Discrete choice models Michel Bierlaire Intelligent Transportation Systems Program Massachusetts Institute of Technology URL: http

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1 Discrete choice models Michel Bierlaire Intelligent Transportation Systems Program Massachusetts Institute of Technology URL: May 23, 1997 Abstract Discrete choice models have played an important role in transportation modeling for the last 25 years. They are namely used to provide a detailed representation of the complex aspects of transportation demand, based on strong theoretical justications. Moreover, several packages and tools are available to help practionners using these models for real applications, making discrete choice models more and more popular. Discrete choice models are powerful but complex. The art of nding the appropriate model for a particular application requires from the analyst both a close familiarity with the reality under interest and a strong understanding of the methodological and theoretical background of the model. The main theoretical aspects of discrete choice models are reviewed in this paper. The main assumptions used to derive discrete choice models in general, and random utility models in particular, are covered in detail. The Multinomial Logit Model, the Nested Logit Model and the Generalized Extreme Value model are also discussed. 1 Introduction In the context of transportation demand analysis, disaggregate models have played an important role these last 25 years. These models consider that the 1

2 demand is the result of several decisions of each individual in the population under consideration. These decisions usually consist of a choice made among a nite set of alternatives. An example of sequence of choices in the context of transportation demand is described in Figure 1: choice of an activity (playyard), choice of destination (6th street), choice of departure time (early), choice of transportation mode (bike) and choice of itinerary (local streets). For this reason, discrete choice models have been extensively used in this context. A model, as a simplied description of the reality, provides a better understanding of complex systems. Moreover, it allows for obtaining prediction of future states of the considered system, controlling or in- uencing its behavior and optimizing its performances. The complex system under consideration here is a specic aspect of human behavior dedicated to choice decisions. The complexity of this \system" clearly requires many simplifying assumptions in order to obtain operational models. A specic model will correspond to a specic This morning, we have decided to go to the play-yard with the children. There are several of them in the city, but the one on 6th street has a very nice climbing structure that the children love. This place is very popular and is usually crowded at late morning. Therefore, we decide to depart early. The day is gorgeous and we prefer to bike to the play-yard. For safety reasons, we decide to avoid the main avenue, and follow a longer itinerary using local streets. Figure 1: A sequence of choices set of assumptions, and it is important from a practical point of view to be aware of these assumptions when prediction, control or optimization is performed. The assumptions associated with discrete choice models in general are detailed in Section 2. Section 3 focuses specically on assumptions related to random utility models. Some of the most used models, the Multinomial Logit Model (Section 4), the Nested Logit Model (Section 5) and the Generalized Extreme value Model (Section 6), are then introduced, with special emphasis on the Nested Logit model. Among the many publications that can be found in the literature, we refer the reader to Ben-Akiva and Lerman (1985), Anderson, de Palma and Thisse (1992) Hensher and Johnson (1981) and Horowitz, Koppelman and Lerman (1986) for more comprehensive developments. 2

3 2 Modeling assumptions In order to develop models capturing how individuals are making choices, we have to make specic assumptions. We will distinguish here among assumptions about 1. the decision-maker: these assumptions dene who is the decision-maker, and what are his/her characteristics; 2. the alternatives: these assumptions determine what are the possible options of the decision-maker; 3. the attributes: these assumptions identify the attributes of each potential alternative that the decision-maker is taking into account to make his/her decision; 4. the decision rules: they describe the process used by the decision-maker to reach his/her choice. In order to narrow down the huge number of potential models, we will consider some of these assumptions as xed throughout the paper. It does not mean that there is no other valid assumption, but we cannot cover everything in this context. For example, even if continuous models will be briey described, discrete models will be the primary focus of this paper. 2.1 Decision-maker As mentioned in the introduction, choice models are referred to as disaggregate models. It means that the decision-maker is assumed to be an individual. In general, for most practical applications, this assumption is not restrictive. The concept of \individual" may easily been extended, depending on the particular application. We may consider that a group of persons (a household or a government, for example) is the decision-maker. In doing so, we decide to ignore all internal decisions within the group, and to consider only the decision of the group as a whole. The example described in Figure 1 reects the decisions of a household, without accounting for all potential negotiations among the parents and the children. We will refer to \decision-maker" and individual" interchangeably throughout the rest of the paper. 3

