20. Spatial and Hedonic Choice Models
Our theory is, if you need the user to tell you what you’re selling, then you
don’t know what you’re selling, and it’s probably not going to be a good
experience.
—Marissa Mayer
In this chapter, we cover models of individual choice over alternatives
represented by their attributes. These models were developed to capture
consumer choices. A person buying a house takes into account its square
footage, the number of bedrooms and bathrooms, and the quality of
construction. These can also be applied to college deans of admissions or
hiring directors as they select among applicants or to voters as they decide
between candidates. An admissions dean considers an applicant’s SAT
scores, grade point average, and extracurriculars. A voter evaluates
candidates based on their positions on education, infrastructure, crime, and
taxes. In addition to helping us understand individual choices, these models
provide insights into why we have the choices we do—for example, why
we have so many choices of breakfast cereals.
In the models we cover, we characterize some attributes as spatial and
others as hedonic. A spatial attribute, such as the color of a jacket or the
thickness of a slice of bread, has no best value. Each individual prefers a
particular amount of the attribute: a consumer of baby back ribs has a
preferred level of spiciness, and an amateur downhill skier has a preferred
angle of descent on a slope. The model assumes that the closer a product’s
attributes are to a person’s ideal point, the more the person values the
product. These ideal points vary across people: one person may prefer
spicier ribs than another.
On hedonic attributes, more (or in some cases less) is always better.
People prefer longer battery life in a smartphone, more square footage in a
house, more durability in the soles of shoes, and better gas mileage in their
cars. Most product choices are hybrids—they contain both spatial hedonic
attributes. A car’s color will be a spatial attribute. Its gas mileage is a
hedonic attribute.
Throughout the chapter, we assume that people choose the alternative
that they value the most. We do this for reasons mentioned in Chapter 4, on
modeling human behavior: rational behavior provides a benchmark, is
analytically tractable, makes a unique prediction, and fits empirically when
the situation is repeated and the stakes are large.
Models of spatial and hedonic competition are widely used in economics
and political science in part because they can be taken to data.1 In this
chapter, we get a hint of their applicability. We begin with a spatial model
of product competition. We then apply the model to politics and show how
it can be used to analyze status quo effects, agenda power, and the influence
of veto players. We then cover the hedonic attribute model and a hybrid
model to reveal insights into price competition. Along the way, we show
how to take data to the models to infer the positions of candidates and
judges based on their votes on bills and legal cases and to infer implicit
prices for unpriced attributes such as cleaner air or a shorter commute.2
The Spatial Competition Model
The spatial competition model assumes alternatives defined by a set of
attributes and consumers defined by ideal points. The simplest version of
the model considers products with a single attribute. In Hotelling’s original
spatial model, that attribute was geographic location.3
Figure 20.1: Hotelling’s Geographic Model of Ice Cream Vendors on a Beach
Hotelling’s model assumes a collection of consumers spread along a
beach, represented by circles in figure 20.1, along with two ice cream
vendors, denoted by A and B. Each customer buys one ice cream from the
nearer vendor. The cut point is an equal distance between the two vendors
and determines who buys from whom. The seven consumers to the left of
the cut point buy from vendor A, and the six consumers to the right of the
cut point buy from vendor B.
Given the idea of consumers preferring closer goods, we can reinterpret
distance more abstractly. For example, we could imagine the ice cream
vendors being in the same location but offering different levels of butterfat
in their ice cream. The same figure can represent vendor B offering a
creamier product than A. In the reinterpretation, the consumers’ locations
are not physical position on the beach but preferred levels of butterfat.
We can again apply one-to-many model thinking and use this same
model to analyze political competition. The Downsian model reinvents
Hotelling’s geographic space as an ideological continuum from left to right.
We can reinterpret figure 20.1 as follows: vendor A represents a liberal
political candidate, vendor B a more conservative candidate, and the circles
represent the ideological ideal points of voters. To extend the analogy, we
assume voters prefer nearer candidates.
