# Fairness, Welfare, and Equity in Personalized Pricing

Nathan Kallus  Cornell University and Cornell Tech<EMAIL_ADDRESS>and
Angela Zhou  Cornell University and Cornell Tech<EMAIL_ADDRESS>
###### Abstract.

We study the interplay of fairness, welfare, and equity considerations in
personalized pricing based on customer features. Sellers are increasingly able
to conduct price personalization based on predictive modeling of demand
conditional on covariates: setting customized interest rates, targeted
discounts of consumer goods, and personalized subsidies of scarce resources
with positive externalities like vaccines and bed nets. These different
application areas may lead to different concerns around fairness, welfare, and
equity on _different_ objectives: price burdens on consumers, price envy, firm
revenue, access to a good, equal access, and distributional consequences when
the good in question further impacts downstream outcomes of interest. We
conduct a comprehensive literature review in order to disentangle these
different normative considerations and propose a taxonomy of different
objectives with mathematical definitions. We focus on observational metrics
that do not assume access to an underlying valuation distribution which is
either unobserved due to binary feedback or ill-defined due to overriding
behavioral concerns regarding interpreting revealed preferences. In the
setting of personalized pricing for the provision of goods with positive
benefits, we discuss how price optimization may provide unambiguous benefit by
achieving a “triple bottom line”: personalized pricing enables expanding
access, which in turn may lead to gains in welfare due to heterogeneous
utility, and improve revenue or budget utilization. We empirically demonstrate
the potential benefits of personalized pricing in two settings: pricing
subsidies for an elective vaccine, and the effects of personalized interest
rates on downstream outcomes in microcredit.

††conference: ACM Conference on Fairness, Accountability, and Transparency;
March 03–10, 2021;

##  1\. Introduction

Personalized pricing, once restricted to the idealized construction of
economic theory, is now squarely within the realm of possibility for firms
newly equipped with a deluge of fine-grained information about individuals and
prediction modeling of demand or willingness to pay based on this information.
Given both the ubiquity of prices and their relevance in important domains
such as hiring, lending, and credit subject to antidiscrimination regulation,
price personalization remains an area of increasing scrutiny and caution, as
well as tentative optimism due to competitive considerations (House, 2015;
Gonzaga, 2018).

The potential of expanded reliance on predictive models in domains affecting
individuals is cause for concern. After all, the extensive study of fairness
considerations in predictive models highlights how the joint structure of
protected attributes and other information can lead algorithmic decisions,
even based only on non-attribute information, to nonetheless lead to disparate
impacts on individuals (Barocas et al., 2018). The setting of personalized
pricing is particularly interesting because it fundamentally involves
considerations of both resource allocation in response to price; as well as
predictive models for such a price response. Auditing challenges arise
precisely because valuations are in general not known or observed; rather only
binary-feedback demand response is observed.

Studying the case of personalized pricing is conceptually challenging because
prices are a shared tool in drastically different domains: we consider
lending/insurance, consumer goods, and public provision. A crucial distinction
is between value-based pricing that offers different prices to customers based
on their estimated willingness to pay, and risk-based pricing which offers
different prices to customers based on their estimated costs, as in lending
and insurance (House, 2015). While discrimination law is strongest in
insurance and lending, in lending, discrimination concerns often arise from
individual agents providing offers from an actuarially-fair securitized rate
sheet (Bartlett et al., 2019). In particular, distributional concerns
regarding price optimization reflect overall concern for differentially
adept/prepared/educated negotiating customers in insurance and lending, but
slight optimism in value-based pricing since low-income individuals may be
more price-sensitive (Bartlett et al., 2019). Hence, the majority of our
analysis will focus on value-based pricing, which lends itself more readily to
price optimization.

In the case of value-based pricing, incidents where price targeting leads to
disparities are often subject to media coverage and public outcry. We
recollect just a few of these incidents: Staples changed prices based on
available brick-and-mortar locations of competitors leading to higher prices
for rural areas (Valentino-Devries et al., ) and Asians faced higher prices as
a result of the Princeton Review’s zip-code based price-targeting (Vafa et
al., 2015; Larson et al., 2015). While these covariate-based pricing schemes
were based on non-group contextual information, nonetheless they induced
disparities in prices along group lines. Although there are not clear anti-
discrimination principles that govern the setting of value-based pricing,
understanding the tradeoffs introduced by considering constraints or fairness
penalties on a myopic price optimization problem can shed light on tradeoffs
between various intuitive notions to inform algorithm design.

In this paper, we study the interplay of fairness and welfare considerations
as they arise in the setting of personalized pricing. Our first task is
conceptual: we square these real-world problem settings alongside previously
expressed concerns regarding price optimization. For example, acknowledging
the empirical reality of economic inequality informs the expected
distributional impacts of covariate-conditional pricing schemes along racial
and economic segments. We then turn to analytical modeling to identify the
multiple objectives which pricing decisions affect with fairness and/or
welfare implications. Prices impose a burden on customers (perhaps only on
those who purchase), result in allocations of the good itself, may be
optimized based on noisy predictors, and the good itself may have downstream
impacts of interest. In Section 4, we taxonomize these considerations into
price parity (marginal and conditional), model error fairness, preferences for
access, allocative efficiency, and actuarial fairness; we offer these
operationalizations (all of which may in themselves not be novel) alongside
contextual discussion. We propose operationalizations of these normative
considerations which avoid classical assumptions of known valuations, focusing
on identifiability based on available information. In Section 5 we provide
further characterization of implications for fairness and welfare
considerations for considering modifications to price optimization in order to
improve on some of these notions, in particular price parity and market share.
Where possible, we provide analytical insights on tradeoffs. In Section 6 we
build empirical case studies on datasets related to the “public-interest
goods” setting based on a study of willingness-to-pay for an elective vaccine
and interest rates for microcredit loans.

##  2\. Related work

The study of algorithmic pricing and revenue management is very extensive and
spans economics, operations research (Gallego et al., 2019), and computer
science. Price discrimination has also been studied for ethical and normative
considerations, especially in relation to privacy (Odlyzko, 2004; Miller,
2014). We now highlight methodological and empirical work of particular
relevance to algorithmic considerations. In Appendix A we discuss domain-level
considerations and more broad related work in greater detail.

We first briefly overview the classical economic taxonomy of price
discrimination (Varian, 1989). First-degree price discrimination offers
individual prices to customers exactly at their willingness to pay, which is
assumed to be known. Second-degree price discrimination depends on the
quantity purchased but does not differ across consumers, such as bulk
discounts. Third-degree price discrimination charges different prices to
different groups of consumers, such as offering senior or student discounts.
We focus on analyzing covariate-conditional prices, which are a form of third-
degree pricing but draw nearer to first-degree pricing (up to the noise of
random valuations).

The economic literature studies welfare (consumer and producer surpluses)
implications of idealized first-degree personalized pricing, assuming
valuation distributions are known. This classical notion of welfare is hence
pegged to the valuation distribution (e.g., consumer welfare is valuation
minus price). (Varian, 1985) studies third-degree price discrimination; using
first-order conditions of valuation distributions, they show that consumer
welfare increases with additional price discrimination as long as total output
increases. (Aguirre et al., 2010) provide analogous conditions for similar
analysis. (Bergemann et al., 2015) shows that a seller can choose various
segmentations that can achieve any combination of increase/decrease in
consumer surplus; (Cummings et al., 2020) study the theoretical computational
efficiency of finding such segmentations.

This classical theory suggests that ideal personalized pricing may improve
welfare relative to uniform monopoly pricing, but that different segmentations
lead to possibly indeterminate outcomes for consumer welfare. The empirical
literature indicates this indeterminacy in important settings. (Fuster et al.,
2018) empirically study implications of machine learning predictors of default
probability on disparities not only for predictive performance, but on using
these predictions to set interest rates for loans via risk-based pricing.
While richer machine learning predictors expand access to credit; they also
result in greater price dispersion for the minority borrower on the margin.
Hence, greater access comes at the cost of a greater price burden. (Dubé and
Misra, 2019) study personalized pricing in a Bayesian setting with posterior
uncertainty quantification in a business-to-business marketing setting; they
find that finer-grained personalized pricing overall increases consumer
welfare; though this is not monotonic in segmentation granularity.

Pricing in the context of mechanism design follows another approach and
assumes elicitation of ontologically valid valuations from strategic
participants due to narrowly bracketed contexts such as auctions, kidney
exchanges, and matching markets. (Dworczak et al., 2019) show that for a two-
sided market, market-clearing competitive equilibrium pricing is not
necessarily optimal if a market designer has distributional preferences.
Noting that classical theory on quasi-linear utility “implicitly embeds the
assumption that each agent values money equally,” they study implications of
dispersion in marginal values for money of market participants for optimal
market mechanisms.

(Cohen et al., 2019) study fairness considerations when each protected group
is assigned one price and the valuation distribution is known. In contrast, we
focus on the prediction setting with rich covariates, and propose metrics that
are completely independent of the valuation distribution. Our analytical
insights focus on covariate-personalized prices and implications of joint
structure of covariate distributions and group variability on fairness
considerations.

Finally, we mention work that studies tensions in fair machine learning,
specifically the role of algorithms in allocating decisions or conferring
utility, to highlight questions of interest that have analogies in the pricing
setting. (Liu et al., 2018; Heidari et al., 2018; Hu and Chen, 2019; Kasy and
Abebe, 2020) broadly study tensions between fairness and welfare when machine
learning enacts allocations, e.g., via classification. An interest of this
work is to study analogous considerations for price optimization, under
corresponding notions of fairness and welfare. Longer-term considerations of
fairness constraints have also been studied (D’Amour et al., 2020; Creager et
al., 2019), typically with the formalism of dynamical systems. In machine
learning, regulatory considerations barring “disparate treatment,” e.g., using
the protected attribute information in the predictor or in algorithmic
interventions, may be in fundamental tension with achieving proposed fairness
notions. (Lipton et al., 2018) studies tensions that arise from interventions
that do not use attribute information. Again, these broad concerns may
additionally be of interest in the setting of price optimization.

##  3\. Problem Setup

We let $X$ denote customer covariates (features) and $A$ the protected
attribute. To simplify discussion we focus on binary comparisons between two
protected groups, $A=a$ and $A=b$. A personalized pricing policy is a function
mapping covariates and possibly protected attribute information to a real-
valued price, $p(X,A)\in\mathbb{R}^{+}$ (or, $p(X)$ for attribute-blind
pricing rules). Each sale instance is associated with a hypothetical and
unknown demand curve representing the demand for each possible price,
$D(p)\in\mathbb{R}^{+}$. Often, when each sale instance is with an individual
customer, $D(p)\in\\{0,1\\}$ is binary where $D(p)=1$ denotes the individual
would purchase or take-up at price $p$. In section 4.4, we further restrict
$D(p)=\mathbb{I}\left[V\leq p\right]$ to be given by a customer valuation.
Otherwise, we do not make this restriction, allowing arbitrary, possibly non-
rational behavior.

We are primarily interested in studying the question of covariate-based
pricing. One approach is to estimate personalized demand at a price, given
covariates, via a parametric or semi/non-parametric model, which we will
denote as $D(p\mid x,a)=\mathbb{E}[D(p)\mid X=x]$.

A personalized pricing function satisfies:

$p^{*}(x,a)\in\operatornamewithlimits{argmax}_{p(x,a)}\mathbb{E}[D(p(X,A))p(X,A)]$

Often, policy or domain-level restrictions may prohibit prices that directly
use the attribute $A$:

$p^{*}(x)\in\operatornamewithlimits{argmax}_{p(x)}\mathbb{E}[D(p(X))p(X)]$

We define $P=p(X,A)$ as the per-individual personalized price; it may
sometimes also refer to $P=p(X)$. We additionally introduce notation for the
revenue, ${R}(P)=PD(P)$, and its covariate-conditional counterpart.

We will also consider cases where the good has an effect on some outcome in
itself, such as repeat purchasing behavior, health benefits, downstream
welfare, so that $D(p)$ is also itself a treatment. We denote the
corresponding potential outcomes as $Y(D(p))$. These represent the causal
outcomes of take-up/non-take-up, e.g., the effect on contracting malaria of
purchasing/not purchasing a bed net.

##  4\. Definitions and metrics

We now introduce definitions of different aspects of fairness or welfare in
personalized pricing, addressing normative considerations motivated by
different contexts, and offer formalisms and operationalizations of these. We
consider a generic personalized price function $P$ which may reflect either
first or third degree price discrimination. In Section 5 we discuss these
operationalizations in more depth and analyze potential trade-offs.

