Nested logit models Michel Bierlaire michel.bierlaire@epfl.ch - PowerPoint PPT Presentation

Nested logit models Michel Bierlaire michel.bierlaire@epfl.ch Transport and Mobility Laboratory Nested logit models – p. 1/23

Red bus/Blue bus paradox • Mode choice example • Two alternatives: car and bus • There are red buses and blue buses • Car and bus travel times are equal: T Nested logit models – p. 2/23

Red bus/Blue bus paradox Model 1 U car = βT + ε car U bus = βT + ε bus Therefore, e βT e βT + e βT = 1 P ( car |{ car , bus } ) = P ( bus |{ car , bus } ) = 2 Nested logit models – p. 3/23

Red bus/Blue bus paradox Model 2 U car = βT + ε car U blue bus = βT + ε blue bus U red bus = βT + ε red bus e βT e βT + e βT + e βT = 1 P ( car |{ car , blue bus , red bus } ) = 3  P ( car |{ car , blue bus , red bus } )  = 1  P ( blue bus |{ car , blue bus , red bus } ) 3 .  P ( red bus |{ car , blue bus , red bus } )  Nested logit models – p. 4/23

Red bus/Blue bus paradox • Assumption of logit: ε i.i.d • ε blue bus and ε red bus contain common unobserved attributes: ◮ fare ◮ headway ◮ comfort ◮ convenience ◮ etc. Nested logit models – p. 5/23

Capturing the correlation ⑦ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ⑦ ⑦ Bus Car ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ⑦ ⑦ Blue Red Nested logit models – p. 6/23

Capturing the correlation If bus is chosen then U blue bus = V blue bus + ε blue bus U red bus = V red bus + ε red bus where V blue bus = V red bus = βT e βT e βT + e βT = 1 P ( blue bus |{ blue bus , red bus } ) = 2 Nested logit models – p. 7/23

Capturing the correlation ⑦ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ⑦ ⑦ Bus Car ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ⑦ ⑦ Blue Red Nested logit models – p. 8/23

Capturing the correlation What about the choice between bus and car? U car = βT + ε car U bus = V bus + ε bus with V bus = V bus ( V blue bus , V red bus ) ε bus = ? Define V bus as the expected maximum utility of red bus and blue bus Nested logit models – p. 9/23

Expected maximum utility For a set of alternative C , define U C = max i ∈C U i = max i ∈C ( V i + ε i ) and V C = E [ U C ] For logit i ∈C U i ] = 1 � e µV i E [max µ ln i ∈C i ∈C e µV i + γ Actually, E [max i ∈C U i ] = 1 µ ln � µ , but the constant term can be ignored. Nested logit models – p. 10/23

Expected maximum utility µ b ln( e µ b V blue bus + e µ b V red bus ) 1 V bus = µ b ln( e µ b βT + e µ b βT ) 1 = 1 = βT + µ b ln 2 where µ b is the scale parameter for the logit model associated with the choice between red bus and blue bus Nested logit models – p. 11/23

Nested Logit Model Probability model: e µV car e µβT 1 P ( car ) = e µV car + e µV bus = µb ln 2 = e µβT + e µβT + µ µ 1 + 2 µb If µ = µ b , then P(car) = 1 3 (Model 2) µ µ b → 0 , and P(car) → 1 If µ b → ∞ , then 2 (Model 1) Nested logit models – p. 12/23

Nested Logit Model Probability model: e µβT + µ µb ln 2 e µV bus 1 P ( bus ) = e µV car + e µV bus = µb ln 2 = e µβT + e µβT + µ 1 + 2 − µ µb If µ = µ b , then P(bus) = 2 3 (Model 2) µ µ b → 0 , then P(bus) → 1 If 2 (Model 1) Nested logit models – p. 13/23

Nested Logit Model 1 P(car) P(bus) 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 mu/mu_b µ µ b Nested logit models – p. 14/23

Solving the paradox µ If µ b → 0 , we have P ( car ) = 1 / 2 P ( bus ) = 1 / 2 P ( red bus | bus ) = 1 / 2 P ( blue bus | bus ) = 1 / 2 P ( red bus ) = P ( red bus | bus ) P ( bus ) = 1 / 4 P ( blue bus ) = P ( blue bus | bus ) P ( bus ) = 1 / 4 Nested logit models – p. 15/23

