Modeling Choice Set Formation Within the GEV Family of
Models
INTRODUCTION
In standard choice models it is assumed that the alternatives among which
choice is exercised can be exogenously specified by the analyst. Thus, in the
most commonly used discrete choice models (e.g. Multinomial Logit – MNL,
Nested MNL, Multinomial Probit – MNP) it is assumed that some set CnÍ
M, where M is the master set of alternatives, is the true set
from which the choice of person n is observed. (The most common
strategy of choice set specification makes all choice sets equal to the master
set, i.e. Cn=M, " n.)
Strictly speaking, however, set Cn is a latent construct to
the analyst since generally nothing is observed about it except the most
preferred alternative.
Choice set imputation, or generation, is clearly relevant from the
perspective of specifying models of choice processes. Swait and Ben-Akiva
(1986) examined theoretically the impacts of choice set mis-specification when
captivity (i.e. being unable to choose anything except the single alternative
in the choice set) is present among a fraction of the population but ignored
by the analyst, who erroneously specifies Cn=M for
the entire population. They report that (1) alternative-specific constants are
downward biased for the alternative exhibiting captivity, and (2) attribute
slope effects become attenuated by the presence of the unrecognized captivity.
One may safely surmise from their analysis that choice set mis-specification
is deleterious to the analyst’s efforts to determine unbiased taste parameter
estimates in any choice model..
Manski (1977) formulated a two-stage characterization of the choice process
as the basis for model development:
, (1)
where C is a choice set in D (M),
the set of subsets of M, Q(C) is the probability that
C is the true choice set, and P(i|C) is the
conditional probability of choice given set C (zero if iÏ
C). The usual inspiration for specifying Q(C) has been the
notion of random constraints (e.g. travel time limits, reservation prices,
restrictions imposed by other agents) acting upon the formation of the set of
actively considered alternatives.
Swait (2001) proposes a new model of choice set generation belonging to the
GEV (Generalized Extreme Value) family of discrete choice models – to our best
knowledge, this is the first such model. An interesting feature of the model
is that the choice set probabilities need make no use of exogenous
information, but are instead taste-driven. (Covariates may be added to the
model, of course, to aid identification of the choice set generation
probabilities. This possibility is addressed subsequently in the discussion of
model extensions.) This differentiates it from Manski’s two-stage framework,
described above.
Swait’s (2001) contributions are two-fold: (1) the class of GEV discrete
choice models is expanded by a new member, denoted the Choice Set Generation
Logit, or GenL, model; and (2) choice set generation models of a
certain specific structure are shown to be consistent with the GEV class and,
by implication, are shown to be consistent with utility maximizing behavior
under certain conditions.
THE GenL MODEL
To those interested, McFadden (1978) presented a theorem characterizing the
Generalized Extreme Value family of probabilistic choice models. The theorem
relates a generation function G(), which must have certain
characteristics, to a corresponding choice probability. The generation
function
, m
³ 0, (2)
where the y’s are non-negative utilities for each of J alternatives,
m is a scale factor, after substituting
, Vi a latent variable without sign
restrictions to guarantee the non-negativity of the arguments of G(),
the choice probability is given by the familiar expression below:
. (3)
Before defining the proposed model, Swait (2001) first addresses the
composition of the set D (M), the set of
possible subsets of M, the master set of alternatives. In any model of
choice set generation, D (M) is part of the
specification of the choice model, just as are the attributes included in the
utility function. Thus, in one model D (M)
may include all possible subsets of M (of which there are 2J-1);
in another, D (M) may be restricted to sets
of size one and the full choice set (this might be called the "captivity"
model); in yet another, it may be restricted to sets of size L or
smaller, where LÎ [1,2J-1].
With this flexibility in mind, define K to be the number of sets
included in D (M), such that 1£
K£ (2J-1).
The GEV generation function for the GenL model is given next:
(4)
which is a valid GEV generating function that satisfies the conditions of
the GEV Theorem if m /m
k£ 1, "
k. The GenL choice probability for iÎ
{1,…,J} is given by
(5a)
where Ki={k|1£ k£
K, iÎ Ck},
(5b)
(5c)
(5d)
Swait (2001) presents proofs for the above expressions.
Model GenL defines some set Ck is the true choice set is
a function of the expected maximum utility derived from the alternatives in
the set. Hence, as alternative j becomes more attractive, all sets
including j will have increased probability of being the true choice
set; this increase in Vj will not impact all sets equally,
however, since the number of and utility levels of other alternatives will
influence how much impact this increase will have on any given subset of
alternatives. This perspective permits an interesting behavioral
interpretation of GenL, namely that decision makers make choices by
eliminating from consideration all alternatives that do not meet some minimum
utility threshold, then pick the highest utility alternative. Then, ceteris
paribus, higher utility means higher chance of being considered, therefore
higher probability of being chosen. Thus, tastes play the fundamental role in
choice set formation, as opposed, for example, to the role of constraints in
eliminating alternatives. Constraint-based theories have generally been the
assumed form of implementing Manski’s (1977) two-stage formulation, but the
experimental and modeling work of Klein and Bither (1987), for example, shows
that minimal utility thresholds are a viable basis for choice set formation.
An interesting behavioral corollary of the endogeneity of choice set
probabilities is that the Q(Ck)’s change at every
point in the attribute space. Thus, policies affecting the attributes of
alternatives generate their impact in two stages, first by impacting choice
set formation, then by impacting competition among alternatives within subsets
of alternatives.
Swait (2001) goes on to examine many properties of the GenL model, as well
as to estimate its parameters for an intercity mode choice situation.
SUMMARY AND CONCLUSION
After a hiatus of some 20 years, it is encouraging to see a revival of
interest in the GEV family of choice models, which has produced two of the
most widely used discrete choice models, MNL and NMNL. Recent progress in
simulation estimation techniques has allowed use of certain complex
specifications for discrete choice modeling (e.g. MNP and random coefficients
versions of GEV models, especially the MNL), some of which are more promising
than others from a practical perspective. Much of the attraction of these more
complex models has been circumventing certain properties of GEV models (e.g.
IIA in the MNL) or capturing more complex cross-substitution behavior than
allowed by others (e.g. NMNL). The literature has not always been completely
frank, however, about the difficulties inherent to estimating these more
complex models, so it is heartening to see the recent burst of renewed energy
in more flexible forms of the GEV family (see Swait 2001 for discussion on
GenL and references for other GEV developments), which has undeniable
computational advantages compared to other model forms.
One feature of the GenL model that is particularly interesting is its
flexibility with respect to testing alternative choice set space
representations. In general, a master set M={1,…,J} of
alternatives has (2J-1) non-empty subsets; the model
requires one scale parameter be estimated for each choice set. This is not
likely to be practical for values of J much greater than 5 or 6. Hence,
empirical applications are more likely to proceed with restricted
representations of the choice set space. GenL is easily adapted for estimating
any non-empty choice set generation process, simply by determining which
subsets of M to exclude. In effect, it becomes possible to selectively
model the choice set space itself using GenL.
REFERENCES