#### Observational metrics.

We introduce the following notions of allocative fairness based on what we may
observe: prices $p$, demand outcome $D(p)$ (e.g., purchase/no purchase), and
potential downstream outcomes due to the good $Y(D(p))$. In Appendix A we
provide further discussion on why, if we are concerned about fairness in the
first place, we might be skeptical about defining fairness relative to
valuations or other _un_ revealed latent preferences/valuations.

###  4.1. Price parity

#### Context.

A customer always benefits from a lower price. The difference between
distributions of prices faced by different groups measures potential
unfairness in price burdens. An extreme example is the so-called “pink tax”:
(De Blasio and Menin, 2015), commissioned by the Mayor’s office in New York
City, studies gender-based pricing differentials and finds an average of 7%
higher prices paid by women; for example, women’s pink razors are more
expensive, for the exact same product111Notably, while legislation has been
proposed to try to address the pink tax, this has not been established as
gender-based discrimination. See (Jacobsen, 2017) for more discussion.. This
highlights the capacity of price optimization to extract consumer value from
behavioral failures of economic regarding valuations. In particular, using
“pinkness” to segment products corresponds to extracting valuation from social
constructions of gender, all other functionality being the same for a product
with no signaling value. While the “pink razor” is an extreme example,
considerations regarding price parity are a common intuitive objection to
personalized pricing.

#### Operationalization.

We introduce a definition based on distributional equivalence of prices for
each group.

###### Definition 0 (Price parity).

$P\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A$

We may also consider parity in moments of the price distribution, which also
enables a simple way to give a scalar metric to disparity.

###### Definition 0 (Marginal price parity).

$\mathbb{E}[P\mid A=a]=\mathbb{E}[P\mid A=b]$

Correspondingly, the marginal price _dis_ parity is

$\mathbb{E}[P\mid A=a]-\mathbb{E}[P\mid A=b].$

The notion of “disparate treatment” in fair machine learning suggests that the
following notion of covariate-conditional price parity is intuitively
appealing.

###### Definition 0 ($x$-conditional price parity).

$p(x,a)=p(x,b)$

Notice, however, that satisfying definition 3 generally does not ensure
satisfying definitions 1 and 2 due to differing distributions of covariates
between groups. Indeed, in general contexts, it is well-understood that equal
treatment need not lead to equal impact; this remains true in personalized
pricing.

Finally, it is often helpful to consider price parity conditional on take-up,
or more generally on demand. Conditioning on take-up reflects that price only
affects consumer utility _if_ the customer purchases.

###### Definition 0 (Take-up-conditional parity).

$P\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A\mid D(P)$

More generally, we may wish to condition on the _effect_ of pricing. Consider
the case where there is a nominal price $p_{0}$ and $P\leq p_{0}$ represents a
personalized potential discount. The event $D(P)>D(p_{0})$ is the event that
demand increases as an _effect_ of the discount. In the binary demand case:
that an individual purchases if and only if given the discount (rather than
purchasing irrespective of discount or not purchasing irrespective), which we
term a _responder_ to the discount. Conditioning on responsiveness accounts
for the possibility that different groups have different valuations or
willingness to pay, and to the extent that one deems it acceptable to
personalize to leverage such differences (and often it is not) parity
conditional on response requires we do not price-discriminate more than is
justified by response to discount.

###  4.2. Model error fairness

#### Context.

Given that fairness in machine learning studies how predictive models may
exhibit differential model performance (predictive accuracy, error
distributions) by group, a natural question is how such disparities in
predictive model performance might affect the suboptimality of different
prices; and whether different groups might experience different price
suboptimality burdens due to error patterns of the predictive model.

#### Operationalization.

The rest of this paper studies pricing based on a true conditional demand
model, $D(p\mid x,a)$. In practice however, only an estimate $\hat{D}(p\mid
x,a)$ is available from observed data. For example, for the pricing problem:

$\displaystyle\hat{p}^{*}(x)$
$\displaystyle\in\operatornamewithlimits{argmin}\mathbb{E}[\hat{D}(p(X))p(X))]$

Price suboptimality fairness is concerned with the decision suboptimality of a
price based on a risk model vs. the price derived from the actuarially fair
“true risk”, $\hat{p}^{*}(x)-p^{*}(x)$. For example, $\hat{p}^{*}(x)$ may
differ from $p^{*}(x)$ when we learn the prediction $\hat{D}(p\mid x,a)$ from
finite samples and differential accuracy thereof could lead to fairness
concerns. In Section 5 we will focus on the revenue objective.

###  4.3. Access and equal access

#### Context.

We are often concerned about _access_ to the good being sold, especially when
the good has benefits that are deemed crucial such as vaccines (see also our
empirical study in section 6.1), loans, or broadband internet. In terms of
welfare, it is important to consider the total access that personalized
schemes lead to, namely the total demand. In terms of fairness, we may be
concerned with allocative parity in the form of parity in market shares or
take-up probabilities by group. Personalized pricing schemes may in fact
enhance both of these measures by allowing revenue extraction from high-
valuation groups to enable offering lower price offers to low-valuation
individuals, hence “pricing people into the market.” High-valuation groups are
usually those with financial means that allows them to have a higher
willingness to pay, and low-valuation groups are usually those with less
financial means.

#### Operationalization.

Total access is simply the marginal demand. When demand is binary, this is the
fraction of individuals who take-up the good.

###### Definition 0 (Total access).

$\displaystyle\mathbb{E}[D(P)].$

The idea of access parity suggests requiring equal allocation of access /
market share / take-up.

###### Definition 0 (Access parity).

$\displaystyle\mathbb{E}[D(P)\mid A=a]=\mathbb{E}[D(P)\mid A=b].$

Correspondingly, access _dis_ parity is $\mathbb{E}[D(P)\mid
A=a]-\mathbb{E}[D(P)\mid A=b]$.

| Insurance | Lending |  Consumer goods |  Public-interest goods
---|---|---|---|---
Moral hazard/Adverse selection | ✓ | ✓ | ✗ | ✗
Revenue-driven price optimization | ✗ | ✗ | ✓ | ✓
Risk-driven price optimization | ✓ | ✓ | ✗ | ✗
Prefer marginal price parity | ✗ | ✗ | ✓ | ✗
Prefer conditional price parity | ✓ | ✓ | ✗ | ✗
Prefer access | ✓ | ✓ | ✓ | ✓
Actuarial fairness | ✓ | ✓ | ✗ | ✗
Allocative efficiency/sorting | ✗ | ✗ | ✓ | ✓
Targeting long-run dynamics | ✓ | ✓ | ✓ | ✓
Table 1. Different problem settings and what fairness/welfare notions are
relevant when.

###  4.4. Allocative efficiency and fairness

#### Context.

In settings where interpreting revealed preferences as rational choices due to
latent valuations, efficiency considerations are concerned with how prices
sort individuals by their valuations and ensure that a good may be targeted
towards those who value it the most. For example, the literature on pricing
health interventions in development is particularly interested in the
difference between free and low-price provisions based on whether prices
better target households who are more likely to use a product (Cohen et al.,
2010). An important further concern is whether errors in sorting individuals
disproportionately affect one group more than another so that, on average,
certain groups are more often incorrectly given priority over others.

#### Operationalization.

We focus on providing observational metrics which assess sorting/targeting
without assuming access to the full valuation distribution. For this section,
we focus on the binary feedback setting where $D(p)\in\\{0,1\\}$.

###### Assumption 1 (Monotonicity).

For any $p,p^{\prime}:$

$p>p^{\prime}\implies D(p)\geq D(p^{\prime})$

Assuming monotonicity of binary demand with respect to price as in Asn. 1 is
equivalent to assuming a random latent threshold model, i.e.,
$D(p)=\mathbb{I}[V\leq p]$. This perspective recognizes that observations of
the binary event $D(p)$, under Asn. 1, are censored observations of the
underlying valuation.

The question is whether a given pricing scheme (for which we have observed the
binary outcomes) appropriately ranks valuations of individuals. A marginal
measure of such efficiency may be _concordance_ :

###### Definition 0 (Concordance).

Given two individuals drawn independently at random, _concordance is_

$\mathbb{P}(V_{1}>V_{2}\mid P_{1}>P_{2}).$

While concordance captures how efficiently prices sort individuals by
valuation, such efficiency may have disparate effects. To capture this
disparity, we propose the class-crossed concordance disparity metric.

###### Definition 0 (Class-crossed concordance disparity).

Given two individuals drawn independently at random from groups $A=a$ and
$A=b$, respectively, _class-crossed concordance disparity_ is

$\mathbb{P}(V_{b}>V_{a}\mid P_{a}<P_{b})-\mathbb{P}(V_{a}>V_{b}\mid
P_{b}<P_{a})$

Class-crossed concordance has the following probabilistic interpretation. The
term $\mathbb{P}(V_{b}>V_{a}\mid P_{a}<P_{b})$ can be interpreted as: of those
whose valuations can be ordered under Asn. 1, what is the probability that
valuations drawn from one group are stochastically greater than valuations
from another group? Class-crossed concordance measures a groupwise disparity
in the difference in these probabilities.

###  4.5. Targeting long-run dynamics

Context. A key domain-level consideration that justifies preferring take-up is
that take-up of the good is itself a treatment with a downstream outcome, such
as future purchases/customer loyalty in e-commerce, net present value of
continued borrowing (Karlan and Zinman, 2019), or usage and downstream health
outcomes of a preventive health intervention in development economics (Dupas,
2014; Cohen et al., 2010; Johnson and Lipscomb, 2019). Therefore, price
impacts an allocation which itself may have heterogeneous effects on longer-
term outcomes for the customer and/or decision-maker: we identify this as
targeting long-run dynamics. This, for example, justifies an overall
preference for expanding market shares.

The possibility of a “triple bottom line”. Price personalization may be
beneficial due to its increasing take-up of a good which is beneficial for
individuals and the decision-maker, targeted price subsidies enable budget-
balanced public provision (in contrast to a complete subsidy), and the access
expansion might particularly those who would benefit the most. Of course,
whether these benefits compound (or whether certain contributors are
irrelevant) need to be assessed in the data of any particular setting, as in
(Dupas, 2009; Cohen et al., 2019).

Operationalization. We recognize the firm’s objective function as population
welfare downstream of a price allocation:

$\mathbb{E}[Y(D(P))].$

###  4.6. Summary of problem settings and relevant notions

Abstractly, we might summarize some of the above considerations by considering
a $\lambda-$scalarized multi-objective optimization problem, which represents
the expansion of considerations beyond myopic revenue maximization:

$\displaystyle\max_{P(\cdot)}$
$\displaystyle\;\;\mathbb{E}[PD(P)]\textstyle+\lambda_{1}(\sum_{a\in\mathcal{A}}\mathbb{E}[D(P)\mid
A=a])+\lambda_{2}\mathbb{E}[Y(P)]$ s.t. $\displaystyle\mathbb{E}[P\mid
A=a]-\mathbb{E}[P\mid A=b]\leq\Gamma$ (1)
$\displaystyle\mathbb{E}[D(P)(P-c)]\geq 0$

Sections 4.1, 4.3 and 4.5 (price parity, access, long-run welfare) might
conceivably be included in a conceptual “multi-objective” version of the
firm’s problem, while Sections 4.2 and 4.4 (price suboptimality, class-crossed
concordance ) are idealized measures.

In table 1 we apply our conceptual taxonomy of problem domains to these
different fairness/welfare notions in pricing. The first category of criteria
summarizes how different problem settings differ in the presence of moral
hazard/adverse selection that justifies risk-based pricing), and the capacity
for price optimization. The second category identifies which notions of
fairness or equity are more or less relevant in different settings.

We caution that the above optimization problem is a conceptual device to
illustrate how these notions might justify revenue sub-optimal allocations.
Table 1 suggests that in any particular application setting, these notions may
not be simultaneously relevant.

##  5\. Analysis of Definitions and Metrics

In this section, we expand further on each definition. Where possible, e.g. by
making additional assumptions, we provide analytical insight on implications
of these fairness notions for price optimization or corresponding
specializations of eq. 1.

###  5.1. Price-parity

We study the price optimization problem with additional price parity
constraints, focusing on highlighting tradeoffs with implications for
algorithm design. To simplify the analysis, we make the following assumptions.
We assume a partially linear demand model with a link function of the non-
price, covariate-driven demand $\overline{D}(x,a)$ and a linear component for
price elasticity.