Comments • A group of similar alternatives is called a nest • Each alternative belongs to exactly one nest • The model is named Nested Logit • The ratio µ/µ b must be estimated from the data • 0 < µ/µ b ≤ 1 (between models 1 and 2) Nested logit models – p. 16/23

Derivation from random utility • Let C be the choice set. • Let C 1 , . . . , C M be a partition of C . • The model is derived as M � P ( i |C ) = Pr( i | m, C ) Pr( m |C ) . m =1 • Each i belongs to exactly one nest m . P ( i |C ) = Pr( i | m ) Pr( m |C ) . • Utility: error components U i = V i + ε i = V i + ε m + ε im . Nested logit models – p. 17/23

Derivation: Pr( i | m ) Pr( i | m ) = Pr( U i ≥ U j , j ∈ C m ) = Pr( V i + ε m + ε im ≥ V j + ε m + ε jm , j ∈ C m ) = Pr( V i + ε im ≥ V j + ε jm , j ∈ C m ) Assumption: ε im i.i.d. EV( 0 , µ m ) e µ m V i Pr( i | m ) = j ∈C m e µ m V j . � Nested logit models – p. 18/23

Derivation: Pr( m |C ) � � Pr( m |C ) = Pr i ∈C m U i ≥ max max j ∈C ℓ U j , ∀ ℓ � = m � � = Pr ε m + max i ∈C m ( V i + ε im ) ≥ ε ℓ + max j ∈C ℓ ( V j + ε jℓ ) , ∀ ℓ � = m , As ε im are i.i.d. EV( 0 , µ m ), i ∈C m ( V i + ε im ) ∼ EV ( ˜ max V m , µ m ) , where 1 � ˜ e µ m V i . V m = ln µ m i ∈C m Nested logit models – p. 19/23

Derivation: Pr( m |C ) Denote i ∈C m ( V i + ε im ) = ˜ V m + ε ′ max m , to obtain Pr( m |C ) = Pr( ˜ m + ε m ≥ ˜ V m + ε ′ V ℓ + ε ′ ℓ + ε ℓ , ∀ ℓ � = m ) . where ε ′ m ∼ EV (0 , µ m ) . Define ε m = ε ′ ˜ m + ε m , to obtain Pr( m |C ) = Pr( ˜ ε m ≥ ˜ V m + ˜ V ℓ + ˜ ε ℓ , ∀ ℓ � = m ) . Nested logit models – p. 20/23

Derivation: Pr( m |C ) Assumption: ˜ ε m i.i.d. EV( 0 , µ ) Pr( m |C ) = Pr( ˜ ε m ≥ ˜ V m + ˜ V ℓ + ˜ ε ℓ , ∀ ℓ � = m ) e µ ˜ V m = V p . � M p =1 e µ ˜ We obtain the nested logit model e µ ˜ e µ m V i V m P ( i |C ) = � j ∈C m e µ m V j � M p =1 e µ ˜ V p � � µ ℓ ∈C m e µ m V ℓ µ m ln � exp e µ m V i = � j ∈C m e µ m V j � � � M µ µ p ln � ℓ ∈C p e µ p V ℓp p =1 exp Nested logit models – p. 21/23

Nested Logit Model µ • If µ m = 1 , for all m , NL becomes logit. • Sequential estimation: • Estimation of NL decomposed into two estimations of logit • Estimator is consistent but not efficient • Simultaneous estimation: • Log-likelihood function is generally non concave • No guarantee of global maximum • Estimator asymptotically efficient • Log likelihood for observation n is ln P ( i n |C n ) = ln P ( i n |C mn ) + ln P ( C mn |C n ) where i n is the chosen alternative. Nested logit models – p. 22/23

Correlation Correlation matrix is block diagonal:  1 if i = j,   1 − µ 2  Corr( U i , U j ) = if i � = j , i and j are in the same nest m, µ 2 m   otherwise . 0  Variance-covariance matrix is block diagonal:  π 2 if i = j,   6 µ 2    6 µ 2 − π 2 π 2 Cov( U i , U j ) = if i � = j , i and j are in the same nest m, 6 µ 2   m   0 otherwise .  Nested logit models – p. 23/23