\section*{Acknowledgements}
This work was supported in part by funding from the Stanford Program on Democracy and the Internet and ARO MURI award \#W911NF-20-1-0252. We thank Jan Overgoor, Amel Awadelkarim, Arjun Seshadri, and Kaitlyn Zhou for helpful feedback and discussions.
\section{Appendix} 
\subsection{Proofs} \label{sec:proofs}
\textbf{Theorem 1.1} In the organic MLE model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykiomle \rightarrow \LimEykiomle$
    \item $\Vykiomle \rightarrow \BigLimVykio$
    \item $\Elomle \rightarrow \left(\frac{\sm}{\sm + \si}-1\right)^2 x_i^2 + \BigLimVykio$.
\end{enumerate}
\begin{proof}
In the organic MLE model, the user makes an estimate of each item position, $\hijmle = \zij$, where $\zij$ is a sample drawn from $N(y_j, \si)$. Since there are an infinite number of items and the support of the item and noise distributions is $\mathbb{R}$, then there must be at least one item estimate located at any given point in $\mathbb{R}$. Thus, when the user chooses the estimate that is closest to her own position, $x_i$, we can expect that the chosen estimate lies at $x_i$. Given the knowledge that the user's chosen item produced a sample at $x_i$, and that the chosen item's position $\ykiomle$ was drawn from $N(0, \sm)$, we can derive a closed form solution for the posterior mean and variance of $\ykiomle$ (since both distributions are Gaussian, forming a conjugate pair), forming the expressions above for $\Eykiomle$ and $\Vykiomle$. Then, the expected squared error for the user is simply the squared bias, $(\Eykiomle - x_i)^2$, added to the variance, $\Vykiomle$.
\end{proof}

\noindent \textbf{Theorem 1.2} In the organic MAP model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykiomap \rightarrow x_i$
    \item $\Vykiomap \rightarrow \BigLimVykio$
    \item $\Elomap \rightarrow \BigLimVykio$.
\end{enumerate}
\begin{proof}
When the user uses MAP, her estimate of item $j$'s position becomes $\hijmap = (\sm/(\sm+\si)) \zij$. Since there are an infinite number of items, she will still choose an estimate that is equal to $x_i$; however, the equation above now indicates that the chosen item must have produced a sample at $((\sm + \si)/\sm)x_i$. Again, we can derive a closed form solution for the posterior distribution of the chosen item's position $\ykiomap$, knowing that it was also drawn from $N(0, \sm)$. We find that the terms cancel out, giving us $\Eykio = x_i$, and the variance does not change compared to the MLE version of the model. Since $\Eykio$ is now unbiased, the user's expected squared error becomes only the variance.
\end{proof}

\noindent \textbf{Theorem 2.1} In the recommender MLE model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykirmle \rightarrow x_i$
    \item $\Vykirmle \rightarrow \sr$
    \item $\Elrmle \rightarrow \sr$.
\end{enumerate}
\begin{proof}
In the recommender MLE model, the recommender system makes an estimate of the user's true position, $\hrimle = \zri$, where $\zri$ is a sample drawn from  $N(x_i, \sr)$. Then, the system finds the item whose position is closest to $\hrimle$. Echoing the logic in Theorem 1, if there are an infinite number of items, then there must be at least one item located at any given point in $\mathbb{R}$. Thus, the chosen item position, $\ykirmle$, must be equal to $\hrimle$, and so the distribution of $\ykirmle$ reduces to the distribution of $\hrimle$, which has an expected value of $x_i$ and variance of $\sr$. Since there is no bias, the expected squared loss for the user is just the variance.
\end{proof}

\noindent \textbf{Theorem 2.2} In the recommender MAP model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykirmap \rightarrow \frac{\su}{\su + \sr} x_i$
    \item $\Vykirmap \rightarrow \left(\frac{\su}{\su + \sr}\right)^2 \sr$
    \item $\Elrmap \rightarrow \left(\frac{\su}{\su + \sr} - 1\right)^2 x_i^2 + \left(\frac{\su}{\su+\sr}\right)^2 \sr$.
\end{enumerate}
\begin{proof}
In the MAP setting, the recommender model's estimate of the user's position becomes $\hrimap = (\su/(\su + \sr)) \zri$. Since there are an infinite number of items, it is still true that the chosen item's position, $\ykirmap$, will match $\hrimap$ perfectly, and so the distribution of $\ykirmap$ reduces to the distribution of $\hrimap$. Since $\hrimap$ is simply a sample drawn from $N(x_i, \sr)$ scaled by $\su/(\su + \sr)$, then $\hrimap$ must be normally distributed with mean $(\su/(\su + \sr))x_i$ and variance $(\su/(\su + \sr))^2 \sr$. This provides the expressions above for $\Eykirmap$ and $\Vykirmap$, and the expected squared loss is the sum of the squared bias, $(\Eykirmap - x_i)^2$, and the variance $\Vykirmap$.
\end{proof}

\noindent \textbf{Theorem 3} In the organic model as $n \rightarrow \infty$, 
\begin{enumerate}
    \item $\VYkomle \rightarrow \frac{m-1}{m}\left[(\frac{\sm}{\sm + \si})^2 \su + \LimVykio\right]$
    \item $\VYkomap \rightarrow \frac{m-1}{m}\left[\su + \LimVykio\right]$.
\end{enumerate}
\begin{proof}
In general, we can think of each chosen item position $\yki \in \Yk$ as the sum of its expected value conditioned on $x_i$, $\Eyki$, and some independent random noise $\epsilon$ with variance $\Vyki$ (which does not depend on $x_i$ in any of our models). 

Using this approach, the process of generating terms in $\Ykomle$ in the organic MLE model can be seen as first drawing a random user position $x$ from $N(0, \su)$, scaling it by $\sm/(\sm + \si)$, then adding this noise term $\epsilon$. We scale $x$ because $\Eykiomle = \sm/(\sm + \si)x_i$ (Theorem 1.1). This output becomes the sum of two random variables, where the first term is the scaled $x$ and the second is $\epsilon$. The variances of these terms are $(\sm/(\sm + \si))^2 \su$ and $\Vykio = \LimVykio$, respectively, so the expected variance of $\Ykomle$ is the sum of these expressions multiplied by $(m-1)/m$, because there are $m$ elements in $\Ykomle$.