###### Assumption 2 (Partially linear demand model).

$D(p\mid x,a)=\beta_{a}p+\overline{D}(x,a).$

###### Assumption 3 (Downward sloping linear demand with respect to price.).

$\beta_{a}<0,\forall a\in\mathcal{A}$

For example, the linear model corresponds to
$\overline{D}(x,a)=\alpha+\gamma^{\top}x$. Linear demand is a common
assumption for contextual pricing (Qiang and Bayati, 2016; Ban and Keskin,
2020; Besbes and Zeevi, 2015). We also assume that price elasticities are
negative.

Without loss of generality, assume group $a$ has higher average price at the
unconstrained solution). The marginal parity222We discuss constraining first
moments of the price distributions (marginal parity) to provide analytical
insights. Constraining higher order moments via e.g. a set constrained by
Kolmogorov-Smirnov statistic (Luo and Mehrotra, 2020), or a moment-based
hierarchy as in (Aswani and Olfat, 2019), is computationally possible.
constrained revenue maximization problem, is a specialization of eq. 1. Let
$p^{*}(x,a;\Gamma)$ denote the corresponding $\Gamma-$parametrized solution.

$\displaystyle p^{*}(x,a;\Gamma)\in$
$\displaystyle\operatornamewithlimits{argmax}_{p(\cdot)}\;\mathbb{E}[p(X,A)D(p(X,A))]$
(2) s.t. $\displaystyle\mathbb{E}[p(X,A)\mid A=a]-\mathbb{E}[p(X,A)\mid
A=b]\leq\Gamma$

The attribute-blind personalized price is $p^{*}(x;\Gamma)$, which restricts
the above optimization to prices which only personalize on $x$. We derive the
parity-constrained optimal price.

###### Theorem 1.

Let

$\xi(A)=(\mathbb{P}[A=a]^{-1}\mathbb{I}[A=a]-\mathbb{P}[A=b]^{-1}\mathbb{I}[A=b]).$

The optimal attribute-based personalized price under marginal parity solving
eq. 2 is:

$\displaystyle p^{*}(x,a;\Gamma)$
$\displaystyle=\frac{-\overline{D}(x,a)+\xi(a)\lambda^{*}_{xa}}{2\beta_{a}},$
$\displaystyle\text{where }\lambda^{*}_{xa}$
$\displaystyle=\frac{\mathbb{E}[\overline{D}(X,A)\xi(A)+2\beta_{A}\Gamma]}{\mathbb{E}[\xi(A)^{2}]}.$

The optimal attribute-blind personalized price is

$\displaystyle p^{*}(x;\Gamma)$
$\displaystyle=\frac{-\overline{D}(x)+\lambda^{*}_{x}\mathbb{E}[\xi(A)\mid
X=x]}{2\mathbb{E}[\beta_{A}\mid X=x]},$ $\displaystyle\text{where
}\lambda^{*}_{x}$
$\displaystyle=\frac{\mathbb{E}[\overline{D}(X)\xi(A)+2\beta_{A}\Gamma]}{\mathbb{E}[\mathbb{E}[\xi(A)\mid
X]^{2}]}.$

∎

The proof is included in Appendix B; the key idea is to study the Lagrangian
dual of the knapsack-constrained quadratic program and solve by swapping the
order of minimum and maximum.

In the following analysis, we focus on an equality constraint in eq. 2.
Interpreting the solution, $p^{*}(x,a)$ differs from the unconstrained
personalized price by a penalty whose size depends on the discrepancy of the
price-independent covariate-based demand within groups.

We highlight some tradeoffs induced by marginal price parity against other
fairness considerations. We first consider a very special setting where the
demand function is invariant across groups; it is only the group-conditional
covariate distribution which induces price disparity.

###### Proposition 0 (Attribute-based vs. attribute-blind pricing under
marginal parity ).

Suppose Asns. 2,3, and further that:

  1. (1)

$\overline{D}(x,a)=\overline{D}(x,b)=\overline{D}(x)$,

  2. (2)

$\overline{D}(x)$ is linear in $x$,

  3. (3)

and $\beta_{a}=\beta_{b}$.

Then we have

$\displaystyle p^{*}(x,a;0)-p^{*}(x;0)<0\iff$
$\displaystyle\frac{\lambda^{*}_{xa}}{\lambda^{*}_{x}}<\mathbb{P}(A=a\mid
X=x)-\frac{\mathbb{P}[A=a]}{\mathbb{P}[A=b]}\mathbb{P}(A=b\mid X=x),$
$\displaystyle\text{ and }p^{*}(x,b;0)-p^{*}(x;0)<0.$

In this very special case, we find that group $b$ would uniformly prefer
attribute-based marginal parity pricing. The relationship is not necessarily
uniform for group $a$. A sufficient condition, for example, is if for some
$x,$ $\mathbb{P}(A=a\mid X=x)>>\mathbb{P}(A=b\mid X=x)$ and
$\mathbb{P}[A=a]=\mathbb{P}[A=b]$: then $p^{*}(x,a;0)-p^{*}(x;0)<0$. Hence,
for $x-$outliers within group $a$ satisfying this condition, attribute-based
pricing is Pareto optimal relative to attribute-blind pricing for both groups.
Since however price disparity in this special case is exactly driven by
variability in $\mathbb{P}(A=a\mid X=x),$ we might also expect that
$p^{*}(x,a;0)-p^{*}(x;0)>0$ for some other $x$.

Building on the results of Theorem 1 and Proposition 2, we also provide bounds
on the revenue loss due to attribute-blind pricing, now in the setting where
the non-price-based demand may differ across groups.

###### Corollary 0 (Revenue loss of $p^{*}(x)$).

Suppose Asn. 3, $\beta_{a}=\beta_{b}$ and
$\overline{D}(x,a)\neq\overline{D}(x,b)$. Then,

$\displaystyle\mathbb{E}[{R}(p^{*}(X,A))-{R}(p^{*}(X))]\geq\frac{1}{4\beta}\mathbb{E}[\overline{D}^{2}(X)-\overline{D}^{2}(X,A)]\geq
0.$

We now use these characterizations from Theorem 1, Proposition 2, and
Corollary 3 to study tradeoffs between price parity and other desiderata, in
particular x-conditional parity, and summarize some implications for algorithm
design.

  1. (1)

A firm may generically prefer attribute-blind pricing to attribute-based
pricing schemes due to regulatory considerations or x-conditional price parity
(definition 3).

  2. (2)

If achieving marginal parity is of interest in view of price disparities,
attribute-blind marginal parity achieves lower firm revenue than attribute-
based marginal parity. We provide a quantitative bound on the gap in a simple
case in Corollary 3.

  3. (3)

Under the special case of Proposition 2, for some $x$, attribute-based
marginal parity is strictly preferable to attribute-blind marginal parity for
both groups.

These considerations might outweigh an intuitive preference for attribute-
blind pricing in the case of marginal parity.

###  5.2. Price suboptimality fairness

We provide a decomposition that gives structural conditions on when the sign
of the prediction error is informative of the sign of the mispricing or
decision error. In particular, this highlights a distinction between analyzing
fairness in data-driven optimal prices vs. fair prediction in machine
learning.

###### Proposition 0 (Price suboptimality error decomposition).

Assume that $\nabla\hat{D}(\hat{p}^{*}(x)\mid x)\neq\nabla\hat{D}(p^{*}(x)\mid
x)$. Up to first order terms,

$\displaystyle\hat{p}^{*}(x)-p^{*}(x)$
$\displaystyle=\frac{{\hat{D}(p^{*}(x)\mid x)-D(p^{*}(x)\mid
x)}}{\nabla\hat{D}(\hat{p}^{*}(x)\mid x)-\nabla\hat{D}(p^{*}(x)\mid x)}$
$\displaystyle+p^{*}\left(1+\frac{\nabla\hat{D}(p^{*}(x)\mid x)-\nabla
D(p^{*}(x)\mid x)}{\nabla\hat{D}(\hat{p}^{*}(x)\mid
x)-\nabla\hat{D}(p^{*}(x)\mid x)}\right)$

The decomposition is not computable from observed data (since some quantities
depend on the unknown $p^{*}(x)$). One implication is that the sign of
$(\hat{p}^{*}(x)-p^{*}(x))$ ultimately depends on the sign of a few
quantities:

$\displaystyle 1$ $\displaystyle\hat{D}(p^{*}(x)\mid x)-D(p^{*}(x)\mid x)$
estimation error $\displaystyle 2$ $\displaystyle\nabla\hat{D}(p^{*}(x)\mid
x)-\nabla D(p^{*}(x)\mid x)$ gradient est. error $\displaystyle 3$
$\displaystyle\nabla\hat{D}(\hat{p}^{*}(x)\mid x)-\nabla\hat{D}(p^{*}(x)\mid
x)$ price elast. subopt.

There are some cases where we may be able to conclude the sign of decision
error $\hat{p}^{*}(x)-p^{*}(x)$: we can conclude $\hat{p}^{*}(x)-p^{*}(x)>0$
if $1,2,3$ are all positive.

However, in general, the main implication of the above proposition is that the
direction of decision disparity is not immediate from prediction error of
$D(p(x)\mid x)$ alone: it also depends on estimation error of the gradient,
and the difference in gradients due to suboptimality. It is more difficult to
conclude implications of pricing decisions (and more broadly, optimization
decisions) based on uncertain nuisance predictions.

Applying the above result to $p^{*}(x,a)-p^{*}(x)$ allows us to make a similar
conclusion for the discrepancy of pricing with respect to attribute-based
$D(p(x)\mid x,a)$ vs. the attribute-blind $D(p(x)\mid x)$ setting.

$\displaystyle p^{*}(x,a)-p^{*}(x)=\frac{{D(p^{*}(x)\mid x,a)-D(p^{*}(x)\mid
x)}}{\nabla D(p^{*}(x,a)\mid x,a)-\nabla D(p^{*}(x)\mid x,a)}$
$\displaystyle+p^{*}\left(1+\frac{\nabla D(p^{*}(x)\mid x,a)-\nabla
D(p^{*}(x)\mid x)}{\nabla D(p^{*}(x,a)\mid x,a)-\nabla D(p^{*}(x)\mid
x,a)}\right)$

###  5.3. Market share

We study a multi-objective version of eq. 1 with additional weights on group-
conditional market share objectives. We consider demand that arises from an
underlying valuation distribution. (This is true without loss of generality
under Asn. 1 of monotonicity.) The most general assumption that admits a
concave price optimization program is log-concavity of valuation
distributions.

###### Assumption 4 (Log-concave valuation distribution).

Suppose that

$V=g(x,a)+\epsilon,$

where $\epsilon$ has a log-concave probability density function.

We assume log-concavity so that the cdf and cumulative cdf of $\epsilon$
(effectively $D(p\mid x,a)$) are also log-concave. Log-concavity is quite
general; log-concave pdfs include the normal, exponential, logistic, extreme
value, Laplace, gamma, Weibull, etc.

Then the population-level market share personalized-price specialization of
eq. 1 is a concave program under Asn. 4, since maximization is equivalent
under the monotonic increasing log transformation:

(3)
$p^{*}(x,a;\lambda)\in\underset{p(\cdot)}{\operatornamewithlimits{argmax}}\;\mathbb{E}[(p(X,A)+\lambda)D(p(X,A))]$

We may consider the attribute-blind restriction of the above:

(4)
$p^{*}(x;\lambda)\in\underset{p(\cdot)}{\operatornamewithlimits{argmax}}\;\mathbb{E}[(p(X)+\lambda)D(p(X))]$

The above problems consider a population-level market share penalty; we also
consider group-conditional market share penalties. These may arise from the
penalty formulation of the market-share constrained problem, where some
$\lambda$ is the optimal dual Lagrange multiplier for the constraints
${\mathbb{E}[D(p)\mid A=a]\geq\Gamma}$.