4 Because of its disaggregate nature, the model has to include the characteristics, or attributes, of the individual. Many attributes, like age, gender, income, eyes color or social security number may be considered in the model 1. The analyst has to identify those that are likely to explain the choice of the individual. There is no automatic process to perform this identication. The knowledge of the actual application and the data availability play an important role in this process. 2.2 Alternatives Analyzing the choice of an individual requires the knowledge of what has been chosen, but also of what has not been chosen. Therefore, assumptions must be made about options, or alternatives, that were considered by the individual to perform the choice. The set containing these alternatives, called the choice set, must be characterized. The characterization of the choice set depends on the context of the application. If we consider the example described in Figure 2, the time spent on each Internet site may be anything, as far as the total time is not more than two hours. The resulting choice set C is represented in Figure 3 on the following page, and is dened by C = f(t Museum ; t News )jt Museum + t News 2; t Museum 0; t News 0g (1) It is a typical example of a continuous choice set, where the alternatives are dened by some constraints and cannot be enumerated. In this paper, we focus on discrete choice sets. A discrete choice set contains a nite number of alternatives that can be explicitly listed. The corresponding choice models are called discrete choice models. The choice of a transportation mode is a typical application leading to a discrete choice set. In this context, the characterization of the choice set consists in the identication of the I have decided to spend a maximum of two hours on Internet, today. There are many interesting services available, but I really like to access the site of the Museum of Fine Arts, Boston, and to read the electronic version of a Belgian newspaper. How much time am I going to spend on each of these sites? Figure 2: Choice on Internet 1 However, if you develop a choice model including eyes color or social security number, please let me know ;-) 4

5 t t Museum + t News = @@ - t Museum Figure 3: Example of a continuous choice set list of alternatives. To perform this task, two concepts of choice set are considered: the universal choice set and the reduced choice set. The universal choice set contains all potential alternatives in the context of the application. Considering the mode choice in the example of Figure 1 on page 2, the universal choice set may contain all potential transportation modes, like walk, bike, bus, car, etc. The alternative plane, which is also a transportation mode, is clearly not an option in this context and, therefore, is not included in the universal choice set. The reduced choice set is the subset of the universal choice set considered by a particular individual. Alternatives in the universal choice set that are not available to the individual under consideration are excluded (for example, the alternative car may not be an option for individuals without a driver license). The awareness of the availability of the alternative by the decisionmaker should be considered as well. The reader is referred to Swait (1984) for more details on choice set generation. In the following, \choice set" will refer to the reduced choice set, except when explicitly mentioned. 5

6 2.3 Attributes Each alternative in the choice set must be characterized by a set of attributes. Similarly to the characterization of the decision-maker described in Section 2.1, the analyst has to identify the attributes of each alternatives that are likely to aect the choice of the individual. In the context of a transportation mode choice, the list of attributes for the mode car could include the travel time, the out-of-pocket cost and the comfort. The list for bus could include the travel time, the out-of-pocket cost, the comfort and the bus frequency. Note that some attributes may be generic to all alternatives, and some may be specic to an alternative (bus frequency is specic to bus). Also, qualitative attributes, like comfort, may be considered. An attribute is not necessarily a directly observed quantity. It can be any function of available data. For example, instead of considering travel time as an attribute, the logarithm of the travel time may be considered. The outof-pocket cost may be replaced by the ratio between the out-of-pocket cost and the income of the individual. The denition of attributes as a function of available data depends on the problem. Several denitions must usually be tested to identify the most appropriate. 2.4 Decision rules At this point, we have identied and characterized both the decision-maker and all available alternatives. We will now focus on the assumptions about the rules used by the decision-maker to come up with the actual choice. Different sets of assumptions can be considered, that leads to dierent family of models. We will describe here three theories on decision rules, and the corresponding models. The neoclassical economic theory, described in Section 2.4.1, introduces the concept of utility. The Luce model (Section 2.4.2) and the random utility models (introduced in Section and developed in Section 3) are designed to capture uncertainty Neoclassical Economic Theory The neoclassical economic theory assumes that each decision-maker is able to compare two alternatives a and b in the choice set C using a preferenceindierence operator &. If a & b, the decision-maker either prefers a to b, or is indierent. The preference-indierence operator is supposed to have the 6

7 following properties: 1. Reexivity: a & a; 8a 2 C: 2. Transitivity: a & b and b & c ) a & c; 8a; b; c 2 C: 3. Comparability: a & b or b & a; 8a; b 2 C: Because the choice set C is nite, the existence of an alternative which is preferred to all of them is guaranteed, that is 9a s.t. a & a 8a 2 C: (2) More interestingly, and because of the three properties listed above, it can be shown that the existence of a function such that U : C?! R : a U(a) (3) a & b, U(a) U(b) 8a; b 2 C (4) is guaranteed. Therefore, the alternative a dened in (2) may be identied as a = arg max U(a): (5) a2c It results that using the preference-indierence operator & to make a choice is equivalent to assigning a value, called utility, to each alternative, and selecting the alternative a associated with the highest utility. The concept of utility associated with the alternatives plays an important role in the context of discrete choice models. However, the assumptions of neoclassical economic theory presents strong limitations for practical applications. Indeed, the complexity of human behavior suggests that a choice model should explicitly capture some level of uncertainty. The neoclassical economic theory fails to do so. 7