The shift from geographic locations of firms and product attributes to
political ideologies involves a transition from physical attributes such as
location and butterfat level to the more abstract concept of ideology. While
we have clear measures of physical attributes, assigning ideologies requires
a method for translating the actions of candidates into numbers. If the
candidates have voting records, we can assign ideologies by first gathering
all of the votes the candidate has cast. We should ignore all votes that lack
an ideological component—unanimous proclamations establishing National
Milk Day and the like. On all other votes, we can rely on expert opinion to
assign a liberal position and a conservative position. A candidate’s spatial
location on the interval can be set equal to the percentage of the time she
votes the conservative position.4 A candidate who always takes the
conservative position is placed to the far right. A candidate who votes
conservative half of the time and liberal half of the time is placed in the
center.
With this model, we can adjudicate claims that American political
parties have become more ideologically distinct by empirical tests. One
analysis, shown in figure 20.2, reveals a marked and increasing polarization
of the average ideal points in each party. This does not prove that
polarization has increased, but it provides evidence. The analysis also
reveals that the polarization is mostly due to a Republican shift to the right.
Figure 20.2: Increasing Ideological Polarization in Congress
Increasing the Number of Attributes
The general spatial competition model includes an arbitrary number of
attributes. A couch can be represented by physical dimensions: length,
width, and depth, type of construction, and type of upholstery. The value (or
utility) that a consumer obtains from a product depends on the product’s
distance to her ideal point across these same dimensions. We can write this
value function as a constant term minus the distance between the alternative
and her ideal point.5
Spatial Competition Model
An alternative consists of N spatial attributes:
= (a1, a2,···, aN).
An individual is represented by an ideal point:
= (x1, x2,···, xN).
The payoff (utility) to an individual from an alternative equals
π( , ) = C − (x1 − a1)2 − (x2 − a2)2 −…− (xN − aN)2
where C > is a constant.
Example:
= (3, 4, 6), = (2, 1, 8), C = 20:
π( , ) = 20 − (3 − 2)2 − (4 − 1)2 − (6 − 8)2 = 6
In the general spatial competition model, two chocolate bars might be
represented by the percentages of cocoa and amounts of sugar they contain,
as shown in figure 20.3. The cut line will be the perpendicular bisector of
the line connecting the two products. Consumers with ideal points to the left
of the cut line prefer A, and consumers to the right prefer B.6
Figure 20.3: Product Attributes of Two Chocolate Bars and the Cut Line
The model can also accommodate any number of products. To add a
third chocolate bar, we add another point in attribute space. To determine
which bar consumers buy, we then draw additional cut lines, as shown in
figure 20.4. The multiple cut lines carve up the space of ideal points into
three regions, known as Voronoi neighborhoods, that partition the space of
ideal points based on their distances to the products.
Figure 20.4: A Spatial Model with Three Products (Voronoi Neighborhoods)
The Downsian Model of Spatial Competition
We next apply the model to explain candidate positioning. To do so, we
assume vote-seeking candidates, who place primary emphasis on winning
elections. We begin with an example to think through the incentives of
candidates. Figure 20.5 shows two scenarios with thirteen voters and two
candidates. Recall that voters prefer the nearer candidate. In the top
diagram, the liberal candidate, denoted by L, receives five votes, while the
conservative candidate, denoted by R, receives eight votes. In the lower
diagram, the liberal candidate moves to the center, attracts seven votes, and
wins the election.
Figure 20.5: Party Moving to Center to Win Election
The liberal candidate has an incentive to move to the center. By the
same logic, so does the conservative candidate. The conservative candidate
could move to the left, though remaining right of L, and also win.
Continuing with this reasoning, the liberal candidate, L, could take a
position even closer to the median voter’s ideal point. If we continue to
apply this logic, we see that candidates should converge on that point. This
result is known as the median voter theorem.
The median voter theorem can be interpreted in strong or weak form. In
strong form, it implies candidates adopt identical positions at the median
voter’s ideal point. That clearly does not hold empirically. In weak form, it
implies that candidates have an incentive to move toward moderate
positions. Empirical evidence suggests that this does happen. During the
course of an election, candidates move toward the center. That movement
need not be a mad rush. Candidates who possess ideological convictions or
benefit from core supporters at the extremes will reposition with caution.
