Maximum Entropy versus
Random Utility Theory in
Discrete Choice Models
Bob Bordley
Adjunct Professor
University of Michigan, Ann Arbor
rbordley@umich.edu
Independence of Irrelevant
Alternatives
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If a new product is introduced or
if a change is made in existing product i,
THEN
the relative probability of buying product j
versus k will be unaffected (for j and k
products different from i)
Logit Model
• X(i) is a vector describing how product I
scores on various attributes
• W is a vector describing the importance
the customer attaches to attributes
• P(i|w) = exp(W’X)/ [ exp(W’ X(k))]
But Logit Model and IIA is
inadequate
• It does not necessarily make sense
because products which compete closely
with the product being changed (or
introduced) should be impacted differently
than products which don’t compete closely
• Note that the assumption may be
adequate for certain choice sets while
being inadequate for others.
McFadden’s Solution
• Random utility theory assumes that individuals
maximize utility but that modelers only observe
noisy estimates of that utility
• A simple form of random utility theory model
leads to the logit model with IIA
• More sophisticated forms of random utility theory
lead to choice models without IIA
• These are often called random utility models
(RUM)
But consumers do not appear to
maximize utility functions!
Vernon Smith, winner of Nobel Prize in Economics, wrote in
his Nobel Prize address:
Psychologists and behavioral economists who
study decision behavior almost uniformly report
Results contrary to rational theory…. imagine the
strain on the brain's resources if at the supermarket
a shopper were required to explicitly evaluate his
preferences for every combination of the tens of
thousands of grocery items that are feasible for a
given budget.
Undermines main theoretical
justification for random utility model
• Random utility models assume that
individuals maximize utility --- although
modelers don’t observe that utility
• If individuals do not maximize utility, it’s
harder to justify empirical models of
individual behavior that impose it
We can still justify these models if they have
empirically plausible properties
An Example of Where They Don’t have
Empirically Plausible Properties
• Mixed Logit Model can closely
approximate virtually any random utility
model (McFadden and Train)
• Mixed Logit Model has the form
– P(i) ~∫ P(i|w) dF(w)
– with P(i|w)=[exp(W’X(i))/[ exp(W’ X(k))]
• Typically dF(W) is considered Gaussian
Our Example
• Consider the Hypothetical Case where the distribution of
attributes X(1),…X(n) has a Gaussian Distribution
• Then it can be shown that
• P(i) ~ exp((X(i)-m)W(X(i)-m)
• Where
– m and W depend on the mean and variance-covariance of
attribute scores
– W is positive-define
– and P(i) violates IIA
• But P(i) increases at an increasing rate as X(i) increases
• Also (X(i)-m)W(X(i)-m) increases at an increasing rate
• Some would say this violates the intuition about
diminishing returns to improvements in an attribute
Conclusion
• The Assumption of Customer Rationality underlying
RUM is untrue
• The properties of RUM models (as reflected in mixed
logit) are questionable in the Gaussian case
• Should we consider an alternative approach?
• How about an approach based on choosing the choice
probability which maximizes entropy subject to
constraints determined by what you observe in the
market?
• Ehsan Soofi proposed such an approach and argued as
equivalent to RUM.
• This paper presents a variation on this approach which is
different from RUM.
Comparison of RUM
• In the case where X(i) was distributed Gaussian, RUM
gave
– P(i) ~ exp((X(i)-m)W(X(i)-m))
– A convex anti-ideal model
– The more a product improves away from m, the better
• In the same case, the Maxent approach gives
– P(i) ~ exp(-(X-m’)W’(X-m’))
– A concave ideal-point model
– The closer a product approaches m, the better.
• Ideal point models are widely used in the literature
• So Maxent properties are, in this case, more intuitive
• Maxent makes no assumption of customer rationality
UNUSUAL FINDING:
Maxent Logit, unlike RUM Logit, Violates
Independence of Irrelevant Alternatives!