8 The exact source of uncertainty is an open question. Some models assume that the decision rules are intrinsically stochastic, and even a complete knowledge of the problem would not overcome the uncertainty. Others consider that the decision rules are deterministic, and motivate the uncertainty from the impossibility of the analyst to observe and capture all dimensions of the problem, due to its high complexity. Anderson et al. (1992) compare this debate with the one between Einstein and Bohr, about the uncertainty principle in theoretical physics. Bohr argued for the intrinsic stochasticity of nature and Einstein claimed that \Nature does not play dice". Two families of models can be derived, depending on the assumptions about the source of uncertainty. Models with stochastic decision rules, like the model proposed by Luce (1959), described in Section 2.4.2, or the \elimination by aspects" approach, proposed by Tversky (1972), assumes a deterministic utility and a probabilistic decision process. Random Utility Models, introduced in Section and developed in Section 3, are based on the deterministic decision rules from the neoclassical economic theory, where uncertainty is captured by random variables representing utilities The Luce model An important characteristic of models dealing with uncertainty is that, instead of identifying one alternative as the chosen option, they assign to each alternative a probability to be chosen. Luce (1959) proposed the choice axiom to characterize a choice probability law. The choice axiom can be stated as follow. Denoting P C (a) the probability of choosing a in the choice set C, and P C (S) the probability of choosing one element of the subset S within C, the two following properties hold for any choice set U, C and S, such that S C U. 1. If an alternative a 2 C is dominated, that is if there exists b 2 C such that b is always preferred to a or, equivalently, P fa;bg (a) = 0, then removing a from C does not modify the probability of any other alternative to be chosen, that is P C (S) = P Cnfag (S n fag): (6) 2. If no alternative is dominated, that is if 0 < P fa;bg (a) < 1 for all a; b 2 C, then the choice probability is independent from the sequence 8

9 of decisions, that is P C (a) = P C (S) P S (a): (7) The independence described by (7) can be illustrated using a example of transportation mode choice, where we consider C = fcar, Bike, Busg. We apply two dierent assumptions to compute the probability of choosing \car" as a transportation mode. 1. The decision-maker may decide rst to use a motorized mode (car or bus, in this case). The probability of choosing \car" is then given by P fcar,bus,bikeg (car) = P fcar,bus,bikeg (fcar,busg) P fcar,busg (car): (8) 2. Alternatively, the decision-maker may decide rst to use a private transportation mode (car or bike, in this case). The probability of choosing \car" is then given by P fcar,bus,bikeg (car) = P fcar,bus,bikeg (fcar,bikeg) P fcar,bikeg (car): (9) Equation (7) of the choice axiom imposes that both assumptions produce the same probability, that is P fcar,bus,bikeg (car) = P fcar,bus,bikeg (fcar,busg) P fcar,busg (car) = P fcar,bus,bikeg (fcar,bikeg) P fcar,bikeg (car): (10) The second part of the choice axiom can be interpreted in a dierent way. Luce (1959) has shown that (7) is a sucient and necessary condition for the existence of a function v : C?! R, such that, for all S C, we have P S (a) = X b2s v(a) v(b) : (11) Also, function v is unique up to a proportionality factor. v 0 : C?! R verifying (11), then If there exists v(a) = kv 0 (a); 8a 2 C; (12) where k 2 R. Similarly to (3), v() may be interpreted as a utility function. We will elaborate more on this result in Section 4. 9

10 2.4.3 Random Utility Models Random utility models assume, as neoclassical economic theory, that the decision-maker has a perfect discrimination capability. In this context, however, the analyst is supposed to have incomplete information and, therefore, uncertainty must be taken into account. Manski (1977) identies four different sources of uncertainty: unobserved alternative attributes, unobserved individual attributes (called \unobserved taste variations" by Manski, 1977), measurement errors and proxy, or instrumental, variables. The utility is modeled as a random variable in order to reect this uncertainty. More specically, the utility that individual i is associating with alternative a is given by U i a = V i a + " i a; (13) where V i a is the deterministic part of the utility, and " i a is the stochastic part, capturing the uncertainty. Similarly to the neoclassical economic theory, the alternative with the highest utility is supposed to be chosen. Therefore, the probability that alternative a is chosen by decision-maker i within choice set C is P i C(a) = P U i a = max b2c U i b : (14) Random utility models are the most used discrete choice models for transportation applications. Therefore, the rest of the paper is devoted to them. 3 Random utility models The derivation of random utility models is based on a specication of the utility as dened by (13). Dierent assumptions about the random term " i a and the deterministic term V i a will produce specic models. We present here the most usual assumptions that are used in practice. In Section 3.1, common assumptions about the random part of the utility are discussed. The deterministic part is treated in Section Assumptions on the random term We will focus here on assumptions about the mean, the variance and the functional form of the random term. 10