(5) $\max\\{\mathbb{E}[p(X,A)D(p(X,A))]\colon\mathbb{E}[D(p(X,A))\mid
A=a]\geq\Gamma_{a}\\}$

Correspondingly, the attribute-based group-conditional market share price
$p^{*}(x,a;\lambda_{a})$ solves, pointwise over $x,a$:

(6)
$p^{*}(x,a;\lambda_{a})\in\underset{p(\cdot)}{\operatornamewithlimits{argmax}}\;(p+\lambda_{a}/\rho_{a})D(p\mid
x,a),\;\;\forall x,a$

We study the sensitivity of the unconstrained-optimal attribute-blind and
attribute-based personalized price to local increases in the penalty
parameter, $\lambda$, relative from the unconstrained optimal price, i.e.
$\nabla_{\lambda}p^{*}(x,a;0).$ This describes how much the price changes in
response to implementing distributional preferences for market share.
Quantifying these sensitivities sheds light on the dependence on
$R^{\prime\prime},$ the second derivative of the (conditional) revenue
function.

###### Lemma 0 (Optimality conditions for different penalties ).

Suppose Asn. 4.

  1. (1)

Population-level market share, attribute-based sensitivity is

$\textstyle\nabla_{\lambda}p^{*}(x,a;0)={R^{\prime\prime}(p^{*}(x,a;0)\mid
x,a)^{-1}}{p^{*}(x,a;0)^{-2}}.$

  2. (2)

Population-level market share, attribute-blind sensitivity is

$\textstyle\nabla_{\lambda}p^{*}(x;0)={R^{\prime\prime}(p^{*}(x;0)\mid
x)^{-1}}{p^{*}(x;0)^{-2}}.$

  3. (3)

Group-level market share, attribute-based sensitivity is

$\textstyle\nabla_{\lambda}p^{*}(x,a;0)=\frac{1}{\rho_{a}}{R^{\prime\prime}(p^{*}(x,a;0)\mid
x,a)^{-1}}{p^{*}(x,a;0)^{-2}}.$

Observe that these sensitivities are negative, under Asn. 4. The proof,
included in the appendix, identifies the (pointwise) optimality conditions of
the constrained optimizations, eq. 4, eq. 6, and applies the implicit function
theorem.

We highlight some implications of Lemma 5 for algorithm design.

  1. (1)

For larger $\left|R^{\prime\prime}\right|$ (greater curvature), the less price
decrease is required to increase market share, and conversely for smaller
$\left|R^{\prime\prime}\right|$, the larger price decrease is required.

  2. (2)

Quantifying these sensitivities in terms of $R^{\prime\prime}$ highlights the
revenue implications of these price fairness changes. Considering a second-
order expansion of the revenue, smaller $\left|R^{\prime\prime}\right|$
suggests that the larger price decrease may not have extreme revenue decrease.

  3. (3)

Curvature also quantifies the rate of convergence of the optimal price, e.g.
if optimizing over a parametrized pricing policy via M-estimation (Kosorok,
2007). Hence, finite-sample variability of the optimal price (which may be
assessed empirically by bootstrapping) may suggest low curvature. This
suggests a robust approach which ensures out-of-sample market share may incur
small revenue tradeoff in the low-curvature regime.

###  5.4. Allocative efficiency: concordance

Assumption 1, of almost sure monotonicity, suggests that the combination of
continuous treatment and binary outcome can be viewed as a censored
observation of the valuation. Again, we do not assume access to the underlying
realizations of valuation distribution, but study what may be concluded about
valuations given that we only observe the censored realizations
$D(P)=\mathbb{I}\left[V>P\right].$

Consider ranking the prices and valuations of two generic price-valuation-
demand triples, $(p_{1},v_{1},D(p_{1})),(p_{2},v_{2},D(p_{2}))$. The only
joint outcome of demands and prices that admits concluding an ordering on the
underlying valuations $v_{1},v_{2}$ is that

$\\{p_{1}<p_{2},D(p_{1})=0,D(p_{2})=1\\}\iff\\{v_{1}<p_{1}<p_{2}<v_{2}\\}$

Using this observation (which is highly dependent on almost sure
monotonicity), we can identify a lower bound on concordance from observational
data.

###### Theorem 6.

Assume Asn. 1 and $D(p)\in\\{0,1\\}$.

$\mathbb{P}(D_{b}(P_{b})>D_{a}(P_{a})\mid
P_{a}<P_{b})\leq\mathbb{P}(V_{a}>V_{b}\mid P_{a}<P_{b})$

Note that $\mathbb{P}(D_{b}(P_{b})>D_{a}(P_{a})\mid P_{a}<P_{b})$ is related
to the concordance index of sensitivity analysis, in particular the
perspective studied by (Steck et al., 2008) who suggest a ranking-based
approach to survival analysis.

In survival analysis, right-censoring occurs when there is a finite horizon
end to data collection for the survival time of patients, so that the observed
survival times are the minimum of the censoring time and the actual survival
time. The concordance score is a generalization of Wilcoxon-Mann-Whitney
statistics and the AUC that applies to continuous output variables, and
accounts for censoring of the data. It is the fraction of all pairs of
subjects whose predicted survival times are correctly ordered among all
subjects that can be ordered. (Steck et al., 2008) observe that two subjects’
survival times can be ordered not only if both of them are uncensored; but
also if the uncensored time of one is smaller than the censored time of
another.

Relative to the concordance index of right-censored survival analysis, the
setting of allocative efficiency is more difficult: we are required to further
restrict attention to pairs $p_{1}<p_{2}$, and we can at best order
$v_{1}<p_{1}<p_{2}<v_{2}$.

###  5.5. Targeting for long-run dynamics: optimal encouragement designs

An important justification for cross-subsidy (preferring take-up) is in
recognizing that take-up of the good, $D(p)$ may itself be a treatment for
downstream outcomes. Such outcomes of interest might include long-term
customer value, social learning, repeated purchase behavior, “compliance”,
attrition, etc. We highlight that one might view $p$ as either a continuous
treatment or instrument.

For example, this is a major focus of (Karlan and Zinman, 2019) which
considers the amortized net present value of customers over a long time-
horizon after they take-up a microcredit loan. Their analysis suggests that
while price discrimination to expand take-up may result in losses in short-
term profits, this can be outweighed by clients “aging into” a loan portfolio
and becoming more profitable. This is also of concern for health interventions
in development: take-up of the good is not the final outcome of interest, but
rather health outcomes are.

In particular, the possibility of such a “triple bottom line” highlights a
situation where non-ideal theory that propagates the effects of known
inequality to the expected failures of classical economic theory may
highlights possible opportunities for personalized pricing to achieve
practical benefit. For example, a plausible narrative recognizes that poorer
households “undervalue” preventive healthcare not because of some fundamental
“underlying preference” for poor health, but due to behavioral considerations
and cognitive burdens which prevent endogenizing the full health benefits of a
product (Bar-Gill, 2018), or for far more practical reasons since they may
simply have lower incomes. As a result, they may be more price sensitive. And,
they may receive outsized additional health benefits from using the preventive
health intervention if they are indeed liquidity constrained due to lack of
other ancillary health interventions. Personalized price offers allow making
lower price offers, increasing take-up, and if indeed these households “priced
in on the margin” receive greater heterogeneous benefits, larger welfare
improvements.

There are two perspectives. In Figure 1(b), price is a continuous instrument
for treatment, and hence, outcome333Estimating a covariate-conditional local
instrumental variable curve, or optimal policy when the price instrument is
the control variable, remains an open problem. See (Kennedy et al., 2019) for
doubly robust estimation of the continuous instrument or (Syrgkanis et al.,
2019) for policy optimization with heterogeneous effects and discrete
instruments.. That is, recognizing that one cannot directly assign the actual
treatment of interest – one can neither force loans upon individuals nor
ethically randomly reject individuals who apply for loans, exogenous price
variation may be the only tool from observational data for assessing the
causal impacts of loans on welfare. Crucially, price satisfies the main
assumptions for instrumental variables; relevance, that it predicts treatment
(loan take-up), and more importantly the exclusion restriction: that
$Y(D(p))=Y(D),\forall p,$ the only effect of the price on outcome is via its
impact on treatment (Imbens and Rubin, 2015). This may be plausible when the
outcome of interest is a quantity such as health impacts of a bednet on
malaria incidence (Bhattacharya and Dupas, 2012), impacts of sanitation on
health outcomes, (Johnson and Lipscomb, 2019), or social learning for
sustained use of the intervention via subsidized first use (Dupas, 2014). In
the lending setting, this may be plausible if it is believed interest rates do
not affect default event, or the amount borrowed.

Another perspective views price as continuous treatment, e.g. Figure 1(a).
There are some posited behavioral economics effects which may lead to a
failure of the exclusion restriction such that price affects outcome, such as
anchoring to reference prices (which attenuates future take-up) or sunk-cost
fallacies (when high prices encourage usage/non-wastage); this is explored in
(Dupas, 2014). Alternatively, interest rates might affect default probability
if individuals are liquidity-constrained. Evidence is mixed in lending:
(Fuster et al., 2018) assumes this, (Alan and Loranth, 2013) finds no effect,
and (Karlan and Zinman, 2008) finds some effect of rates on default.
Alternatively, interest rates might have an effect on the extensive demand
margin (amount borrowed). In this setting, we might instead consider price as
a continuous treatment with a composite outcome of take-up and observed
outcome, conditional on take-up (such as amount borrowed or default outcome).

From the perspective of optimization, we generally view price as a treatment
and optimize for corresponding downstream outcomes, e.g. conduct an “intention
to treat” analysis.

$X$$P$$D$$Y$ (a) Price as treatment

$X$$P$$D$$Y$ (b) Price as instrument

##  6\. Case Studies

###  6.1. Willingness to pay for elective vaccine

We build a case study from (Slunge, 2015), a willingness-to-pay study for
vaccination against tick-borne encephalitis in Sweden. The vaccine for tick-
borne encephalitis (TBE) is elective and the study is interested in assessing
determinants of willingness to pay to inform health policy. Demand is
associated with price and income; as well as individual contextual factors
such as age, geographic risk factors, trust, perceptions and knowledge about
tick-borne disease. The health policy recommendation uses the learned demand
model to estimate the vaccination rate under a free, completely subsidized
vaccine. This setting corresponds to the setting of public provision, where a
decision-maker has a preference for higher take-up due to dynamic
externalities of vaccines (which are nonetheless difficult to precisely
estimate or target). The study was a contingent valuation study which asked
individuals about take-up at a random price of 100, 250, 500, 750, or 1000
SEK. The study finds that “The current market price of the TBE vaccine deters
a substantial share of at-risk people with low incomes from getting
vaccinated.”

(a) Revenue curves for
$A=\\{a,b\\},a,b$ segments

(b) Distribution of
$p^{*}(X)\mid A=a,p^{*}(X)\mid A=b$

(c) Estimated revenue
comparison

(d) Comparing take-up

Figure 2. Willingness to pay for elective vaccine (Figures 2(b) and 2(a)) and
Microcredit (Figures 2(d) and 2(c)).

In Figures 2(a) and 2(b), we compare distributional considerations of
segmented vs personalized pricing. Let $A=b$ indicate low-income. We follow
(Slunge, 2015) and learn a logistic regression model of binary demand by
simply appending the price covariate with the other covariates, so that
$D(x,p)=\sigma(\gamma^{\top}x+\beta p)$. A natural approach in the setting
where a free subsidy is not feasible due to budget constraints, is third-
degree price discrimination: segment based on income and offer a price to low-
income and high-income groups separately. We consider such a group-segmented
approach in Figure 2(a). The blue curve (“agg”) represents the revenue curve
for a uniform price. The red (“maj”) and green (“min”) curves are revenue
curves from the majority (high-income) and minority (low-income) groups,
respectively. We indicate the resulting optimal prices for each of these
curves with vertical dashed lines. Notably, the $A=a$ revenue curve has
greater second-degree curvature than the $A=b$ revenue curve. Because of the
flat revenue curve, the market share of the minority group can be
substantially increased without much extra cost.

We next consider a covariate-driven personalization approach. In Figure 2(b),
we plot histograms of the group-conditional distributions of these optimal
prices, ${p^{*}(X)\mid A=a}$ and ${p^{*}(X)\mid A=b}$. The optimal group-based
prices are indicated by the vertical lines for reference. We solve

$p^{*}(x)\in\underset{p(\cdot)}{\operatornamewithlimits{argmax}}\;\;\mathbb{E}[D(p(X))p].$

Notably, the optimal prices for the low-income group are overall lower than
those for the high-income group. Expected take-up in this segment increases to
$27.5\%$ from $24.2\%$ under the uniform monopoly price or $26.7\%$ under
$p^{*}(b)$, the optimal group-based price of Figure 2(a). Compared to the
uniform monopoly price which obtains expected optimal revenue of $313.6\cdot
10^{3}$, the group-based segment scheme $p^{*}(A)$ obtains expected revenue of
$282.2\cdot 10^{3}$, and the personalized pricing scheme $p^{*}(X)$ obtains
expected revenue of $318.8\cdot 10^{3}$. In this setting, covariate-based
personalized pricing is strictly beneficial in terms of (mild) revenue
benefits that are also able to achieve greater market share for the minority
group. While the group-based segmentation results in a lower price for the
minority group, it is overall not incentive-compatible for a decision-maker to
use this segmentation because it attains less revenue than even uniform
monopoly pricing.