• Standard Logit
– P(i) ~ exp(W’X)
– W reflects customer preferences and is independent
of the choice set
• Maxent Logit
– P(i) ~ exp(K’X)
– K is a Lagrangian multiplier which is chosen, for some
vector M, to enforce the constraint
• P(1)X(1)+….P(n)X(n) = M
– Changes in attributes or addition of new products will
violated this constraint unless K changes
– If K changes when the choice set changes, P(i) can
violate independence of irrelevant alternatives
Example:Red-Bus/Blue-Bus Paradox
• Suppose m is proportion of customers that take the bus
• Suppose we only have one red bus and one car and
50% of people take the red bus.
• Let I=1 if bus and I=0 else.
• Then logit model predicts 50% of people drive bus. But
if we then add a blue bus (so that we have a red bus, a
blue bus and a car), the probability of driving a bus goes
to (2/3)
• But the weight, K, in the maxent model automatically
adjust to ensure that only 50% of the people drive the
bus. Hence adding a blue bus causes the weight to
chance so that the probability of driving a bus remains at
50%
• Thus the celebrated `red-bus/blue-bus’ problem which
plagues the logit model is avoided by the maxent logit
model.
1st Choice/Second Choice with
Maxent
• Let P(i|j) be the fraction of buyers of product j with I as a
first choice
• Let P(j) be the fraction of buyers of product j
• Let P(ij) be the fraction of buyers with i as a second choice
and j as a first choice.
• Define M1 to be the average score of first choice products
on the various attributes
• Define M2 to be the average score of second choice
products on the various attributes
• Define V(ij) to be the covariance between
– the attributes, X(j) of the first choice and
– the attributes X(i), of the second choice
Maxent Model
• Choose a distribution for P(ij) to maximize
– Σ P(ij) ln(P(ij))
• subject to
– Σ Σ P(ij) = 1
– Σ Σ P(ij) X(i) = M2 (mean score of 2nd choice)
– Σ Σ P(ij) X(i) = M1 (mean score of 1st choice)
– Σ Σ P(ij) (X(i)-M2)(X(j)-M1)=V
Maxent Result
• P(ij) ~ exp(K’X(i)+K”X(j)+X(j) Q X(i))
• P(j) ~ Σ P(ij)
Questions?
Setting up the Problem
• Let x be a particular set of attribute scores (e.g., a fuel economy of 20
mpg and a price of $20K)
• Let P(x,w) be the joint probability of an individual buying a product with
attribute scores x and having importance weights w
• Let f(x) be probability that a a product with attribute score x is available
for purchase to the individual
• We observe the average attribute scores of products purchased on the
market (e.g., the US vehicle fleet might have an average mpg of
23mpg with the average vehicle price being $25K)
• We also observe the variance-covariance matrix of those attribute
scores (e.g., we know that more fuel efficient vehicles are generally
smaller)
• Suppose we believe that we can also estimate the mean and variancecovariance of the parameters w as well as their correlation with
attribute scores
– Thus we expect that individuals who weight fuel economy more highly tend
to buy vehicles that score well on fuel economy
What we Know (or could Know)
• We observe the average attribute scores of products
purchased on the market (e.g., the US vehicle fleet might
have an average mpg of 23mpg with the average vehicle
price being $25K)
• We also observe the variance-covariance matrix of those
attribute scores (e.g., we know that more fuel efficient
vehicles are generally smaller)
• Suppose we believe that we can also estimate the mean
and variance-covariance of the parameters w
• We also believe there is a correlation between the
importance an individual puts on an attribute and the
attribute score of the vehicle they buy
– Thus we expect that individuals who weight fuel economy more
highly tend to buy vehicles that score well on fuel economy
Maxent Preliminary Solution
• P(x,w) is Gaussian
Independence of Irrelevant
Alternatives is Violated
• Introducing a new product changes the
shadow price of that combination of
attributes.
• In a Gaussian model, this changes the
choice probability for all attribute scores
(and especially for those products whose
scores are similar to the product which
was newly introduced.)
But you can’t always get what you want…
• The logit model describes your probability of choose
product I in terms of the ratio of the desirability of product
I to the sum of the desirability of all other products
• This is often called the choice set. You cannot buy a
product that is not available
• Another way of modeling availability is to assign a large
cost to products that are unavailable
• This is useful in that we can also distinguish between
products that are easy to obtain, somewhat harder to
obtain as well as products that are impossible to obtain.