In the organic MAP case, the process of generating terms in $\Ykomap$ can also be seen as first drawing a random user position $x$ from $N(0, \su)$, then adding a noise term $\epsilon$. We do not scale $x$ in this case, since $\Eykiomap = x_i$ (Theorem 1.2). The output is again the sum of two random variables, where the first term is $x$ and the second is $\epsilon$. Their variances are $\su$ and $\LimVykio$, respectively, so the expected variance of $\Ykomap$ is their sum scaled by $(m-1)/m$. 
\end{proof}

\noindent \textbf{Theorem 4} In the recommender model as $n \rightarrow \infty$, 
\begin{enumerate}
    \item $\VYkrmle \rightarrow \frac{m-1}{m}(\su + \sr)$
    \item $\VYkrmap \rightarrow \frac{m-1}{m}\left[\frac{\su}{\su+\sr} \su \right]$.
\end{enumerate}
\begin{proof}
We can use the same approach from Theorem 3 to derive $\VYk$ here. In the recommender MLE model, the process of generating terms in $\Ykrmle$ can be seen as drawing a random user position $x$ from $N(0, \su)$ then adding a noise term $\epsilon$ that has variance $\Vykirmle = \sr$ (Theorem 2.1). We do not scale $x$ here either because $\Eykirmle = x_i$. The sum of these terms' variances is $\su + \sr$, and again we scale this by $(m-1)/m$. 

In the recommender MAP model, to generate terms in $\Ykrmap$, 
we draw a random user position $x$ from 
$N(0, \su)$, scale it by $\su/(\su + \sr)$, 
then add a noise term $\epsilon$ (Theorem 2.2). 
The variance of the scaled $x$ is 
$(\su/(\su + \sr))^2 \su$ and the variance of 
$\epsilon$ is $\VYkrmap = (\su/(\su + \sr))^2 \sr$. 
Adding these terms together, 
simplifying, and multiplying by $(m-1)/m$ yields 
expression for $\VYkrmap$ in the theorem statement.
\end{proof}




\section{Conclusion}
We have introduced two contrasting models of item-matching processes: one describing a generic personalized recommender system, and the other describing a setting where users search for items organically without the mediation of any system.
In both cases, we have a decision-making agent trying to match users to items with limited information, but the difference lies in who/what is doing the matching, and what information they have access to vs.\ what is being estimated. Comparing the two, we have seen that this simple switch in perspective results in dramatic differences at both the individual and population levels.
For example, in the MLE versions of the models, the recommender model forms unbiased item matches for the users, while the organic model does not, and the organic model favors central users, while the recommender model does not. 
Applying shrinkage has notably diverging effects on the two models: while it can reduce MSE in both cases, shrinkage leads the recommender model to choose increasingly similar sets of items, while leading the organic model to diversify its selections. 
These findings, which are robust to changes in user and item distribution, single and multiple dimensions, and asymptotic and finite settings, indicate pervasive differences between recommended and organic processes of item-matching.

We have worked to study one of the core counterfactual questions about recommender systems: if there were no recommender system, how would the world be different? Our analyses provide evidence that the use of recommender systems fundamentally alters how humans interact with content. Recommender systems are usually touted as reducing search frictions in markets, but through our work we find that such reductions in search friction comes with myriad subtle trade-offs.
We hope that future studies can build upon the framework established in this paper, using our models as building blocks to analyze diverse potential effects of recommender systems, such as polarization, filter bubbles, user retention rates, or fairness across users or products. 
As a specific direction, by modeling repeated choices and incorporating learning dynamics where users learn from the items they consume \citep{diehl2015persuasion}, it should be possible to analyze how the phenomena we document here, pertaining to single choices, compound over time and affect the evolution of users' opinions and behaviors. 
Finally, we also hope in future work to tie the theory of our model to real-world experiments; for example, through randomized trials where participants are assigned to unmediated organic search versus system-mediated search processes, to understand the differences most broadly.

\subsection{Empirical analysis with MovieLens data} \label{sec:empirics}
In this section, we further extend our simulations by incorporating multidimensional user and item positions fitted on real ratings data. We use the MovieLens 1M dataset\footnote{https://grouplens.org/datasets/movielens/1m/.}, which contains 1,000,209 ratings from approximately 6,000 MovieLens users on 3,900 movies \citep{harper2015movielens}. First, we filter the ratings matrix to only keep users that have rated at least 50 movies, then filter to keep movies with at least 50 ratings.
After filtering, we are left with a rating matrix $R \in \mathbb{R}^{m' \times n}$, where we have $m' = 4,297$ users and $n = 2,514$ movies remaining. 

\paragraph{Learning user and item distributions from data.}
We apply a collaborative filtering algorithm to $R$ in order to infer latent embeddings $\mathcal{U} = [u_i]_{i=1}^{m'}$, $u_i \in \mathbb{R}^{d}$, for each user $i$, and $\mathcal{V} = [v_j]_{j=1}^n$, $v_j \in \mathbb{R}^d$, for each movie $j$. Collaborative filtering uses the history of interactions between all users and items to infer latent representations, where users and items are encoded into low-dimensional spaces such that if a user has given an item a high rating, their representations should be ``similar'', and if a user has given an item a low rating, their representations should be further apart. Similarity can be measured in different ways (e.g., inner products, Euclidean distance); since our models take a distance-based perspective to utility, we implement a collaborative filtering procedure \citep{khoshneshin2010recsys} that embeds users and items into a unified Euclidean space where items that are closer to users are more attractive to them.

