DOMINANCE AMONG ALTERNATIVES IN RANDOM UTILITY MODELS:
A GENERAL FRAMEWORK AND AN APPLICATION TO DESTINATION
CHOICE
Ennio Cascetta
Andrea Papola
University of Naples “Federico II” - Italy
1 INTRODUCTION
Random utility (r.u.) models simulate the choice of a decision maker among a
set of known alternatives (choice set). More in detail they reproduce the
choice probability of any alternative as the probability the utility of that
alternative to be the maximum among the utilities of all the alternatives
included in the choice set.
Therefore, simulating the choice of an alternative by using r.u. models
requires the knowledge of the choice set of any decision maker. If this is
unknown, the analyst should first simulate the choice set of each individual
and then his/her choice of within the choice set. Choice set simulation is not
an easy task: few models are available, the theoretically most appealing
models are quite complex and sometimes it is not straightforward to find
significant attributes to reproduce the availability/perception of an alternative
that is the probability of its belonging to the choice set. Due to these reasons,
very often r.u. models are used without a preliminary choice set simulation,
also in those choice contexts where the real choice set of any decision maker
is quite unknown to the analyst. Many examples can be reported within the
transport demand modelling literature; perhaps the most significant is the
destination choice model where the same choice set including hundreds of
traffic zones is generally assumed for any decision maker. Obviously any
decision maker knows and considers only a much more limited number of
traffic zones in his/her destination choice but the problem is how to model this
individual choice sets and with which attributes?
In this context, this paper has a twofold objective:
• to propose a general framework to introduce the concept of dominance
among alternatives in choice set modelling i.e. how to define dominance
attributes and how to use them in choice set modelling;
• to propose operational destination choice models including dominance
variables, (taking into account also of spatial effects) as well as spatial
variables reproducing the better knowledge of zone with a privileged spatial
position.
The paper is organized as follows: in section 2 the different choice set
simulation approaches are briefly reviewed. In section 3 the use of dominance
in choice set simulation is treated. In section 4 a destination choice model
involving dominance variables as well as spatial variables is proposed and the
relative estimation results presented.
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2 CHOICE SET SIMULATION
The first choice set simulation approach proposed in literature is the Mansky’s
formulation [1977]:
N
p i ( j ) = ∑ p i ( j / C )[ X i ] ⋅ p i (C )[Y i ]
(1)
C =1
where the probability of choosing an alternative j by user i is obtained as the
sum over all the N possible choice sets C definable within a given universal
set of alternatives, of the probability of C to be the choice set for individual i,
pi(C), multiplied by the probability of j to be the alternative chosen by
individual i within C, pi(j/C). In equation (1), Xi and Yi are, respectively, the
vectors of attributes influencing the utility and the availability/perception of all
the alternatives for individual i.
Mansky approach to choice set simulation requires the explicit enumeration of
all the N possible choice sets, where N increases exponentially with the
number of the alternatives as well as the computational effort to apply this
approach. Furthermore it does not provide a specific model to simulate p(C) 1.
In 1987 Swait and Ben-Akiva provide the following way to simulate p(C) within
the Mansky formulation:
p (C ) =
1
⋅ ∏ p( j ∈ C ) ⋅ ∏ p( j ∉ C )
1 − p (∅) j∈C
j∉C
where p(j∈C), p(j∉C) represent the probability of any alternative j respectively
to belong and not to belong to C and they are assumed to be independent
among each other. Moreover p(∅) is the probability of the empty set and the
term 1/(1-p(∅)) is needed in order to have:
N
∑ p(C ) = 1
C =1
In 1995 Morikawa proposes a different formulation reproducing the Mansky
approach where the single p(j∈C) are differently aggregated; but the real
computational advantages introduced by this approach should still be
investigated (Papola 2005).