• It’s often referred to as a shadow price in constrained
optimization
Choice Models with Choice Sets
• So we can eliminate the need to explicitly enumerate the
alternatives in the choice set by simply introduce an attribute that
reflects the relative availability of a set of attribute scores
• We distinguish between
– The availability scores we assign to a product. For example, the vehicle
we currently own is highly available (as long as it still works). Buying a
new vehicle that exists at a dealership is somewhat more difficult.
Buying a new vehicle that doesn’t exist is impossible unless we’re rich
enough to have it custom-built.
– The unobserved weight which is attached to availability --- which is
designed to reflects this `shadow’ price
• The correlation between the probability of purchasing a product and
its unavailability is obviously negative
• So we need to supplement P(x,w) to include this new attribute and
its relative importance.
Relationship to Random Utility Models
• The most widely used random utility model which generalizes logit is
the mixed logit.
• This model
– Defines a logit model with importance weights w
– Integrates the logit model over some distribution (generally Gaussian) in w
to get the choice model
• It can be shown that if the distribution of attributes, x, in the choice set
is Gaussian, then
– The mixed logit model is equivalent to
– Defining a logit model with a quadratic utility
• Where the quadratic utility for each choice alternative i depends on the mean
and variance-covariance of attribute scores across the choice set
• Quadratic utility is in the form of an anti-ideal point model where scores that
exceed the reference score attract higher marketshare
– The normality assumption can be made to hold by Box-Cox
transformations as long as attributes are somewhat continuous. It will not
hold for categorical (zero-one) variables
• Nonetheless it provides a simple point of comparison with the maxent
model
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Relationship to Random Utility Models(cont.)
• In contrast, the maxent model is equivalent to
– Defining a logit model with a quadratic utility
• Where the quadratic utility for each choice alternative I depends on
the mean and variance-covariance of attribute scores across the
choice set
• Quadratic utility is in the form of an ideal-point model where scores
that are closer to the reference score attract higher marketshare.
• Thus given a Gaussian distribution of alternatives
– Mixed logit uses a quadratic anti-ideal point model
• In this model, incremental improvements on an attribute increase
utility at an accelerating rate.
– Maxent uses a quadratic ideal point model
• In this model, incremental improvements on an attribute increase
utility at a decreasing rate.
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Most people would consider the ideal point (and thus
the maxent model) more intuitive. However further work
is needed.
Estimation(1995 MY Data)
• 4525 Observations of Vehicles and their Variants in a given Year
• Gaussian Model: Logarithm of share is Quadratic in Attributes
• Relevant Variables
– Categorical
• Is Vehicle 4-Wheel Drive or All-Wheel Drive?
• Internal Features (Base, Standard, Deluxe)
• Transmission (Manual, Automatic)
– Non-Categorical
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Total Price (and its Square)
Rear Knee Room (and its Square)---`measures size’
Acceleration
Quality (and its square)
--- `measures workmanship’
Incentives (and its square)
Repair Frequency (and its square)
Model Year (and its square) --- `measures newness’
Dealer location (average miles to a dealer)
# of Seats
Application:Corporate Strategy
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If π is variable profit per unit sold
If S is the potential sales in a segment
Let P = exp(U)/[exp(U)+1]
If U* =U/N, i.e., the perceived value of the firm’s
portfolio per new product program
If N is the number of new product products
If C is the cost per new product program, then
the optimal number of new product programs, N,
satisfies
π exp(U*N)/[1+exp(U*N)] - CN
so that N*=[ln( C) – ln[U* π – C]]/U*
Optimal profit is C[1 – U*N*]/U*
Conclusion
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Random Utility Models are an almost universally accepted way of
addressing discrete choice models
But the justification for random utility models assumes that consumers
actually do maximize a utility function even though analysts cannot observe
it exactly
Empirical evidence strongly suggests this assumption is false
Since extensive tools have been developed to apply random utility models,
they will undoubtedly remain the tools of choice in the near-term
But in the long-term, we need discrete choice models that are not based on
faulty assumptions
Where should we look?
– The building block for discrete choice mdoels based on random utility theory is
the logit model.
– Soofi showed that the building block for discrete choice models based on maxent
is also the logit model.
– Hence it’s natural to focus on maxent for the next-generation of discrete choice
models
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This paper present the results of an application to the automotive segment