Choice Theory. Matthieu de Lapparent

Transcription

1 Choice Theory Matthieu de Lapparent Transport and Mobility Laboratory, School of Architecture, Civil and Environmental Engineering, Ecole Polytechnique Fédérale de Lausanne Transport and Mobility Laboratory Choice Theory 1 / 37

2 Outline 1 Choice theory foundations 2 Consumer theory 3 Simple example 4 Random utility theory Transport and Mobility Laboratory Choice Theory 2 / 37

3 Choice theory foundations Choice theory 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, bicycle, walk... Evaluation of the attributes of the alternatives: Price, time, flexibility, reliability, comfort Choice: Decision rule Implementation: Travel Transport and Mobility Laboratory Choice Theory 3 / 37

4 Choice theory foundations Building the theory A choice theory defines 1 Decision maker 2 Alternatives 3 Attributes of alternatives 4 Decision rule Transport and Mobility Laboratory Choice Theory 4 / 37

5 Choice theory foundations Decision maker Unit of analysis Individual Socio-economic characteristics: age, gender, income, education, etc. A group of persons (ignoring within-group peer effects) Household, firm, government agency Group characteristics Notation: subscript n Transport and Mobility Laboratory Choice Theory 5 / 37

6 Choice theory foundations Alternatives Choice set Mutually exclusive, finite, exhaustive set of alternatives Universal choice set C Individual n: choice set C n C Availability, awareness, feasibility, consideration Example: Choice of transport mode C = {car, bus, metro, walk}...traveller has no drivers licence, trip is 12km long C n = {bus, metro} Swait, J. (1984) Probabilistic Choice Set Formation in Transportation Demand Models Ph.D. dissertation, Department of Civil Engineering, MIT, Cambridge, Ma. Transport and Mobility Laboratory Choice Theory 6 / 37

7 Choice theory foundations Continuous choice set Microeconomic demand analysis Commodity bundle q 1 : quantity of milk q 2 : quantity of bread q 3 : quantity of butter q 3 p1q1 + p2q2 + p3q3 = I q 2 Unit price: p i Budget: I q 1 Transport and Mobility Laboratory Choice Theory 7 / 37

8 Choice theory foundations Discrete choice set Discrete choice analysis List of alternatives Brand A Brand B Brand C C B A Transport and Mobility Laboratory Choice Theory 8 / 37

9 Choice theory foundations Alternative attributes Characterize each alternative i for each individual n cost travel time walking time comfort bus frequency etc. Nature of the variables Generic or specific Quantitative or qualitative Measured or perceived Transport and Mobility Laboratory Choice Theory 9 / 37

10 Choice theory foundations Decision rules Standard microeconomic decision-maker Full knowledge of options, context and environment Utility Organized and stable system of preferences Evaluates each alternative and assigns precise pay-off (measured through the utility index) Selects alternative with highest pay-off Captures attractiveness of alternative Allows ranking (ordering) of alternatives What decision maker optimizes Transport and Mobility Laboratory Choice Theory 10 / 37

11 Choice theory foundations A matter of viewpoints Individual perspective Individual possesses perfect information and discrimination capacity Modeler perspective Modeler does not have full information about choice process Treats the utility as a random variable At the core of the concept of random utility Transport and Mobility Laboratory Choice Theory 11 / 37

12 Consumer theory Consumer theory Neoclassical consumer theory Underlies mathematical analysis of preferences Allows us to transform attractiveness rankings... into operational demand functions Keep in mind Utility is a latent concept It cannot be directly observed Figure: Jeremy Bentham Transport and Mobility Laboratory Choice Theory 12 / 37

13 Consumer theory Consumer theory Continuous choice set Consumption bundle Budget constraint Q = No attributes, just quantities q 1. q L ; p = L p l q l I. l=1 p 1. p L Transport and Mobility Laboratory Choice Theory 13 / 37

14 Consumer theory Continuity : if Q a is preferred to Q b and Q c is arbitrarily close to Q a, then Q c is preferred to Q b. Transport and Mobility Laboratory Choice Theory 14 / 37 Preferences Operators,, and Q a Q b : Q a is preferred to Q b, Q a Q b : indifference between Q a and Q b, Q a Q b : Q a is at least as preferred as Q b. To ensure consistent ranking Completeness: for all bundles a and b, Q a Q b or Q a Q b or Q a Q b. Transitivity: for all bundles a, b and c, if Q a Q b and Q b Q c then Q a Q c.

