Analysis of implicit choice set generation using the Constrained Multinomial Logit model

Transcription

1 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 1/27 Analysis of implicit choice set generation using the Constrained Multinomial Logit model Michel Bierlaire, Ricardo Hurtubia and Gunnar Flötterröd Transport and Mobility Laboratory

2 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 2/27 Introduction Choice model: P n (i C n ) Common practice: C n characterized by deterministic rules Modeling the choice set generation (Manski, 1977): P n (i) = Combinatorial complexity C m C P n (i C m )P n (C m ) Operational instances: Random constraints (Swait and Ben-Akiva, 1987, Ben-Akiva and Boccara, 1995) MEV framework (Swait, 2001)

3 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 3/27 Introduction Heuristics: Implicit Availability/Perception model (Cascetta and Papola, 2001) Constrained Multinomial Logit model (Martinez et al., 2009) Objective: analyze the quality of the CMNL as a choice set generation process.

4 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 4/27 Deterministic Choice Set Generation Availability conditions Exogenous variables A in = { Choice model 1 if alternative i is considered by individual n, 0 otherwise. P n (i C n ) = Pr (U in U jn, j C n ) = Pr (U in + lna in U jn + lna jn, j C). Note: a choice model with deterministic choice set generation can always be written in terms of the universal choice set

5 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 5/27 Deterministic Choice Set Generation Logit model: P n (i) = e V in+ln A in j C ev jn+ln A jn = A in e V in j C A jne V jn What if variables A in are not exogenously given?

6 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 6/27 Probabilistic Choice Set Generation Approaches: Correct model: Manski (1977) most of the time impractical Sampling of alternatives: Assume C n = C, n Sample a subset for estimation see Frejinger, Bierlaire and Ben-Akiva (forthcoming) for route choice Replace A in by a probability distribution Availability/Perception (Cascetta and Papola, 2001) Cutoffs (Martinez et al., 2009)

7 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 7/27 Cutoffs Optimization problem of rational consumer n: max δ ni δ ni U in (X i ) i C subject to δ ni = 1, i C δ ni {0, 1}, i C But attributes are meaningful only within some bounds l nk X ik u nk i C, k An alternative i with one of its attributes is out of bounds is not considered

8 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 8/27 Cutoffs Examples: Item too expensive Traveling by train involves a too long walking distance to the station etc. If these rules are deterministic, the variables A in can be derived If not, what can be done?

9 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 9/27 Cutoffs Idea: relax the constraint in a probabilistic way Example: constraint l X V not considered = l + ε 1 V considered = X + ε 2 P(considered) = Example: constraint X u e ρx e ρx + e ρl = e ρ(l X) P(considered) = e ρx e ρx + e ρu = e ρ(x u)

10 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 10/27 Cutoffs Example: 2 X rho=1 rho=5 rho=

11 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 11/27 Cutoffs Example: X rho=1 rho=5 rho=

12 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 12/27 Cutoffs Constraint l X u P(considered) = We denote this quantity by φ n (X) e ρ(l X) 1 + e ρ(x u)

13 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 13/27 Cutoffs Example: 2 X rho=1 rho=5 rho=

14 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 14/27 Cutoffs The utility function now becomes V i = k β k X ik + k 1 ρ lnφ n(x ik ) where k ranges only on constrained attributes. Note that lnφ(x) = ln(1 + e ρ(l X) ) ln(1 + e ρ(x u) ) = ln(1 + e ρl e ρx ) ln(1 + e ρx e ρu ) Can be estimated, although it is difficult

15 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 15/27 Comparison of CMNL and Manski Simple example: Binary logit: C = {1, 2} Alternative 1 is always available Alternative 2 is considered with probability φ 2 We have P(C n = {1}) = 1 φ 2 P(C n = {2}) = 0 P(C n = {1, 2}) = φ 2

16 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 16/27 Comparison of CMNL and Manski Manski s model Root Choice sets {1} {2} {1,2} Alternatives 1 2

17 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 17/27 Comparison of CMNL and Manski Manski s model P (1) = P(C n = {1}) ev 1 e V 1 + P(C n = {2})0 + P(C n = {1, 2}) e V 1 e V 1+e V 2 CMNL model = (1 φ 2 ) + φ 2 e V 1 e V 1+e V 2 P(1) = e V 1 e V 1 + e V 2 +ln φ 2. Note: for given V s, Manski is linear in φ 2, not CMNL

18 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 18/27 Comparison of CMNL and Manski Equal utility 1 V_1=V_2 CMNL Manski P_ phi_2

19 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 19/27 Comparison of CMNL and Manski Alt. 1 is dominant 1 V_2-V_1= P_ phi_2 CMNL Manski

20 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 20/27 Comparison of CMNL and Manski Alt. 2 is dominant 1 V_2-V_1=2 CMNL Manski P_ phi_2

21 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 21/27 Comparison of CMNL and Manski Alt. 2 is even more dominant 1 V_2-V_1=4 CMNL Manski P_ phi_2

22 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 22/27 Comparison of CMNL and Manski CMNL underestimates the choice probability for alternative 1 When alt. 1 is dominant, it makes no difference if it is preferred because of a high utility, or if because 2 is not even considered. When alt. 2 is dominant, the CMNL may be completely off Clearly, the model parameters could be adjusted to attenuate that error

23 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 23/27 Synthetic data Swissmetro data set, 5607 observations 1. Driving a car (CAR) 2. Regular train (TRAIN) 3. Swissmetro, the future high speed train (SM) Exogenous variables come from the data set Synthetic choice set TRAIN and SM always available CAR available depending on travel time φ CAR = exp(ω(tt CAR /60 a)) Synthetic choice

24 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 24/27 Synthetic data Postulated model Parameter Value Car Train Swissmetro ASC CAR ASC SM β cost Cost (CHF) Cost (CHF) Cost (CHF) β tt In veh. travel time (minutes) In veh. travel time (minutes) In veh. travel time (minutes) β he Headway (minutes) Headway (minutes) a 3 Consideration threshold of car (hours) ω 1,2,3,5,10 Consideration dispersion of car

25 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 25/27 Synthetic data 100 choice data sets are simulated for each value of ω Results: mean of each parameter over 100 estimations t-test against the true value, based on the empirical std. deviation.

26 Estimation results for Manski s model real ω value parameter real value estimate t-test estimate t-test estimate t-test estimate t-test ASC CAR ASC SM β cost β he β time a ω see top

27 Estimation results for CMNL model real ω value parameter real value estimate t-test estimate t-test estimate t-test estimate t-test ASC CAR ASC SM * * β cost * * * β he β time * * * * a * * * ω see top (* indicates an insignificant parameter)

28 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 26/27 Synthetic data Manski model performs well, as expected CMNL may significantly bias the estimates The more deterministic the constraint, the better the CMNL

29 Analysis of implicit choice set generation using the Constrained Multinomial Logit model p. 27/27 Conclusion CMNL is not adequate to model the choice set generation It is a model on its own, derived from semi-compensatory arguments Its complexity is linear in the number of alternatives, while Manski s model is exponential. Research question: how can we modify the CMNL to be a better approximation of Manski s model?