11 For all practical purposes, the mean of the random term is usually supposed to be zero. It can be shown that this assumption is not restrictive. We do it here on a simple example. Considering the example described in Figure 4, we denote the mean of the error term of each alternative by We consider a choice between two alternatives, that is C = f1; 2g. The probability for a given decision-maker to choose alternative 1 is P f1;2g (1) = P[U 1 U 2 ] = P[V 1 + " 1 V 2 + " 2 ] = P[V 1? V 2 " 2? " 1 ]: Figure 4: A binary model m 1 = E[" 1 ] and m 2 = E[" 2 ], respectively. Then, the error terms can be specied as " 1 = m 1 + " 0 1 (15) and " 2 = m 2 + " 0 2; (16) where " 0 1 and " 0 2 are random variables with zero mean. Therefore, P f1;2g (1) = P[V 1? V 2 " 2? " 1 ] = P[V 1? V 2 (m 2 + " 0 2)? (m 1 + " 0 1)] = P[(V 1 + m 1 )? (V 2 + m 2 ) " 0 2? " 0 1] (17) The terms m 1 and m 2, called Alternative Specic Constants (ASC), are capturing the mean of the error term. Therefore, it can be assumed without loss of generality, that the error terms have zero mean if the model specication includes these ASCs. The zero mean assumption is valid if the deterministic part of the utility function of each alternative includes an Alternative Specic Constant. In practice, it is impossible to estimate the value of all ASCs from observed data. Considering again the example of Figure 4, the probability of choosing alternative 1, say, is not modied if an arbitrary constant K is added to both utilities. Therefore, only the dierence between the two ASCs 11

12 can be identied. Indeed, from (17), we have P f1;2g (1) = P[(V 1 + m 1 )? (V 2 + m 2 ) " 0 2? " 0 1] = P[V 1 + m 1 + " 0 1 V 2 + m 2 + " 0 2] = P[V 1 + m 1 + " K V 2 + m 2 + " K]; (18) for any K 2 R. If K =?m 1, we obtain P f1;2g (1) = P[V 1 + " 0 1 V 2 + (m 2? m 1 ) + " 0 2]; or, equivalently, dening M = m 2? m 1, P f1;2g (1) = P[V 1 + " 0 1 V 2 + M + " 0 2]: Dening K =?m 2 produces the same result. This property can be generalized easily to models with more than two alternatives, where only dierences between ASCs can be identied. It is common practice to constrain one ASC in the model to zero. From a modeling viewpoint, the choice of the particular alternative whose ASC is constrained is purely arbitrary. However, Bierlaire, Lotan and Toint (1997) have shown that the estimation process is inuenced by this choice. They propose a dierent technique of ASC specication which is optimal from an estimation perspective. To derive assumptions about the variance of the random term, we observe that the scale of the utility may be arbitrarily specied. Indeed, for any 2 R; > 0, we have P f1;2g (1) = P[U 1 U 2 ] = P[U 1 U 2 ] = P[V 1? V 2 (" 2? " 1 )]: (19) The arbitrary decision about is equivalent to assuming a particular variance v of the distribution of the error term. Indeed, if Var[(" 2? " 1 )] = v; (20) we have also = v p Var[("2? " 1 )] : (21) 12

13 We will illustrate this relationship with several examples in the remaining of this section. Once assumptions about the mean and the variance of the error term distribution have been dened, the focus is now on the actual functional form of this distribution. We will consider here three dierent distributions yielding to three dierent families of models: linear, probit and logit models. The linear model is obtained from the assumption that the density function of the error term is given by f(x) = 8 < : 1 2L if x 2 [?L; L] 0 elsewhere (22) where L 2 R; L 0, is an arbitrary constant. This density function is used to derive the probability of choosing one particular alternative. Considering the example presented in Figure 4 on page 11, the probability is given by (23) (see Figure 5). P f1;2g (1) = 8 >< >: 0 if V 1? V 2 <?L V 1? V 2 + L 2L if? L V 1? V 2 L 1 if V 1? V 2 > L (23)!!! 6P f1;2g (1) 1 0.5!!!!!!!!!!!!!!!!!!!!!!!?L 0 L - V 1? V 2 Figure 5: Linear model The linear model presents some problem for real applications. First, the probability associated with extreme values (jv 1? V 2 j L in the example) 13