###  6.2. Credit Elasticities

(Karlan and Zinman, 2008) randomize prices for a microfinance lender for
repeat borrowers. An extensive literature on microfinance sought to assess
whether microcredit was able to provide longer term benefits in improving
outcomes for household. The rise of the sector led to partially subsidized
lenders as well as interest from the private sector. The question of the study
was to leverage price randomization in the microcredit setting and assess the
effects of lower, or higher, interest rates on revenue for the lender.
Overalal, the findings suggest lower rates could decrease profits by a small
amount. But the paper considers that at the domain level, since microfinance
initiatives may have targeting preferences, e.g. for financial inclusion for
women or lower-income individuals, such potential mild profit losses could be
offset by expanded inclusion of these target groups due to heterogeneity in
take-up.

This setting could present an opportunity for personalized pricing to
differentially lower interest rates and expand revenue to targeted groups.
Adopting an “intention to treat” analysis, we use the method of (Kallus and
Zhou, 2018) to consider off-policy evaluation and optimization of a continuous
linear personalized pricing policy from the randomized controlled trial data.
The policy parametrization is linear in the covariates, which include income,
demographics, location, and loan history information. The method of (Kallus
and Zhou, 2018) considers a kernel-based estimator of the counterfactual value
of a pricing policy. We use the Epanechnikov kernel and a bandwidth of $0.3$;
the optimization is non-convex. Because of the fundamental problem of causal
inference, we lack the ability to directly assess outcomes.

Nonetheless, we provide some comparison of the estimated revenue and market
shares under the personalized policy. We consider a 50/25/25 training/nuisance
estimation/validation split, training a random forest on the nuisance
estimation split, and learning an optimal policy on the training data with a
doubly robust estimator. We use the random forest to estimate the revenue of
the personalized policy in comparison to constant interest rates on the
validation set.444This is in general a biased “direct method”, and we take
care to avoid extrapolation from interest rates. However, for the sake of
comparing against constant treatment assignment, using the direct method
reduces variance. Finally, to indicate the sampling variation in our
comparison induced by training the benchmark model, we repeat draws of the
nuisance/validation sets, and report the sampling variation in revenue
estimates via confidence bands of one standard error.

We include the results in Figure 2(c) and Figure 2(d). Figure 2(c) plots the
random-forest imputed revenue of the personalized allocation (indicated in
black dashed, plotting one standard error), comparing against the imputed
revenue via the random forest model of constant interest rates (on the
x-axis), in blue. The personalized allocation rule increases estimated revenue
(as expected). We assess some of the distributional characteristics of the
resulting allocation. In Figure 2(d), we compare the access properties of the
personalized decision rules for subgroups of interest, namely female and non-
female borrowers and low-income and non-low-income borrowers. The estimates of
access in Figure 2(d) are based on a logistic regression of demand, learned on
the nuisance estimation dataset.

In horizontal dashed lines, we plot the access estimates for these subgroups
under the personalized allocation rule. Note the achievable subgroup access
levels correspond to intersections of vertical lines with the demand curves.
In comparison, the personalized pricing allocation rule is able to increase
access for female and low-income borrowers, relative to the optimal constant
interest rate (around 7.2). The unconstrained optimal personalized price
however achieves lower takeup for non-low income borrowers (i.e. it increased
interest rates for them). Stronger distributional guarantees may be possible
by further constraining the price optimization problem. Overall, the goal is
to highlight that personalized pricing can improve firm revenue, as well as
increase access and improving targeting abilities.

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#### Overview of the appendix

  * •

Appendix A provides a literature review of how considerations regarding price
personalization have arisen in different problem settings.

  * •

Appendix B provides proofs of the analysis in Section 5.

  * •

Appendix C provides further details on the empirics.

##  Appendix A Literature review

###  A.1. Different problem settings (summary)

#### Lending

Discrimination in lending is an important problem subject to regulation by the
CFPB. Not only do banks decide whether or not to offer loans or extend credit
(previously considered as classification problems), they also decide on the
terms, i.e. interest rates, of the loan or credit, using risk-based pricing.
These rates have significant welfare implications for individuals if
discrimination leads to differences in rate terms. Standard discretionary
pricing practices, where individual loan agents have some discretion to set
rates above/below securitized rates based on local operating costs and
competitive factors, may lead to disparate impacts due to, e.g. differential
consumer bargaining leverage or discrimination.

There is extensive literature studying household finance, including credit
constraints of subprime borrowers with heterogeneity. For example, firms may
find via price optimization that they are able to raise prices on subprime
borrowers without affecting demand for credit-constrained consumers.

Price optimization in the lending setting is primarily based on discretionary
markups/discounts, e.g. negotiated on an individual basis with consumers.
There are subsets of loans that are securitized by the government, which are
presumed to be actuarially fair prices. Price dispersion beyond this can be
explained a combination of individual discretionary pricing (negotiation) or
market concentration (price optimization). A common strategy in papers that
study potential discriminatory pricing in lending (e.g. [Bartlett et al.,
2019]) is to regress deviations from the Fannie Mae-securitized rate schedule
on controls (e.g. for location-varying costs of lending); the coefficient on
race in such a regression corresponds to unexplained disparate impact in
pricing.

#### Insurance

Insurance is the one of the originating domains of risk-assessment. However;
it has the least available data. Our analysis correspondingly does not focus
on or shed light on important questions in this area.

#### Consumer products

There are growing concerns regarding the use of price optimization for
consumer goods. Many concerns in consumer product pricing are complicated due
to the difficulty of disambiguating consumer valuation based on “legitimate”
aspects such as individual preferences (which may be culturally conditioned;
hence associated with categories) vs. aspects that may seem “illegitimate” or
repugnant to extract rent from consumers on the basis of. A different source
of concern arises when universal provision of a good is expected.

However, studies show that consumers react strongly negatively to perceptions
of price unfairness (perception is an important mediator, because different
formats for the same posted price tend to have different fairness
perceptions). This consumer backlash may be so strong it is posited as a
reason that retailers do not wish to conduct extremely fine first-degree price
discrimination. Therefore, we might be interested in general notions of price
equity.

#### Goods with public externalities

In this setting which arises in development, public, or health economics, a
centralized decision-maker (DM) has some utility or preference for individuals
receiving the good in addition to individuals’ idiosyncratic distributions of
willingness to pay. Price optimization is beneficial because it can subsidize
participation in the market and take-up of the good by pricing at willingness-
to-pay for individuals with high valuations to subsidize lower price offers to
lower willingness-to-pay [Cohen et al., 2010, Dupas, 2009, Johnson and
Lipscomb, 2019]. Cross-subsidy is particularly beneficial when understanding
willingness-to-pay is in part related to ability to pay, which is commonly
discussed in development economics with regards to credit-constrained
consumers.

###  A.2. Implications of price optimization in lending

#### Lending and implications of regulation

In this setting, such as risk-based pricing in insurance and discretionary
pricing in mortgage lending, discretionary pricing is highly regulated and the
leeway for discretionary pricing or “price optimization” (rent extraction from
consumers) is severely limited. Limited discretion is given to lending agents,
for example to match competing offers or possibly (under interpretations of
fair lending law) to cover operating costs (for example in certain geographic
regions), e.g. “justified business necessity”. [Bartlett et al., 2019] study
mortgage lending for loans bought by Fannie Mae, which are subject to price
schedules based on creditworthiness, and find that while financial tech
companies that engage in algorithmic pricing reduce discrimination on LatinX
and African-American borrowers by 40%, nonetheless bias still persists (as
measured by residual coefficients on race in a regression of interest rates on
controls).

Nonetheless, concerns about actuarial risk may still arise when actuarial risk
is estimated from covariates. [Bartlett et al., 2019] summarizes their
interpretation of discrimination law and its implications for the legitimacy
of personalized pricing:

> (a) Scoring or pricing loans explicitly on credit-risk macro-fundamental
> variables is legitimate; (b) Scoring or pricing on a proxy variable that
> only correlates with race or ethnicity through hidden fundamental variables
> is legitimate; (c) Scoring or pricing on a proxy variable that has
> significant residual correlation with race or ethnicity after
> orthogonalizing with respect to hidden fundamental credit-risk variables is
> illegitimate.

#### Distributional concerns of interest in insurance/lending and risk-based
pricing.

In a recent case “Association belge des Consommateurs Test-Achats ASBL and
Others v. Conseil des ministres” ([eui, 2011]), the EU banned the use of sex
for pricing in insurance policies. [Avraham et al., 2013] points out some
aspects in which the redistributive transfers which result may be preferred to
transfers implemented through the typical public economics perspective of tax-
and-transfer because the magnitude is pegged to the difference in marginal
risk by group; and the transfer does not distort labor incentives as do
changes in tax schedules.

[Bartlett et al., 2019] points out that regulators may have distributional
concerns about price discretion in lending that differ from other settings
(such as revenue management and pricing). In revenue management and pricing,
and more generally settings with high fixed costs and lower marginal costs,
price inelastic customers such as business-class travelers may subsidize
access to a service via fare or price discrimination. In contrast, price-
sensitive consumers in insurance and lending may be more fiscally savvy and/or
“shop around” more for better terms; there is some concern that
underprivileged borrowers may also be less equipped to leverage competition as
such. This points to the importance of using non-ideal theory and domain-
specific considerations in motivating considerations of fairness.

#### Underlying normative considerations barring risk classification based on
protected attribute.

[Avraham et al., 2013] points out that normative grounds for arguments against
risk classification based on protected attribute include luck egalitarian
arguments (not holding individuals responsible for factors out of their
control, which is difficult with risk to operationalize), as well as the
ontological invalidity of social classifications for absolute risk except
insofar as social categories designate the effects of inequities in history.
Additionally, the consequences of inequity in accuracy of risk classification,
arguments regarding causality (which however are more effectively levied along
the lines of differential accuracy due to the presence of unobserved
confounders) and privacy.

#### Concerns about price optimization: rent extraction from behavioral bias

A key concern in household finance is in explaining empirical puzzles that
contradict predictions from conventional economic theory. To this end,
behavioral household finance considers implications of behavioral economics
for explaining some of these puzzles.

However, the expansion of the consumer credit market in the 1970s and 1980s
led to concerns regarding rising debt, and many puzzles in household finance
where empirical phenomena contradicted the assumptions of classical economic
theory. [Block-Lieb and Janger, 2005] studies the “myth of the rational
borrower” in the context of the discussion about bankruptcy law. The central
controversy was about whether or not the law provides incentives to declare
bankruptcy, or if increases in the number of bankruptcies are instead driven
by circumstances typically out of a borrower’s control such as income shocks.
They argue that a key counter-argument is “the myth of the rational borrower”;
that behavioral biases from economic theory lead suboptimally rational
borrowers to overborrow relative to their actual marginal returns to credit.
Whether or not consumers are behaviorally biased, or suboptimal in ways
predicted by behavioral economics, is a research topic: [Wright, 2006]
counter-argues that empirical evidence is not so strong that deviations from
predictions of economic theory are explained by behavioral economics.
Behavioral household finance remains an active area of study; although a meta-
analysis on the evidence for the effectiveness of financial education
interventions suggests that they are overall ineffective [Fernandes et al.,
2014]. We refer to [Beshears et al., 2018] for a fuller discussion.

We argue that the main takeaways to guide our analysis are therefore:

  * •

Conventional pricing theory and welfare analysis that assumes revealed
purchasing behavior reflects valuation may be inappropriate under strong
evidence of behaviorally suboptimal consumers, and in particular concerns
about credit- and liquidity-constrained customers, especially poorer
households.

  * •

Price optimization based on behavioral considerations may be profit-
maximizing, but value-based pricing in this setting may be suspect (or may
introduce distributional concerns opposite the slight favor in revenue
management for price optimization). Competitive considerations may support the
utility of price optimization, but we will abstract away from most competitive
considerations and focus on implications for an idealized actuarially-fair
pricing problem.