In this framework, given a user embedding $u_i$ and movie embedding $v_j$, their predicted rating $\hrij$ is
\begin{align}
    \hrij = \mu + b_i - ||u_i - v_j||^2,
\end{align}
where $\mu$ is the global average rating in $R$, $b_i$ is the user bias term capturing users that tend to rate higher or lower, and $||u_i - v_j||^2$ is the squared distance between the two embeddings. This framework naturally integrates our definition of user loss---the squared distance between a user and her matched item---since a user's predicted rating is exactly the negative squared distance, translated by $\mu + b_i$ (which remains constant per user). Furthermore, this formulation fits with the choice function in our models: just as we assume that the recommender system will choose the item that minimizes distance to its estimate of the user, that very same item here would be the one that maximizes predicted rating.  

To learn these embeddings, we define the following objective function, where we aim to minimize the regularized loss over all observed ratings in $R$:
\begin{align}
    \min_{\mathcal{U}, \mathcal{V}, \mathcal{B}} \sum_{i, j} w_{ij} [(\rij - \hrij)^2] + \lambda(||u_i - v_j||^2 + b_i^2). \label{eqn:cf_loss}
\end{align}
Here $\mathcal{B}$ represents the set of all user biases $b_i$, and $w_{ij} \in \{0, 1\}$ indicates whether the rating $\rij$ is observed in $R$. We then use gradient descent to update the latent parameters $\mathcal{U}$, $\mathcal{V}$, and $\mathcal{B}$ with respect to the regularized loss. After learning $\mathcal{U}$, i.e., the embeddings of the users in the MovieLens dataset, we sample from this empirical distribution to generate $m$ ``test'' users. This creates a new matrix of user positions, $\mathcal{X} \in \mathbb{R}^{m \times d}$, which we use in our simulations to represent unseen users. Our simulations do not rely on unseen movies, however, so we can directly set $\mathcal{Y}$, the item positions in our simulations, to $\mathcal{V}$, the movie positions directly inferred from the MovieLens dataset.

\subsubsection{Simulations with MovieLens embeddings.} 
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/movielens_distribution.png}
    \caption{Distribution of learned user (top) and movie (bottom) embeddings from the MovieLens 1M dataset.}
    \label{fig:embeddings}
\end{figure}

Using this approach, we learn user and movie embeddings of dimension $d=5$ from the MovieLens dataset. In Figure \ref{fig:embeddings}, we display the learned user and movie distributions. We see that the user distribution is less dispersed than the movie distribution in each dimension, and the movie distribution is sometimes asymmetric (clearly non-Gaussian). For both embeddings, the covariance between dimensions is low.

We then simulate the organic and recommender models with $m = 300$ users and $n = 2,514$ movies (the number of movies in the filtered MovieLens dataset). For the organic model, we fix each user's noise covariance $\Si$ to $0.5 \cdot \Sm$, where $\Sm$ was the empirical covariance fitted on $\mathcal{Y}$, the movie embeddings. Similarly, for the recommender model, we fixed the system's noise covariance $\Sr$ to $0.5 \cdot \Su$, where $\Su$ was the empirical covariance fitted on $\mathcal{X}$, the user embeddings. We scaled the user / movie covariance in this way, as opposed to using spherical noise, so that the size of the noise per dimension would scale with the variance of the estimated population in that dimension, which we believed was more realistic. When shrinkage is applied within each model, we no longer have the user or system construct MAP estimates with Gaussian priors: the prior distribution is neither Gaussian nor known. Instead, we implemented shrinkage parameterized by a scalar $\alpha \in [0, 1]$, which interpolates between the MLE estimate and the mean over all MLE estimates. That is, user $i$'s shrunken estimate $\hijshr$ for movie $j$ becomes
\begin{align}
\label{eq:js}
    \hijshr = (1 - \alpha)\hijmle + \alpha (\frac{1}{n} \sum_{k=1}^n \hat{y}_{k}^{(i, \text{MLE})}),
\end{align}
and the recommender system's shrunken estimate of user $i$ is the equivalent interpolation between $\hrimle$ and the mean MLE estimate over all users. This family of estimators generalizes James--Stein shrinkage and Empirical Bayes estimators, which correspond to specific recipes for choosing $\alpha$ (or $\alpha_i$, different for each user). 

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/movielens_no_shrinkage_2.png}
    \caption{Individual-level results from MovieLens experiments. We plot each user's average squared distance to their 10 nearest movies (as a measure of centrality) vs. the user's loss, averaged over 500 trials. In order to see data points more clearly, we truncated outliers beyond the 95th percentile of the x-axis (showing 285 out of 300 users).}
    \label{fig:movielens-individual}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/shrinkage_alpha.pdf}
    \caption{Population-level results from MovieLens experiments. We assess the effect of varying the shrinkage parameter $\alpha$ on the MSE and variance of matched item positions $\VYk$, running 500 trials for each model and value of $\alpha$.}
    \label{fig:movielens-population}
\end{figure}


Despite the lopsided movie distribution and expanded number of dimensions, we find that the key qualitative results from our theoretical analyses hold. 
In Figure \ref{fig:movielens-individual}, we see that the organic MLE model continues to strongly favor more central users, compared to the recommender MLE model, which is much more even-handed across user positions. 
In our earlier setting with items drawn from $N(0, \sm)$, the centrality of a user could be summarized by the absolute value of $x_i$, but here we need a more empirical measure; for example, we use the user's average squared distance to their 10 nearest movies (where higher average distance corresponds to lower centrality). 

Secondly, we simulate different amounts of shrinkage ($\alpha \in \{0.05, 0.1, \cdots, 0.95\}$) and evaluate the impacts on our population-level metrics, the mean user loss (MSE) and the variance of matched item positions $\VYk$. In multiple dimensions, we compute the generalized variance as the product of the eigenvalues of the covariance matrix of $\Yk$; we log-transform this quantity to make it more interpretable. As we increase the amount of shrinkage, at first this improves MSE for both models, but eventually MSE begins to increase; the optimal level of shrinkage seems to fall around $\alpha \approx 0.4$ for both models. However, even though the MSE curves look similar, the $\VYk$ curves completely diverge: the more we increase $\alpha$, the more $\VYk$ grows in the organic model, and the more it shrinks in the recommender model. This matches our earlier findings that shrinkage has a ``diversifying'' effect on the organic model but a ``homogenizing'' effect on the recommenders.




