Does Machine Learning Amplify Pricing Errors in the Housing Market? — The
Economics of Machine Learning Feedback Loops

Nikhil Malik∗ and Emaad Manzoor† ‡USC Marshall School of Business,
<EMAIL_ADDRESS>
†Cornell SC Johnson College of Business<EMAIL_ADDRESS>

Abstract

Machine learning algorithms are increasingly employed to price or value homes
for sale, properties for rent, rides for hire, and various other goods and
services. Machine learning-based prices are typically generated by complex
algorithms trained on historical sales data. However, displaying these prices
to consumers anchors the realized sales prices, which will in turn become
training samples for future iterations of the algorithms. The economic
implications of this machine learning “feedback loop” — an indirect human-
algorithm interaction — remain relatively unexplored. In this work, we develop
an analytical model of machine learning feedback loops in the context of the
housing market. We show that feedback loops lead machine learning algorithms
to become overconfident in their own accuracy (by underestimating its error),
and leads home sellers to over-rely on possibly erroneous algorithmic prices.
As a consequence at the feedback loop equilibrium, sale prices can become
entirely erratic (relative to true consumer preferences in absence of ML price
interference). We then identify conditions (choice of ML models, seller
characteristics and market characteristics) where the economic payoffs for
home sellers at the feedback loop equilibrium is worse off than no machine
learning. We also empirically validate primitive building blocks of our
analytical model using housing market data from Zillow. We conclude by
prescribing algorithmic corrective strategies to mitigate the effects of
machine learning feedback loops, discuss the incentives for platforms to adopt
these strategies, and discuss the role of policymakers in regulating the same.
Algorithmic Price, Economics of AI, Bias-Variance, Housing Market, Zillow,
Zestimate.

##  1 Introduction

Machine learning-based algorithmic pricing (“ML pricing” henceforth) is
increasingly used to facilitate transactions in markets for housing
(zillow2022zestimate; realtor2022realestimate), property rentals
(redfin2022rent; airbnb2022smart), peer-to-peer loans
(lendingclub2022modelrank), and fine art (liveart2022estimate), among others
(pandey2021disparate). By providing consumers accurate and on-demand estimates
of product values without a labor-intensive appraisal process, ML prices111ML
pricing could dictate (such as for ride-shares on Uber, for example), suggest
(such as for rents on Airbnb, for example), or simply display a price to
consumers. On Zillow, for example, the machine learning-based “Zestimate” is
simply displayed as an estimate of the current value of a home. We will
formally define how value of a home relates to transaction or sale price if
homeowner was in market to make a sale. efficiently reduce pricing uncertainty
and friction for all buyers and sellers (forbes2021price) and democratize
access to information for those who lack pricing experience (huang2021seller;
kehoe2018dynamic). For example, a rideshare driver and rider avoid the
friction of negotiating the price for every trip because the price is set by
an algorithm. An investor with optimism about the art market but no artistic
expertise can purchase art pieces for an ML price, benefiting both the
investor and the artist (Bailey 2020).

ML prices are typically generated by algorithms that capture high-dimensional
product characteristics (bertini2021pitfalls) and dynamically adapt to
evolving market conditions (brown2021competition). These algorithms are
trained (and periodically re-trained) to maximize the accuracy of estimated
prices by uncovering patterns in historical realized prices in the market.

ML prices are widely believed to anchor realized sales prices
(era2019zestimate; oxford2021avm; agentfire2022zestimate), which are in turn
used to train future iterations of the ML pricing algorithms. This creates a
feedback loop between the ML pricing algorithm and its own training data; the
figure above illustrates a feedback loop in the context of the housing market
and Zillow’s ML price. Such feedback loops have been reported to limit the
ability of machine learning algorithms to learn from their errors, among other
undesirable outcomes (chaney2018algorithmic; jiang2019degenerate). However,
the economic effects of ML pricing on consumers in the presence of such
feedback loops remain unexplored.

In this work, we analytically characterize how markets are affected by machine
learning-based pricing algorithms when the algorithms influence and learn from
consumer behaviors in a feedback loop. The key novelty is how we model the
interplay between the algorithm’s self-reported confidence (displayed as a
confidence interval, for example) and the consumers reliance (sensitivity of
consumers’ beliefs or actions to the ML price) in the algorithm. We show that
these reinforce each other other due to the feedback loop resulting in over-
confidence and over-reliance. Extensive prior research has modeled the
dynamics of consumer behavior (under a static algorithm) or of algorithms
(under static consumer behavior) separately, we model and identify a joint
equilibrium.

Our analytical model is grounded in institutional details of one specific
context: the housing market and the ML prices (called Zestimates) on Zillow,
which is currently the dominant platform for listing and discovering homes for
sale. It consists of two components. In the first component, we model how
sellers’222We explicitly model sellers’ belief construction, choice and
payoffs. We only model buyers as a crowd. We qualitatively argue that formal
results for sellers’ extend to buyers as well. beliefs drive realized sales
prices and their economic payoffs. In the second component, we model how the
algorithm learns from historical sales. These components are fused with
sellers’ beliefs dependent on ML prices and algorithm trained on realized sale
prices. It is important to note that any significant mistakes in the ML price
of a home is propagated to sale prices but gets slowly attenuated over the
feedback loop because seller also rely on external signals. Further, any
temporary price mistakes are uncorrelated across homes, this there are no
price bubbles. The innovation in this paper is to reveal the reinforcing
feedback between aggregate quantities (instead of individual prices) - how
seller reliance on ML prices increases with algorithms’ reported confidence
and the confidence calculation improves with reliance. At equilibrium —
attained after the algorithm and the market evolve simultaneously until
convergence — we characterize sellers’ private valuations, and economic
payoffs, sales prices, the algorithm’s accuracy and self-reported confidence,
and sellers’ reliance on the algorithm.

Our model reveals three key findings. First, presence of ML prices can
increase the deviation (in both positive and negative directions) of realized
sales prices from the “true” home value (grounded in true underlying consumer
preferences for home features without interference from ML pricing). Second,
we show conditions where this adversely effect the economic payoffs for
sellers. If left unchecked, this deviation amplifies until realized sales
prices and ML prices are entirely random (and uncorrelated with the “true”
home value) at equilibrium. At this equilibrium : (i) the buyers and sellers
fully rely on the ML price, (ii) the ML price are identical to the eventual
realized sales price (akin to a self fulfilling prophecy), (iii) the
algorithm’s self-reported confidence is maximal, and (iv) sellers’ may be
worse off than in the absence of ML algorithm. These findings are counter to
conventional wisdom that ML prices (by crunching large amounts of revealed
preference data) are useful in inferring underlying preferences. Third, we
identify seller characteristics (impatience or cost, risk aversion, ability to
price home in absence of ML prices) where this equilibrium and its adverse
economic implications are worse. We also discuss role of the exogenous factors
such as level of ML adoption, the ML algorithm capacity333A high capacity
algorithm has access to more training sample and trainable parameters to
better fit data patterns. For example, a deep neural network with thousands of
parameters and millions of training samples has a higher capacity than a
degree-2 polynomial. and other market characteristics.