A different approach requiring a computational effort which is really much
more limited with respect to the Mansky approach is the IAP (Implicit
Availability Perception) r.u. model (Cascetta, Papola 2001) where the
individual choice set is implicitly simulated by adding to the utility of each
alternative j the logarithm of its “availability/perception” APj:
U *j = U j + ln APj
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where APj may assume all figures between 0 (not available/perceived) and 1
(completely available and perceived). Consequently, according with the r.u.
theory:
p[ j ] = prob[U i + ln APi ≤ U j + ln APj
∀i ∈ C ]
With some assumptions (U*j independently and identically Gumbel distributed,
E[lnAPj]= lnE[APj]=ln p(j∈C)) the (2) becomes:
p[ j ] =
exp(V j )[ X j ] ⋅ p( j ∈ C )[Y j ]
∑ h exp(Vh )[ X h ] ⋅ p(h ∈ C )[Yh ]
(2)
which is formally analogous to Foteringham 1983. Obviously a further
simplification consists in reproducing the availability/perception of an
alternative j by introducing the availability/perception attributes Yj directly in
the systematic utility of that alternative. For example, if using a multinomial
logit (MNL) r.u. model:
p[ j ] =
with
exp(V j )
∑h exp(Vh )
V j = ∑ n β n X jn + ∑ k γ kY jk
(3)
(4)
where βn and λk are coefficients of utility and availability/perception attributes.
In other words, the lower (greater) probability of an alternative j to be
perceived (i.e. to belong to the choice set) is simulated by reducing
(increasing) its systematic utility with the availability/perception attributes.
Analogously with models (1) and (2) the effect will be to reduce the over all
choice probability of the less available/perceived alternatives with respect to
those which are more available/perceived.
3 THE ROLE OF DOMINANCE IN CHOICE SET SIMULATION
As described in the previous section, any choice set simulation approach
need some availability/perception attributes Yj in order to simulate the
probability of any alternative to belong to the choice set, p(j∈C). In this paper
the main idea is to take account of dominance features in order to define
dominance attributes to be used as perception attributes in the choice set
simulation.
Indeed, in many choice contexts, it can be observed that some alternatives
are not taken into account since they are “dominated” by other alternatives. In
general, an alternative i is dominated by another alternative j if i is “worse”
than j, with respect to one or more characteristics, without being better with
respect to any characteristic. The concept of dominance among alternatives is
widely recognized in the project evaluation, e.g. through Multi-criteria
Decision-Making (Chankong and Haimes 1983, Haimes and Chankong 1985)
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where dominated alternatives are excluded from the “choice set” on the basis
of the principle of rationality (transferability of preferences). Instead, they have
never been used (in author knowledge) within the discrete choice theory.
The formal simulation of dominance implies the specification of:
(a) a rule to define comparable alternatives;
(b) a rule to define a dominated (or dominant) alternative between a pair of
comparable alternatives;
(c) a method of using such information.
In this paper an approach to model dominance within the random utility theory
is proposed, specifying the above three points under the general assumptions
of the r.u. theory. As regards (a), two alternatives, i and j, are defined
comparable if they are characterized by the same attributes (i.e. if their utility
functions are specified in the same way).
With regard to (b), a simple rule is to define i dominating j if all utility attributes
are larger (not smaller) in i than in j and all disutility attributes are smaller (not
larger) in i than in j, with at last one disequality strictly satisfied. Stronger
dominance rules can be generated, for example by introducing some
thresholds in these comparisons between attribute values (e.g. i dominates j if
the utility (disutility) attributes assumes in i a value greater (less) than that
assumed in j by some threshold) or some specific factors generated by the
comparison between i and j (e.g the spatial effects, see next section).
Regarding point (c), dominance information can be used both in deterministic
and in probabilistic choice set simulation. First of all, the list of all the
dominated alternatives has to be identified. Then a dominance degree of each
alternative j, that is the number of alternatives within C dominating j, can be
identified. Successively, all dominated alternatives (perhaps with a dominance
degree greater than a certain threshold) can be deterministically excluded by
the choice set or, alternatively, dominance attributes can be generated for
each alternative (e.g. the dominance degree itself) and used as perception
attributes Y in any of the probabilistic choice set simulation approaches
described in the previous section.
4 A DESTINATION CHOICE MODEL WITH IMPLICIT CHOICE SET
SIMULATION INVOLVING DOMINANCE AND SPATIAL VARIABLES
As already mentioned in the introduction, in destination choice contexts it is
generally assumed that the choice set is composed by all traffic zones
included in the study area. This is usually a very large number (hundreds of
elements) and consequently the assumed choice set is generally unrealistic.