14 is exactly zero. Therefore, if any extreme event happens in the reality, the model will never capture it. Second, the discontinuity of the derivatives at?l and L causes problems to most of the estimation procedures. We conclude the presentation of the linear model by emphasizing that the constant L determines the scale of the distribution. For the binary example, Var(" 2? " 1 ) = L 2 =3. Using (21), we have that assuming Var((" 2? " 1 )) = 1 is equivalent to assuming = p 3=L. A common value for L is 1=2, that is = 2 p 3. The Normal Probability Unit, or Probit, model is derived from the assumption that the error terms are normally distributed, that is f(x) = 1 p 2 e? 1 2 ( x ) 2 ; (24) where 2 R; > 0 is an arbitrary constant. This density function is used to derive the probability of choosing one particular alternative. Considering the example presented in Figure 4 on page 11, and assuming that " 1 and " 2 are normally distributed with zero mean, variances 2 1 and 2 2 respectively, and covariance 12, the probability is given by (25) (see Figure 6 on the next page). P f1;2g (1) = Z V 1?V2 x=?1 1 p 2 e? 1 2 ( x ) 2 dx; (25) where 2 = ? 2 12 is the variance of (" 2? " 1 ) The probit model is motivated by the Central Limit Theorem 2, assuming that the error terms are the sum of independent unobserved quantities. Unfortunately, the probability function (25) has no closed analytical form, which limits practical use of this model. We refer the reader to Daganzo (1979) for a comprehensive development of probit models. We conclude this short introduction of the probit model by looking at the scale parameter. Considering again the binary example presented in Figure 4 on page 11 in the probit context, we have Var(" 2? " 1 ) = 2. Using (21), we have that assuming Var((" 2? " 1 )) = 1 is equivalent to assuming = 1=. It is common practice to arbitrary dene = 1, that is = 1. 2 The Central Limit Theorem states that the sum of a large number of independent random variables approximates the asymptotically normal distribution. It has been proved by Markov (1900). 14

15 1 0.9 N(x) Figure 6: Probit model Despite its complexity, the probit model has been applied to many practical problems (see Whynes, Reedand and Newbold, 1996, Bolduc, Fortin and Fournier, 1996, Yai, Iwakura and Morichi, 1997 among recent publications). However, the most widely used model in practical applications is probably the Logistic Probability Unit, or Logit, model. The error terms are now assumed to be independent and identically Gumbel distributed. The density function of the Gumbel distribution is given by (26) (see Figure 7 on the next page). f(x) = e?(x?) e?e(x?) ; (26) where 2 R is the location parameter, and 2 R; > 0 is the scale parameter. The mean of the Gumbel distribution is + ; (27) 15

16 0.4 f(x) Figure 7: Gumbel distribution where = lim n!1 nx i=1 1 i? ln(n) 0:5772 (28) is the Euler constant. The variance is : (29) The Gumbel distribution is an approximation of the Normal law, as shown in Figure 8 on the following page, where the plain line represents the Normal distribution, and the dotted line the Gumbel distribution. We derive the probability function for the binary example of Figure 4 on page 11 from the following property of the Gumbel distribution. If " 1 is Gumbel distributed with location parameter 1 and scale parameter, and " 2 is Gumbel distributed with location parameter 2 and scale parameter, then " = " 2? " 1 follows a Logistic distribution with location parameter 16

17 1 N(x) g(x) Figure 8: Comparison between Normal and Gumbel distribution 2? 1 and scale parameter (the name of the Logit model comes from this property). The density function of the Logistic distribution is given by f(x) = e?x (e?x + 1) 2 ; (30) where 2 R; > 0 is the scale parameter. As a consequence, we have, or, equivalently, P f1;2g (1) = 1 e?(v 1?V2) + 1 (31) P f1;2g (1) = e V 1 e V 1 + e V2 : (32) In order to determine the relationship between the scale parameter and the variance of the distribution, we compute Var(" 2? " 1 ) = Var(" 2 ) + Var(" 1 ) = 2 2 =6 2. Using (21), we have that assuming Var((" 2? " 1 )) = 1 is equivalent to assuming = p 3=. It is common practice to arbitrary dene = 1, that is = p 3=. 17

18 In most cases, the arbitrary decision about the scale parameter does not matter and can be safely ignored. But it is important not to completely forget its existence. Indeed, it may sometimes play an important role. For example, utilities derived from dierent models can be compared only if the value of is the same for all of them. It is usually not the case with the scale parameters commonly used in practice, as shown in Table 1. Namely, a utility estimated with a logit model has to be divided by p 3= before being compared with a utility estimated with a probit model. Arbitrary Estimated Model value utility Linear L = 1 2 p 3 V 2 Probit = 1 1 V Logit = 1 V p 3 Table 1: Model comparison The list of models presented here above is not exhaustive. Other assumptions about the distribution of the error term will lead to other families of models. For instance, Ben-Akiva and Lerman (1985) cite the arctan and the truncated exponential models. These models are not often used in practice and we will not consider them here. 3.2 Assumptions on the deterministic term The utility of each alternative must be a function of the attributes of the alternative itself and of the decision-maker identied in Sections 2.1 and 2.3. We can write the deterministic part of the utility that individual i is associating with alternative a as V i a = V i a (x i a); (33) where x i a is a vector containing all attributes, both of individual i and alternative a. The function dened in (33) is commonly assumed to be linear in the parameters, that is, if n attributes are considered, V i a (x i a ) = 1x i a (1) + 2x i a (2) + + nx i a (n) = 18 nx k=1 k x i a(k); (34)

