Logit with multiple alternatives Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique F´ ed´ erale de Lausanne M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 1 / 78
Outline Outline Random utility 1 Choice set 2 Error term 3 Systematic part 4 Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity A case study 5 Maximum likelihood estimation 6 Simple models 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 2 / 78
Random utility Random utility For all i ∈ C n U in = V in + ε in What is C n ? What is ε in ? What is V in ? M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 3 / 78
Choice set Outline Random utility 1 Choice set 2 Error term 3 Systematic part 4 Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity A case study 5 Maximum likelihood estimation 6 Simple models 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 4 / 78
Choice set Choice set Universal choice set Mode choice All potential alternatives for driving alone the population sharing a ride Restricted to relevant taxi alternatives motorcycle bicycle walking transit bus rail rapid transit M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 5 / 78
Choice set Choice set Individual’s choice set Mode choice No driver license driving alone No auto available sharing a ride Awareness of transit services taxi Transit services unreachable motorcycle Walking not an option for bicycle long distance walking transit bus rail rapid transit M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 6 / 78
Choice set Choice set Choice set generation is tricky How to model “awareness”? What does “long distance” exactly mean? What does “unreachable” exactly mean? We assume here deterministic rules Car is available if n has a driver license and a car is available in the household Walking is available if trip length is shorter than 4km. M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 7 / 78
Error term Outline Random utility 1 Choice set 2 Error term 3 Systematic part 4 Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity A case study 5 Maximum likelihood estimation 6 Simple models 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 8 / 78
Error term Error terms Main assumption ε in are extreme value EV(0, µ ), independent and identically distributed. Comments Independence: across i and n . Identical distribution: same scale parameter µ across i and n . Scale must be normalized: µ = 1. M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 9 / 78
Error term Derivation of the logit model Assumptions C n = { 1 , . . . , J n } U in = V in + ε in ε in ∼ EV(0 , µ ) ε in i.i.d. Choice model P ( i |C n ) = Pr( V in + ε in ≥ j =1 ,..., J n V jn + ε jn ) max Assume without loss of generality (wlog) that i = 1 P (1 |C n ) = P ( V 1 n + ε 1 n ≥ j =2 ,..., J n V jn + ε jn ) max M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 10 / 78
Error term Derivation of the logit model Composite alternative Define a composite alternative: “anything but alternative one” Associated utility: U ∗ = j =2 ,..., J n ( V jn + ε jn ) max From a property of the EV distribution J n 1 U ∗ ∼ EV � e µ V jn , µ µ ln j =2 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 11 / 78
Error term Derivation of the logit model Composite alternative From another property of the EV distribution U ∗ = V ∗ + ε ∗ where J n V ∗ = 1 � e µ V jn µ ln j =2 and ε ∗ ∼ EV(0 , µ ) M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 12 / 78
Error term Derivation of the logit model Binary choice P (1 |C n ) = P ( V 1 n + ε 1 n ≥ max j =2 ,..., J n V jn + ε jn ) P ( V 1 n + ε 1 n ≥ V ∗ + ε ∗ ) = ε 1 n and ε ∗ are both EV(0, µ ). Binary logit e µ V 1 n P (1 |C n ) = e µ V 1 n + e µ V ∗ where J n V ∗ = 1 � e µ V jn µ ln j =2 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 13 / 78
Error term Derivation of the logit model We have J n e µ V ∗ = e ln � Jn j =2 e µ Vjn = � e µ V jn j =2 and e µ V 1 n P (1 |C n ) = e µ V 1 n + e µ V ∗ e µ V 1 n = e µ V 1 n + � J n j =2 e µ V jn e µ V 1 n = � J n j =1 e µ V jn M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 14 / 78
Error term Scale parameter The scale parameter µ is not identifiable: µ = 1. Warning: not identifiable � = not existing µ → 0, that is variance goes to infinity µ → 0 P ( i | C n ) = 1 lim ∀ i ∈ C n J n M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 15 / 78
Error term Scale parameter µ → + ∞ , that is variance goes to zero 1 lim µ →∞ P ( i | C n ) = lim µ →∞ j � = i e µ ( Vjn − Vin ) 1+ � � 1 if V in > max j � = i V jn = 0 if V in < max j � = i V jn What if there are ties? V in = max j ∈C n V jn , i = 1 , . . . , J ∗ n P ( i |C n ) = 1 i = 1 , . . . , J ∗ i = J ∗ n and P ( i |C n ) = 0 n + 1 , . . . , J n J ∗ n M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 16 / 78
Systematic part Outline Random utility 1 Choice set 2 Error term 3 Systematic part 4 Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity A case study 5 Maximum likelihood estimation 6 Simple models 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 17 / 78
Systematic part Systematic part of the utility function V in = V ( z in , S n ) z in is a vector of attributes of alternative i for individual n S n is a vector of socio-economic characteristics of n M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 18 / 78
Systematic part Linear utility Functional form: linear utility Notation x in = ( z in , S n ) Linear-in-parameters utility functions � V in = V ( z in , S n ) = V ( x in ) = β k ( x in ) k k Not as restrictive as it may seem M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 19 / 78
Systematic part Continuous variables Outline Random utility 1 Choice set 2 Error term 3 Systematic part 4 Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity A case study 5 Maximum likelihood estimation 6 Simple models 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 20 / 78
Systematic part Continuous variables Explanatory variables: alternatives attributes Numerical and continuous ( z in ) k ∈ R , ∀ i , n , k Associated with a specific unit Examples Auto in-vehicle time (in min.) Transit in-vehicle time (in min.) Auto out-of-pocket cost (in cents) Transit fare (in cents) Walking time to the bus stop (in min.) Straightforward modeling M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 21 / 78
Systematic part Continuous variables Explanatory variables: alternatives attributes V in is unitless Therefore, β depends on the unit of the associated attribute Example: consider two specifications = β 1 TT in + · · · V in β ′ 1 TT ′ V in = in + · · · If TT in is a number of minutes, the unit of β 1 is 1/min If TT ′ in is a number of hours, the unit of β ′ 1 is 1/hour Both models are equivalent, but the estimated value of the coefficient will be different = β ′ ⇒ TT in β 1 TT in = β ′ 1 TT ′ 1 in = = 60 TT ′ β 1 in M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 22 / 78
Systematic part Continuous variables Explanatory variables: alternatives attributes Generic and alternative specific parameters = β 1 TT auto + · · · V auto V bus = β 1 TT bus + · · · or = β 1 TT auto + · · · V auto V bus = β 2 TT bus + · · · Modeling assumption: a minute has/has not the same marginal utility whether it is incurred on the auto or bus mode M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 23 / 78
Systematic part Continuous variables Explanatory variables: socio-eco. characteristics Numerical and continuous ( S n ) k ∈ R , ∀ n , k Associated with a specific unit Examples Annual income (in KCHF) Age (in years) Warning: S n do not depend on i M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 24 / 78
Systematic part Continuous variables Explanatory variables: socio-eco. characteristics They cannot appear in all utility functions V ′ V 1 = β 1 x 11 + β 2 income = β 1 x 11 1 V ′ V 2 = β 1 x 21 + β 2 income ⇐ ⇒ = β 1 x 21 2 V ′ = β 1 x 31 + β 2 income = β 1 x 31 V 3 3 In general: alternative specific characteristics = β 1 x 11 + β 2 income + β 4 age V 1 V 2 = β 1 x 21 + β 3 income + β 5 age V 3 = β 1 x 31 M. Bierlaire (TRANSP-OR ENAC EPFL) Logit with multiple alternatives 25 / 78
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