In particular, in a sufficiently large study area like the urban area of a large
town, most of the residents know only a limited part of the city with sufficient
detail to evaluate its attractive capacity for commercial, sporting, recreational
activities and the effective cost to reach it. Therefore, when simulating the
destination choice context through r.u. models, the analyst, generally, would
strongly need a preliminary choice set simulation for any decision maker.
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Moreover, in r.u. destination choice models, the same attributes are generally
used for each alternative zone, trip purpose and demand segment (individual
category). These can be divided into attractiveness attribute of the zone
(depending on the trip purpose and on the individual category) and an
impedance attribute reproducing the user cost to reach that zone from his/her
origin. Hence, according to the definition rules given in the previous section,
all the alternatives of this choice context are comparable by definition.
For all these reasons, in this paper the general methodology proposed in the
previous section in order to introduce dominance features within the choice
set simulation of r.u. models is applied first to this choice context. Obviously,
simulating the “perception” of an alternative zone makes sense for those trip
purposes (e.g. shopping, sport and leisure) in which the user actually chooses
the destination zone, i.e. for those trips that are generally called “nonsystematic”.
The model proposed and developed in this paper with reference to destination
choice in the context of travel demand modelling can easily be extended to
other spatial interaction models (e.g. residential or economic activity location
models) as well as to other travel related choices (e.g. mode, departure time,
parking location and type, scheduled service and road network path) by
adapting the proposed methodology to the specific choice context.
4.1 Specification of the model
The high number of alternatives generally involved in destination choice
suggests the use of implicit choice set simulation approaches both Logit (see
eqns. (3) and (4)) and IAP Logit (2).
As “utility” attributes Xj of the generic destination zone j for non-systematic
trips, the natural logarithm of the number of workplaces in services and in
commerce located in j was used as attractiveness attribute of j and the logsum
on the MNL mode choice 2 relative to the oj relation (o being the user origin)
was used as impedance attribute of j.
As “availability/perception” attributes Yj, different dominance attributes were
generated and used together with other spatial availability/perception
attributes for each zone j. In particular, two different dominance degrees were
generated using distance instead of logsum as an impedance attribute:
(a) a simple dominance degree of each alternative j that is the number of
zones containing a number of workplaces larger than that in j and, at the
same time, closer, with respect to j, from the user origin o;
(b) a spatial dominance degree involving the spatial position of the zones. In
particular it was assumed that i spatially dominates j if it dominates j in
relation to point (a) and i is “along the path” to reach j from the user origin
o. In this case, i represents an intervening (and better) destination
opportunity along the path, or bundle of paths, towards j (see Fig.1).
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These dominance attributes were generated sequentially and used
simultaneously. In other words, a first spatial dominance degree was
generated as in point (b) representing the number of zones dominating j most
strongly; then a second dominance degree was generated as in point (a)
representing the number of zones dominating j non spatially.
D1
O = origin
D = destinations
WP = workplaces
dOD = OD distance
O
D2
D3
D4
WP1 = WP2 = WP3 = WP4
dOD1 = dOD2 = dOD3 < dOD4
D1,D2,D3 dominate D4
D2 spatially dominates D4
area of possible zones
spatially dominating D4
Fig.1 - Example of spatial and non spatial dominance
Together with dominance variables, other perception attributes has been
identified in order to simulate the perception of a destination alternative.
These are spatial variables reproducing the greater knowledge and the
consequent greater perception probability of zones with a privileged spatial
position (Improta Papola 2000). In more detail, each individual is assumed to
know better some zones of the area in term of opportunities, attractiveness,
time and cost to reach them. These are mainly the zones where he/she
carries out activities systematically or, however, frequently: typical examples
are the residential zone, the “working” zone, the historical and the commercial
centre. Around these spatial positions and the shortest paths connecting them
among each other, a number of more known, sometime more convenient
(when doing chain trips) and therefore more perceived zones can be
identified. The greater perception of this zones can easily be simulated within
the r.u. models through dummy variables.