19 where 1 ; : : : ; n are parameters to be estimated. This assumption simplies the formulation and the estimation of the model, and is not as restrictive as it may seem. Indeed, nonlinear eects can still be captured in the attributes denition, as mentioned in Section 2.3. Nonlinear eects can be captured with a linear-in-parameters utility function, using an appropriate denition of attributes. 4 Multinomial logit model As introduced in the previous section, the logit model is derived from the assumption that the error terms of the utility functions are independent and identically Gumbel distributed. These models were rst introduced in the context of binary choice models, where the logistic distribution is used to derive the probability. Their generalization to more than two alternative is referred to as multinomial logit models. If the error terms are independent and identically Gumbel distributed, with location parameter 0 and scale parameter, the probability that a given individual choose alternative i within C is given by P C (i) = e V i P k2c ev k : (35) The derivation of this result is attributed to Holman and Marley by Luce and Suppes (1965). We refer the reader to Ben-Akiva and Lerman (1985) and Anderson et al. (1992) for additional details. It is interesting to note that the multinomial logit model can also be derived from the choice axiom dened by (6) and (7). Indeed, dening S = C and v(a) = e Va, we have that (11) is equivalent to (35). The multinomial logit model can be derived both from random utility theory and from the choice axiom. An important property of the multinomial logit model is the Independence from Irrelevant Alternatives (IIA). This property can be stated as follows. The ratio of the probabilities of any two alternatives is independent from the choice set. That is, for any choice sets S and T such that 19

20 S T C, for any alternative a 1 and a 2 in S, we have P S (a 1 ) P S (a 2 ) = P T (a 1 ) P T (a 2 ) : (36) This result can be proven easily using (35). Ben-Akiva and Lerman (1985) propose an equivalent denition: The ratio of the choice probabilities of any two alternatives is entirely unaected by the systematic utilities of any other alternatives. The IIA property of multinomial logit models is a limitation for some practical applications. This limitation is often illustrated by the red bus/blue bus paradox (see, for example, Ben-Akiva and Lerman, 1985) in the modal choice context. We prefer here the path choice example presented in Figure 9. We consider a commuter traveling from an origin O to a destination D. He/she is confronted with the path choice problem described below, where the choice set is f1; 2a; 2bg and the only attribute considered for the choice is travel time. We assume furthermore that the travel time for any alternative is the same, that is V 1 = V 2a = V 2b = T. Finally, the travel time on the small sections a and b is supposed to be signicantly smaller than the total travel time T. As a result, we expect the probability of choosing path 1 or path 2 to be almost 50%, irrespectively of the choice between a and b. O H HHHHH 1H HHHHH a 2 b D Figure 9: A path choice example The probability provided by the multinomial logit model (35) for this example are P f1;2a;2bg (1) = P f1;2a;2bg (2a) = P f1;2a;2bg (2b) = e T P k2f1;2a;2bg et = 1 3 ; (37) which is not consistent with the intuitive result. This situation appears in choice problems with signicantly correlated alternatives, as it is clearly the case in the example. Indeed, alternatives 2a and 2b are so similar that their 20

21 utilities share many unobserved attributes of the path and, therefore, the assumption of independence of the random part of these utilities is not valid in this context. The Nested Logit Model, presented in the next section, partly overcomes this limitation of the multinomial logit model. 5 Nested logit model The nested logit model, rst derived by Ben-Akiva (1973), is an extension of the multinomial logit model designed to capture correlations among alternatives. It is based on the partitioning of the choice set C into several nests C k such that C = [ k C k ; (38) and C k \ C l = ;; 8k 6= l: (39) The utility function of each alternative is composed of a term specic to the alternative, and a term associated with the nest. If i 2 C k, we have U i = V i + " i + V Ck + " Ck : (40) The error terms " i and " Ck are supposed to be independent. As for the multinomial logit model, error terms " i are supposed to be independent and identically Gumbel distributed, with scale parameter k. The distribution of " Ck is such that the random variable max j2ck U j is Gumbel distributed with scale parameter. Each nest within the choice set is associated with a pseudo-utility, called composite utility, expected maximum utility, inclusive value or accessibility in the literature. The composite utility for nest C k is dened as V 0 C k = V Ck + 1 k ln X j2c k e kv j ; (41) where V Ck is the component of the utility which is common to all alternatives in the nest C k. 21

22 where The probability model is then given by P C (i) = P C (C k ) P Ck (i); (42) and P C (C k ) = P Ck (i) = 0 ev C k P n ; (43) 0 l=1 ev C l e kv i Pj2C k e kv j : (44) The parameters and k reect the correlation among alternatives in the nest C k. Indeed, if i; j 2 C k, we have q = 1? corr(u i ; U j ): (45) k Clearly, we have 0 k 1: (46) Ben-Akiva and Lerman (1985) derive condition (46) directly from utility theory. Note also that if k = 1, we have corr(u i ; U j ) = 0. When k = 1 for all k, the nested logit model is equivalent to the multinomial logit model. The parameters and k are closely related in the model. Actually, only their ratio is meaningful. It is not possible to identify them separately. A common practice is to arbitrarily constrain one of them to a value (usually 1). The impacts of this arbitrary decision on the model are briey discussed in Section 5.1. We illustrate here the Nested Logit Model with the path choice example described in Figure 9 on page 20. First, the choice set C = f1; 2a; 2bg is divided into C 1 = f1g and C 2 = f2a; 2bg. The deterministic components of the utilities are V C1 = T, V 1 = 0, V C2 = T? and V 2a = V 2b =. The composite utilities of each nest are V 0 C 1 = V C1 = T; (47) 22