###  A.3. Perceptions of unfairness from consumers due to different prices
(equal treatment)

It is unclear whether there is sufficient regulation to merit equal treatment
as a normative rule for pricing, or to understand this squarely within the
sphere of discrimination law. Some consumer research such as [Haws and
Bearden, 2006] studies consumer perceptions of fairness. Since in fact
consumer reaction may differ to personalized pricing based on salience – e.g.
showing prices vs. showing discounts – the inconsistency of these reactions
themselves suggests that intuitive perceptions of fairness in pricing do not
reflect rational economic behavior; but rather that behavioral considerations
interact with the predictions of standard expected utility frameworks.

There are many high-profile instances where differential pricing appears
exploitative but we would argue that what is doing the normative work is more
broadly related to concerns regarding perpetuating historical injustices or
overt extraction of customer rent for reasons beyond their agency and
correlated with historical inequity. In that light, the normative force is
really in the ”leakage” of structural considerations and the correlation
between price optimization and these specific aspects. Conversely, it is
difficult to reason abstractly about these settings without further grounded
context.

We posit some examples. Consider the razor tax. The razor tax is objectionable
because it extracts customer welfare on the grounds of gender roles for
functionally the same product. At some level, our objection is far more
difficult to levy on the level of homophily or taste associated with
“pinkness”, than it would be to levy on the objection that societal
expectations regarding grooming have led to endogenizing higher willingness to
pay. Another example is higher prices for shipping/delivery to certain zip
codes (or not providing service at all: Here, there is some expectation of
equitable service provision, or not reifying historical disadvantages due to
location; even though antitrust provisions would allow charging different
prices based on differential costs of doing business. Another important point
contrasts price optimization based on first party vs. third-party data. Here,
there is a violation of privacy norms. Consumers may object to certain kinds
of data being used for price optimization vs. other kinds of data.

Disentangling aspects of valuations driven by the social constructions of race
and gender, e.g. via mediation analysis or other counterfactual notions, is
likely a very difficult task to operationalize.

###  A.4. Personalized pricing and public provision

In this specialized setting, price optimization allows significant benefit for
cross-subsidizing allocation of a good. Cross-subsidization is particularly
beneficial when a central decision-maker has a utility preference for more
individuals obtaining the good vs. not; for example in situations with
positive externalities.

These pricing schemes are of particular relevance for provision of health
interventions in development, where targeting can ensure take-up of households
of distributional interest [Johnson and Lipscomb, 2019]. A natural question
is, why not provide the good for free? The answer is due to a combination of
scarce resources and efficacy that requires take-up/compliance. In these
scarce-resource settings in health and development, if free provision to all
were possible and feasible, that would be optimal. However, when resource
constraints are binding, and the effects of health interventions are also
realized only by “compliance” of the consumer in using it (preventive
healthcare, bednets, contraceptives), the DM prefers subsidies to increase
access to those who would use the intervention if offered. So there are
additional targeting concerns that prefer allocation to individuals who would
take-up and use the health intervention; that is, the DM is interested in
balancing over-provision and over-exclusion [Cohen et al., 2015]. To this end,
price discrimination is considered a tool to screen-in individuals who would
use the good, rather than simply allocating for free.

This setting is common in development and public economics. [Johnson and
Lipscomb, 2019] conceive a pricing mechanism to price individuals into a
market for mechanical desludging in Kenya; a health intervention that has
positive externalities but which may be out of reach for the poorest
households. Estimating a demand and reservation price model, they learn
personalized prices, estimate the personalized price solution via a
multinomial logistic model, and run a RCT to evaluate out-of-sample the
effects of personalized pricing on take-up as well as overall health outcomes.

###  A.5. Research questioning the ontological stability of valuations

#### Ontological stability of valuations.

While a typical economic rejoinder to these concerns highlights that first-
degree price discrimination achieves a Pareto-efficient allocation of goods to
individuals at their valuation and universal access, we would note the actual
distributional implications of personalized pricing depend on how consumer
behavior does or does not reflect an implicitly assumed ontologically stable
and valid idiosyncratic “valuation”.

While there is strong mechanism design theory for soliciting individual
valuations in, for example, auctions, we argue that many of the domains where
consumer-facing personalized pricing has faced pushback are different from the
restricted domains where auctions are deployed in practice. We propose a
taxonomy of types of failures of the “ideal theory” of willingness to pay
which raise concerns for fairness. These settings are not merely pathological
or point failures of economic theory, but rather exactly how structural
inequities manifest in price optimization settings. A first concern is
behavioral biases (and price optimization which extracts rent from them). For
example, credit card contracts with promotional offers but high terms
afterwards. This motivates the use of mandatory disclosure notices in that
setting. A second concern is that of differential endowments which surface via
different incomes or credit constraints, budget constraints, and liquidity
constraints. This is a concern when individuals’ reflected purchase behavior
is not a realization of their ideal valuation or willingness to pay, of the
item, but rather of their ability to pay. [Liscow, 2018] discusses a similar
argument against Kaldor-Hicks efficiency arguments for welfare analysis in
empirical legal studies. In settings with credit constraints, individuals are
credit or liquidity-constrained, and therefore borrow at high (perhaps even
irrationally high) interest rates because of lack of other options. [Alan and
Loranth, 2013]

##  Appendix B Proofs

###  B.1. Proofs for price parity

###### Proof of theorem 1 .

Optimizing over attribute-based personalized prices $p^{*}(x)$:

We consider the Lagrangian dual of eq. 2. We then applying Sion’s minimax
theorem to swap the order of min and max operations, which is valid under
compactness [Boyd and Vandenberghe, 2004]:

(7)
$\min_{p(x,a)}\max_{\lambda}\mathbb{E}\left[-pD(p)+\lambda(p\xi(A)-\Gamma)\right]=\max_{\lambda}\min_{p(x)}\mathbb{E}\left[-pD(p)+\lambda(p\xi(A)-\Gamma)\right]$

For a linear demand model, observing that when $p^{*}(x,a)$ is unrestricted,
we let $p^{*}(x,a;\lambda)$ be the optimal solution parameterized by
$\lambda$:

(8)
$p^{*}(x,a;\lambda)\in\operatornamewithlimits{argmax}\mathbb{E}[-pD(p)+\lambda(p\xi(A)-\Gamma)\mid
X=x,A=a].$

The optimal price with the $\lambda$ penalty is computable in closed form:

$p^{*}(x,a;\lambda)=\frac{-\overline{D}(x,a)+\lambda\xi(a)}{2\beta}.$

Plugging in this solution:

$\displaystyle\max_{\lambda}L(\lambda,p^{*}(X,A;\lambda))=\max_{\lambda}\mathbb{E}\left[-p^{*}(X,A;\lambda)D(p^{*}(X,A;\lambda))+\lambda(p^{*}(X,A;\lambda)\xi-\Gamma)\right]$

We maximize the above over $\lambda$. Taking derivatives with respect to
$\lambda$, we obtain the first order necessary condition for optimality,
letting $p^{*}(\lambda)=p^{*}(X,A;\lambda)$, $\overline{D}=\overline{D}(X,A)$
for brevity is:

$\displaystyle 0$
$\displaystyle=\mathbb{E}\left[-p^{*}(\lambda)\frac{d}{d\lambda}D(p^{*}(\lambda))+-(\frac{d}{d\lambda}p^{*}(\lambda))D(p^{*}(\lambda))+(p^{*}(\lambda)\xi-\Gamma)+\lambda\frac{\xi^{2}}{2\beta}\right]$
$\displaystyle=\mathbb{E}\left[-\left(\frac{-\overline{D}+\lambda\xi}{2\beta}\right)\frac{\xi}{2}-\frac{\xi}{2\beta}(\overline{D}+\frac{1}{2}(-\overline{D}+\lambda\xi))+(p^{*}(\lambda)\xi-\Gamma)+\lambda\frac{\xi^{2}}{2\beta}\right]$
(9)
$\displaystyle=\mathbb{E}\left[\frac{\overline{D}\xi}{4\beta}-\frac{\overline{D}\xi}{4\beta}-\lambda\frac{\xi^{2}}{2\beta}+(p^{*}(\lambda)\xi-\Gamma)+\lambda\frac{\xi^{2}}{2\beta}\right]$
(10)
$\displaystyle=\mathbb{E}\left[\left(\frac{-\overline{D}+\lambda\xi}{2\beta}\xi-\Gamma\right)\right]$

From eq. 11 we conclude

$\lambda^{*}=\frac{\mathbb{E}[\overline{D}(X,A)\xi(A)+2\beta\Gamma]}{\mathbb{E}[\xi^{2}(A)]}.$

Note that further taking the derivative of eq. 11 with respect to $\lambda$
verifies $L(\lambda,p^{*}(\lambda))$ is concave in $\lambda$,
$\frac{d^{2}L(\lambda,p^{*}(\lambda))}{d\lambda^{2}}=\mathbb{E}[\frac{\xi^{2}}{2\beta}]<0$
under Asn. 3. Therefore the first-order necessary condition is also
sufficient.

Optimizing over attribute-blind personalized prices $p^{*}(x)$:

The counterpart of eq. 8 is, scaling by a constant $f(x)$, the covariate
density of $x$, to simplify

$\Delta f(x\mid a)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathbb{P}(X=x\mid
A=a)-\mathbb{P}(X=x\mid A=b)=\mathbb{E}[\xi(A)\mid
X=x]f(x)=(\rho_{a}^{-1}\mathbb{P}(A=a\mid X=x)-\rho_{b}^{-1}\mathbb{P}(A=b\mid
X=x))g(x)$ $\displaystyle p^{*}(x;\lambda)$
$\displaystyle\in\operatornamewithlimits{argmin}-pD(p\mid
x)+\lambda(p\mathbb{E}[\xi(A)\mid X=x]-\Gamma)$ $\displaystyle
p^{*}(x;\lambda)$ $\displaystyle\in\operatornamewithlimits{argmin}-pD(p\mid
x)f(x)+\lambda(p\mathbb{E}[\xi(A)\mid X=x]-\Gamma)f(x)$ $\displaystyle\iff
p^{*}(x;\lambda)$ $\displaystyle\in\operatornamewithlimits{argmin}-pD(p\mid
x)f(x)+\lambda p\Delta f(x\mid a)$

Note $D(p\mid x)=D-\alpha+\mathbb{E}[\beta_{A}\mid X=x]+\mathbb{E}[g(x,A)\mid
x]$.

The optimal $\lambda-$ parametrized price is

$p^{*}(x;\lambda)=\frac{-\overline{D}(x)+\lambda\mathbb{E}[\xi(A)\mid
X=x]}{2\mathbb{E}[\beta_{A}\mid
X=x]}=\frac{-\overline{D}(x)+\lambda\frac{\Delta f(x\mid
a)}{f(x)}}{2\mathbb{E}[\beta_{A}\mid X=x]}.$

We correspondingly solve (analogous to the previous):

$\displaystyle\max_{\lambda}L(\lambda,p^{*}(X;\lambda))=\max_{\lambda}\mathbb{E}\left[-p^{*}(X;\lambda)D(p^{*}(X;\lambda))+\lambda(p^{*}(X;\lambda)\xi-\Gamma)\right]$

with the corresponding first-order conditions

$\displaystyle 0$
$\displaystyle=\mathbb{E}\left[-p^{*}(\lambda)\frac{d}{d\lambda}D(p^{*}(\lambda))+-(\frac{d}{d\lambda}p^{*}(\lambda))D(p^{*}(\lambda))+(p^{*}(\lambda)\xi-\Gamma)+\lambda\frac{\xi^{2}}{2\mathbb{E}[\beta_{A}\mid
X=x]}\right]$ (11)
$\displaystyle=\mathbb{E}\left[\left(\frac{-\overline{D}+\lambda\mathbb{E}[\xi(A)\mid
X=x]}{2\mathbb{E}[\beta_{A}\mid X=x]}\mathbb{E}[\xi(A)\mid
X=x]-\Gamma\right)\right]$

Analogously we conclude

$\lambda^{*}=\frac{\mathbb{E}[\overline{D}(X)\mathbb{E}[\xi(A)\mid
X]+2\mathbb{E}[\beta_{A}\mid X]\Gamma]}{\mathbb{E}[\mathbb{E}[\xi(A)\mid
X]^{2}]}.$

∎

###### Proof of Proposition 2.

We would like to derive conditions that might inform of the sign of
$p^{*}(x)-p^{*}(x,a)$.