This model reduces each candidate and each voter to a single ideological
point, a strong assumption. Czech writer and politician Václav Havel takes
exception to one-dimensional ideological projections: “To wandering
humankind, it [ideology] offers an immediately available home: all one has
to do is accept it, and suddenly everything becomes clear once more, life
takes on new meaning, and all mysteries, unanswered questions, anxiety,
and loneliness vanish. Of course, one pays dearly for this low-rent home:
the price is abdication of one’s own reason, conscience, and responsibility,
for an essential aspect of this ideology is the consignment of reason and
conscience to a higher authority.7” Havel makes a good point. We should
not abdicate reason for ideology. Models provide us with tools to reason
better. This particular model helps us to understand why politicians act as
they do. Using data, we can determine the confidence of each politician’s
placement on the left-right interval. A politician who always takes a
moderate position can be placed in the middle of the interval with high
confidence.
Incidentally, Havel’s denial that he can be reduced to a point could be
tested with data. If his criticism holds, if we cannot pin down his ideology
based on votes, we need not abandon the model. We could represent the
uncertainty over Havel’s ideology by assigning him an interval rather than a
point. Or, we could construct a time series of his measured ideology to see
if he remains consistent. Studies of the ideological positions of Supreme
Court justices show that some justices become more liberal as they spend
more time on the bench.8
Last, we could increase the dimensionality of the model. A twodimensional model could distinguish between social policies and fiscal
policies. The model can then capture a politician who advocates liberal
positions on social policies and conservative positions on fiscal policies.
For the United States Congress, one dimension explains approximately 83%
of the variation in votes. Adding a second dimension adds only another
4%.9
In addition to allowing us to model preferences with greater accuracy,
adding dimensions also changes our theoretical findings. We start with a
two-dimensional model. From the one-dimensional model, we know that if
a candidate is not located at the median on either issue, then the other
candidate could win an election by matching the first candidate’s position
on one issue (thus making that issue irrelevant) and taking the median
position on the other issue. Similarly, if one candidate takes the median
position on one issue but does not on the second issue, then the other
candidate could take the median position on both issues and win the
election. It follows that the only position that has the possibility of not
being beatable is the two-dimensional median. Figure 20.6 shows how the
two-dimensional median, represented by a circle, can be defeated. If the
square candidate positions to the left on issue one and to the right on issue
two, she produces a cut line in which she wins three votes.
Building intuition from this example, we see that the two-dimensional
median will be unbeatable only if voter ideal points are arranged such that
fewer than half lie in every direction from the median, a condition called
radial symmetry. Radial symmetry would be satisfied if voters’ ideal points
were uniformly distributed across a disc or a square, a very strong
assumption. This result, that any position can be defeated, is known as
Plott’s no-winner result. It holds in two or more dimensions.10
Figure 20.6: The Two-Dimensional Median Loses to a Challenger
The difference in results is stark. A one-dimensional model implies that
candidates position at the median. Higher-dimensional models imply that
they should not. Which type of model do we believe? We should place
complete faith in neither model, but instead gain insights from both. The
one-dimensional model reveals a powerful incentive for vote-seeking
candidates to move toward moderate policies. The higher-dimensional
models demonstrate the limits to those incentives. No position guarantees a
win, so we should not expect an equilibrium. We should instead expect
complexity, an endless dance of competition for votes through coalition
building.11
Status Quo Effects, Agenda Control, and Veto Players
We can also apply the Downsian model as a lens for interpreting the
ideological dimensions of bills passed by committees, councils, legislatures,
and presidential systems. Here again, the key will be to map pieces of
legislation onto the same single ideological dimension as committee
members. We consider three strategic effects here: the influence of current
policy (status quo effects), the power of agenda control, and the effect of
adding veto players.
Throughout, we rely on an example involving a committee of three
people with ideal points at 40, 60, and 80, in which the committee member
with an ideal point at 40 gets to propose a policy that must be approved by a
majority. Figure 20.7 shows the effect of the status quo on final policy. If
the status quo is at 80, she needs legislation that the median voter, the
committee member at 60, prefers to the status quo. In this case, the median
voter would accept a proposal at 40, the proposer’s ideal point, as it is just
as good as the status quo.12 If we move the status quo to 70, the median
voter would reject a proposal of 40. The proposer has to offer a policy at 50.