\section{Models and Metrics}
Our models describe two contrasting processes through which users could be matched with items of choice, e.g., movies, news articles, or consumer products. In Section \ref{sec:model-definition}, we introduce the notation and formal dynamics of our models. In Section \ref{sec:metrics}, we define the individual and population-level metrics we will use to compare the models' outcomes.

\subsection{Model definition} \label{sec:model-definition}
In both models, we will have $m$ users and $n$ items. Each user and each item has a latent position; for example, this could represent movie attributes in the movie context or ideology in the context of news articles. We denote user positions as $x_i \in \mathbb{R}^{d}$ and item positions as $y_j \in \mathbb{R}^{d}$, where $d$ represents the dimensionality of the latent space. 
Overall, the set $\X = \{x_i\}_{i=1}^{m}$ represents all user positions and the set $\Y = \{y_j\}_{j=1}^n$ represents all item positions. Our analysis will make very weak distance-based assumptions about the interest/utilities of users for items: we simply assume that user $i$'s enjoyment of item $j$ is monotonically decreasing to the distance between $x_i$ and $y_j$. Thus, under both models, the decision-making agent (either the user or the recommender system) wants to match the user to the item whose position is closest to hers.

\paragraph{Organic model.}
In this model, the user will ``organically'' search through the collection of items and choose one for herself (Figure \ref{fig:model}a). When surveying each item $j$, she only has access to a noisy sample of its true position, $y_j$. She draws her sample $\zij$ from $N(y_j, \Si)$, where $\Si$ represents the covariance in her noise.
Then, the user makes an estimate $\hij$ of item $j$'s true position.
In the simplest case, the user takes the maximum likelihood estimate (MLE) based on her single sample of $y_j$:
\begin{align}
    \hijmle = \zij.
\end{align}
We compare this MLE to an alternate case where the user applies some form of shrinkage \citep{james1961estimation} to her estimates. For example, if the user assumes a Gaussian prior on the item positions, she could take a maximum a posteriori (MAP) estimate of each, shrinking the estimate towards the item mean: 
\begin{align}
    \hijmap = (\Sm^{-1} + \Si^{-1})^{-1} (\Sm^{-1} \mm + \Si^{-1}\zij),
\end{align}
where $\mm$ and $\Sm$ are the mean and covariance of the item distribution, respectively. This style of shrinkage can also be thought of as regularization towards the item mean. In an empirical setting without a clear prior, Empirical Bayes shrinkage \citep{efron1976multivariate} would be preferred.

After surveying all $n$ items, the user will have constructed estimates for every item's position.
Since we assume utility is monotonically decreasing in distance, the user will then choose the item whose estimate is closest to her own position, $x_i$. Let $\kio$ represent the item chosen by user $i$ under the organic setting. Then,
\begin{align}
    \kio = \argmin_{j \in [n]} ||x_i - \hij||. \label{eqn:user-decision}
\end{align}
We let $\ykio$ denote the position of the chosen item $\kio$, and will later study properties of $\ykio$ as a random variable that inherits its randomness from the user's estimates, $\hij$.

\begin{figure*}
    \centering
    \includegraphics[width=\linewidth]{images/model_schematic.pdf}
    \caption{Model schematics. (a) In the organic model, the user samples and estimates item positions, and chooses the estimate closest to her own position. When shrinkage is applied, the user's estimates of the items shifts inward, and the user's choice switches from the inner to outer item. (b) In the recommender model, the system samples and estimates the user's position, and chooses the item closest to its estimate. When shrinkage is applied, the system's estimate of the user shifts inward, and the system's choice switches from the outer to inner item.}
    \label{fig:model}
\end{figure*}

\subsubsection{Recommender model.}
In this model, a recommender system takes on the burden of search instead of the user (Figure \ref{fig:model}b). When user $i$ comes onto the platform, the system gathers a noisy sample of $i$'s true position $x_i$. This sample $\zri$ is drawn from $N(x_i, \Sr)$, where $\Sr$ represents the covariance in the recommender's noise. Then, just as the user estimated item positions based on her noisy samples, the system makes an estimate $\hri$ of user $i$'s true position based on its sample of the user. Again, we first consider the case where the recommender system takes the MLE:
\begin{align}
    \hrimle = \zri.
\end{align}
We will again compare the MLE to the case where the recommender system applies shrinkage to its estimates of user positions. With a Gaussian prior, the MAP estimate of user $i$'s position is
\begin{align}
    \hrimap = (\Su^{-1} + \Sr^{-1})^{-1} (\Su^{-1} \mus + \Sr^{-1} \zri),
\end{align}
where $\mus$ and $\Su$ are the mean and covariance of the user distribution, respectively. In the absence of a prior, Empirical Bayes shrinkage would again be preferred.

As a reversal of the organic model, we assume that the recommender system has perfect knowledge of $\Y$, the set of true item positions, but can only estimate the user's true position.
Let $\kir$ represent the chosen item for user $i$ under the recommender model. Similar to the organic model, the system will choose the item whose position is closest to $\hri$:
\begin{align}
    \kir &= \argmin_{j \in [n]} ||\hri - y_j||.
\end{align}
Here we again let $\ykir$ denote the position of the matched item $\kir$, where $\ykir$ is a random variable that inherits its randomness from the system's estimates, $\hri$.

We deliberately set up the recommendation process to be as similar as possible to the organic process---the structure of the noise and logic in the choice function are identical---so that we can compare the models on the basis of who/what controls the matching and what information they have access to. We do not, for example, model the process of users reacting to recommendations. Instead, we assume that $\kir$ will be the user's match, comparing it directly to the match $\kio$ under the organic model. Furthermore, we note that both models are made up of distinct modules (sampling, estimation, choice), each of which could be swapped out for more complex processes. For example, in the organic model, one might be interested in other decision-making rules for the user, such as those that make explore/exploit trade-offs or capture risk aversion (in the case of non-uniform sample noise), or in the recommender model, the user estimation process could be derived from a specific algorithm. Thus, our models describe flexible item-matching processes that can be extended to encompass many real-world systems.