There are a few extreme cases which might be informative (of one regime or
another).

  1. (1)

$\beta_{a}=\beta_{b},\overline{D}(x,a)=\overline{D}(x,b)$

Disparities are solely due to covariate distributions across groups.

  2. (2)

$\beta_{a}>\beta_{b},\overline{D}(x,a)=\overline{D}(x,b)=\overline{D}(x)$

Disparities are solely due to group-level differences in price elasticity or
differences in covariate distribution across groups.

We only included conclusions for case 1 in the main text. In this appendix we
also provide a sufficient condition for case 2 (though this is less
interpretable; hence less useful).

Under case 1,

$\displaystyle p^{*}(x,a)-p^{*}(x)$
$\displaystyle=\frac{-\overline{D}(x)+\lambda^{*}(x,a)\xi(a)}{\beta}-\frac{-\overline{D}(x)+\lambda^{*}(a)\mathbb{E}[\xi(A)\mid
X=x]}{\beta}=\frac{\xi(a)\lambda^{*}(x,a)-\mathbb{E}[\xi(A)\mid
X=x]\lambda^{*}(x))}{\beta}$
$\displaystyle=\frac{\xi(a)\lambda^{*}(x,a)-\xi(a)\lambda^{*}(x)+\xi(a)\lambda^{*}(x)-\mathbb{E}[\xi(A)\mid
X=x]\lambda^{*}(x))}{\beta}$ (12)
$\displaystyle=\frac{1}{\beta}\left(\xi(a)(\lambda^{*}(x,a)-\lambda^{*}(x))+\lambda^{*}(x)(\xi(a)-\mathbb{E}[\xi(A)\mid
X=x])\right)$

We conclude the signs of the following terms:

  1. (1)

$(\lambda^{*}(x,a)-\lambda^{*}(x))\leq 0,$

by assumption of linearity of demand, Jensen’s inequality, since $f(x)=x^{2}$
is a convex function, and by iterated expectation:

(13) $\mathbb{E}[\mathbb{E}[\xi(A)\mid
X]^{2}]\leq\mathbb{E}[\mathbb{E}[\xi(A)^{2}\mid X]]=\mathbb{E}[\xi(A)^{2}]$

Note that
$\lambda^{*}(x,a)\leq\lambda^{*}(x)\iff\mathbb{E}[\mathbb{E}[\xi(A)\mid
X]^{2}]\leq\mathbb{E}[\xi(A)^{2}]$.

  2. (2)

$\lambda^{*}(x)<0,$ under assumption for theorem 1 that group $a$ faces the
higher unrestricted personalized prices.

  3. (3)

$(\xi(a)-\mathbb{E}[\xi(A)\mid X=x])>0\text{ and
}(\xi(b)-\mathbb{E}[\xi(A)\mid X=x])<0.$

This may be verified by observing

$\textstyle\frac{1}{\rho_{a}}>\frac{\mathbb{P}(A=a\mid
X=x)}{\rho_{a}}-\frac{\mathbb{P}(A=b\mid X=x)}{\rho_{b}}\iff
1-\mathbb{P}(A=a\mid X=x)>-\mathbb{P}(A=b\mid
X=x)\frac{\rho_{a}}{\rho_{b}}\iff\rho_{b}>-\rho_{a},$

and concluding based on nonnegativity of $\rho_{a},\rho_{b}$.

Therefore, identifying signs of terms in eq. 12:

$\displaystyle\textstyle p^{*}(x,a)-p^{*}(x)$
$\displaystyle=\textstyle\underbrace{\frac{1}{\beta}}_{<0}\left(\underbrace{\xi(a)(\lambda^{*}(x,a)-\lambda^{*}(x))}_{\leq
0}+\underbrace{\lambda^{*}(x)}_{\leq
0}\underbrace{(\xi(a)-\mathbb{E}[\xi(A)\mid X=x])}_{\geq 0}\right)$ (14)
$\displaystyle\textstyle p^{*}(x,b)-p^{*}(x)$
$\displaystyle=\textstyle\underbrace{\frac{1}{\beta}}_{<0}\left(\underbrace{\xi(b)}_{<0}\underbrace{(\lambda^{*}(x,a)-\lambda^{*}(x))}_{\leq
0}+\underbrace{\lambda^{*}(x)}_{\leq
0}\underbrace{(\xi(b)-\mathbb{E}[\xi(A)\mid X=x])}_{\leq 0}\right)$

The claim for $p^{*}(x,a)-p^{*}(x)<0$ follows by simplifying and factoring out
$\rho_{a}$ and $\nicefrac{{\lambda^{*}(x)}}{{\beta_{a}}}>0$:

$\displaystyle p^{*}(x,a)-p^{*}(x)<0$
$\displaystyle\iff\frac{1}{\beta}\left(\rho_{a}^{-1}(\lambda^{*}(x,a)-\lambda^{*}(x))+\lambda^{*}(x)(\rho_{a}^{-1}(1-(\mathbb{P}(A=a\mid
X=x)-\nicefrac{{\rho_{a}}}{{\rho_{b}}}\mathbb{P}(A=b\mid X=x))\right)<0$
$\displaystyle\textstyle\iff\left(\frac{\mathbb{E}[\mathbb{E}[\xi(A)\mid
X=x]^{2}]}{\mathbb{E}[\xi(A)^{2}]}-1\right)+((1-(\mathbb{P}(A=a\mid
X=x)-\nicefrac{{\rho_{a}}}{{\rho_{b}}}\mathbb{P}(A=b\mid X=x))<0$
$\displaystyle\textstyle\iff\frac{\mathbb{E}[\mathbb{E}[\xi(A)\mid
X=x]^{2}]}{\mathbb{E}[\xi(A)^{2}]}<(\mathbb{P}(A=a\mid
X=x)-\nicefrac{{\rho_{a}}}{{\rho_{b}}}\mathbb{P}(A=b\mid X=x))$

The claim follows for $p^{*}(x,b)-p^{*}(x)$ directly from eq. 14.

To interpret this condition, denote $\Delta f(x;a,b)$ as the covariate-driven
group divergence:

$\Delta f(x;a,b)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathbb{P}(A=a\mid
X=x)-\frac{\rho_{a}}{\rho_{b}}\mathbb{P}(A=b\mid
X=x)=\frac{\mathbb{E}[\xi(A)\mid X=x]}{\rho_{a}^{-1}}.$

Observe that

$\Delta f(x;a,b)<0\iff\frac{\mathbb{P}(A=a\mid X=x)}{\mathbb{P}(A=b\mid
X=x)}<\frac{\rho_{a}}{\rho_{b}}\iff\frac{\mathbb{P}(X=x\mid
A=a)}{\mathbb{P}(X=x\mid A=b}<1,$

i.e. the sign of $\Delta f(x;a,b)$ depends on the covariate likelihood under
the two classes (and the magnitude depends on the magnitude of this covariate-
based divergence relative to the covariate-uninformed ratio,
$\nicefrac{{\rho_{a}}}{{\rho_{b}}}$ ).

Under case 2:

Let $\Delta\beta_{a}(x)=\frac{\beta_{a}}{\mathbb{E}[\beta_{A}\mid X=x]}$ and
recall that
$\frac{\lambda^{*}_{xa}}{\lambda^{*}_{x}}=\frac{\mathbb{E}[\mathbb{E}[\xi(A)\mid
X=x]^{2}]}{\mathbb{E}[\xi(A)^{2}]}\leq 0$. Adding and subtracting
$\frac{-\overline{D}(x)+\xi(a)\lambda^{*}(x)}{\beta_{a}}$,

$\displaystyle p^{*}(x,a)-p^{*}(x)>0$ $\displaystyle\iff$
$\displaystyle\left(\frac{-\overline{D}(x)+\lambda^{*}_{xa}\xi(a)}{\beta_{a}}-\frac{-\overline{D}(x)+\xi(a)\lambda^{*}_{x}}{\beta_{a}}\right)+\left(\frac{-\overline{D}(x)+\xi(a)\lambda^{*}_{x}}{\beta_{a}}-\frac{-\overline{D}(x)+\mathbb{E}[\xi(A)\mid
X=x]\lambda^{*}_{x}}{\mathbb{E}[\beta_{A}\mid X=x]}\right)>0$
$\displaystyle\iff$
$\displaystyle\frac{\xi(a)}{\beta_{a}}(\lambda^{*}(x,a)-\lambda^{*}(x))+-\overline{D}(x)\left(\frac{1}{\beta_{a}}-\frac{1}{\mathbb{E}[\beta_{A}\mid
X=x]}\right)+\left(\frac{\xi(a)\lambda^{*}_{xa}}{\beta_{a}}-\frac{\mathbb{E}[\xi(A)\mid
X=x]\lambda^{*}_{x}}{\mathbb{E}[\beta_{A}\mid X=x]}\right)>0$
$\displaystyle\iff$
$\displaystyle{\xi(a)}\underbrace{\left(\frac{\mathbb{E}[\mathbb{E}[\xi(A)\mid
X=x]^{2}]}{\mathbb{E}[\xi(A)^{2}]}-1\right)}_{<0\text{ by
}\lx@cref{creftype~refnum}{prop1-eq-
jensen}}+-\frac{\overline{D}(x)}{\lambda^{*}_{x}}\left({1}-\Delta\beta_{a}(x)\right)+\left(\xi(a)\frac{\mathbb{E}[\mathbb{E}[\xi(A)\mid
X=x]^{2}]}{\mathbb{E}[\xi(A)^{2}]}-\mathbb{E}[\xi(A)\mid
X=x]\Delta\beta_{a}(x)\right)>0$

where in the last line, we factor out
$\nicefrac{{\lambda^{*}(x)}}{{\beta_{a}}}>0$. Correspondingly for $A=a,b$
respectively:

$\displaystyle p^{*}(x,a)-p^{*}(x)>0$
$\displaystyle\iff\underbrace{\rho_{a}^{-1}\left(\frac{\lambda^{*}_{xa}}{\lambda^{*}_{x}}-1\right)}_{<0}+-\frac{\overline{D}(x)}{\lambda^{*}_{x}}\left({1}-\Delta\beta_{a}(x)\right)+\rho_{a}^{-1}\left(\frac{\lambda^{*}_{xa}}{\lambda^{*}_{x}}-\Delta
f(x;a,b)\cdot\Delta\beta_{a}(x)\right)>0$ $\displaystyle
p^{*}(x,b)-p^{*}(x)>0$
$\displaystyle\iff\underbrace{-\rho_{b}^{-1}\left(\frac{\lambda^{*}_{xa}}{\lambda^{*}_{x}}-1\right)}_{>0}+-\frac{\overline{D}(x)}{\lambda^{*}_{x}}\left({1}-\Delta\beta_{b}(x)\right)+\rho_{b}^{-1}\left(-1\cdot\frac{\lambda^{*}_{xa}}{\lambda^{*}_{x}}-\rho_{b}\mathbb{E}[\xi(A)\mid
X=x]\Delta\beta_{b}(x)\right)>0$

Unlike the previous case, this case does not admit determinate conclusions on
signs.