23 and V 0 C 2 = T? ln(e 2 + e 2 ) = T ln 2: (48) The probability of choosing each nest is then P C (C 1 ) = e V 0C 1 e V 0C 1 + e V 0C 2 = e T = 1 e T (T + ln 2 + e 2 ) 1 + e ln 2 2 = ; (49) and P C (C 2 ) = 1? P C (C 1 ) = ; (50) where the value of has been assumed to be 1, without loss of generality. The probability of each alternative is then computed. We obtain and P C (1) = P C (C 1 ) = ; (51) P C (2a) = P C (2b) = e 2 e 2 + e 2 P C(C 2 ) = : (52) The values of P C (1), P C (2a) and P C (2b) as a function of 1 are plotted on 2 Figure 10 on the next page. From (46), we have that because 2 has been arbitrarily dened as 1. We observe that, when 1 = 1, the nested 2 logit model produces the same results as the multinomial logit model (37), and all probabilities are 1. On the other hand, when 3 2 goes to innity, and 1= 2 goes to 0, the probability of each nest is closer and closer to 1/2. At the limit, the model is becoming a binary choice model, where the small detours a and b are ignored in the choice process. 23

24 P C (1) P C (2a) and P C (2b) = 2 Figure 10: Probability of each alternative as a function of Normalization of nested logit models In order to compute the probabilities in the previous example, we have arbitrarily decided to constraint to 1. Alternatively, we could have decided to constraint 2 to 1. It is easy to show that, in this case, we have P C (1) = 1 (53) and P C (2a) = P C (2b) = ; (54) which is equivalent to (51) and (52), replacing 1= 2 by. A model where the scale parameter is arbitrarily constrained to 1 is said to be \normalized from the top". A model where one of the parameters k is constrained to 1 is said to be \normalized from the bottom". The latter may produce a simpler formulation of the model. We illustrate it using the example of Figure 11. We have e 1 utbus e 1 ln(e 1 ut bus+e 1 ut metro ) P C (bus) = e 1 utbus + e 1 utmetro e 1 ln(e 1 ut bus+e 1 ut metro ) + e 2 ln(e 2 r tcar +e 2 r t bike) 24

25 In the context of a mode choice with C = fbus, metro, car, bikeg, we consider a model with two nests: C 1 = fbus,metrog contains the public transportation modes and C 2 = fcar,bikeg contains the private transportation modes. For the sake of the example, we consider the following deterministic terms of the utility functions: V bus = u t bus, V metro = u t metro, V car = r t car and V bike = r t bike, where t i is the travel time using mode i, r and u are parameters to be estimated. Note that we have one parameter for private and one for public transportation, and we have not included the alternative specic constants in order to keep the example simple. Figure 11: A mode choice example and P C (car) = e 2 rtcar e 2 rtcar + e 2 rtbike e 2 ln(e 2 r tcar +e 2 r t bike) e 1 ln(e 1 ut bus+e 1 ut metro) + e 2 ln(e 2 r tcar +e 2 r t bike) : If we impose 1 = 1, we can dene 1 =, 2 = = 2, ~ u = u and ~ r = 2 r to obtain the following expressions. and P C (bus) = e ~ utbus e 1 ln(e ut ~ bus+e utmetro) ~ e ~ utbus + e utmetro ~ e 1 ln(e ut ~ bus+e utmetro) ~ + e 2 ln(e r ~ tcar +e r ~ t bike) P C (car) = e ~ rtcar e 2 ln(e ~ r tcar +e ~ r t bike) e ~ rtcar + e rtbike ~ e 1 ln(e ut ~ bus+e utmetro) ~ + e 2 ln(e r ~ tcar +e r ~ t bike) This formulation, proposed by Daly (1987), simplies the estimation process. For this reason, it has been adopted in estimation packages like ALOGIT (Daly, 1987) or HieLoW (Bierlaire, 1995, Bierlaire and Vandevyvere, 1995). ALOGIT (Daly, 1987) and HieLoW (Bierlaire, 1995, Bierlaire and Vandevyvere, 1995) estimate nested logit models normalized from the bottom. We emphasize here that this formulation should be used with caution when the same parameters are present in more than one nest. In this case, specic techniques, inspired from articial trees proposed by Bradley and Daly (1991) must be used to obtain a correct specication of the model. The description of these techniques is out of the scope of this paper. 25