In our model, buyers and sellers’ do not correct over-reliance and the
platform does not correct algorithms’ over-confidence. Buyers and sellers do
not correct because they trust the algorithms’ confidence
calculation444Platforms’ like Zillow broadly publicize
zillow2020zestimateaccuracy that their algorithms’ confidence calculation is
tied to typical data science practice of measuring out of sample errors. Our
analytical formulation is tied to this definition. In the absence of the
feedback loop this would in fact be the correct way to measure confidence. Our
findings are moderated but not eliminated if buyers and sellers are fully
rational about the presence of the feedback loop phenomenon and (correctly)
calibrate their reliance of ML prices. The overconfidence of ML algorithm and
increased deviation of realized sales prices from the “true” home value are
not eliminated. We do not model platform as a strategic agent to correct the
over-confidence or benefit from it. We identify various strategies that
platforms could employ to correct the over-confidence. But all strategies
effectively limit the visibility of ML prices to buyers and sellers. Our model
enables analyzing the trade-off between direct positive impact of making ML
prices visible to a single home and indirect negative impact of pervasive
influence of ML prices (via adverse feedback loop). We qualitatively discuss
why these corrective strategies may not be in line with platforms typical
revenue streams from ad sales and iBuying.

Our model is built on two primitive assumptions: (i) that buyers and sellers
indeed rely on the ML price, and (ii) that the machine learning algorithm is
periodically re-trained with data from recent sales555The feedback loop would
be absent if the pricing algorithm were driven instead by rules coded by
domain experts. The feedback loop would be too slow to have practical
implications if the ML price were reliant on older sales and thus relatively
static.. We provide empirical evidence to support these assumptions using
real-world housing market transactions from Zillow. To support the first
assumption, we collect data from Zillow (Appendix A.1) and use updates to the
Zestimate algorithm by Zillow as an instrument to quantify the reliance on
Zestimate (Appendix A.2). Reliance is defined as the sensitivity of the sale
price (or sellers’ list price) to change in Zestimate visible to buyers and
sellers. We find an average reliance of 15%; for example, if the Zestimate
visible to a home was increased by $10,000 the expected sale price would
increase by $1,500 (+15% $\times$ $10,000). We further find that the reliance
on the Zestimate varies with the width of the displayed Zestimate range (a
confidence interval which quantifies the algorithm’s self-reported
confidence). To support the second assumption, we measure empirical
correlations between changes in the Zestimate for a home and new sales in that
home’s neighborhood (Appendix A.3). We find that if the sale price of a home
were $10,000 higher, the Zestimate of (approximately) 25 peer homes (that are
similar in characteristics) increases by $2,000 or more. Thus, empirical
evidence supports the analytical model assumptions666It does not confirm the
mechanism or findings of our model..

A key intermediate finding of our model is the over-confidence of machine
learning-based pricing algorithms in their ML prices. This over-confidence is
also evident anecdotally777It does not confirm our model but simply says the
algorithm over-confidence (a finding of our model) is empirically plausible.,
as illustrated in Figure 1. Figure 1 (bottom) shows that the Zestimate is
increasingly more accurate over time, likely driven by continuous improvements
to the underlying algorithm by Zillow. However, the algorithm’s confidence (as
measured by the width of the Zestimate range or confidence interval) does not
show a similar trend. In fact, the algorithm’s confidence has a discontinuous
reduction in June 2021, when Zillow announced a major update to the algorithm.
A plausible explanation for this discontinuous reduction is that the reported
Zestimates were overly-confident before being fixed in June 2021. Until the
eventual fix by Zillow, the overly-confident algorithm was active in
production, without any oversight to limit its potentially adverse effects.

Figure 1: Zillow announced a major update to its Zestimate algorithm on June
15, 2021 (red line). (Top Left) Zillow’s self-reported confidence in its
Zestimate (as quantified by the width of the displayed Zestimate range)
increased discontinuously from 22.69% (3 months before June 2021) to 24.66% (3
months after June 2021). (Bottom Left) In the same time period (3 months
before and after June 2021), empirical average error in the Zestimate
(relative to the realized sales price) decreased from 4.36% to 3.41%, and the
95% error in the Zestimate (relative to the realized sales price) decreased
from 23.41% to 16.37%. (Right) A snapshot of Zestimate range of 14% ($384,000
$\pm$ 7%) for a home on Zillow.

ML prices that both influences and learn from consumer behaviors are
increasingly being used to democratize access to information in a variety of
markets (zillow2022zestimate; realtor2022realestimate; forbes2021price;
airbnb2022smart; lendingclub2022modelrank; liveart2022estimate). To the best
of our knowledge, our work is the first to jointly model the interdependent
dynamics of machine learning based-algorithms and consumer behavior. As such,
we contribute to the literature on machine learning feedback loops
(bottou2013counterfactual; perdomo2020performative; wager2014feedback;
sinha2016deconvolving), which does not consider consumer behavior and economic
outcomes, and to the literature on dynamic consumer behavior, which does not
consider the dynamics of machine learning algorithms. This paper is one of the
first (i) to empirically measure the impact of ML pricing on home sale prices
(in the context of Zillow and the Zestimate) and (ii) to begin to disentangle
Zillow’s proprietary Zestimate algorithm. These two smaller contributions
should spark more academic research on and policymaker scrutiny of the
dominant role of Zillow’s ML pricing algorithm in the housing market.

More broadly, our work showcases the perils of deploying algorithms that are
myopically optimized for statistical objectives such as accuracy, at the
expense of long run economic objectives. Such algorithms, as a consequence of
feedback loops, could reinforce non-diverse and self-fulfilling preferences
(such as recommending fashion content to women and sports content to men, or
recommending higher interest rates for loan applications from historically
under-served groups due to higher predicted risks). If left unchecked, in the
long run, these reinforced preferences could fully diverge from the “ground
truth” i.e., preferences in absence of algorithmic interference. We unpack how
this divergence from the “ground truth” can make consumers worse off. As such,
our work suggests that policymakers monitor and regulate ML pricing. Our work
also suggests that intermediary platforms consider the long-term implications
of their deployed pricing algorithms when subject to feedback loops. The
eventual divergence from ground-truth predicted by our analytical model poses
risks to platforms’ reputation and brand perception, which partly depends on
the accuracy of their deployed pricing algorithms.