15 Consumer theory Preferences, cont. further non necessary assumptions to have a well-behaved utility function: monotonicity, non satiety, convexity Transport and Mobility Laboratory Choice Theory 15 / 37

16 Consumer theory Utility Utility function Parametrized function: Ũ = Ũ(q 1,..., q L ; θ) = Ũ(Q; θ) Consistent with the preference indicator: Ũ(Q a ; θ) Ũ(Q b; θ) is equivalent to Q a Q b. Unique up to an order-preserving transformation Transport and Mobility Laboratory Choice Theory 16 / 37

17 Consumer theory Optimization problem Optimization Decision-maker solves the optimization problem max q R L U(q 1,..., q L ) subject to the budget (available income) constraint L p i q i = I. i=1 Demand Quantity is a function of prices and budget q = f (I, p; θ) Transport and Mobility Laboratory Choice Theory 17 / 37

18 Consumer theory Optimization problem subject to Lagrangian of the problem: max U = β 0 q β 1 q 1,q 1 qβ p 1 q 1 + p 2 q 2 = I. L(q 1, q 2, λ) = β 0 q β 1 1 qβ 2 2 λ(p 1q 1 + p 2 q 2 I ). Necessary optimality condition L(q 1, q 2, λ) = 0 where λ is the Lagrange multiplier and β s are the Cobb-Douglas preference parameters Transport and Mobility Laboratory Choice Theory 18 / 37

19 Consumer theory Framework Optimality conditions Lagrangian is differentiated to obtain the first order conditions We have L/ q 1 = β 0 β 1 q β q β 2 2 λp 1 = 0 L/ q 2 = β 0 β 2 q β 1 1 qβ λp 2 = 0 L/ λ = p 1 q 1 + p 2 q 2 I = 0 β 0 β 1 q β 1 1 qβ 2 2 λp 1 q 1 = 0 β 0 β 2 q β 1 1 qβ 2 2 λp 2 q 2 = 0 Adding the two and using the third optimality condition λi = β 0 q β 1 1 qβ 2 2 (β 1 + β 2 ) Transport and Mobility Laboratory Choice Theory 19 / 37

20 Consumer theory Framework Equivalent to β 0 q β 1 1 qβ 2 2 = λi (β 1 + β 2 ) As β 0 β 2 q β 1 1 qβ 2 2 = λp 2q 2, we obtain (assuming λ 0) q 2 = I β 2 p 2 (β 1 + β 2 ) Similarly, we obtain q 1 = I β 1 p 1 (β 1 + β 2 ) Transport and Mobility Laboratory Choice Theory 20 / 37

21 Consumer theory Demand functions Product 1 Product 2 q 1 = I p 1 β 1 β 1 + β 2 q 2 = I p 2 β 2 β 1 + β 2 Comments Demand decreases with price Demand increases with budget Demand independent of β 0, which does not affect the ranking Transport and Mobility Laboratory Choice Theory 21 / 37

22 Consumer theory Marginal rate of substitution Factoring out λ from first order conditions we get p 1 = U(q )/ q 1 p 2 U(q = MU(q 1) )/ q 2 MU(q 2 ) MRS Ratio of marginal utilities (right) equals... ratio of prices of the 2 goods (left) Holds if consumer is making optimal choices Transport and Mobility Laboratory Choice Theory 22 / 37

23 Consumer theory Discrete goods Discrete choice set Calculus cannot be used anymore U = U(q 1,..., q L ) with and { 1 if product i is chosen q i = 0 otherwise q i = 1. i Transport and Mobility Laboratory Choice Theory 23 / 37

24 Consumer theory 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. Transport and Mobility Laboratory Choice Theory 24 / 37