Finally, if the status quo sits at the median voter’s ideal point of 60, the
proposer has no power. We can thus draw the following inference: the
proposer has the most power when the status quo has an extreme value.
Figure 20.7: The Effect of the Status Quo on Final Policy
This insight applies in any context where people vote and opinions can
be mapped to one dimension. The head of a nonprofit making a proposal to
her board of directors to increase spending on affordable housing efforts has
little agenda power if current spending levels align with the wishes of her
median board member. She can have power if current policies do not align
with that board member’s ideal.
To show the power of proposing, we consider the case where the status
quo sits at 70, shown in figure 20.8. The top diagram shows the previous
case, where the proposer has an ideal point at 40 and proposes a policy at
50. The middle diagram shows the case in which the median voter can
propose his ideal point of 60 and also obtain the vote of the committee
member at 40. The bottom panel shows the case where the proposer has an
ideal point of 80. She cannot offer a policy that both she and the median
voter prefer to the status quo, so she accepts the status quo.
This exercise reveals the limit of proposer power. Legislation may move
in the direction of the ideal point of the person in power, but as we have
learned, the extent of that power will be mitigated by the representativeness
of the status quo.13
Last, we can use this same model to consider multiple levels of
government and an increased number of veto players. Here, we interpret the
three committee members as the median voters in the House and the Senate,
and the President. For legislation to pass, each of these three must prefer it
over the status quo. In this scenario, each has veto power. Refer again to
figure 20.8, where the status quo sits at 70. If all three voters must approve
any change, then no proposal can defeat the status quo. Any policy to the
left of 70 will be vetoed by the voter at 80. Any policy to the right of 80
will be vetoed by both of the other voters.14 If all three voters can veto
legislation, there will be no new legislation unless the status quo lies outside
the interval from 40 to 80—that is, if the current status quo is to the left or
right of anyone’s ideal point. The model reveals a tight link between the
number and ideological diversity of veto players and the extent of gridlock.
That insight holds more generally. Organizations with diverse veto players
will be unable to take action.
Figure 20.8: The Effect of the Proposer on Final Policy
The Hedonic Competition Model
The hedonic model of competition also represents the alternatives (typically
products) by attributes. However, in this model, the attributes consist of
quality, efficiency, or price, in which more (or, in the case of price, less) of
the attribute is always preferred. To capture heterogeneity, the model allows
individuals to attach different weights to the dimensions.
Using linear regression, we can infer implicit values for the attributes of
goods using the hedonic competition model (also known as the Lancaster
model). The application is straightforward. The model assumes the payoff
to be a linear function of the product’s attributes and the person’s weights.
If we have data on the selling prices of thousands of homes as well as the
relevant attributes of those homes (square footage, number of bedrooms and
bathrooms, quality of local schools, size of yard, and quality of
construction), a regression will produce the average weight (in dollars) of
each attribute for those people who bought the houses. This is known as
hedonic regression.
Hedonic Competition Model
An alternative consists of N valence attributes:
= (v1, v2,…, vN).
Individual preferences are captured by weights
assigned to the attributes.
= (w1, w2,…, wN)
The payoff (utility) to an individual from an alternative equals
π( , ) = w1 · v1 + w2 · v2 + ··· + wN · vN
.
Example:
= (3, 1, 2), = (4, 2, 5):
π( , ) = 4 · 3 + 2 · 1 + 5 · 2 = 24
.
Some of those attributes, such as a swimming pool or a new kitchen,
have market prices. As a check on the model, we can compare the estimated
prices to the market costs. If the regression shows that a swimming pool
adds $150,000 to the price of a house and swimming pools cost $15,000,
we know the model is missing attributes. If the regression shows the added
value to be only $8,000, then this likely means that people do not recover
the full costs of adding pools.
Other attributes, such as the commute time from the house to the city
center, do not have market prices. In those cases, the regression produces an
implicit price for that attribute, and that implicit price can be informative.
The table below shows hypothetical price data for six houses.