##  2 Related Literature

A stream of literature has looked at pricing in the housing market before
introduction of ML. linneman1986empirical showed a large variance buyers and
sellers home value estimates, which is unsurprising given that most buyers and
sellers transact infrequently (say, once in 10 years), and, unlike some other
assets, houses have a large diversity of features. Field surveys and
experimental research tried to estimate valuation errors before the advent of
pricing algorithms; error estimates include 14% (goodman1992accuracy), 5.3%
(kiel1999accuracy), and 16% (ihlanfeldt1986alternative). One would expect that
agents, brokers, and other market experts could correct valuation errors
(han2015microstructure), but sellers and expert agents’ contract under
information asymmetry. The seller’s inability to observe his agent’s efforts
creates a moral hazard in the principal (seller) – agent contract
anglin1991residential. When searching (for buyers) is costly, the agent has an
incentive to undervalue the home and save on search costs; when the market is
competitive, the agent has an incentive to overvalue the home to outbid
competing agents. Further, the adverse selection problem prevents the seller
from accurately judging whether the agent is knowledgeable about the state of
the market. The challenges faced by buyers and sellers in pricing homes is
further supported by WakeField research survey (Melcher 2021) which finds that
the average US home-buyer tours 15 homes and makes offers on 10 homes,
spending a cumulative $845 million of work time on home search and pricing.
85% of first-time buyers say that it is challenging to make offers and
stressful to be rejected or outbid.

In this context it is not surprising that ML prices have influence.
Participants are likely to find ML pricing attractive because (i) it is free
and thus highly accessible, unlike an appraiser; (ii) it seems impartial,
unlike agents; and (iii) the platforms (e.g., Zillow and RedFin) report low
error rates between ML price and eventual sale price to convey accuracy. Some
experts have been highly critical of Zillow’s Zestimate (or Redfins’ Estimate)
for being inaccurate, using outdated data, missing local non digitized
information and even altogether unusable (houwzer2020zillow). Zillow’s
received customer ratings of 3.8, 2.8 and 1.6 on consumeraffairs.com,
sitejabber.com and trustpilot.com. Given these concerns we empirically
validate that buyers and sellers are in-fact sensitive to changes in
Zestimate.

Another stream of literature has looked at feedback loops in a wide range of
online learning settings where an ML algorithm learns by making mistakes
(barocas2017fairness). Such feedback designs are innocuous in settings where
the ML predictions do not contaminate the ground truth label, but elsewhere,
the feedback design can slowly accrue a technical debt (sculley2015hidden)
that eventually has profound effects (amodei2016concrete).
perdomo2020performative identify conditions under which a feedback loop will
converge to a stable point. Our analytical framework roughly concurs, but we
are less concerned with statistical properties and more concerned with the
payoffs at the equilibrium. We discern how much of the covariate distribution
shift comes from the evolution of intrinsic housing preferences versus from
the ML feedback itself. If the latter dominates, the social surplus may be
lost even as the ML algorithm reaches optimal accuracy.

ML feedback loops have been documented in settings such as - ad placement
(bottou2013counterfactual), search engine rankings (wager2014feedback) and
recommender systems (chaney2018algorithmic; sinha2016deconvolving). Our model
of sellers has similarities with the model of (schmit2018human), who assume
that users are naïve in believing that ML recommendations are unbiased, and
users are myopic and honest about their current action without regard to its
impact on future states via the feedback loop. Importantly, in all these
examples, individuals interact with ML predictions without an outside option.
For example, a user who is seeking relevant search results does not have an
alternative mechanism besides the ML algorithm; she cannot realistically
achieve the same outcome by interacting with a crowd of her peers. In the
housing market, however, buyers and sellers can interact to determine prices
in the absence of ML pricing. This interaction may happen indirectly, for
example when a seller observes lack of visits or offers from buyers. Thus, the
introduction of ML pricing to the housing market is unique in that it replaces
the wisdom of the crowd with a single correlated signal. To our knowledge, we
are one of the first to evaluate the impacts of ML feedback loops on a market
as a whole.

The business literature has identified consequences of ML algorithmic pricing
besides the feedback loop. For example, bias propagated by the algorithmic
pricing of hotels, car insurance, loans (israeli2020algorithmic), and ride-
hailing (pandey2021disparate). (bertini2021pitfalls) find that customers
misperceive the motives of firms that offer algorithmic pricing.
(assad2020algorithmic) and (brown2021competition) study whether algorithms
provide competitive advantages or lead to collusive outcomes.
(huang2021seller) identifies settings in which algorithmic pricing may
increase market friction. (yu2020algorithmic) argue that algorithmic price
might mitigate racial disparities in the housing market. The present paper is
unique in this literature because it does not model ML pricing or its
statistical properties as static. Instead, we model ML algorithm’s learning in
conjunction with the evolving market as both move toward equilibrium.

##  3 Model

In this section, we model the interdependent evolution of consumer behavior
and a machine learning-based pricing algorithm in the context of a housing
market. We decompose our model into two components. In the first component, we
model how a seller determines the price at which to list their home, buyer
offers, the realized sale prices and we define true value of a home (Section
3.1). In the second component, we model how a machine learning-based pricing
algorithm estimates home values (Section 3.2). In Section 4, we allow both
model components to evolve simultaneously and interdependently, and
characterize the equilibrium of the feedback loop between the market and the
pricing algorithm.

###  3.1 First Component: Market Participants

Modeling home-buyers. Consider a home for sale on the market in time period
$\tau$. We focus on modeling an “exemplar buyer” of this home: the buyer who
has the highest willingness to offer among all buyers interested in this home.
Let the willingness to offer of the exemplar buyer be a random variable
$y_{\tau}$ drawn from a distribution with mean $\mu$ (assume to be unique for
each home) and variance $\sigma_{b}^{2}$ (assumed to be identical across
homes), where the variance captures the heterogeneity in buyers’ preferences.
Since the seller (the current homeowner) only entertains the highest offer in
each time period, it is sufficient to model the exemplar buyer instead of all
prospective buyers.

Let the home be listed at price $l_{\tau}$. The exemplar buyer will then offer
$\min(y_{\tau},l_{\tau})$, without revealing their willingness to offer
$y_{\tau}$888We assume that “highest and best offer” negotiations are absent..
The home sells to the exemplar buyer at the end of time period $\tau$ for
$\min(y_{\tau},l_{\tau})=l_{\tau}$ if $y_{\tau}\geq l_{\tau}$, and remains
unsold otherwise. We denote by $P(\textrm{sale}|l_{\tau})=P(y_{\tau}\geq
l_{\tau};\mu,\sigma_{b}^{2})$ the probability that the home sells in time
period $\tau$. While individual buyers can update their valuations during
their home search, their aggregate offer distribution $(\mu,\sigma_{b}^{2})$
is assumed to be static999Modeling buyers’ aggregate offer distribution as
declining over time on market $\tau$ does not change any of our conclusions..

Figure 2: (Left) The home sale process as a single-period game with the seller
as the focal agent. The game repeats until the home is sold or the seller
decides to exit the market. (Right) A hypothetical distribution of the list
price, best offer distribution, and probability of sale.

