Choice theory Michel Bierlaire michel.bierlaire@epfl.ch Transport - PowerPoint PPT Presentation

Choice theory Michel Bierlaire michel.bierlaire@epfl.ch Transport and Mobility Laboratory Choice theory – p. 1/26

Framework Choice: outcome of a sequential decision-making process • Definition of the choice problem: How do I get to EPFL? • Generation of alternatives: car as driver, car as passenger, train • Evaluation of the attributes of the alternatives: price, time, flexibility, comfort • Choice: decision rule • Implementation: travel Choice theory – p. 2/26

Framework A choice theory defines 1. decision maker 2. alternatives 3. attributes of alternatives 4. decision rule Choice theory – p. 3/26

Framework Decision-maker : • Individual or a group of persons • If group of persons, we ignore internal interactions • Important to capture difference in tastes and decision-making process • Socio-economic characteristics: age, gender, income, education, etc. Choice theory – p. 4/26

Framework Alternatives : • Environment: universal choice set ( U ) • Individual n : choice set ( C n ) Choice set generation: • Availability • Awareness Swait, J. (1984) Probabilistic Choice Set Formation in Transportation Demand Models Ph.D. dissertation, Department of Civil Engineering, MIT, Cambridge, Ma. Choice theory – p. 5/26

Framework Continuous vs. discrete Continuous choice set: q Milk ✻ p Beer q Beer + p Milk q Milk = I ✠ C n ✲ q Beer Discrete choice set: C n = { Car, Bus, Bike } Choice theory – p. 6/26

Framework Attributes ➜ cost ✔ Generic vs. specific ➜ travel time ➜ walking time ✔ Quantitative vs. qualita- ➜ comfort tive ✔ Perception ➜ bus frequency ➜ etc. Choice theory – p. 7/26

Framework Decision rules Neoclassical economic theory Preference-indifference operator � (i) reflexivity a � a ∀ a ∈ C n (ii) transitivity a � b and b � c ⇒ a � c ∀ a, b, c ∈ C n (iii) comparability a � b or b � a ∀ a, b ∈ C n Choice theory – p. 8/26

Framework Decision rules Neoclassical economic theory (ctd) ☞ Numerical function ∃ U n : C n − → R : a � U n ( a ) such that a � b ⇔ U n ( a ) ≥ U n ( b ) ∀ a, b ∈ C n ✞ ☎ Utility ✝ ✆ Choice theory – p. 9/26

Framework Decision rules • Utility is a latent concept • It cannot be directly observed Choice theory – p. 10/26

Framework Continuous choice set • Q = { q 1 , . . . , q L } consumption bundle • q i is the quantity of product i consumed • Utility of the bundle: U ( q 1 , . . . , q L ) • Q a � Q b iff U ( q a 1 , . . . , q a L ) ≥ U ( q b 1 , . . . , q b L ) • Budget constraint: L � p i q i ≤ I. i =1 Choice theory – p. 11/26

Framework Decision-maker solves the optimization problem q ∈ R L U ( q 1 , . . . , q L ) max subject to L � p i q i = I. i =1 Example with two products... Choice theory – p. 12/26

Framework q 1 ,q 2 U = β 0 q β 1 1 q β 2 max 2 subject to p 1 q 1 + p 2 q 2 = I. Lagrangian of the problem: L ( q 1 , q 2 , λ ) = β 0 q β 1 1 q β 2 2 + λ ( I − p 1 q 1 − p 2 q 2 ) . Necessary optimality condition ∇ L ( q 1 , q 2 , λ ) = 0 Choice theory – p. 13/26

Framework Necessary optimality conditions β 0 β 1 q β 1 − 1 q β 2 − = 0 λp 1 1 2 β 0 β 2 q β 1 1 q β 2 − 1 − = 0 λp 2 2 p 1 q 1 + p 2 q 2 − I = 0 . We have β 0 β 1 q β 1 1 q β 2 − = 0 λp 1 q 1 2 β 0 β 2 q β 1 1 q β 2 − λp 2 q 2 = 0 2 so that λI = β 0 q β 1 1 q β 2 2 ( β 1 + β 2 ) Choice theory – p. 14/26