\subsection{Metrics} \label{sec:metrics}
We are interested in comparing the outcomes of these models through two lenses, at the individual and at the population levels. At the individual level, we will treat the squared distance between the user and their matched item as a user-level loss, since we model utility as monotonically decreasing in distance. Let $\Yk = \{y_{k_1}, y_{k_2}, \cdots, y_{k_m}\}$ represent the multiset of matched item positions over all users; note that $|\Yk| = |\X| = m$, and that each item position $y_j$ might appear 0 times, once, or multiple times in $\Yk$. Then, the loss for user $i$ is defined as
\begin{align}
    \lossi = ||\yki - x_i||^2.
\end{align}
As we analyze $\lossi$ in Section \ref{sec:theorems}, we will also derive $\Eyki$ and $\Vyki$ along the way; that is, the expected position and variance of the matched item for user $i$. We choose these metrics to study because they form the building blocks of many downstream phenomena of interest. For example, if the expected match for a user is not the user's own position, the matching process might eventually shift user preferences or opinions, and if different users systematically experience different losses (as we will see happens under certain settings of our models), this implies inequities in the model as it provides matches of differential quality to users on the basis of their preferences.

The population-level analysis, meanwhile, is concerned with average user loss and the overall collection of matched items. Observe that the average loss over users simply becomes the mean squared error (MSE) of the matches:
\begin{align}
    \text{MSE}(\X, \Yk) = \frac{1}{m} \sum_{i = 1}^m \lossi = \frac{1}{m} \sum_{i = 1}^m ||\yki - x_i||^2.
\end{align}
These matches are based on estimates, where shrinkage is known to reduce the MSE of an estimator. As such, there are opportunities in both models to reduce mean user loss by having the user / system apply shrinkage during estimation. However, while shrinkage might reduce MSE, that is not its only effect. In Figure \ref{fig:model}, we illustrate a basic intuition for a key result: due to the reversal in who is doing the estimating and what is being estimated, shrinkage acts as a ``diversifying'' force in the organic model, but it serves as a ``homogenizing'' force in the recommender model. We see that shrinkage in the organic model shifts the user's estimates of the items inward (i.e., towards 0), so items that are further out now have a better chance of being chosen. Meanwhile, shrinkage in the recommender model shifts the system's estimate of the {\it user} inward, so now items closer to the center are likelier to be chosen.

To quantify this intuition, we also study $\VYk$ as another population-level metric, the variance of the matched item positions $\Yk$. This variance has numerous interpretations in the real world: for example, seeing what kinds of items are being selected may guide content creators as they decide what to generate next. Furthermore, there is evidence that users adjust their preferences to better align with content that they are matched with, whether because they were persuaded by the content \citep{diehl2015persuasion}, or by the very fact that it was recommended to them \citep{summers2016ad}. Such dynamics suggest that users may become more heterogeneous if they are matched with a more diverse set of items, and more homogeneous if they are all matched to similar content. With repeated rounds of matching and opinion formation, a great level of heterogeneity in users could lead to polarization or radicalization; conversely, increasing homogeneity could result in filter bubbles and a lack of diverse perspectives for those on the platform.
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\item Your .tex file must compile in PDF\LaTeX{} --- \textbf{ no .ps or .eps figure files.}
\item All fonts must be embedded in the PDF file --- \textbf{ this includes your figures.}
\item Modifications to the style sheet (or your document) in an effort to avoid extra page charges are NOT allowed.
\item No type 3 fonts may be used (even in illustrations).
\item Your title must follow US capitalization rules.
\item \LaTeX{} documents must use the Times or Nimbus font package (do not use Computer Modern for the text of your paper).
\item No \LaTeX{} 209 documents may be used or submitted.
\item Fonts that require non-English language support (CID and Identity-H) must be converted to outlines or removed from the document (even if they are in a graphics file embedded in the document). 
\item Two-column format in AAAI style is required for all papers.
\item The paper size for final submission must be US letter. No exceptions.
\item The source file must exactly match the PDF.
\item The document margins must be as specified in the formatting instructions.
\item The number of pages and the file size must be as specified for your event.
\item No document may be password protected.
\item Neither the PDFs nor the source may contain any embedded links or bookmarks.
\item Your source and PDF must not have any page numbers, footers, or headers.
\item Your PDF must be compatible with Acrobat 5 or higher.
\item Your \LaTeX{} source file (excluding references) must consist of a \textbf{single} file (use of the ``input" command is not allowed.
\item Your graphics must be sized appropriately outside of \LaTeX{} (do not use the ``clip" command) .
\end{itemize}

If you do not follow the above requirements, it is likely that we will be unable to publish your paper.

\section{What Files to Submit}
You must submit the following items to ensure that your paper is published:
\begin{itemize}
\item A fully-compliant PDF file.
\item Your  \LaTeX{}  source file submitted as a \textbf{single} .tex file (do not use the ``input" command to include sections of your paper --- every section must be in the single source file). The only exception is the bibliography, which you may include separately. Your source must compile on our system, which includes the standard \LaTeX{} support files.
\item All your graphics files.
\item The \LaTeX{}-generated files (e.g. .aux and .bib file, etc.) for your compiled source.
\item All the nonstandard style files (ones not commonly found in standard \LaTeX{} installations) used in your document (including, for example, old algorithm style files). If in doubt, include it.
\end{itemize}

Your \LaTeX{} source will be reviewed and recompiled on our system (if it does not compile, you may incur late fees).   \textbf{Do not submit your source in multiple text files.} Your single \LaTeX{} source file must include all your text, your bibliography (formatted using aaai.bst), and any custom macros. Accompanying this source file, you must also supply any nonstandard (or older) referenced style files and all your referenced graphics files. 

Your files should work without any supporting files (other than the program itself) on any computer with a standard \LaTeX{} distribution. Place your PDF and source files in a single tar, zipped, gzipped, stuffed, or compressed archive. Name your source file with your last (family) name.