Modeling home-sellers. Our model of a home-seller is illustrated in Figure 2.
At the start of each time period $\tau$, the seller decides whether to list
her home for sale, and determines the list price $l_{\tau}$. At the end of the
time period, the home (if listed) sells at the list price $l_{\tau}$ with
probability $P(\textrm{sale}|l_{\tau})$, based on our model of home-buyers. If
the home does not sell at the list price, the seller returns to the decision
of whether to continue listing her home for sale101010In Appendix
LABEL:app:bargaining, we discuss why we do not model the seller’s choice to
accept an offer below the list price.. In each time period, the seller earns a
flow payoff (regardless of whether or not her home sells) from retaining
ownership of the home (such as rental income $r$) less market participation
costs $c$ (such as maintaining the home for open-houses). If the seller
decides to exit the the market, she receives a lifetime payoff $x$ from
retaining ownership of the home (such as via rental income $r$ at discount
$\gamma$). We treat this as a terminal state: the seller cannot change her
decision to exit the market.

As an example, consider a seller who stays in the market for time periods
$1,\dots,T$ and lists her home at a sequence of prices
${l_{1},...,l_{\tau},...,l_{T}}$, until her home sells in period $T$. Her
payoff $\pi$ and the outside option value $x$ (had she not entered the market
at all) are given by111111Note that $\gamma\approx 1$ if each period is 1
month long; a 3% yearly interest rate (or 0.97 yearly discounting) is
equivalent to 0.997 monthly discounting. Since homes are typically listed for
a few months before selling, $\gamma^{T}$ is also close to 1. We will use
these approximations to simplify our calculations.,

$\displaystyle\pi=\frac{1-\gamma^{T}}{1-\gamma}\times(r-c)+\gamma^{T}\times
l_{T}\text{ \quad; \quad}x=\frac{r}{1-\gamma}$ (1)

The seller enters the market if her estimate of the expected payoff
$\tilde{\mathbb{E}}[\pi]$ is greater then the outside option value $x$. The
expectation $\tilde{\mathbb{E}}[\pi]$ is over all possible realizations of
sale prices, on-market durations, and whether or not the home sells. The
seller lists her home at the optimal price $l_{\tau}^{*}$ at the beginning of
every period $\tau$ to maximize her expected payoff
$\tilde{\mathbb{E}}[\pi|l_{\tau}]$ from period $\tau$ onwards.

We use the notation $\tilde{\mathbb{E}}[\cdot]$ to distinguish between the
seller’s estimate of the expected payoff and the actual expected payoff
$\mathbb{E}[\cdot]$. The seller’s estimate of the expected payoff depends on
her estimate of the sale probability
$\tilde{\mathbb{P}}(\textrm{sale}|l_{\tau})$, which in turn depends on her
estimate of the offer distribution parameter $\tilde{\mu}_{\tau}$ in period
$\tau$. The seller also has some uncertainty (variance) about her estimate,
which we denote by $\sigma^{2}_{s,\tau}$. The seller’s uncertainty and the
estimation errors arising from $\tilde{\mu}_{\tau}\neq\mu$,
$\tilde{\mathbb{P}}(\textrm{sale}|l_{\tau})\neq\mathbb{P}(\textrm{sale}|l_{\tau})$,
and $\tilde{E}[\pi|l_{\tau}]\neq E[\pi|l_{\tau}]$) will lower payoffs, which
platforms like Zillow attempt to alleviate with their machine learning-based
pricing algorithms.

We denote the seller’s initial estimates as $\tilde{\mu}=\tilde{\mu}_{\tau=0}$
and $\sigma_{s}^{2}=\sigma^{2}_{s,\tau=0}$. In each subsequent time period
$\tau$, the seller learns and updates her initial estimates and uncertainty
($\tilde{\mu}_{\tau},\sigma^{2}_{s,\tau}$). We model these updates as a
Bayesian learning process in which the seller combines her current estimate
$(\tilde{\mu}_{\tau},\sigma^{2}_{s,\tau})$ with the observable market signal
$(\mu,\sigma^{2}_{\textrm{signal}})$ to develop an updated estimate
$(\tilde{\mu}_{\tau+1},\sigma^{2}_{s,\tau+1})$. The market signal is observed
when the seller interacts with prospective buyers, some of whom implicitly of
explicitly reveal the price they are willing to offer for the seller’s home.
We assume that the market signal is noisy, but unbiased with respect to the
buyer’s offer distribution parameter $\mu$. With time on the market, we assume
that the seller’s estimate of $\mu$ improves and her uncertainty reduces:
$\tilde{\mu}_{\tau}\rightarrow\mu,\sigma^{2}_{s,\tau}\rightarrow 0$ as
$\tau\rightarrow\infty$.

Home values. Intuitively, a home’s value $v$ is its market clearing price.
This is easy to formalize for, say, a single IBM stock, with millions of
identical units transacted among thousands of buyers and sellers every day. In
contrast, a single home is bought and sold only a few times in many years.
Hence, we consider a thought experiment where the same home is sold thousands
of times. The value of the home $v$ in this thought experiment is its sale
price $p$ (which depends on the buyer offer distribution and the seller’s
estimate of this parameter) averaged over all the times it is sold:

$\displaystyle
v=\mathbb{E}[\mathbb{E}[p|\mu,\tilde{\mu},\sigma_{b},\sigma_{s},\sigma_{signal}]]$
(2)

Next we will consider machine learning based estimation of this home value.
Note that the machine learning algorithm does not model the market structure
(offers $(\mu,\sigma_{b})$, beliefs $(\tilde{\mu},\sigma_{s}$, and signals
$\sigma_{signal}$).

###  3.2 Second Component: Machine Learning-Based Pricing Algorithm

We assume that the home value $v_{k,t}=G_{t}(\mathbf{X}_{k})$ of a home $k$ at
time $t$ is a time-evolving function of the home’s characteristics
$\mathbf{X}_{k}$. This can be interpreted as home buyers’ preferences for the
home’s characteristics evolving over time, while the home’s characteristics
remain unchanged. We further assume that the resulting home value evolves as a
random walk:

$\displaystyle v_{k,t}=v_{k,t-1}+e^{\textrm{rw}}_{k,t},\qquad
v_{k,t}=G_{t}(\mathbf{X}_{k}),~{}e^{\textrm{rw}}_{k,t}\sim
N(0,\sigma^{2}_{\textrm{rw}})$ (3)

In this section, we propose a general machine learning framework for home
value estimation. For analytical simplification we assume that the machine
learning model parameters $\theta_{t}$ are retrained every period $t$ to
approximate the underlying true home preferences
$G_{t}(\mathbf{X}_{k})$121212This simplifies the analytical expressions
without changing the results.. Finally we derive analytical expressions for
the machine learning-based prices as a function of observed sale prices.

General machine learning framework. Our framework assumes that the machine
learning model of home values is trained to (i) accurately estimate the value
of each home in the training data, and (ii) estimate similar home values for
homes with similar features. Satisfying objective (i) increases in-sample
accuracy, but decreases out-of-sample accuracy due to over-fitting.
Essentially, the inclusion of objective (ii) regularizes the model to increase
its generalization power. Our framework is a generalization of the network
lasso framework (hallac2015network) to non-convex optimization objectives.