In this paper, in particular, since information about the working zone of each
individual was not available, dummy variables have been proposed in order to
distinguish the residential zone, the zones within a certain radius from the
residential zone and the zones along the path between the residential zone
and the city centre (respectively intra_j/o, clos1_j/o, clos2_j/o, centre_j/o). In
Tab.1 the proposed specification is reported.
MNL,
IAP
Logit
∑βX
∑γ Y
non-systematic trips
n
n
nj
k
k kj
= β wp ln wp _ ser − comm _ j + β log sum log sum _ oj
= γ dom 1dom1 _ j / o + γ dom 2 dom 2 _ j / o + γ centrecentre _ j / o +
+ γ intra intra _ j / o + γ clos1clos1 _ j / o + γ clos2clos 2 _ j / o
Tab.1 - Specification of the model
where:
ln wp_ser-comm_j = natural logarithm of workplaces in services and
commerce of zone j;
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logsum_oj = mode choice logsum relative to the oj pair;
dom1_j/o = spatial dominance degree of zone j: it indicates the number of
zones i for which the following occurs simultaneously:
(a) i contains a number of workplaces larger than that in j;
(b) the distance from the user origin to i (distoi) is shorter than that
to j (distoj);
(c) i is along the path o-j: distoi + disti j<1.2⋅distoj.
dom2_j/o = simple dominance degree; it indicates the number of zones for
which conditions (a) and (b) simultaneously occurs regardless
condition (c);
centre_j/o = dummy variable equal to one for all zones j, along the path from
the user residence o to the city centre c:
distoj + distjc <1.2⋅distoc and zero otherwise.
intra_j/o = dummy variable equal to one for the user residence zone and
zero otherwise;
clos1_j/o = dummy variable equal to one for all zones within a given radius (1
kilometer) of the user residence and zero otherwise;
clos2_j/o = dummy variable equal to one for all zones whose distance from
the user residence is included in the interval [1km-2km] and zero
otherwise.
All used threshold values were estimated on the basis of the obtained
estimation results.
4.2 Estimation of the model
In order to estimate the proposed model, the Maximum Likelihood (ML)
method was used. In other words “optimal” β and γ values are those
maximizing the likelihood function, or rather, its natural logarithm:
[ β , γ ] ML = arg max ln L( β , γ ) = arg max ∑i=1...n ln p i [ j (i )]( X i , Y i , β , γ )
(5)
where L(β,γ) is the likelihood function, j(i) the alternative actually chosen by
decision-maker i, pi[j(i)] the relative choice probability given by the model (both
MNL and IAP Logit) and the sum is extended to all the sample. To estimate
MNL models, the Alogit program [Daly, 1987] was used while to estimate IAP
logit models an ad hoc estimation procedure was implemented (with the Excel
software).
The database used to estimate the model consists of 7182 home based nonsystematic trips (shopping, sport, leisure, etc.) selected among 90,000 daily
trips of residents in the metropolitan area of Rome. The study area is divided
into 463 traffic zones that represent the complete choice set of the model.
In Tab.2 the estimation results of the MNL model specified as in Tab.1 are
reported; those relative to the IAP Logit model are not reported since they are
very similar to MNL results. In particular, five specifications are reported in
increasing order of attributes used. For each specification, the estimated value
of each coefficient and, in parenthesis, the relative t-statistic values is
reported.
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As can be seen, all coefficient signs are consistent with expectations: utility
attributes (βwp, βlogsum) and positive perception attributes (βcentre, βintra, βclos1,
βclos2) have a positive coefficient while negative perception attributes (βdom1,
βdom2) have a negative coefficient. Moreover, all coefficients in all
specifications are very significant and there is a considerable improvement in
the goodness of fit statistic when passing from one specification to the next.
This result is confirmed by the likelihood ratio test results on comparisons
between pairs of models reported in Tab.3. In particular, a substantial
improvement in the goodness of fit of the model is achieved by using
dominance attributes that are always very significant, thereby confirming the
importance of this kind of approach in simulating user destination choice.
Moreover, in specifications (3), (4) and (5) it can be observed that the
coefficient relative to the stronger dominance degree (βdom1) is always much
more negative than that relative to the simple dominance degree (βdom2). This
confirms the importance of considering zone spatial positions, when defining
and identifying the dominated alternatives and in general in the choice set
simulation of a destination choice context. Furthermore a large significance
can be noted for all the spatial perception attributes used to simulate the
better knowledge of the zones with a privileged spatial position.
