Discrete Choice Analysis I Moshe Ben-Akiva 1.201 / 11.545 / ESD.210 Transportation Systems Analysis: Demand & Economics Fall 2008Outline of 2 Lectures on Discrete Choice ● Introduction ● A Simple Example ● The Random Utility Model ● Specification and Estimation ● Forecasting ● IIA Property ● Nested Logit 2Outline of this Lecture ● Introduction ● A simple example – route choice ● The Random Utility Model – Systematic utility – Random components ● Derivation of the Probit and Logit models – Binary Probit – Binary Logit – Multinomial Logit 3Continuous vs. Discrete Goods Continuous Goods Discrete Goods x2 Indifference curves u1 u2 u3 auto x1 bus 4Discrete Choice Framework ● Decision-Maker – Individual (person/household) – Socio-economic characteristics (e.g. Age, gender,income, vehicle ownership) ● Alternatives – Decision-maker n selects one and only one alternative from a choice set Cn={1,2,…,i,…,Jn} with Jn alternatives ● Attributes of alternatives (e.g.Travel time, cost) ● Decision Rule – Dominance, satisfaction, utility etc. 5Choice: Travel Mode to Work • Decision maker: an individual worker • Choice: whether to drive to work or take the bus to work • Goods: bus, auto • Utility function: U(X) = U(bus, auto) • Consumption bundles: {1,0} (person takes bus) {0,1} (person drives) 6Consumer Choice • Consumers maximize utility – Choose the alternative that has the maximum utility (and falls within the income constraint) If U(bus) > U(auto) �choose bus If U(bus) < U(auto) �choose auto U(bus)=? U(auto)=? 7Constructing the Utility Function ● U(bus) = U(walk time, in-vehicle time, fare, …) U(auto) = U(travel time, parking cost, …) ● Assume linear (in the parameters) U(bus) = β1×(walk time) + β2 ×(in-vehicle time) + … ● Parameters represent tastes, which may vary over people. – Include socio-economic characteristics (e.g., age, gender, income) – U(bus) = β1 ×(walk time) + β2 ×(in-vehicle time) + β3 ×(cost/income) + … 8Deterministic Binary Choice ● If U(bus) - U(auto) > 0 , Probability(bus) = 1 If U(bus) - U(auto) < 0 , Probability(bus) = 0 0 U(bus)-U(auto) P(bus) 0 1 9Probabilistic Choice ● Random utility model Ui = V(attributes of i; parameters) + epsiloni ● What is in the epsilon? Analysts’ imperfect knowledge: – Unobserved attributes – Unobserved taste variations – Measurement errors – Use of proxy variables ● U(bus) = β1 ×(walk time) + β2 ×(in-vehicle time + β3 ×(cost/income) + … + epsilon_bus 10Probabilistic Binary Choice 0 V(bus)-V(auto) P(bus) 0 1 11A Simple Example: Route Choice – Sample size: N = 600 – Alternatives: Tolled, Free – Income: Low, Medium, High 12 Route choice Income Low (k=1) Medium (k=2) High (k=3) Tolled (i=1) 10 100 90 200 Free (i=2) 140 200 60 400 150 300 150 600A Simple Example: Route Choice Probabilities ● (Marginal) probability of choosing toll road P(i = 1) Pˆ(i = 1) = 200 / 600 = 1/3 ● (Joint) probability of choosing toll road and having medium income: P(i=1, k=2) Pˆ(i = 1, k = 2) = 100 /600 = 1/6 2 3 ∑∑ ( , ) = 1P i k i=1 k =1 13Conditional Probability P(i|k) P(i, k) = P(i) ⋅ P(k | i) = P(k) ⋅ P(i | k) Independence P(i | k) = P(i) P(k | i) = P(k) P i ( ) = ∑P i k ( , ) k P k ( ) = ( , )∑P i k i P i k P k ( | ) i = ( , ), P i ( ) ≠ 0 P i ( ) P i k ( , )P i k ( | ) = , P k ( ) ≠ 0 ( ) P k 14Model : P(i|k) ● Behavioral Model~ Probability (Route Choice|Income) = P(i|k) ● Unknown parameters P(i = 1| k = 1) =π1 P(i = 1| k = 2) =π2 P(i = 1| k = 3) =π3 15Example: Model Estimation ● Estimation 6.0333 .00.067 5 3 33 1 215 1 1 ˆ,ˆ,ˆ === === πππ ( ) )15/11 ===s 020 .0150 1(15 /1ˆ1ˆ 1 11 −⋅−⋅ N ππ 1/3 N=600 2ˆπ Sampling distribution frequency πˆ ⋅ (1−πˆ ) 1/ 3⋅ (1−1/ 3) = 0.027 2 2 =s2 = 300 N2 πˆ ⋅ (1−πˆ ) 3 / 5⋅ (1− 3 / 5) = 0.040 3 3 =s3 = 150 N3 Standard errors 16c o ceree =str ut onExample: Forecasting ● Toll Road share under existing income distribution: 33% ● New income distribution Route h i Income Low (k=1) Medium (k=2) High (k=3) Tolled (i=1) 1/15*45=3 1/3*300=100 3/5*255=153 256 43% F (i 2) 42 200 102 344 57% New income distribution 45 300 255 600 Existing income di ib i 150 300 150 600 ● Toll road share: 33%�43% 17The Random Utility Model ● Decision rule: Utility maximization – Decision maker n selects the alternative i with the highest utility Uin among Jn alternatives in the choice set Cn. Uin = Vin + εin Vin =Systematic utility : function of observable variables εin =Random utility 18The Random Utility Model ● Choice probability: P(i|C ) = P(Uin ≥Ujn, ∀j ∈C ) n n = P(Uin - Ujn ≥0, ∀j ∈C )n = P(Uin = maxj Ujn,∀j ∈C )n ● For binary choice: Pn(1) = P(U1n ≥U2n) = P(U1n– U2n≥0) 19The Random Utility Model Routes Attributes Utility (utils) Travel time (t) Travel cost (c) Tolled (i=1) t1 c1 U1 Free (i=2) t2 c2 U2 U1 = −β1t1 −β2c1 +ε1 U2 = −β1t2 −β2c2 +ε2 β1,β2 > 0 20The Random Utility Model ● Ordinal utility - Decisions are based on utility differences - Unique up to order preserving transformation U1 = (−β1 1 t −β2c1 +ε1 + K )λ U = (−βt −βc +ε + K)λ2 1 2 2 2 2 1, 2,β β λ > 0 21The Random Utility Model + ++β + ++ • • • V1 > V2 V1 = V2 V1 < V2 + • c1-c2 Alt. 2 is dominant + ++++ + + β1• + ++ U1 = − β ⋅t1 − c1 +ε1• • + • •• + •• + + t1-t22 1U2 = − β⋅t2 − c2 +ε2• ++ β2 ••• • 1 • + • β= β1 = "value of time" •• •• + β2 • Alt. 1 is dominant • Choice = 1 + Choice = 2 β1U1 −U2 = − β2 ⋅ (t1 − t2) − (c1 − c2) + (ε1 −ε) 22 2The Systematic Utility ● Attributes: describing the alternative – Generic vs. Specific • Examples: travel time, travel cost, frequency – Quantitative vs. Qualitative • Examples: comfort, reliability, level of service – Perception – Data availability ● Characteristics: describing the decision-maker – Socio-economic variables • Examples: income,gender,education 23Random Terms ● Capture imperfectness of information ● Distribution of epsilons ● Variance/covariance structure – Correlation between alternatives – Multidimensional decision • Example: Mode and departure time choice ● Typical models – Logit model