Formally, let $\mathcal{H}_{t}$ be a set of $N$ homes used to train the
machine learning model at time $t$131313The machine learning training period
duration $t$ (say 1 year) is typically different than the home listing period
duration $\tau$ (say 1 week or 1 month). We also skip the details of hold out
sample for validation within the training data.. For each home
$k\in\mathcal{H}_{t}$, let $v_{k,t}$ be the true value of the home at time
$t$, and $\mathbf{X}_{k}\in\mathbb{R}^{\bar{Q}}$ be a $\bar{Q}$-dimensional
vector of the home’s characteristics (such as its number of bedrooms and year
of construction). We assume that the machine learning model parameters
$\theta_{t}$ are given by:

$\displaystyle\theta_{t}=\textrm{argmin}_{\theta}~{}\left[~{}\underbrace{\sum_{k\in\mathcal{H}_{t}}f_{k}(\mathbf{X}_{k}|\theta)}_{\textrm{objective
(i)}}\quad+\quad\underbrace{\sum_{m,n\in\mathcal{H}_{t}}g_{mn}(\mathbf{X}_{m},\mathbf{X}_{n}|\theta)}_{\textrm{objective
(ii)}}~{}\right]~{}$ (4)

Here, $f_{k}(\mathbf{X_{k}}|\theta)$ is a loss function capturing the first
objective of accurate in-sample estimation for each home $k\in\mathcal{H}_{t}$
in the training data, and $g_{mn}(\mathbf{X_{m}},\mathbf{X_{n}}|\theta)$ is a
loss function capturing the second objective of making similar estimations for
similar homes. Note that we make no assumptions about the functional form of
either loss function, or about the architecture of the machine learning model.
As such, our framework generalizes a variety of common machine learning
models, including deep neural networks.

Example. As a specific example of our framework, let
$\hat{v}_{k}=F(\mathbf{X}_{k},\Theta_{k})$ be the estimated price of each home
$k$ derived using a neural network $F(\cdot)$ with weights $\Theta_{k}$. While
the weights are different for each home $k$, a single neural network
$F(\cdot)$ is trained to learn the weights $\Theta_{1},\dots,\Theta_{N}$ of
all $N$ homes in the training data $\mathcal{H}_{\tau}$; these weights can be
viewed as homes’ embeddings. To maximize in-sample accuracy, we minimize the
mean squared error by setting
$f_{k}(\mathbf{X_{k}}|\theta)=[\hat{v}_{k}-v_{k,t}]^{2}=[F(\mathbf{X}_{k},\Theta_{k})-v_{k,t}]^{2}$.
To improve out-of-sample accuracy, we penalize differences between the learned
weights for pairs of similar homes. Specifically, we set
$g_{mn}(\mathbf{X}_{m},\mathbf{X}_{n}|\theta)=\lambda\|\mathbf{X}_{m}-\mathbf{X}_{n}\|_{2}\|\Theta_{m}-\Theta_{n}\|_{2}$,
where $\|\cdot\|$ is the Euclidean distance or $L_{2}$ norm, and $\lambda$ is
a regularization hyperparameter. Note that the parameters
$\theta=\\{F(\cdot),\lambda\\}\cup\\{\Theta_{k}\\}_{k\in\mathcal{H}_{t}}$
include the neural network architecture $F(\cdot)$, the regularization
hyperparameter $\lambda$, and the weights $\Theta_{k}$ for each home
$k\in\mathcal{H}_{t}$.

Regularization and home clustering. Solving for $\theta_{\tau}$ in Eq. 4
essentially performs joint home value estimation and home clustering, where a
pair of homes $i,j$ with similar learned weights $\Theta_{i},\Theta_{j}$ can
be assigned to the same cluster by discretizing the weight values (by rounding
them to the nearest integer, for example). After discretization, $Q$ homes in
a cluster $\mathcal{C}_{i}$ for $i=1,\dots,Q$ have the same learned weights
$\Theta_{k}$ and similar characteristics $\mathbf{X}_{k}$ (and hence, similar
estimated home values). The regularization parameter $\lambda$ controls the
clustering granularity: $\lambda=0$ permits each home to form its own cluster,
while $\lambda\rightarrow\infty$ leads to a single cluster containing all
homes in the training data. We denote by $Q$ the number of clusters, and
assume (for analytical simplification) that each cluster contains exactly
$N/Q$ homes.

Generating algorithmic prices. The clustering of homes facilitates making out-
of-sample predictions. Let $\bar{\mathbf{X}}_{\mathcal{C}}$ denote the
centroid of cluster $\mathcal{C}$ (the average of $\mathbf{X}_{k}$ over all
homes $k\in\mathcal{C}$), and let $\bar{v}_{\mathcal{C},t}$ be the average
value of all homes $k\in\mathcal{C}$ at time $t$. We assign each out-of-sample
home $q$ with characteristics $\mathbf{X}_{q}$ to its nearest cluster
$\mathcal{C}(q)\in\\{\mathcal{C}_{1},\dots,\mathcal{C}_{J}\\}$ with the
smallest Euclidean distance
$\|\mathbf{X}_{q}-\bar{\mathbf{X}}_{\mathcal{C}}\|_{2}$, and could
subsequently use $\bar{v}_{\mathcal{C}(q),t}$ as its predicted home value.
However, true home values are unobservable in practice. Hence, a common proxy
for home values is their sales price, which are observable in the past. Using
this proxy plays an important role in the machine learning errors that we
discuss later in this section. Let $p_{k,t}$ be the sale price of home $k$ at
time $t$, and let $\bar{p}_{\mathcal{C},t}$ be the average sale price of all
homes in cluster $\mathcal{C}$. Then the predicted price of an out-of-sample
home $q$ assigned to cluster $\mathcal{C}(q)$ (as described above) at time
$t+1$ is given by:

$\displaystyle
z_{q,t+1}=\bar{p}_{\mathcal{C}(q),t}=\frac{Q}{N}\times\sum_{k\in\mathcal{C}(q)}p_{k,t}$
(5)

Table 1: Assumptions about the distributions in our simple and full models. |  | Simple Model
---
| Full Model
---
Buyer Offer Distribution $P(y_{\tau})$ | $U[\mu-\sigma,\mu+\sigma]$ | $N(\mu,\sigma_{b}^{2})$
Seller Estimate $P(\tilde{\mu})$ | $P(\tilde{\mu}=\mu-2\sigma)=P(\tilde{\mu}=\mu)=P(\tilde{\mu}=\mu+2\sigma)=1/3$ | $N(\mu,\sigma_{s}^{2})$
Seller Learning $\sigma_{\textrm{signal}}^{2}$ | 0 | $\kappa\sigma_{b}^{2}$
Table 2: Table of notation and definitions

Exogenous Variables

Notation | Description
---|---
$v_{t}$ | True value (market clearing price) of home at time $t$
$\sigma_{rw}^{2}=Var_{t}[v_{t+1}-v_{t}]$ | Variance of true value changes of a home over time.
$\sigma_{v}^{2}=Var_{k}[v_{t}]$ | Variance of true value across all homes in one period.
$\sigma_{e}^{2}=Var_{i}[\tilde{v}_{i,t}-v_{t}]$ | Variance of participant $i$ valuations (before introducing ML)
$N$ | Total number of homes sold in one period.

