queation 1,4,5,6
philxu1988
choicemodels13.pdf
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Discrete Choice Models
Guillermo Gallego Department of Industrial Engineering and Operations Research
Columbia University
Spring 2013
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Outline
I Introduction
I Neoclassical Theory
I Luce Model
I Random Utility Models
I Applications to Revenue Management
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Outline
I Introduction
I Neoclassical Theory
I Luce Model
I Random Utility Models
I Applications to Revenue Management
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Outline
I Introduction
I Neoclassical Theory
I Luce Model
I Random Utility Models
I Applications to Revenue Management
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Outline
I Introduction
I Neoclassical Theory
I Luce Model
I Random Utility Models
I Applications to Revenue Management
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Outline
I Introduction
I Neoclassical Theory
I Luce Model
I Random Utility Models
I Applications to Revenue Management
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Outline
I Introduction
I Neoclassical Theory
I Luce Model
I Random Utility Models
I Applications to Revenue Management
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Introduction
Choice Modeling is a general purpose tool for making probabilistic predictions about human decision making behavior. It is regarded as the most suitable method for estimating consumers willingness to pay in multiple dimensions. The Nobel Prize for economics was awarded to Daniel McFadden in 2000 for his work in the area. We will consider three theories about about how a rational decision maker decides among a discrete number of alternatives:
I Neoclassical choice model
I Luce choice model
I Random utility choice models
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Introduction
Choice Modeling is a general purpose tool for making probabilistic predictions about human decision making behavior. It is regarded as the most suitable method for estimating consumers willingness to pay in multiple dimensions. The Nobel Prize for economics was awarded to Daniel McFadden in 2000 for his work in the area. We will consider three theories about about how a rational decision maker decides among a discrete number of alternatives:
I Neoclassical choice model
I Luce choice model
I Random utility choice models
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Introduction
Choice Modeling is a general purpose tool for making probabilistic predictions about human decision making behavior. It is regarded as the most suitable method for estimating consumers willingness to pay in multiple dimensions. The Nobel Prize for economics was awarded to Daniel McFadden in 2000 for his work in the area. We will consider three theories about about how a rational decision maker decides among a discrete number of alternatives:
I Neoclassical choice model
I Luce choice model
I Random utility choice models
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Neoclassical Choice Model
This theory assumes there is a preference operator, say � to compare alternatives in a choice set S. The operator is assumed to satisfy the following three properties:
I i � i ∀i ∈ N I i � j and j � k implies i � k I i � j or j � i ∀i, j ∈ N
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Neoclassical Choice Model
This theory assumes there is a preference operator, say � to compare alternatives in a choice set S. The operator is assumed to satisfy the following three properties:
I i � i ∀i ∈ N I i � j and j � k implies i � k I i � j or j � i ∀i, j ∈ N
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Neoclassical Choice Model
This theory assumes there is a preference operator, say � to compare alternatives in a choice set S. The operator is assumed to satisfy the following three properties:
I i � i ∀i ∈ N I i � j and j � k implies i � k I i � j or j � i ∀i, j ∈ N
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Neoclassical Choice Model
Theorem: There exists a function U : N →< such that i � i if and only if U(i) ≥ U(j). Moreover, U is unique up to positive affine transformations.
Decision makers act to maximize expected utility.
If U is concave then EU(W ) ≤ U(EW ). Justifies insurance.
Difficult to specify utility functions
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
The Luce Model (1959)
Let πi (S) be the probability of choosing i ∈ S ⊂ N and let
πT (S) = ∑ i∈T
πi (S) for all T ⊂ S.
Luce proposed the following axioms:
I If there is an j ∈ S such that πi ({i, j}) = 0 then πT (S) = πT−i (S − i).
I If πi ({i, j}) ∈ (0, 1) for all i, j, then πi (S) = πi (T )πT (S) for all i ∈ T .
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
The Luce Model (1959)
Let πi (S) be the probability of choosing i ∈ S ⊂ N and let
πT (S) = ∑ i∈T
πi (S) for all T ⊂ S.
Luce proposed the following axioms:
I If there is an j ∈ S such that πi ({i, j}) = 0 then πT (S) = πT−i (S − i).
I If πi ({i, j}) ∈ (0, 1) for all i, j, then πi (S) = πi (T )πT (S) for all i ∈ T .