We may simplify the condition to obtain that if $-\Delta
f(x;a,b)\lambda^{*}_{x}+\overline{D}(x)\rho_{a}>0,$

$\textstyle
p^{*}(x,a)-p^{*}(x)>0\iff\Delta\beta_{a}(x)<\frac{\lambda^{*}_{x}-2\lambda^{*}_{xa}+\overline{D}(x)\rho_{a}}{-\Delta
f(x;a,b)\lambda^{*}_{x}+\overline{D}(x)\rho_{a}},$

with the inequality on $\Delta\beta_{a}(x)$ holding in the opposite direction
if instead $-\Delta f(x;a,b)\lambda^{*}_{x}+\overline{D}(x)\rho_{a}<0$. ∎

###  B.2. Model error fairness

###### Proof of proposition 4 .

We omit dependence on fixed $x$ and denote $D(p)=D(p\mid x)$. Gradient
$\nabla$ is with respect to $p$. We assume price elasticity of demand is
nonpositive, $\nabla_{p}\eta<0$. We specialize to a revenue setting by
observing that $\nabla h=\eta(p)+\nabla\eta\cdot p$, so that
$\hat{p}^{*},p^{*}$ satisfy the first order optimality conditions:

$\displaystyle\hat{p}^{*}$
$\displaystyle=-\frac{\hat{D}(\hat{p}^{*})}{\nabla\hat{D}(\hat{p}^{*})},\qquad
p^{*}=-\frac{\eta(p^{*})}{\nabla\eta(p^{*})}.$

Taylor expanding $\hat{D}(\hat{p}^{*})$ around $p^{*}$:

$\displaystyle\hat{p}^{*}-p^{*}$
$\displaystyle=-\frac{\hat{D}(\hat{p}^{*})}{\nabla\hat{D}(\hat{p}^{*})}+\frac{D(p^{*})}{\nabla
D(p^{*})}=\frac{\hat{D}(p^{*})+\nabla\hat{D}(p^{*})(\hat{p}^{*}-p^{*})}{\nabla\hat{D}(\hat{p}^{*})}+\frac{D(p^{*})}{\nabla
D(p^{*})}$

so that

$\displaystyle(\hat{p}^{*}-p^{*})\left(1-\frac{\nabla\hat{D}(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}\right)$
$\displaystyle=-\frac{\hat{D}(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}+\frac{D(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}\frac{\nabla\hat{D}(\hat{p}^{*})}{\nabla
D(p^{*})}+o((\hat{p}^{*}-p^{*})^{2})$
$\displaystyle=-\frac{\hat{D}(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}+\frac{D(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}\left(1+\frac{\nabla\hat{D}(\hat{p}^{*})-\nabla
D(p^{*})}{\nabla D(p^{*})}\right)+o((\hat{p}^{*}-p^{*})^{2})$
$\displaystyle=\frac{\hat{D}(p^{*})-D(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}+\frac{D(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}\left(\frac{\nabla\hat{D}(\hat{p}^{*})-\nabla
D(p^{*})}{\nabla D(p^{*})}\right)+o((\hat{p}^{*}-p^{*})^{2})$

Therefore,

$\displaystyle(\hat{p}^{*}-p^{*})$
$\displaystyle=\left(\frac{\nabla\hat{D}(\hat{p}^{*})}{\nabla\hat{D}(\hat{p}^{*})-\nabla\hat{D}(p^{*})}\right)\left(\frac{\hat{D}(p^{*})-D(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}+\frac{D(p^{*})}{\nabla\hat{D}(\hat{p}^{*})}\left(\frac{\nabla\hat{D}(\hat{p}^{*})-\nabla
D(p^{*})}{\nabla D(p^{*})}\right)\right)+o((\hat{p}^{*}-p^{*})^{2})$
$\displaystyle=(\nabla\hat{D}(\hat{p}^{*})-\nabla\hat{D}(p^{*}))^{-1}\left({\hat{D}(p^{*})-D(p^{*})}+\frac{D(p^{*})}{\nabla
D(p^{*})}\left({\nabla\hat{D}(\hat{p}^{*})-\nabla
D(p^{*})}\right)\right)+o((\hat{p}^{*}-p^{*})^{2})$
$\displaystyle\overset{1}{=}(\nabla\hat{D}(\hat{p}^{*})-\nabla\hat{D}(p^{*}))^{-1}\left({\hat{D}(p^{*})-D(p^{*})}+\frac{D(p^{*})}{\nabla
D(p^{*})}\left({\nabla\hat{D}(\hat{p}^{*})-\nabla\hat{D}(p^{*})+\nabla\hat{D}(p^{*})-\nabla
D(p^{*})}\right)\right)..$
$\displaystyle=\left(\frac{{\hat{D}(p^{*})-D(p^{*})}}{\nabla\hat{D}(\hat{p}^{*})-\nabla\hat{D}(p^{*})}+p^{*}\left(1+\frac{\nabla\hat{D}(p^{*})-\nabla
D(p^{*})}{\nabla\hat{D}(\hat{p}^{*})-\nabla\hat{D}(p^{*})}\right)\right)+o((\hat{p}^{*}-p^{*})^{2})$

where in $1$ we expand $\left({\nabla\hat{D}(\hat{p}^{*})-\nabla
D(p^{*})}\right)$; then simplify. ∎

###  B.3. Market share

The main tool for analyzing local sensitivities of the optimal prices is an
implicit function theorem to differentiate the optimal solution with respect
to the parameter. We state it for completeness.

###### Theorem 1 (Dini Classical Implicit Function Theorem (Thm. 1B.1 of
[Dontchev and Rockafellar, 2009]).

Consider a function
$f:\mathbb{R}^{d}\times\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ with values
$f(\lambda,p)$ with $\lambda$ the parameter and $p$ the variable to solve for.
The equation $f(\lambda,p)$ is associated with the solution mapping

$S\colon\lambda\mapsto\\{p\in\mathbb{R}^{n}\mid f(\lambda,p)=0\\},\text{ for
}\lambda\in\mathbb{R}^{d}$

Let $f$ be continuously differentiable in a neighborhood of
$(\overline{\lambda},\overline{p})$ such that
$f(\overline{\lambda},\overline{p})=0$, and let the partial Jacobian of $f$
with respect to $p$ at $(\overline{\lambda},\overline{p})$, namely
$\nabla_{x}f(\overline{\lambda},\overline{p})$.

Then the solution mapping S has a single valued localization s around
$\overline{\lambda}$ for $\overline{p}$ which is continuously differentiable
in a neighborhood $Q$ of $\overline{p}$ with Jacobian satisfying

$\nabla
s(\lambda)=-\nabla_{p}f(\lambda,s(\lambda))^{-1}\nabla_{\lambda}f(\lambda,s(\lambda))\text{
for every }\lambda\in Q$

Using the implicit function theorem, we can characterize the sensitivities of
solutions under attribute-blind vs. attribute-based, and group market share
vs. population market share penalties. We restate an expanded versio of Lemma
5.

Lemma 1[Optimality conditions for different penalties ]

The sensitivities of price with respect to $\lambda$, $\frac{\partial
p^{*}(x)}{\partial\lambda}$.

  1. (1)

$p^{*}(x)$ with population market share penalty satisfies
$\frac{1}{p^{*}(x)+\lambda}+\frac{D^{\prime}(p^{*}(x)\mid x)}{D(p^{*}(x)\mid
x)}=0,\;\;\forall x$ so that

$\nabla_{\lambda}p^{*}(x;0)=\frac{R^{\prime\prime}(p^{*}(x)\mid
x))^{-1}}{p^{*}(x)^{2}}.$

  2. (2)

$p^{*}(x,a)$ with population market share satisfies
$\frac{1}{p^{*}(x,a)+\lambda}+\frac{D^{\prime}(p^{*}(x,a)\mid
x,a)}{D(p^{*}(x,a)\mid x,a)}=0,\;\;\forall x,a$ so that

$\nabla_{\lambda}p^{*}(x,a;0)=\frac{R^{\prime\prime}(p^{*}(x,a)\mid
x,a))^{-1}}{p^{*}(x,a)^{2}}.$

  3. (3)

$p^{*}(x,a)$ with group-level market share
$\frac{1}{p^{*}(x,a)+\lambda_{a}/\rho_{a}}+\frac{D^{\prime}(p^{*}(x,a)\mid
x,a)}{D(p^{*}(x,a)\mid x,a)}=0$ so that

$\nabla_{\lambda}p^{*}(x,a;0)=\frac{1}{\rho_{a}}\frac{R^{\prime\prime}(p^{*}(x,a)\mid
x,a))^{-1}}{p^{*}(x,a)^{2}}.$

###### Proof of Lemma 5.

  1. (1)

$p^{*}(x,a)$ with population market share

$\displaystyle
p^{*}(x,a)\in\operatornamewithlimits{argmax}\mathbb{E}[pD(p)]+\lambda\mathbb{E}[D(p)]$
$\displaystyle\iff$ $\displaystyle
p^{*}(x,a)\in\operatornamewithlimits{argmax}\log((p+\lambda)D(p\mid x,a))$

Therefore $p^{*}(x,a)$ satisfies the following:

$\displaystyle\frac{1}{p^{*}(x,a)+\lambda}+\frac{D^{\prime}(p^{*}(x,a)\mid
x,a)}{D(p^{*}(x,a)\mid x,a)}=0,\;\;\forall x,a$

The expression for $\nabla_{\lambda}p^{*}(x)(0)$ follows by applying the
Implicit function theorem on the optimality condition.

  2. (2)

$p^{*}(x)$ with population market share

$\displaystyle
p^{*}(x)\in\operatornamewithlimits{argmax}\mathbb{E}[pD(p)]+\lambda\mathbb{E}[D(p)]$
$\displaystyle\iff$ $\displaystyle
p^{*}(x)\in\operatornamewithlimits{argmax}\mathbb{E}[\mathbb{E}[(p+\lambda)D(p)\mid
X]],\forall x$ $\displaystyle\iff$ $\displaystyle
p^{*}(x)\in\operatornamewithlimits{argmax}(p+\lambda)D(p\mid x),\forall x$
$\displaystyle\iff$ $\displaystyle
p^{*}(x)\in\operatornamewithlimits{argmax}\log((p+\lambda)D(p\mid x)),\forall
x$

Therefore $p^{*}(x)$ satisfies the following:

$\displaystyle\frac{1}{p^{*}(x)+\lambda}+\frac{D^{\prime}(p^{*}(x)\mid
x)}{D(p^{*}(x)\mid x)}=0,\;\;\forall x$

  3. (3)

$p^{*}(x,a)$ with group-level market share

$\displaystyle\iff$ $\displaystyle
p^{*}(x,a)\in\operatornamewithlimits{argmax}\mathbb{E}[(p+\lambda_{a}/\rho_{a})D(p)\mid
X=x,A=a]$ $\displaystyle\iff$ $\displaystyle
p^{*}(x,a)\in\operatornamewithlimits{argmax}(p+\lambda_{a}/\rho_{a})D(p\mid
x,a)$ $\displaystyle\iff$ $\displaystyle
p^{*}(x,a)\in\operatornamewithlimits{argmax}\log((p+\lambda_{a}/\rho_{a})D(p\mid
x,a))$

Therefore $p^{*}(x,a;\lambda)$ is such that

$\displaystyle\frac{1}{p+\lambda_{a}/\rho_{a}}+\frac{D^{\prime}(p\mid
x,a)}{D(p\mid x,a)}=0$

∎

###  B.4. Allocative efficiency: Concordance

###### Proof of Theorem 6.

Let $\mathcal{I}\\{A=a\\}$ denote index sets for data points within group $a$,
etc.

$\displaystyle\mathbb{P}(D(p_{i})<D(p_{j})\mid P_{a}<P_{b})$
$\displaystyle=\frac{1}{|\\{(i,j)\colon
p_{i}<p_{j}\\}|}\frac{1}{n^{2}}\sum_{i\in\mathcal{I}\\{A=a\\}}\sum_{j\in\mathcal{I}\\{A=b\\}}\mathbb{I}[D(p_{i})=0,D(p_{j})=1,p_{i}<p_{j}]$
$\displaystyle=\frac{1}{|\\{(i,j)\colon
p_{i}<p_{j}\\}|}\frac{1}{n^{2}}\sum_{i\in\mathcal{I}\\{A=a\\}}\sum_{j\in\mathcal{I}\\{A=b\\}}\mathbb{I}[v_{i}<p_{i}<p_{j}<v_{j}]$
$\displaystyle\leq\frac{1}{|\\{(i,j)\colon
p_{i}<p_{j}\\}|}\frac{1}{n^{2}}\sum_{i\in\mathcal{I}\\{A=a\\}}\sum_{j\in\mathcal{I}\\{A=b\\}}\mathbb{I}[\\{v_{i}<v_{j}\\}\cap\\{p_{i}<p_{j}\\}]$
$\displaystyle=\mathbb{P}(V_{a}>V_{b}\mid P_{a}<P_{b})$

where the first equality holds because under Asn. 1 (a.s. monotonicity), the
following events are a.s. equivalent:

$\\{p_{i}<p_{j},D(p_{i})=0,D(p_{j})=1\\}\iff v_{j}>v_{i}.$

The second inequality holds because
$\\{v_{i}<p_{i}<p_{j}<v_{j}\\}\subset\\{\\{v_{i}<v_{j}\\}\cap\\{p_{i}<p_{j}\\}\\}$.

∎

##  Appendix C Dataset details

#### Details about the study [Slunge, 2015]

We omit concerns about non-response. The survey was distributed online in
2013. $N=1116$. There are 28 data columns with information including
categorical age values, gender, geographic factors and risk factors, and
information about knowledge and trust about vaccines.