Choice Theory Amanda Stathopoulos amanda.stathopoulos@epfl.ch Transport and Mobility Laboratory, School of Architecture, Civil and Environmental Engineering, Ecole Polytechnique F´ ed´ erale de Lausanne Transport and Mobility Laboratory Choice Theory 1 / 35
Outline Choice theory foundations 1 Consumer theory 2 Simple example 3 Random utility theory 4 Transport and Mobility Laboratory Choice Theory 2 / 35
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 Evaluation of the attributes of the alternatives: Price, time, flexibility, comfort Choice: Decision rule Implementation: Travel Transport and Mobility Laboratory Choice Theory 3 / 35
Choice theory foundations Building the theory A choice theory defines Decision maker 1 Alternatives 2 Attributes of alternatives 3 Decision rule 4 Transport and Mobility Laboratory Choice Theory 4 / 35
Choice theory foundations Decision maker Unit of analysis Individual Socio-economic characteristics: age, gender, income, education, etc. A group of persons (we ignore internal interactions) Household, firm, government agency Group characteristics Notation: n Transport and Mobility Laboratory Choice Theory 5 / 35
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 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 / 35
Choice theory foundations Continuous choice set Microeconomic demand analysis q 3 Commodity bundle q 1 : quantity of p 1 q 1 + p 2 q 2 + p 3 q 3 = I milk q 2 : quantity of bread q 3 : quantity of q 2 butter Unit price: p i Budget: I q 1 Transport and Mobility Laboratory Choice Theory 7 / 35
Choice theory foundations Discrete choice set Discrete choice analysis • C List of alternatives Brand A Brand B B • Brand C • A Transport and Mobility Laboratory Choice Theory 8 / 35
Choice theory foundations Alternative attributes Characterize each alternative i for each individual n Nature of the variables ➜ cost ✔ Generic or specific ➜ travel time ✔ Quantitative or ➜ walking time qualitative ➜ comfort ✔ Measured or perceived ➜ bus frequency ➜ etc. Transport and Mobility Laboratory Choice Theory 9 / 35
Choice theory foundations Decision rules Economic man Grounded in global rationality Relevant knowledge of options/environment 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 Utility Captures attractiveness of alternative Allows ranking (ordering) of alternatives What decision maker optimizes Transport and Mobility Laboratory Choice Theory 10 / 35
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 / 35
Consumer theory Consumer theory Neoclassical consumer theory Underlies mathematical analysis of preferences Allows us to transform ’attractiveness rankings’... into an operational demand functions Keep in mind Figure : Jeremy Bentham Utility is a latent concept It cannot be directly observed Transport and Mobility Laboratory Choice Theory 12 / 35
Consumer theory Consumer theory Continuous choice set Consumption bundle q 1 p 1 . . . . Q = ; p = . . q L p L Budget constraint L � p ℓ q ℓ ≤ I . ℓ =1 No attributes, just quantities Transport and Mobility Laboratory Choice Theory 13 / 35
Consumer theory 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 . “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 / 35
Consumer theory Utility Utility function Parametrized function: U = � � U ( q 1 , . . . , q L ; θ ) = � U ( Q ; θ ) Consistent with the preference indicator: U ( Q a ; θ ) ≥ � � U ( Q b ; θ ) is equivalent to Q a � Q b . Unique up to an order-preserving transformation Transport and Mobility Laboratory Choice Theory 15 / 35
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 16 / 35
Consumer theory Optimization problem 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 ( p 1 q 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 17 / 35
Consumer theory Framework Optimality conditions Lagrangian is differentiated to obtain the first order conditions ∂ L /∂ q 1 = β 0 β 1 q β 1 − 1 q β 2 − λ p 1 = 0 1 2 ∂ L /∂ q 2 = β 0 β 2 q β 1 1 q β 2 − 1 − λ p 2 = 0 2 ∂ L /∂λ = p 1 q 1 + p 2 q 2 − = 0 I We have β 0 β 1 q β 1 1 q β 2 − λ p 1 q 1 = 0 2 β 0 β 2 q β 1 1 q β 2 − λ p 2 q 2 = 0 2 Adding the two and using the third optimality condition λ I = β 0 q β 1 1 q β 2 2 ( β 1 + β 2 ) Transport and Mobility Laboratory Choice Theory 18 / 35
Consumer theory Framework Equivalent to λ I 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 2 = λ p 2 q 2 , we obtain (assuming λ � = 0) I β 2 q ∗ 2 = p 2 ( β 1 + β 2 ) Similarly, we obtain I β 1 q ∗ 1 = p 1 ( β 1 + β 2 ) Transport and Mobility Laboratory Choice Theory 19 / 35
Consumer theory Demand functions Product 1 β 1 1 = I q ∗ p 1 β 1 + β 2 Product 2 2 = I β 2 q ∗ p 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 20 / 35
Consumer theory Marginal rate of substitution Factoring out λ from first order conditions we get p 1 = ∂ U ( q ∗ ) /∂ q 1 = MU ( q 1 ) p 2 ∂ U ( q ∗ ) /∂ 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 21 / 35
Consumer theory Discrete goods Discrete choice set Calculus cannot be used anymore U = U ( q 1 , . . . , q L ) with � 1 if product i is chosen q i = 0 otherwise and � q i = 1 . i Transport and Mobility Laboratory Choice Theory 22 / 35
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 23 / 35
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 U = � � U ( y 1 , y 2 ) , where we impose the restrictions that, for i = 1 , 2, � 1 if travel alternative i is chosen , = y i 0 otherwise; and that only one alternative is chosen: y 1 + y 2 = 1 . Transport and Mobility Laboratory Choice Theory 24 / 35
Simple example Simple example: mode choice Utility functions = − β t t 1 − β c c 1 , U 1 U 2 = − β t t 2 − β c c 2 , where β t > 0 and β c > 0 are parameters. Equivalent specification U 1 = − ( β t /β c ) t 1 − c 1 = − β t 1 − c 1 U 2 = − ( β t /β c ) t 2 − c 2 = − β t 2 − c 2 where β > 0 is a parameter. Choice Alternative 1 is chosen if U 1 ≥ U 2 . Ties are ignored. Transport and Mobility Laboratory Choice Theory 25 / 35
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