26 The scale parameters have to be normalized in nested logit models. Indeed, only their ratio is meaningful. All possible normalizations produce an equivalent model, but the estimation process depends on the particular specication. A direct extension of the nested logit model consists in partionning some or all nests into sub-nests, which can, in turn, be divided into sub-nests. Because of the complexity of these models, their structure is usually represented as a tree, as suggested by Daly (1987). Clearly, the number of potential structures, reecting the correlation among alternatives, can be very large. No technique has been proposed thus far to identify the most appropriate correlation structure directly from the data. We conclude our introduction of nested logit models by mentioning their limitations. These models are designed to capture choice problems where alternatives within each nest are correlated. No correlation across nests can be captured by the Nested Logit Model. When alternatives cannot be partitioned into well separated nests to reect their correlation, Nested Logit Models are not applicable. This is the case for most route choice problems. Several models within the \logit family" have been designed to capture specic correlation structures. For example, Cascetta (1996) captures overlapping paths in a route choice context using commonality factors, Koppelman and Wen (1997) capture correlation between pair of alternatives, and Vovsha (1997) proposes a cross-nested model allowing alternatives to belong to more than one nest. The two last models are derived from the Generalized Extreme Value model, presented in the next section. 6 Generalized extreme value model The Generalized Extreme Value (GEV) model has been introduced by Mc- Fadden (1978) in the context of residential location. This general model actually consists in a large family of models that are consistent with random utility theory. The probability of choosing alternative i within C = f1; : : : ; ng is given by P C (i) = i (e V 1 ; : : : ; e Vn ) G(e V 1 ; : : : ; e V n ) ; (55) where G : R n +! R is a dierentiable function with the following properties. 26

27 1. G(x) 0 for all x 2 R n +, 2. G is homogeneous of degree 3 > 0, that is G(x) = G(x), for all x 2 R n +, 3. lim xi!+1 G(x 1 ; : : : ; x i ; : : : ; x n ) = +1 for all i such that 1 i n, and 4. the kth partial derivative with respect to k distinct x i is non-negative if k is odd, and non-positive if k is even, that is, 8i 1 ; : : : ; i k such that 1 i j n if 1 j k and i j 6= i l if 1 j; l k and j 6= l, we have 8x 2 R n +; As an example, we k i 1 : : ik (x) G(x) = nx i=1 0 0 if k is odd; if k is even: x i ; (56) which has the required properties, as it can be easily veried. Then, = ev P i e(?1)v i e V i n = P i=1 e V n ; (57) i i=1 e V i which is the multinomial logit model. Similarly, the nested logit model can be derived with P C (i) = i (e V 1 ; : : : ; e Vn ) G(e V 1 ; : : : ; e V n ) G(x) = nx k=1 X i2c k e kxi! k : (58) It can be shown that property 4 holds if 0 k 1, which is consistent with condition (46). The multinomial logit model and the nested logit model are both specic Generalized Extreme Value models. The Generalized Extreme Value model provides a nice theoretical framework for the development of new discrete choice models, like Koppelman and Wen (1997) and Vovsha (1997). 3 McFadden (1978) assumed = 1. Ben-Akiva and Francois (1983) generalized the result to any > 0. 27

28 7 Conclusion We have covered in this paper the main theoretical aspects of discrete choice models in general, and random utility models in particular. A good awareness of underlying assumptions is necessary for an ecient use of these models for practical applications. In particular, we have focused on the location parameters and the scale parameters in multinomial and nested logit models. Despite its importance, the role of these parameters tend to be underestimated by practitioners. This may lead to incorrect specications of the models, or incorrect interpretation of the results. 8 Acknowledgments This paper is based on a lecture given at the NATO Advanced Studies Institute Operations Research and Decision Aid Methodologies in Trac and Transportation Management, Balatonfured, Hungary, March Comments from the students and other lecturers of the ASI have been very useful to write this paper. Moreover, I am very grateful to Moshe Ben-Akiva and John Bowman for their valuable discussions and comments. References Anderson, S. P., de Palma, A. and Thisse, J.-F. (1992). Discrete Choice Theory of Product Dierentiation, MIT Press, Cambridge, Ma. Ben-Akiva, M. and Francois, B. (1983). homogeneous generalized extreme value model, Working paper, Department of Civil Engineering, MIT, Cambridge, Ma. Ben-Akiva, M. E. (1973). Structure of passenger travel demand models, PhD thesis, Department of Civil Engineering, MIT, Cambridge, Ma. Ben-Akiva, M. E. and Lerman, S. R. (1985). Discrete Choice Analysis: Theory and Application to Travel Demand, MIT Press, Cambridge, Ma. Bierlaire, M. (1995). A robust algorithm for the simultaneous estimation of hierarchical logit models, GRT Report 95/3, Department of Mathematics, FUNDP. 28

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