\textbf{Do not send files that are not actually used in the paper.} We don't want you to send us any files not needed for compiling your paper, including, for example, this instructions file, unused graphics files, and so forth.  A shell script (created by an AAAI member --- it might not work without modification on your system) that might help you create the \LaTeX{} source package is included in the Author Kit.

\section{Using \LaTeX{} to Format Your Paper}

The latest version of the AAAI style file is available on AAAI's website. Download this file and place it in a file named ``aaai.sty" in the \TeX\ search path. Placing it in the same directory as the paper should also work. You must download the latest version of the complete author kit so that you will have the latest instruction set.

\subsection{Document Preamble}

In the \LaTeX{} source for your paper, you \textbf{must} place the following lines as shown in the example in this subsection. This command set-up is for three authors. Add or subtract author and address lines as necessary, and uncomment the portions that apply to you. In most instances, this is all you need to do to format your paper in the Times font. The helvet package will cause Helvetica to be used for sans serif, and the courier package will cause Courier to be used for the typewriter font. These files are part of the PSNFSS2e package, which is freely available from many Internet sites (and is often part of a standard installation).

Leave the setcounter for section number depth commented out and set at 0 unless you want to add section numbers to your paper. If you do add section numbers, you must uncomment this line and change the number to 1 (for section numbers), or 2 (for section and subsection numbers). The style file will not work properly with numbering of subsubsections, so do not use a number higher than 2.


\begin{quote}
\begin{small}
\textbackslash documentclass[letterpaper]{article}\\
\% \textit{Required Packages}\\
\textbackslash usepackage\{aaai\}\\
\textbackslash usepackage\{times\}\\
\textbackslash usepackage\{helvet\}\\
\textbackslash usepackage\{courier\}\\
\textbackslash setlength\{\textbackslash pdfpagewidth\}\{8.5in\}
\textbackslash setlength\{\textbackslash pdfpageheight\}\{11in\}\\
\%\%\%\%\%\%\%\%\%\%\\
\% \textit{PDFINFO for PDF\LaTeX{}}\\
\% Uncomment and complete the following for metadata (your paper must compile with PDF\LaTeX{})\\
\textbackslash pdfinfo\{\\
/Title (Input Your Paper Title Here)\\
/Author (John Doe, Jane Doe)\\
/Keywords (Input your paper's keywords in this optional area)\\
\}\\
\%\%\%\%\%\%\%\%\%\%\\
\% \textit{Section Numbers}\\
\% Uncomment if you want to use section numbers\\
\% and change the 0 to a 1 or 2\\
\% \textbackslash setcounter\{secnumdepth\}\{0\}\\
\%\%\%\%\%\%\%\%\%\%\\
\% \textit{Title, Author, and Address Information}\\
\textbackslash title\{Title\}\\
\textbackslash author\{Author 1 \textbackslash and Author 2\textbackslash\textbackslash \\ 
Address line\textbackslash\textbackslash\\ Address line\textbackslash\textbackslash \\
\textbackslash And\\
Author 3\textbackslash\textbackslash\\ Address line\textbackslash\textbackslash\\ Address line\}\\
\%\%\%\%\%\%\%\%\%\%\\
\% \textit{Body of Paper Begins}\\
\textbackslash begin\{document\}\\
\textbackslash maketitle\\
...\\
\%\%\%\%\%\%\%\%\%\%\\
\% \textit{References and End of Paper}\\
\textbackslash bibliography\{Bibliography-File\}\\
\textbackslash bibliographystyle\{aaai\}\\
\textbackslash end\{document\}
\end{small}
\end{quote}

\subsection{Inserting Document Metadata with \LaTeX{}}
PDF files contain document summary information that enables us to create an Acrobat index (pdx) file, and also allows search engines to locate and present your paper more accurately. \textbf{Document Metadata  for Author and Title are REQUIRED.} 

If your paper includes illustrations that are not compatible with PDF\TeX{} (such as .eps or .ps documents), you will need to convert them. The epstopdf package will usually work for eps files. You will need to convert your ps files to PDF however.

\textit{Important:} Do not include \textit{any} \LaTeX{} code or nonascii characters (including accented characters) in the metadata. The data in the metadata must be completely plain ascii. It may not include slashes, accents, linebreaks, unicode, or any \LaTeX{} commands. Type the title exactly as it appears on the paper (minus all formatting). Input the author names in the order in which they appear on the paper (minus all accents), separating each author by a comma. You may also include keywords in the Keywords field.



\subsection{Preparing Your Paper}

After the preamble above, you should prepare your paper as follows:

\begin{quote}
\begin{small}
\textbackslash begin\{document\}\\
\textbackslash maketitle\\
...\\
\textbackslash bibliography\{Bibliography-File\}\\
\textbackslash bibliographystyle\{aaai\}\\
\textbackslash end\{document\}\\
\end{small}
\end{quote}
\subsection{Incompatible Packages}
The following packages are incompatible with aaai.sty and/or aaai.bst and must not be used (this list is not exhaustive --- there are others as well):
\begin{itemize}
\item hyperref
\item natbib
\item geometry
\item titlesec
\item layout
\item caption
\item titlesec
\item T1 fontenc package (install the CM super fonts package instead)
\end{itemize}

\subsection{Illegal Commands}
The following commands may not be used in your paper:
\begin{itemize}
\item \textbackslash input
\item \textbackslash vspace (when used before or after a section or subsection)
\item \textbackslash addtolength 
\item \textbackslash columnsep
\item \textbackslash top margin (or text height or addsidemargin or even side margin)
\end{itemize}

\subsection{Paper Size, Margins, and Column Width}
Papers must be formatted to print in two-column format on 8.5 x 11 inch US letter-sized paper. The margins must be exactly as follows: 
\begin{itemize}
\item Top margin: .75 inches
\item Left margin: .75 inches
\item Right margin: .75 inches
\item Bottom margin: 1.25 inches
\end{itemize} 


The default paper size in most installations of \LaTeX{} is A4. However, because we require that your electronic paper be formatted in US letter size, you will need to alter the default for this paper to US letter size. Assuming you are using the 2e version of \LaTeX{}, you can do this by including the [letterpaper] option at the beginning of your file: 
\textbackslash documentclass[letterpaper]{article}. 