Key Endogenous Variables

Notation | Description
---|---
${l_{\tau}}_{t}$ | Sequence of list prices set by seller for home at time $t$
$p_{t}$ | Realized sale price
$\tilde{v}_{i,t}$ | Participant $i$ valuation of home at time $t$
$z_{t}$ | ML (Machine Learning) price of home at time $t$
$Q$ | ML model hyperparameter controlling number of clusters

Outcomes Variables (as function of reliance $\alpha$)

Notation | Description
---|---
$\alpha(\hat{\sigma}^{2}_{z})$ | Participant reliance on ML price
$\sigma^{2}_{z}(\alpha)$ | True ML price error ($\sigma^{2}_{z}(\alpha=0)$ denoted by $\sigma^{2}_{z}$)
$\hat{\sigma}^{2}_{z}(\alpha)$ | Estimated ML price error ($\hat{\sigma}^{2}_{z}(\alpha=0)$ denoted by $\hat{\sigma}^{2}_{z}$)
$\pi,\Pi(\alpha)$ | Realized payoff and expected risk averse payoff

##  4 Results

In Section 3.1, we described our model of buyer-seller interactions grounded
in the structure of the housing market — the offer distribution parameters
($\mu,\sigma_{b}^{2}$), the seller’s estimate of the offer distribution mean
$\tilde{\mu}$, her uncertainty $\sigma^{2}_{s}$, the seller’s learning aided
by a market signal $(\mu,\sigma^{2}_{\textrm{signal}})$, and the seller’s
outside option value $x$ and market participation costs $c$. We further
formalized the sellers’ optimal listing price choice $l_{\tau}$ in terms of
these parameters. However, a signal such as the ML price $z$ does not convey
information about the offer distribution $\mu$ or the list price $l_{\tau}$,
nor is it tailored to an individual sellers characteristics. Instead, the ML
price $z$ is an estimate of a home’s value $v$ (equation 2) summarized over
these market structure details and individual heterogeneity.

In section 4.1, we derive the home value $v$ and distribution of sale prices
($E[p],Var[p]$). We consider two sets of assumptions (listed in Table 1) about
the distributions of offers $y_{\tau}$, the seller’s estimate
$\tilde{\mu}_{\tau}$, and the market signal $\sigma_{\textrm{signal}}^{2}$. In
our results described in Section 4, we use the “simple” model for analytical
closed-form solutions. In Appendix B.2, we employ numerical simulations under
the “full” model to verify that our findings are consistent with the “simple”
model. In section 4.2, we will derive machine learning pricing errors (true
$\sigma_{z}^{2}=Var[v-z]$ and estimate $\hat{\sigma}_{z}^{2}=Var[p-z]$). In
section 4.3, we will identify equilibrium of the feedback loop between the
machine learning pricing error estimate $\hat{\sigma}_{z}^{2}(\alpha)$ and
reliance on machine learning prices $\alpha(\hat{\sigma}_{z}^{2})$ which
depend on each other. Finally, in section 4.4, we will use these equilibrium
expressions to formulate payoffs for sellers at the equilibrium. Table 2 and 3
summarize the notation and key results.

###  4.1 Home Value and Sale Prices

Consider an oracle who knows a home’s true offer distribution
($\mu,\sigma_{b}^{2}$), and knows that the sellers have potentially erroneous
estimates. The oracle could calculate the home’s value $v(\mu)$ by integrating
over the sellers’ estimates of the buyer offer distribution $\tilde{\mu}$, and
over the stochasticity in buyers’ offers (embedded in
$\mathbb{E}[p|\mu,\tilde{\mu}]$), as follows:

$\displaystyle
v(\mu)=\mathbb{E}[\mathbb{E}[p|\mu,\tilde{\mu}]]=\int_{\tilde{\mu}}P(\tilde{\mu}|\mu,\sigma)\times\mathbb{E}[p|\mu,\tilde{\mu}]$
(6)

Similarly, we can formalize the seller’s estimate $\tilde{v}$ of her home’s
value in time period $\tau$ (where we drop the subscript $\tau$). The seller
does not know her home’s true offer distribution ($\mu,\sigma_{b}^{2}$),
However, the seller knows that her estimate $\tilde{\mu}$ is drawn from a
distribution with mean $\mu$ and variance $\sigma^{2}_{s}$. The seller’s
valuation of her home $\tilde{v}$ is then given by
$\tilde{v}=\tilde{\mathbb{E}}[v(\mu)]$, where the expectation is over all
possible estimates of $\mu$ by the seller. Now consider an external signal
from an ML price $z$ that also claims to estimate the home’s value:
$z=\hat{E}[p]=\hat{v}$, where we use $\hat{E}[\cdot]$ to differentiate between
the algorithm’s estimate and the seller’s estimate (denoted by
$\tilde{\mathbb{E}}[\cdot]$). After observing this signal, the seller updates
their valuation $\tilde{v}$ by combining their prior valuation (before
observing and independent of the ML price) and the ML price $z$ as follows:

$\displaystyle\tilde{v}=(1-\alpha)\times\tilde{\mathbb{E}}[v(\mu)]+\alpha\times
z\quad;\quad\textrm{Var}[\tilde{v}]:=(1-\alpha)^{2}\textrm{Var}[\tilde{v}]$
(7)

This formulation for impact of ML price $z$ condenses complex details on how
sellers absorb the ML price $z$. The seller (jointly with their agent)
receives informative signals from a lot of sources (agent, appraisers,
neighbors and market experts) all embedded into their private valuation
$\tilde{\mathbb{E}}[v]$. The ML price $z$ is yet another informative signal,
specially treated in our model because we want to isolate its impact, to
construct a final valuation $\tilde{v}$. The seller uses the valuation
($\tilde{v}$ e.g., $520k) to infer likely offers ($\tilde{\mu}\pm\sigma_{b}$
e.g., $500k to $550k) and a corresponding good list price ($l^{*}$ e.g.,
$540k). A similar influence occurs for buyers. Individual buyers (jointly with
any buyer agent) incorporate ML price $z$ as yet another informative signal
into constructing a valuation $\tilde{v}$. This in turn updates individual
buyers’ willingness to offer and consequently the offer distribution from the
exemplar buyer every period $\mu$. We do not explicitly model the estimation
and willingness to offer for individual buyer. But implicitly the ML price $z$
impacts valuations of both buyers and sellers, and thereby impacts the
realized sale price $p$. Going forward we will use the phrase buyer-seller
when discussing impact of ML price.

Using the definitions above, we can derive expressions for home value $v$,
buyer-seller valuation $\tilde{v}$ and the sale price $p$ 141414Proofs in
Appendix LABEL:app:_proofs..

