Choice theory Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique F´ ed´ erale de Lausanne M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 1 / 55
Outline Outline Preferences Theoretical foundations 1 Utility maximization Choice theory Indirect utility Decision maker Microeconomic results Characteristics Discrete goods 3 Choice set Utility maximization Alternative attributes Simple example 4 Decision rule Probabilistic choice theory 5 Microeconomic consumer theory The random utility model 2 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 2 / 55
Theoretical foundations Outline Preferences Theoretical foundations 1 Utility maximization Choice theory Indirect utility Decision maker Microeconomic results Characteristics Discrete goods 3 Choice set Utility maximization Alternative attributes Simple example 4 Decision rule Probabilistic choice theory 5 Microeconomic consumer theory The random utility model 2 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 3 / 55
Theoretical foundations Choice theory Choice theory Choice: outcome of a sequential decision-making process defining the choice problem generating alternatives evaluating alternatives making a choice, executing the choice. Theory of behavior that is descriptive: how people behave and not how they should abstract: not too specific operational: can be used in practice for forecasting M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 4 / 55
Theoretical foundations Choice theory Building the theory Define 1 who (or what) is the decision maker, 2 what are the characteristics of the decision maker, 3 what are the alternatives available for the choice, 4 what are the attributes of the alternatives, and 5 what is the decision rule that the decision maker uses to make a choice. M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 5 / 55
Theoretical foundations Decision maker Decision maker Individual a person a group of persons (internal interactions are ignored) household, family firm government agency notation: n M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 6 / 55
Theoretical foundations Characteristics Characteristics of the decision maker Disaggregate models Individuals face different choice situations have different tastes Characteristics income sex age level of education household/firm size etc. M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 7 / 55
Theoretical foundations Choice set Alternatives Choice set Non empty finite and countable set of alternatives Universal: C Individual specific: C n ⊆ C Availability, awareness Example Choice of a transportation model C = { car, bus, metro, walking } If the decision maker has no driver license, and the trip is 12km long C n = { bus , metro } M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 8 / 55
Theoretical foundations Choice set 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 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 9 / 55
Theoretical foundations Choice set Discrete choice set Discrete choice analysis • C List of alternatives Brand A Brand B B • Brand C • A M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 10 / 55
Theoretical foundations Alternative attributes Alternative attributes Characterize each alternative i Nature of the variables for each individual n Discrete and continuous price Generic and specific travel time Measured or perceived frequency comfort color size etc. M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 11 / 55
Theoretical foundations Decision rule Decision rule Homo economicus Rational and narrowly self-interested economic actor who is optimizing her outcome Utility U n : C n − → R : a � U n ( a ) captures the attractiveness of an alternative measure that the decision maker wants to optimize Behavioral assumption the decision maker associates a utility with each alternative the decision maker is a perfect optimizer the alternative with the highest utility is chosen M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 12 / 55
Microeconomic consumer theory Outline Preferences Theoretical foundations 1 Utility maximization Choice theory Indirect utility Decision maker Microeconomic results Characteristics Discrete goods 3 Choice set Utility maximization Alternative attributes Simple example 4 Decision rule Probabilistic choice theory 5 Microeconomic consumer theory The random utility model 2 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 13 / 55
Microeconomic consumer theory Microeconomic consumer theory Continuous choice set Consumption bundle q 1 p 1 . . . . Q = ; p = . . q L p L Budget constraint L � p T Q = p ℓ q ℓ ≤ I . ℓ =1 No attributes, just quantities M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 14 / 55
Microeconomic consumer theory Preferences 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 . Rationality 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 . M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 15 / 55
Microeconomic consumer theory Utility maximization 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 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 16 / 55
Microeconomic consumer theory Utility maximization Optimization Optimization problem � max U ( Q ; θ ) Q subject to p T Q ≤ I , Q ≥ 0 . Demand function Solution of the optimization problem Quantity as a function of prices and budget Q ∗ = f ( I , p ; θ ) M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 17 / 55
Microeconomic consumer theory Utility maximization Example: Cobb-Douglas 25 20 15 , q 2 ) = θ 0 q θ 1 1 q θ 2 2 10 5 0 0 0 5 10 15 20 20 q 1 15 10 5 q 2 0 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 18 / 55
Microeconomic consumer theory Utility maximization Example 20 A 15 10 q 2 B 5 0 0 5 10 15 20 q 1 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 19 / 55
Microeconomic consumer theory Utility maximization Example Optimization problem U ( q 1 , q 2 ; θ 0 , θ 1 , θ 2 ) = θ 0 q θ 1 � 1 q θ 2 max 2 q 1 , q 2 subject to p 1 q 1 + p 2 q 2 = I . Lagrangian of the problem: L ( q 1 , q 2 , λ ) = θ 0 q θ 1 1 q θ 2 2 + λ ( I − p 1 q 1 − p 2 q 2 ) . Necessary optimality condition ∇ L ( q 1 , q 2 , λ ) = 0 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 20 / 55
Microeconomic consumer theory Utility maximization Example Necessary optimality conditions θ 0 θ 1 q θ 1 − 1 q θ 2 − λ p 1 = 0 ( × q 1 ) 1 2 θ 0 θ 2 q θ 1 1 q θ 2 − 1 − λ p 2 = 0 ( × q 2 ) 2 p 1 q 1 + p 2 q 2 − = 0 . I We have θ 0 θ 1 q θ 1 1 q θ 2 − λ p 1 q 1 = 0 2 θ 0 θ 2 q θ 1 1 q θ 2 − λ p 2 q 2 = 0 . 2 Adding the two and using the third condition, we obtain λ I = θ 0 q θ 1 1 q θ 2 2 ( θ 1 + θ 2 ) or, equivalently, λ I θ 0 q θ 1 1 q θ 2 2 = ( θ 1 + θ 2 ) M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 21 / 55
Microeconomic consumer theory Utility maximization Solution From the previous derivation λ I θ 0 q θ 1 1 q θ 2 2 = ( θ 1 + θ 2 ) First condition θ 0 θ 1 q θ 1 1 q θ 2 2 = λ p 1 q 1 . Solve for q 1 I θ 1 q ∗ 1 = p 1 ( θ 1 + θ 2 ) Similarly, we obtain I θ 2 q ∗ 2 = p 2 ( θ 1 + θ 2 ) M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 22 / 55
Microeconomic consumer theory Utility maximization Optimization problem q 2 I / p 2 Income constraint q ∗ 2 q 1 q ∗ I / p 1 1 M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 23 / 55
Microeconomic consumer theory Utility maximization Demand functions Product 1 1 = I θ 1 q ∗ p 1 θ 1 + θ 2 Product 2 2 = I θ 2 q ∗ p 2 θ 1 + θ 2 Comments Demand decreases with price Demand increases with budget Demand independent of θ 0 , which does not affect the ranking Property of Cobb Douglas: the demand for a good is only dependent on its own price and independent of the price of any other good. M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 24 / 55
Microeconomic consumer theory Utility maximization Demand curve (inverse of demand function) 4000 Good 1, Low income (1000) 3500 Good 1, High income (10000) Good 2, Low income (1000) 3000 Good 2, High income (10000) 2500 Price 2000 1500 1000 500 0 0 5 10 15 20 Quantity consumed M. Bierlaire (TRANSP-OR ENAC EPFL) Choice theory 25 / 55
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