This command is usually sufficient to change the format. Sometimes, however, it may not work. Use PDF\LaTeX{} and include
\textbackslash setlength\{\textbackslash pdfpagewidth\}\{8.5in\}
\textbackslash setlength\{\textbackslash pdfpageheight\}\{11in\}
in your preamble. 

\textbf{Do not use the Geometry package to alter the page size.} Use of this style file alters aaai.sty and will result in your paper being rejected. 


\subsubsection{Column Width and Margins.}
To ensure maximum readability, your paper must include two columns. Each column should be 3.3 inches wide (slightly more than 3.25 inches), with a .375 inch (.952 cm) gutter of white space between the two columns. The aaai.sty file will automatically create these columns for you. 

\subsection{Overlength Papers}
If your paper is too long, turn on \textbackslash frenchspacing, which will reduce the space after periods. Next,  shrink the size of your graphics. Use \textbackslash centering instead of \textbackslash begin\{center\} in your figure environment. If these two methods don't work, you may minimally use the following. For floats (tables and figures), you may minimally reduce \textbackslash floatsep, \textbackslash textfloatsep, \textbackslash abovecaptionskip, and \textbackslash belowcaptionskip. For mathematical environments, you may minimally reduce \textbackslash abovedisplayskip, \textbackslash belowdisplayskip, and \textbackslash arraycolsep. You may also alter the size of your bibliography by inserting \textbackslash fontsize\{9.5pt\}\{10.5pt\} \textbackslash selectfont
right before the bibliography. 

Commands that alter page layout are forbidden. These include \textbackslash columnsep, \textbackslash topmargin, \textbackslash topskip, \textbackslash textheight, \textbackslash textwidth, \textbackslash oddsidemargin, and \textbackslash evensizemargin (this list is not exhaustive). If you alter page layout, you will be required to pay the page fee \textit{plus} a reformatting fee. Other commands that are questionable and may cause your paper to be rejected include  \textbackslash parindent, and \textbackslash parskip. Commands that alter the space between sections are also questionable. The title sec package is not allowed. Regardless of the above, if your paper is obviously ``squeezed" it is not going to to be accepted. Before using every trick you know to make your paper a certain length, try reducing the size of your graphics or cutting text instead or (if allowed) paying the extra page charge. It will be cheaper in the long run.

\subsection{Figures}
Your paper must compile in PDF\LaTeX{}. Consequently, all your figures must be .jpg, .png, or .pdf. You may not use the .gif (the resolution is too low), .ps, or .eps file format for your figures.

When you include your figures, you must crop them \textbf{outside} of \LaTeX{}. The command \textbackslash includegraphics*[clip=true, viewport 0 0 10 10]{...} might result in a PDF that looks great, but the image is \textbf{not really cropped.} The full image can reappear when page numbers are applied or color space is standardized. 

\subsection{Type Font and Size}
Your paper must be formatted in Times Roman or Nimbus. We will not accept papers formatted using Computer Modern or Palatino or some other font as the text or heading typeface. Sans serif, when used, should be Courier. Use Symbol or Lucida or Computer Modern for \textit{mathematics only. } 

Do not use type 3 fonts for any portion of your paper, including graphics. Type 3 bitmapped fonts are designed for fixed resolution printers. Most print at 300 dpi even if the printer resolution is 1200 dpi or higher. They also often cause high resolution imagesetter devices and our PDF indexing software to crash. Consequently, AAAI will not accept electronic files containing obsolete type 3 fonts. Files containing those fonts (even in graphics) will be rejected. 

Fortunately, there are effective workarounds that will prevent your file from embedding type 3 bitmapped fonts. The easiest workaround is to use the required times, helvet, and courier packages with \LaTeX{}2e. (Note that papers formatted in this way will still use Computer Modern for the mathematics. To make the math look good, you'll either have to use Symbol or Lucida, or you will need to install type 1 Computer Modern fonts --- for more on these fonts, see the section ``Obtaining Type 1 Computer Modern.")

If you are unsure if your paper contains type 3 fonts, view the PDF in Acrobat Reader. The Properties/Fonts window will display the font name, font type, and encoding properties of all the fonts in the document. If you are unsure if your graphics contain type 3 fonts (and they are PostScript or encapsulated PostScript documents), create PDF versions of them, and consult the properties window in Acrobat Reader. 

The default size for your type should be ten-point with twelve-point leading (line spacing). Start all pages (except the first) directly under the top margin. (See the next section for instructions on formatting the title page.) Indent ten points when beginning a new paragraph, unless the paragraph begins directly below a heading or subheading.

\subsubsection{Obtaining Type 1 Computer Modern for \LaTeX{}.}

If you use Computer Modern for the mathematics in your paper (you cannot use it for the text) you may need to download type 1 Computer fonts. They are available without charge from the American Mathematical Society:
http://www.ams.org/tex/type1-fonts.html. 

\subsection{Title and Authors}
Your title must appear in mixed case (nouns, pronouns, and verbs are capitalized) near the top of the first page, centered over both columns in sixteen-point bold type (twenty-four point leading). This style is called ``mixed case." Author's names should appear below the title of the paper, centered in twelve-point type (with fifteen point leading), along with affiliation(s) and complete address(es) (including electronic mail address if available) in nine-point roman type (the twelve point leading). (If the title is long, or you have many authors, you may reduce the specified point sizes by up to two points.) You should begin the two-column format when you come to the abstract. 