###### Lemma 1

The true home value $v$ is given by,

$\displaystyle v=(\mu+2\sigma/3-\sqrt{2c\sigma})$ (8)

Each seller’s valuation (and the variance of this valuation) is given by,

$\displaystyle\tilde{v}(\alpha=0)=\tilde{\mathbb{E}}[p|z]=\tilde{\mu}+2\sigma/3-\sqrt{2c\sigma}\text{
; \quad}\sigma_{e}^{2}(\alpha=0)=\sigma_{e}^{2}=Var[\tilde{v}]=8\sigma^{2}/3$
$\displaystyle\tilde{v}(\alpha)=(1-\alpha)\times\Big{(}\tilde{\mu}+2\sigma/3-\sqrt{2c\sigma}\Big{)}+\alpha\times
z\text{ ; \quad}\sigma_{e}^{2}(\alpha)=(1-\alpha)^{2}\sigma_{e}^{2}$ (9)

The distribution of sale price is given by,

$\displaystyle\mathbb{E}[p|z]=(1-\alpha)\times\Big{(}\mu+2\sigma/3-\sqrt{2c\sigma}\Big{)}+\alpha\times
z\text{ ;
\quad}\sigma_{\epsilon}^{2}(\alpha)=Var[p]=\delta\sigma_{e}^{2}(\alpha)\text{
; \quad}\delta=(1/6)$ (10)

Reliance on ML price ($\alpha>0$) adds bias in valuations and sale prices
i.e., $v\neq\mathbb{E}[p]=\mathbb{E}[\tilde{v}]$ if $z\neq p$. It also reduces
variance in valuations and sale prices i.e., $Var[p]$ and $Var[\tilde{v}]$ are
decreasing in $\alpha$.

Let the subscript $i$ index each seller. We can express each seller’s
valuation as $\tilde{v}=\mathbb{E}[\tilde{v}_{i}]+e_{i}$ where
$e_{i}=(1-\alpha)(\tilde{\mu}-\mu)$ is the noise in valuation across
individual sellers with $E[e_{i}]=0$. We can also express the realized sale
price as $p=\mathbb{E}[p]+\epsilon$ where $\epsilon$ is the noise in realized
sale prices (across multiple hypothetical sale instances) with
$E[\epsilon]=0$. There are two intermediate results worth highlighting here.
First, the true expected sale price is equal to the expected seller valuation:
$\mathbb{E}[p]=\mathbb{E}[\tilde{v}_{i}]$. At $\alpha=0$ we have
$v=\mathbb{E}[p]=\mathbb{E}[\tilde{v}_{i}]$ i.e., prior valuations (before
observing ML price) are unbiased with respect to the true value $v$. At
$\alpha>0$, the valuation are not unbiased anymore i.e.,
$\mathbb{E}[\tilde{v}_{i}]\neq v$ if $z\neq p$. Second, the variance in sale
prices $Var[p]$ is proportional to variance (disagreement) of seller’s
valuation $Var[\tilde{v}]$151515Variance (disagreement) in individual seller’s
valuation is a theoretical measure in the housing market setting because only
one homeowner can be the seller for a unique home.. The constant $\delta$ can
be interpreted as the degree to which disagreement in private valuations is
reduced by participating in the market (buyers and sellers learning and
attaining consensus). The variance in valuations $Var[\tilde{v}]$, and
consequently variance in sale prices $Var[p]$ are diminished by a fraction
$(1-\alpha)^{2}$. So overall, ML price is adding some bias but removing some
variance from valuations and sale prices. The payoff implications of this will
be expressed in Lemma 2 and Lemma 3 in section 4.4.

###  4.2 Machine Learning Pricing Errors

We can use equations 5 (ML price $z$ as function of sale prices $p$), and
Lemma 1 (sale prices $p$ as function of ML price $z$) to formulate the
feedback loop. Figure 3 visually illustrates this loop. In this section, we
will start by calculate the true Machine Learning pricing errors
$\sigma_{z}^{2}=Var[v-z]$ and its empirical estimate
$\hat{\sigma}_{z}^{2}=Var[p-z]$ (because true value $v$ is not observed, only
sale prices $p$ are observed).

Figure 3: Buyer-seller $i$ of home $k$ construct home valuation $\tilde{v}_{i,k,t}$ using ML price $z_{k,t}$ and private estimation with weights $\alpha$ and $1-\alpha$ respectively. The market interaction among buyer-sellers with similarly constructed valuations produces a sale prices $p_{k,t}$. $N$ home sales similar to home $k$ form training sample for Machine Learning model. In the next period, a home $j$ is influenced by ML price $z_{j,t+1}$. The machine learning error can be estimated $\hat{\sigma}_{z}^{2}$ using ML price $z_{j,t+1}$ and realized sale price $p_{j,t+1}$. This is an approximation since the true home value $v_{j,t+1}$ is not observed. Table 3: At a glance view of Lemma and Propositions Lemma 1 | Expression for home value and sale price distribution
---|---
Lemma 2 | Expression for expected payoff
Lemma 3 | Expression for variance of payoffs
Proposition 1 | Confounded ML error is inflated i.e., $\sigma^{2}_{z}(\alpha)\geq\sigma^{2}_{z}(\alpha=0)$
Proposition 2 | Confounded ML error estimate is deflated i.e., $\hat{\sigma}^{2}_{z}(\alpha)\leq\sigma^{2}_{z}(\alpha)$ if $\alpha$ large enough
Proposition 3 | Confounded ML error estimate $\hat{\sigma}^{2}_{z}(\alpha)$ is decreasing in $\alpha$ if $\alpha$ large enough
Proposition 4 | Proposition 2 and 3 always true at equilibrium $\alpha^{*}$
Proposition 5 |  $\alpha^{*}=1$ always an equilibrium and only equilibrium if $\sigma_{z}^{2}/\sigma_{e}^{2}$ is small enough
Proposition 6 | Variance in payoff $Var[\pi]$ increasing in $\alpha$ if $\alpha$ is large enough
Proposition 7 | Risk averse payoff $\Pi(\alpha^{*}=1)<\Pi(\alpha=0)$ if $\sigma_{z}^{2}/\sigma_{e}^{2}$ is large enough

The ML price $z_{k,t+1}$ for a focal home $k$ is given by
$z_{k,t+1}=\bar{p}_{K,t}$ from equation 5, where $K=\mathcal{C}(k)$ is the
focal home’s peer cluster at time $t$. We now focus on the focal home $k$ and
drop the subscript $k$. The error in ML price is the difference between the
actual realized home value $v_{t+1}$ and ML price $z_{t}$. This error can be
decomposed into three components as,

$\displaystyle v_{t+1}-z_{t+1}=\underbrace{v_{t+1}-v_{t}}_{\text{Random
Walk}}+\underbrace{v_{t}-v_{K,t}}_{\text{Unpriced
Features}}+\underbrace{v_{K,t}-\bar{p}_{K,t}}_{\text{Finite Sample Error}}$
(11)

Home values in the current period $v_{t}$ can not forecast the random walk of
preferences and values into the next period $v_{t+1}$. We denote variance of
random walk error as an exogenous and constant quantity
$Var[v_{t+1}-v_{t}]=\sigma^{2}_{rw}$.