25 Simple example Simple example: mode choice Attributes Attributes Alternatives Travel time (t) Travel cost (c) Car (1) t 1 c 1 Train (2) t 2 c 2 Utility Ũ = Ũ(y 1, y 2 ), where we impose the restrictions that, for i = 1, 2, y i = { 1 if travel alternative i is chosen, 0 otherwise; and that only one alternative is chosen: y 1 + y 2 = 1. Transport and Mobility Laboratory Choice Theory 25 / 37

26 Simple example Simple example: mode choice Utility functions U 1 = β t t 1 β c c 1, U 2 = β t t 2 β c c 2, where β t > 0 and β c > 0 are parameters. Equivalent specification where β > 0 is a parameter. U 1 = (β t /β c )t 1 c 1 = βt 1 c 1 U 2 = (β t /β c )t 2 c 2 = βt 2 c 2 Choice Alternative 1 is chosen if U 1 U 2. Ties are ignored (note: the probability that it occurs is uniformly equal to 0 because Us are continuous functions). Transport and Mobility Laboratory Choice Theory 26 / 37

27 Simple example Simple example: mode choice Choice Alternative 1 is chosen if or βt 1 c 1 βt 2 c 2 β(t 1 t 2 ) c 1 c 2 Alternative 2 is chosen if βt 1 c 1 βt 2 c 2 or β(t 1 t 2 ) c 1 c 2 Dominated alternative If c 2 > c 1 and t 2 > t 1, U 1 > U 2 for any β > 0 If c 1 > c 2 and t 1 > t 2, U 2 > U 1 for any β > 0 Transport and Mobility Laboratory Choice Theory 27 / 37

28 Simple example Simple example: mode choice Trade-off Assume c 2 > c 1 and t 1 > t 2. Is the traveler willing to pay the extra cost c 2 c 1 to save the extra time t 1 t 2? Alternative 2 is chosen if β(t 1 t 2 ) c 1 c 2 or β c 2 c 1 t 1 t 2 β is called the willingness to pay or value of time Transport and Mobility Laboratory Choice Theory 28 / 37

29 Simple example Dominated choice example 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 Transport and Mobility Laboratory Choice Theory 29 / 37

30 Simple example Illustration 4 train is chosen 2 cost by car-cost by train 0-2 car is chosen time by car-time by train Transport and Mobility Laboratory Choice Theory 30 / 37

31 Simple example Illustration with real data 4 train is chosen 2 cost by car-cost by train 0-2 car is chosen time by car-time by train Transport and Mobility Laboratory Choice Theory 31 / 37

32 Simple example Is utility maximization a behaviorally valid assumption? Assumptions Decision-makers have preferences in line with classical consumer theory are able to process full information have perfect discrimination power have perfect knowledge are perfect maximizer are always consistent Transport and Mobility Laboratory Choice Theory 32 / 37

33 Random utility theory Introducing probability Constant utility Human behavior is inherently random Utility is deterministic Consumer does not maximize utility Probability to use inferior alternative is non zero Niels Bohr Nature is stochastic Random utility Decision-maker are rational maximizers Analysts have no access to the utility used by the decision-maker Utility becomes a random variable Einstein God does not throw dice Transport and Mobility Laboratory Choice Theory 33 / 37

34 Random utility theory Assumptions Sources of uncertainty Unobserved attributes Unobserved taste variations Measurement errors Instrumental variables Manski 1973 The structure of Random Utility Models Theory and Decision 8: Transport and Mobility Laboratory Choice Theory 34 / 37

35 Random utility theory Random utility maximization Probabilistic setup Use a probabilistic approach: what is the probability that alternative i is chosen? What is the probability that alternative i is the one that gives maximum utility? Transport and Mobility Laboratory Choice Theory 35 / 37

36 Random utility theory Random utility model Probability model P(i C n ) = Pr(U in U jn, all j C n ), Random utility U in = V in + ε in. Random utility model P(i C n ) = Pr(V in + ε in V jn + ε jn, all j C n ), or P(i C n ) = Pr(ε jn ε in V in V jn, all j C n ). Transport and Mobility Laboratory Choice Theory 36 / 37

37 Random utility theory Over to the lab: CM1 112 Further Introduction to Biogeme Binary Logit Model Estimation Transport and Mobility Laboratory Choice Theory 37 / 37