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
The Luce Model (1959)
Theorem: There is a function v : N →<+ such that for all i ∈ S ⊂ N
πi (S) = vi
V (S) where V (S) =
∑ j∈S
vj
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Random Utility Models
Random utility models assume that the the decision maker can perfectly discriminate among alternatives but the decision maker may have incomplete information. The utility for alternative i is modeled as Ui = µi + �i where �i is a mean zero random variable. Then
πi (S) = P(Ui ≥ Uj ∀j ∈ S)
To put the model to good use we need to make assumptions about the deterministic term µi and the distribution of the random term �i for each i ∈ S.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Modeling the Deterministic Term
If each alternative can be characterized by a vector, say x, of attributes then we can model the deterministic part of the utility as a linear function of a vector of attributes, e.g.,
µi = β ′xi
Common attributes used in Airline Revenue Management include the fare, fare restrictions, travel time, number of stops, carrier, type of plane, etcetera. Some of the attributes may be in log space, e.g., log of fares.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Modeling the Random Term
I Uniform random term: Linear model
I Normal random term: Probit model
I Gumbel random term: Multinomial Logit model (our focus). A Gumbel random variable X has cdf
F (x|φ) = e−e −(γ+φx)
−∞ < x < ∞
with mean zero and variance σ2 = π 2
6φ2 , where γ = 0.57721 is
the Euler constant.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Modeling the Random Term
I Uniform random term: Linear model
I Normal random term: Probit model
I Gumbel random term: Multinomial Logit model (our focus). A Gumbel random variable X has cdf
F (x|φ) = e−e −(γ+φx)
−∞ < x < ∞
with mean zero and variance σ2 = π 2
6φ2 , where γ = 0.57721 is
the Euler constant.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Modeling the Random Term
I Uniform random term: Linear model
I Normal random term: Probit model
I Gumbel random term: Multinomial Logit model (our focus). A Gumbel random variable X has cdf
F (x|φ) = e−e −(γ+φx)
−∞ < x < ∞
with mean zero and variance σ2 = π 2
6φ2 , where γ = 0.57721 is
the Euler constant.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Multinomial Logit Model
Theorem: If the error terms are independent Gumbel-φ then
πi (S) = Pr(Ui ≥ Uj ∀j ∈ S) = vi
V (S)
where vi = e φµi , ∀i ∈ S.
Consistent with Luce’s model; with neoclassical when φ is large
Winbugs: http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml NLOGIT: http://www.limdep.com/
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Independence of Irrelevant Alternatives (IIA)
The MNL suffers from the IIA property:
πi (S)
πj (S) = πi (T )
πj (T ) ∀ i, j ∈ S ⊂ T
Example: S = {c,b1} and T = C = {c,b1,b2}, with v(c) = v(b1) = v(b2) = 1. Then
πc (S) = 1/2 πc (T ) = 1/3.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Nested Multinomial Logit Model
Under this model, the choice set is partitioned into non-overlapping subsets S = S1 ∪ . . .∪Sk and for i ∈ Sj , the choice is modeled as
πi (S) = πi (Sj )πSj (S)
where each component on the right is modeled by a MNL model.
Example: S1 = {c},S2 = {b1,b2}, πS1 (S) = πS2 (S) = 1/2, then
πc (S) = 1/2 and πb1 (S) = πb2 (S) = 1/4.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Mixture of MNLs
I All random utility models can be approximated to any degree of accuracy by a mixture of MNLs.
I The mixture is specified by (θk,vk ) where θk is the probability that an arriving customer belongs to market segment k = 1, . . . ,K , and vk is the vector of weights vk (a) a ∈ C is the vector of attraction values that governs the MNL for market segment k.
I The problem with the mixture model is that it is difficult to estimate the parameters and it is difficult to do optimization, e.g., to find the assortment that maximizes revenues.
Guillermo Gallego Discrete Choice Models
Overview Introduction
Neoclassical Choice Model Luce Model
Random Utility Models Applications to Revenue Management
Applications to Revenue Management
I There set of products is N = {1, . . . ,n}. I The no purchase alternative i = 0 is always available.
I For each S ⊂ N
πi (S) = vi
v0 + V (S) π0(S) =
v0 v0 + V (S)
.
I There is a fare pj associated with each product j ∈ N. I r(S) =
∑ i∈S piπi (S) is the expected revenue associated with
S ⊂ N. I Often interested in finding S ⊂ N to maximize r(S) − zπ(S),
where π(S) = ∑
j∈S π(S).
Guillermo Gallego Discrete Choice Models
