor from the root node and with smaller values of the parameterθp(i,j )associated with this node. Further- more, the covariance between the perceived utilities of two alternatives i and j whose first common ancestor is their mutual parent(p(i, j )=a(i)=a(j ))is not less than the covariance between either of them and any other alternative. Continu- ing with the example of Fig.3.8, the covariance betweenSTandFT will be greater than or equal to that of either of the two elementary alternatives with any other elementary alternative.
Choice probabilities are significantly affected by the values of parameters θr, and therefore by the levels of correlation between alternatives. Figure3.9shows the values of choice probabilities for the alternatives in Fig.3.8, for different pa- rametersθr and assuming that all systematic utilities have the same value: VAI= VCD=VCP=VBS=VST =VFT. If the alternatives are independent (specifica- tion 1:θr/θo=1∀r), the model becomes a multinomial logit and all the alternatives have equal choice probabilities. As the correlation increases, that is, as parameters θCR, θLT, and θRW decrease, the choice probability of the most correlated alterna- tives tends to decrease. For example, in specification 3, the alternatives belonging to the two groups car (CD,CP) and public land transport (BS,ST,FT) are strongly correlated with a correlation coefficientρ=0.9775. They tend to be seen as a single alternative and their choice probabilities tend to be equal shares of the probability of a single alternative associated with each group. For the same reasons, the choice probability of alternativeAI, which is not correlated with any other alternative, in- creases with increases in the correlation of the alternatives belonging to the various groups (specifications 2 and 3).
From the previous results, it can easily be demonstrated that multinomial logit and single-level hierarchical logit models are special cases of the multilevel hierar- chical logit. Two different approaches can be used to show this for the multinomial logit model. In the first approach, the tree is that of the multinomial logit model described in Fig.3.1. In this case, there are no intermediate nodes and the ancestor a(j )of every leafj∈I is the rooto. It then follows thatθa(j )=θo, Aj= ∅and, by applying expression (3.3.38), that
p[j] =exp(Vj/θo) exp(Yo)
which, by developing the term exp(Yo), gives rise to expression (3.3.6) for the multinomial logit.
Alternatively the multinomial logit model can be obtained from a tree of any form in which the parametersθr of all the intermediate nodes are the same and equal toθo. In this case, it follows from (3.3.44) that the covariance between any pair of alternatives is equal to zero (the residuals are independent), the coefficients δr=θr/θa(r)are all equal to one, and (3.3.38) reduces to the MNL expression.
3.3 Some Random Utility Models 115
Specification No. 1 2 3 4 5 6 7
θLT/θo 1.000 0.900 0.150 1.000 1.000 0.800 0.400 θCR/θo 1.000 0.900 0.150 0.800 0.800 0.600 0.200 θRW/θo 1.000 0.900 0.150 0.600 0.200 0.600 0.200
p[AI] 0.166 0.180 0.304 0.190 0.205 0.212 0.280
p[CD] 0.166 0.168 0.169 0.166 0.178 0.161 0.161
p[CP] 0.166 0.168 0.169 0.166 0.178 0.161 0.161
p[BS] 0.166 0.161 0.120 0.190 0.205 0.174 0.165
p[FT] 0.166 0.161 0.120 0.144 0.117 0.146 0.117
p[ST] 0.166 0.161 0.120 0.144 0.117 0.146 0.117
Fig. 3.9 Choice probabilities of the multilevel hierarchical logit model of Fig.3.8for varying parameters
The single-level hierarchical logit model described in the previous section can be considered as a special case of a tree with only one level of intermediate nodes
a a(j )
=o ∀j∈I
Furthermore, the parametersθr are all equal toθ whereas the parameter associated with the root is still indicated byθo. It can easily be demonstrated that the choice probability (3.3.19) obtained for the single-level hierarchical logit model results as a special case of expression (3.3.38).
Finally, as in the case of single-level hierarchical logit model, a systematic util- ity can be assigned to structural or intermediate nodes. This could be the part of the systematic utility common to all the alternatives connected by an intermediate node. In this case, ifris a structural node andVr the systematic utility assigned to it, (3.3.35) becomes
p r/a(r)
=exp(Vr/θa(r)+δrYr′) exp(Ya(r))
whereYr′ is the logsum variable associated with a noder calculated without the systematic utilityVr, “transferred” to the structural node. Specifications of this type are used in Chap. 4.