Logit
specifications
βwp
(t-statistic)
βlogsum
(t-statistic)
βdom1
(t-statistic)
βdom2
(t-statistic)
βcentre
(t-statistic)
βintra
(t-statistic)
βclos1
(t-statistic)
βclos2
(t-statistic)
ln (0)
ln (β)
ρ2
(1)
(2)
(3)
(4)
0.911
(37.5)
3.204
(114.8)
0.367
(15.7)
2.232
(63.6)
0.392
(16.6)
2.209
(62.6)
-0.154
(-16.7)
-0.060
(-14.7)
0.343
(14.4)
1.732
(36.4)
-0.163
(-17.8)
-0.065
(-15.7)
0.925
(14.3)
(5)
0.407
(16.6)
0.794
(14.8)
-0.168
(-17.8)
-0.087
-0.033
(-28.4)
(-9.1)
0.547
(12.2)
0.628
(8.7)
0.385
(5.3)
1.407
(22.7)
-19994.97 -19994.97 -19994.97 -19994.97 -19994.97
-12582.78 -11754.37 -11718.43 -11620.01 -11204.30
0.3707
0.4121
0.4139
0.4189
0.4396
Tab.2 - Different specifications of the destination choice Logit model with
implicit choice set simulation
Interestingly, a progressive decrease in the logsum coefficient (βlog) towards
“correct” 3 values can be obtained by introducing perception attributes. This
result shows how crucial is choice set simulation in order to avoid problems of
coefficient mis-specification (Williams H. and Ortuzar J., 1982).
In Tab.3 the generic element ij represents the result of the likelihood ratio test
on comparisons between specification i and specification j. In the same table
the 95th percentiles of a χ2 random variable with a number of degrees of
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freedom equal to the difference between the number of coefficients used in
specification i and that used in specification j are also reported.
As already stated, the test is largely satisfied in all cases.
specifications
(1)
(1)
-
(2)
(3)
(4)
(5)
(2)
(3)
(4)
1656.82
(3.84)
1728.70 71.88
(5.99)
(3.84)
1925.54 268.72 196.84
(7.81)
(5.99)
(3.84)
2756.96 1100.14 1028.26 831.42
(12.59) (11.07) (9.49) (7.81)
(5)
-
Tab.3 - Likelihood ratio tests results on comparisons between pairs of models
5 CONCLUSIONS
In this paper the concept of dominance among alternatives is introduced in the
choice set simulation and in particular in destination choice models for nonsystematic trips. Indeed, in this choice dimension and for this kind of trips, it is
particularly important to simulate the individual choice set because of the very
large size of the universal choice set. For the same reason, implicit choice set
simulation approaches are also suggested. Moreover, the way in which the
destination choice model is generally specified makes it possible and
theoretically convincing to introduce dominance features in the choice set
simulation.
Different dominance variables were defined and used in implicit choice set
simulation models together with spatial variables reproducing the better
knowledge of zone with a privileged spatial position. The estimation results
show a generally high significance of these attributes and a considerable
improvement in the goodness of fit statistics of the models. This result
confirms both the importance of choice set simulation in this choice context
and the improvements that can be achieved in order to do it, by considering
dominance and spatial variables. Further empirical evidence supporting the
validity of the proposed model will have to be found in successive application
phases, which will be the first step in future research.
Acknowledgments
We are grateful to STA (Società Trasporti Automobilistici) S.p.A. for its
willingness to make household surveys available and to Paolo Quarantotto for
his help in estimating the model.
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1
In the following, for simplicity, the reference i to the generic individual will be omitted.
The logsum on the mode choice relative to an oj relation represents the expected value of
the maximum among the utilities perceived by users in going from o to j using different
transport modes (m) and for the MNL it is:
log sum _ oj = E[max{U m / oj }] = ln ∑ exp(Vm / oj )
2
m
3
m
The logsum coefficient, consistently with the random utility theory, should assume a value
included in the [0-1] interval (see Ben-Akiva and Lerman 1985, Cascetta 2001).
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