Framework Therefore λI β 0 q β 1 1 q β 2 = 2 ( β 1 + β 2 ) As β 0 β 2 q β 1 1 q β 2 = λp 2 q 2 , we obtain (assuming λ � = 0 ) 2 Iβ 2 q 2 = p 2 ( β 1 + β 2 ) Similarly, we obtain Iβ 1 q 1 = p 1 ( β 1 + β 2 ) Choice theory – p. 15/26

Framework Iβ 1 = q 1 p 1 ( β 1 + β 2 ) Iβ 2 = q 2 p 2 ( β 1 + β 2 ) Demand functions Choice theory – p. 16/26

Framework Discrete choice set • Similarities with Knapsack problem • Calculus cannot be used anymore U = U ( q 1 , . . . , q L ) with � if product i is chosen 1 q i = 0 otherwise and � q i = 1 . i Choice theory – p. 17/26

Framework • Do not work with demand functions anymore • Work with utility functions • U is the “global” utility • Define U i the utility associated with product i . • It is a function of the attributes of the product (price, quality, etc.) • We say that product i is chosen if U i ≥ U j ∀ j. Choice theory – p. 18/26

Framework Example: two transportation modes = − βt 1 − γc 1 U 1 = − βt 2 − γc 2 U 2 with β , γ > 0 U 1 ≥ U 2 iff − βt 1 − γc 1 ≥ − βt 2 − γc 2 that is − β γ t 1 − c 1 ≥ − β γ t 2 − c 2 or c 1 − c 2 ≤ − β γ ( t 1 − t 2 ) Choice theory – p. 19/26

Framework Obvious cases: • c 1 ≥ c 2 and t 1 ≥ t 2 : 2 dominates 1. • c 2 ≥ c 1 and t 2 ≥ t 1 : 1 dominates 2. • Trade-offs in over quadrants Choice theory – p. 20/26

Framework 4 train is chosen 2 cost by car-cost by train 0 -2 car is chosen -4 -4 -2 0 2 4 time by car-time by train Choice theory – p. 21/26

Framework 4 train is chosen 2 cost by car-cost by train 0 -2 car is chosen -4 -4 -2 0 2 4 time by car-time by train Choice theory – p. 22/26

Assumptions Decision rules Neoclassical economic theory (ctd) Decision-maker Analyst ✔ perfect discriminating ca- ✔ knowledge of all attributes pability ✔ perfect knowledge of � (or ✔ full rationality U n ( · ) ) ✔ permanent consistency ✔ no measurement error Choice theory – p. 23/26

Assumptions Uncertainty Source of uncertainty? ☞ Decision-maker: stochastic decision rules ☞ Analyst: lack of information ☞ Bohr: “Nature is stochastic” ☞ Einstein: “God does not play dice” Choice theory – p. 24/26

Assumptions Lack of information: random utility models Manski 1973 The structure of Random Utility Models Theory and Decision 8:229–254 Sources of uncertainty: ☞ Unobserved attributes ☞ Unobserved taste variations ☞ Measurement errors ☞ Instrumental variables For each individual n , U in = V in + ε in and P ( i |C n ) = P [ U in = max j ∈C n U jn ] = P ( U in ≥ U jn ∀ j ∈ C n ) Choice theory – p. 25/26

Random utility models U in = V in + ε in • Dependent variable is latent • Only differences matter P ( i |C n ) = P ( U in ≥ U jn ∀ j ∈ C n ) = P ( U in + K ≥ U jn + K ∀ j ∈ C n ) ∀ K ∈ R P ( i |C n ) = P ( U in ≥ U jn ∀ j ∈ C n ) = P ( λU in ≥ λU jn ∀ j ∈ C n ) ∀ λ > 0 Choice theory – p. 26/26