The Constrained Multinomial Logit: A semi compensatory choice model

Transcription

1 The Constraned Multnomal Logt: A sem compensatory choce model F. Martnez (Uversty of Chle F. Agula (Uversty of Chle R. Hurtuba (TRANSP-OR EPFL August 28, 2008

2 Introducton Tradtonal logt models assume a compensatory utlty functon (trade-off between attrbutes. Ths approach fals to recogze attrbute thresholds n consumer behavor. A mxed strategy s proposed, usng compensatory utltes wth cutoff factors that that restran choces to the avalable doman.

3 Contents of presentaton Dscrete choce problem Constraned random utlty Constraned multnomal logt Applcaton examples Calbraton ssues Conclusons

4 The constraned dscrete choce problem Consumer s problem: max δ C δ U n ( X, p s. a C δ 1, δ {0,1} C Requres to: Specfy a utlty functon able to nclude constrants or Specfy a predefned set of avalable alternatves (C

5 The constraned dscrete choce problem Ratonal behavor: Max utlty s.t. constrants: max δ C δ U n ( X, p s. t C δ 1, δ {0,1} C a nk X k b nk C, k {1,..., K 1} a nk p b nk C

6 Approaches Non compensatory utlty, e.g. elmnaton by aspects Tversky, 1972 Two stage approach. Generate each consumer s feasble choce set. Dffculty: large choce sets Heurstc to reduce choce sets Mansk, 77 Swat - BenAkva, 87 BenAkva - Boccara, 95 Cantllo - Ortúzar, 04 Morkawa, 95 One step approach: reduced utlty Determstc model: lnear penaltes ncluded n utlty. Contnuous but non-dfferentable Smulate avalablty/percepton mplctly n the extended utlty. Bnomal logt Swat, 01 Cascetta - Papola, 01

7 Constraned random utlty V n ( Z 1 V ( Z + lnφ ( Z C n μ + ε Compensatory (ndrect utlty Utlty penalty Random term φ K k 1 φ L nk ( a φ nk U nk ( b nk

8 Constraned random utlty Lower and upper cutoffs: φ L nk 1+ exp 1 ( ω ( a Z + ρ η f a k 1 nk k k k f a nk nk << Z k Z k φ U nk 1+ exp 1 ( ( ω Z b + ρ k k nk k 1 η k f f b b nk nk >> Z k Z k ρ k 1 ω k 1 ηk ln ηk

9 Constraned random utlty L φ nk U φ nk ank Z k bnk

10 Constraned Multnomal Logt C n n Z Z V Z V ε φ μ + + ( ln 1 ( ( C j C nj nj C V V P exp( exp( μ φ μ φ Gumbel dstrbuted (0,μ

11 Constraned Multnomal Logt 1 Probablty of the compensatory utlty Compensatory utlty fucton Probablty of the constraned utlty η a b Z

12 Constraned Multnomal Logt Propertes: Preserves the closed logt formula Represents a jont logt model Modelng compensatory choce Modelng constrant volaton Applcatons: Real estate supply: planng regulatons Consumers: ncome and tme budgets, attrbute percepton, externaltes and agglomeraton economes Transport: congeston

13 Applcaton example 1 Land-use model (Martnez et al, 2008: P n / φ exp( B φ exp( B g C g g B α + β x n nk k k φ 1+ exp 1 ( ω ( δ I n Average ncome n zone To estmate: δ + ω, n a ρ n n β nk

14 Applcaton example 1

15 Applcaton example 1 Smlar log-lkelhood Cutoff parameters were possble to dentfy Constants are lower n the CMNL (behavor explaned by cutoffs Dfferent forecastng results when constraned attrbute changes sgfcantly

16 Applcaton example 2 Land-use model (MUSSA 2008 P n / φ H n exp( B φ H exp( B g C g g g Number of bddng households φ 1+ 1 η η 1 exp ( a I ( ω n ρ 1 ω 1 η ln η To estmate

17 Applcaton example 2 Mn tolerated average zone-ncome (US$ Income levels (US$ < > 2371

18 Applcaton example 2

19 Calbraton ssues Explct exogenous constrants (budget, capacty are useful when forecastng demand. Problems: Every observaton complyng wth restrctons. Correlaton between parameters n the cutoff and the compensatory utlty functon.

20 Conclusons The CMNL enhances the dscrete choce models by mposng a realstc doman avodng the choce set generaton. Preserves the closed logt formula. Allows to nclude multple constrants It can be used to model both endogenous and exogenous constrants. Requres further research on calbraton methods

21 Questons?