If we assume the houses are identical on all other attributes and we run a
hedonic regression, we obtain the following equation:
Price ($) = 100(Square Ft) + 20,000(# Bedrooms) − 2000(Commute Time)
The regression equation estimates that people value each additional
square foot at $100, each bedroom at $20,000, and each minute saved
commuting at $2,000 over the period of home ownership. A person who
lives in a house for twenty years spends 4,000 to 5,000 days commuting. If
we take the lower number, each extra minute of daily commute time results
in 4,000 minutes, or over 60 hours, commuting. The $2,000 estimate
equates to around $30 an hour. In other words, people pay for proximity “as
if” they are paying themselves $30 an hour to sit in traffic.15
A Hybrid Model of Product Competition
The spatial and hedonic models differ in how they represent preferences
over attributes. In the spatial model of competition, each person has a
preferred level of each attribute, and his value for an alternative increases as
it gets nearer his ideal point on those dimensions. In the hedonic model,
people prefer either more or less of each attribute.
Many of the choices we model—over consumer goods and services,
ideal life partners, public policies, religions, and job applicants—include
both spatial and hedonic attributes. We may each have our own preferred
level of crispiness for french fries, yet we all prefer to pay less per serving.
Crispiness is spatial. Price is hedonic. Employers likely differ in the
personality characteristics they look for in potential employees. Some firms
prefer extroverts. Others may prefer introverts. All firms prefer more
honesty and integrity. Thus, personality type is a spatial attribute, while
honesty is a hedonic attribute.
We can thus create a hybrid model in which the alternatives contain both
spatial and hedonic attributes. This model can be used to analyze market
entry, product differentiation, and the extent of price competition. If we
return to our example of the chocolate bars, before choosing a new
product’s attributes, an entrant might first place the three existing products
in attribute space and then survey consumers to learn about the distribution
of their ideal points. The entrant could then estimate the Voronoi
neighborhoods for her proposed product. If that neighborhood contains few
consumers, she should not expect substantial sales. Any entrepreneur
considering entering a market can take this approach. A boot designer can
plot existing designs of insulated boots, of which there may be dozens, and
find that none come in shiny patent leather. Someone designing a
smartphone app for making to-do lists can map the features of existing
apps, measure total market demand, and project possible sales.
We can visualize a price reduction in the spatial model of competition as
a movement of the cut line. Refer back to figure 20.3 showing the two
chocolate bars. The cut line corresponds to the ideal points of consumers
who are indifferent between A and B. If the firm producing B lowers its
price, and if consumers prefer to pay less for candy bars, then this will shift
the cut line toward A and increase B’s market share. We do not need the
model to know that B lowering its price should increase its market share.
We do need the model to estimate the magnitude of that effect. The key will
be to distinguish between crowded markets, with a large number of
products in a low-dimensional attribute space, and a sparse market, where
there are few competitors. In a crowded market, each product has a small
Voronoi neighborhood. In a sparse market, the Voronoi neighborhoods are
huge.
image
Figure 20.9: Price Competition in Sparse and Crowded Markets
Price changes have different effects in the two types of markets. Figure
20.9 shows the effects of a hypothetical 10% price reduction in candy bar B,
from $2.00 to $1.80. The diagram on the left shows a sparse market.
Lowering the price for B shifts the cut line between product A and B and
increases B’s market share from 50% to 54%, an 8% increase in B’s market
share. The 10% price drop and 8% sales increase reduces revenue by 3%.
Lowering prices would be a bad idea. The diagram on the right shows a
crowded market with seven types of candy bars. Here, the price drop has a
smaller effect on B’s absolute market share, an increase of 5% from 15% to
20%, but this 5% represents a larger proportional increase (33%) in B’s
market share. The overall effect is a 20% increase in revenue.16 Thus, the
model predicts stronger price competition in crowded markets than in
sparse markets, and extreme competition for commodities: products that are
indistinguishable, like crude oil, pork bellies, and red wheat #2. It predicts
less price competition for high-end fashion goods, where designers can
sustain substantial price markups because product dimensionality creates a
sparse market.