We can interpret the clustering of homes into $Q$ clusters (described in
Section 3.2) in terms of matching on homes’ features (home characteristics
such as its age and size). Specifically, given a total number of features
$\bar{Q}$, we can view the homes within a cluster as being identical or
matching on $Q$ features. In using the peer cluster’s mean sale price,
$\bar{Q}-Q$ unique features of the focal home are left unpriced i.e., an error
$v_{t}-v_{K,t}$. Intuitively, the variance of unpriced features should depend
on variance of all features and the number of priced features $Q$. The
variance of all features, also the heterogeneity in housing stock,
$\sigma^{2}_{v}$ is treated as an exogenous constant. The choice of priced
features $Q$ explains increasingly greater proportion of the total variance.
This is captured by monotonically decreasing function $h(Q)$ (possibly with
positive second derivative because of diminishing returns). Thus we have
$Var[v_{t}-v_{K,t}]=h(Q)\times\sigma^{2}_{v}$. Intuitively, as the number of
“clustering features” (features used to place homes in the same cluster)
increases, clusters will have fewer, very similar homes. Hence, the variance
of home values in a cluster will be low. Similarly, as the number of
“clustering features” decreases, clusters will have more, dissimilar homes.
Hence, the variance of home values in a cluster will be high.

The ML price is effectively the sample mean of cluster sale prices. The finite
sample error is the difference between the mean of cluster sale prices and the
true cluster value i.e., $v_{K,t}-\bar{p}_{K,t}$. Since there are $N/Q$ home
sales in the cluster, we can express $\bar{p}_{K,t}$ as,

$\displaystyle\bar{p}_{K,t}=(Q/N)\times\sum_{k}\Big{[}p_{k,t}\Big{]}$
$\displaystyle=(Q/N)\times\sum_{k}\Big{[}\mathbb{E}[v_{i,k,t}]+\epsilon_{k,t}\Big{]}$
$\displaystyle=(Q/N)\times\sum_{k}\Big{[}\mathbb{E}[(1-\alpha)\times\tilde{v}_{i,k,t}+\alpha\times
z_{k,t}]+\epsilon_{t}\Big{]}$
$\displaystyle=(Q/N)\times\sum_{k}\Big{[}(1-\alpha)\times
v_{k,t}+\alpha\times(v_{k,t}+e^{z}_{k,t})+\epsilon_{k,t}\Big{]}$
$\displaystyle=(Q/N)\times\sum_{k}\Big{[}v_{k,t}+\alpha
e^{z}_{k,t}+\epsilon_{k,t}\Big{]}$
$\displaystyle=v_{K,k,t}+\alpha(Q/N)\times\sum_{k}e^{z}_{k,t}+(Q/N)\times\sum_{k}\epsilon_{k,t}$
(12)

Note that under $\alpha=0$, the second components disappears. Under
$\alpha>0$, this additional component captures the confounding of the sale
price from the ML price.

We can now express the variance of the finite sample error as,

$\displaystyle
Var[v_{K,t}-\bar{p}_{K,t}]=Var\Big{[}(Q/N)\times\sum_{k}(\alpha\mathbb{E}[e^{z}_{k,t}]+\epsilon_{k,t})\Big{]}$
$\displaystyle=\alpha^{2}Var[(Q/N)\times\sum_{k}e^{z}_{k,t}]+Var[(Q/N)\sum_{k}\epsilon_{k,t}]$
$\displaystyle=\alpha^{2}Q\sigma_{z}^{2}/N+\delta
Q(1-\alpha)^{2}\sigma^{2}_{e}/N$ (13)

We can now write the full ML price error variance as,

$\displaystyle\sigma_{z}^{2}(\alpha)=Var\Big{[}(v_{t+1}-v_{t})+(v_{t}-v_{K,t})+(v_{K,t}-\bar{p}_{K,t})\Big{]}$
$\displaystyle=\sigma^{2}_{rw}+h(Q)\times\sigma^{2}_{v}+\alpha^{2}Q\sigma_{z}^{2}(\alpha)/N+(1-\alpha)^{2}\delta
Q\sigma^{2}_{e}/N$
$\displaystyle=\Big{(}\frac{1}{1-Q\alpha^{2}/N}\Big{)}\times(\sigma^{2}_{rw}+h(Q)\times\sigma^{2}_{v}+(1-\alpha)^{2}\delta
Q\sigma^{2}_{e}/N)$
$\displaystyle=\sigma^{2}_{rw}+h(Q)\times\sigma^{2}_{v}+(1-\alpha)^{2}\delta
Q\sigma^{2}_{e}/N$ (14)

The denominator $(1-Q\alpha^{2}/N)$ is very close to 1 since $Q<<N<<\infty$
and set to 1 going forward as a conservative assumption for analytical
simplification. The ML error is increasing in random walk of home preferences
$\sigma^{2}_{rw}$, heterogeneity in housing stock $\sigma^{2}_{v}$ and error
in private valuations $\sigma^{2}_{e}$. The ML error sensitivity to $Q$ is
mixed - the unpriced feature error component $h(Q)\times\sigma^{2}_{v}$ is
decreasing in $Q$ while the finite sample error component
$(1-\alpha)^{2}\delta Q\sigma^{2}_{e}/N$ is increasing. We will discuss
endogenization of $Q$ later in this section. The ML error is increasing in
$\alpha$. In fact, we can express the ML error in terms of the un-confounded
ML error $\sigma_{z}^{2}=\sigma_{z}^{2}(\alpha=0)$ as,

$\displaystyle\sigma_{z}^{2}(\alpha)=\sigma_{z}^{2}+\alpha(2-\alpha)\delta
Q\sigma^{2}_{e}/N$ $\displaystyle\text{where
\quad}\sigma_{z}^{2}=\sigma_{z}^{2}(\alpha=0)=\sigma^{2}_{rw}+h(Q)\times\sigma^{2}_{v}+\delta
Q\sigma^{2}_{e}/N$ (15)

The un-confounded ML error $\sigma_{z}^{2}$ are valid in a limited setting
where the platform does not reveal the ML price or the buyers-sellers do not
use the ML price at all. In the rest of the paper, we will continue to compare
confounded results at $\alpha>0$ with unconfounded results at $\alpha=0$ to
highlight impact of the confounding over the feedback loop. In comparing
expressions for confounded ($\sigma_{z}^{2}(\alpha>0)$) and unconfounded
($\sigma_{z}^{2}=\sigma_{z}^{2}(\alpha=0)$) ML errors, the additive term
$\alpha(2-\alpha)\delta Q\sigma^{2}_{e}/N$ captures the amplification in ML
error because of the confounding.