This relationship between the number of attributes and the extent of
price competition suggests that a good strategy would be to add new
attributes. This would make the market more sparse, reduce price
competition, and lead to higher profits. Even if that inference is correct, the
strategy may be easier stated than accomplished. People must value the new
attribute. For each successful attribute—cordless stereo speakers—one can
find multiple failed attempts—Bic’s ill-fated disposable underwear.
Summary
The spatial model of competition, the hedonic model of competition, and a
hybrid of the two provide a framework within which we can represent
different products, political candidates, or even job applicants. These
models can measure ideological positions, price implicit attributes, and
evaluate potential market entry positions. They generate insights into how
market competition creates an incentive for differentiation, how political
competition creates an incentive for convergence, and how price
competition should be more intense for products with fewer attributes.
In the models, we make rather strong and empirically dubious
assumptions. For example, we assume that people do not change their
preferences, and that they do not succumb to social pressures. If that were
so, why do firms and politicians spend enormous amounts of money trying
to change preferences? We could shrug off this criticism by referring again
to Box’s dictum that all models are wrong.
We can also construct a more nuanced response that distinguishes
between fundamental preferences, the outcomes that a person desires, and
her instrumental preferences, the person’s preferences over the attributes
that produce the fundamental outcomes. A student’s fundamental
preferences may strike a balance between being popular, healthy, and
scholarly. She may pursue these fundamental ends through instrumental
actions—waking early, going to the juice bar, and completing her
homework so as to have time to be social in the evenings. Her choice of the
fruit shake helps her to achieve a fundamental preference for good health. It
is an instrumental preference. If she comes across a scientific paper
revealing the high sugar content in fruit shakes, she may switch to drinking
water. If so, her instrumental preferences change even though her
fundamental preferences do not. Once again, we see how a model is not an
end in itself but provides an architecture to structure our thinking.
Many Models of Value
In a market economy, we can measure an individual’s value for a good
by the price she is willing to pay. An individual might value a
pastrami sandwich at $7, a painting by Goya at $3 million, and a oneacre lot in Ocala, Florida, at $75,000. Many economic models ignore
the source of these valuations. George Stigler famously quoted the
sensualist Mitya from Dostoyevsky’s The Brothers Karamazov, who
said, “De gustibus non est disputandum” (In matters of taste, there can
be no disputes). The models we cover say less about tastes than about
how tastes translate to the monetary values people assign to goods.
Hedonic attribute model: This model explains a good’s value based
on intrinsic attributes. Differences in valuations depend on different
underlying preferences over the good’s attributes. These attributes
could be the physical components of the good. A pastrami sandwich
consists of rye bread, six ounces of pastrami, Swiss cheese, mustard,
pickles, and onions. Its value can then be written as a weighted linear
combination of those components. More elaborate hedonic models can
include interaction terms. The pastrami may be even more valuable if
served on grilled rye bread.
Coordination model: This model explains prices as socially
constructed. The value of a Goya painting depends on what other
people believe its value to be. Initially, people have beliefs or opinions
about the value of the painting. They then interact with other people in
their social network and update their beliefs. Two people could both
set their values equal to the mean of their two values; one person
could change her value to match the other’s value; or each person
could move their valuation in the direction of the others. Given any of
these three assumptions, valuations converge locally. People
connected to one another will have similar valuations. The ultimate
value assigned to a good will depend on the initial distribution of
values, the social network, and the order in which pairings occurs.
Predictive models: This model explains prices as forecasts of future
value. The value of the one-acre lot in Ocala, Bitcoin, or a stock
depends on how much people will pay for them in the future. These
valuations depend on predictive models, which in turn depend on
attributes and categories. We might categorize Ocala as warm, lowtax, and inland. Variation in people’s valuations arises from different
predictive models. Investors use multiple predictive models. These
models may rely on attributes or, as in the case of valuing Bitcoin,
also make assumptions about coordination.
These three models provide three distinct explanations for the value of
a good. No one model will be best in all cases. Each will have cases
where it works best. The models function as arrows in our quiver. The
value of a pastrami sandwich most likely derives from its intrinsic
properties. The value of the Goya painting may be largely socially
constructed: a painting has value if people believe it has value. The
price of the land in Florida likely depends on valuations derived from
predictions of future real estate values.