\subsubsection{Formatting Author Information}
Author information can be set in a number of different styles, depending on the number of authors and the number of affiliations you need to display. For several authors from the same institution, use \textbackslash and:

\begin{quote}
\begin{small}
\textbackslash author\{Author 1 \textbackslash and  ...  \textbackslash and Author \textit{n}\textbackslash \textbackslash  \\
Address line \textbackslash \textbackslash ~... \textbackslash \textbackslash ~Address line\}
\end{small}
\end{quote}

\noindent If the names do not fit well on one line use:

\begin{quote}
\begin{small}
\textbackslash author\{Author 1\}\textbackslash \textbackslash \\ \{\textbackslash bf Author 2\}\textbackslash \textbackslash ~ 
... \textbackslash \textbackslash ~\{\textbackslash bf Author \textit{n}\}\textbackslash \textbackslash \\
Address line \textbackslash \textbackslash ~ ... \textbackslash \textbackslash ~ Address line\}
\end{small}
\end{quote}

\noindent For authors from different institutions, use \textbackslash And:

\begin{quote}
\begin{small}
\textbackslash author\{Author 1\textbackslash \textbackslash ~ Address line \textbackslash \textbackslash ~...  \textbackslash \textbackslash ~ Address line
\textbackslash And  ...  
\textbackslash And
Author \textit{n}\textbackslash \textbackslash \\ Address line\textbackslash \textbackslash ~
... \textbackslash \textbackslash ~
Address line\}
\end{small}
\end{quote}

\noindent To start a separate ``row" of authors, use \textbackslash AND:
\begin{quote}
\begin{small}
\textbackslash author\{Author 1\textbackslash\textbackslash ~ Address line \textbackslash\textbackslash ~
...  \textbackslash \textbackslash ~ Address line\textbackslash\textbackslash \\
\textbackslash AND\\
Author 2 \textbackslash\textbackslash ~ Address line \textbackslash\textbackslash ~
...  \textbackslash \textbackslash ~ Address line\textbackslash\textbackslash \\
\textbackslash And\\
Author 3 \textbackslash\textbackslash ~ Address line \textbackslash\textbackslash ~
...  \textbackslash \textbackslash ~ Address line\textbackslash\textbackslash \\\}
\end{small}
\end{quote}

\noindent If the title and author information does not fit in the area allocated, place
\textbackslash setlength\textbackslash titlebox\{\textit{height}\}
after the \textbackslash documentclass line where \{\textit{height}\} is something like 2.5in.

\subsection{\LaTeX{} Copyright Notice}
The copyright notice automatically appears if you use aaai.sty. If you are creating a technical report, it is not necessary to include this notice. You may disable the copyright line using the \verb+\+nocopyrightcommand. To change the entire text of the copyright slug, use:
\textbackslash copyrighttext \{\emph{text}\}.
Either of these must appear before \textbackslash maketitle. Please be advised, however, that \textit{if you disable or change the copyright line and transfer of copyright is required, your paper will not be published.}

\subsection{Credits}
Any credits to a sponsoring agency should appear in the acknowledgments section, unless the agency requires different placement. If it is necessary to include this information on the front page, use
\textbackslash thanks in either the \textbackslash author or \textbackslash title commands.
For example:
\begin{quote}
\begin{small}
\textbackslash title\{Very Important Results in AI\textbackslash thanks\{This work is
 supported by everybody.\}\}
\end{small}
\end{quote}
Multiple \textbackslash thanks commands can be given. Each will result in a separate footnote indication in the author or title with the corresponding text at the botton of the first column of the document. Note that the \textbackslash thanks command is fragile. You will need to use \textbackslash protect.

Please do not include \textbackslash pubnote commands in your document.

\subsection{Abstract}
The abstract must be placed at the beginning of the first column, indented ten points from the left and right margins. The title ÒAbstractÓ should appear in ten-point bold type, centered above the body of the abstract. The abstract should be set in nine-point type with ten-point leading. This concise, one-paragraph summary should describe the general thesis and conclusion of your paper. A reader should be able to learn the purpose of the paper and the reason for its importance from the abstract. The abstract should be no more than two hundred words in length. (Authors who are submitting short one- or two-page extended extracts should provide a short abstract of only a sentence or so.) \textbf{Do not include references in your abstract!}

\subsection{Page Numbers}

Do not \textbf{ever} print any page numbers on your paper. 

\subsection{Text }
The main body of the paper must be formatted in ten-point with twelve-point leading (line spacing).

\subsection{Citations}
Citations within the text should include the author's last name and year, for example (Newell 1980). Append lower-case letters to the year in cases of ambiguity. Multiple authors should be treated as follows: (Feigenbaum and Engelmore 1988) or (Ford, Hayes, and Glymour 1992). In the case of four or more authors, list only the first author, followed by et al. (Ford et al. 1997).

\subsection{Extracts}
Long quotations and extracts should be indented ten points from the left and right margins. 

\begin{quote}
This is an example of an extract or quotation. Note the indent on both sides. Quotation marks are not necessary if you offset the text in a block like this, and properly identify and cite the quotation in the text. 

\end{quote}

\subsection{Footnotes}
Avoid footnotes as much as possible; they interrupt the reading of the text. When essential, they should be consecutively numbered throughout with superscript Arabic numbers. Footnotes should appear at the bottom of the page, separated from the text by a blank line space and a thin, half-point rule. 

\subsection{Headings and Sections}
When necessary, headings should be used to separate major sections of your paper. Remember, you are writing a short paper, not a lengthy book! An overabundance of headings will tend to make your paper look more like an outline than a paper.

First-level heads should be twelve-point Times Roman bold type, mixed case (initial capitals followed by lower case on all words except articles, conjunctions, and prepositions, which should appear entirely in lower case), with fifteen-point leading, centered, with one blank line preceding them and three additional points of leading following them. Second-level headings should be eleven-point Times Roman bold type, mixed case, with thirteen-point leading, flush left, with one blank line preceding them and three additional points of leading following them. Do not skip a line between paragraphs. Third-level headings should be run in with the text, ten-point Times Roman bold type, mixed case, with twelve-point leading, flush left, with six points of additional space preceding them and no additional points of leading following them.

\subsubsection{Section Numbers}
The use of section numbers in AAAI Press papers is optional. To use section numbers in \LaTeX{}, uncomment the setcounter line in your document preamble and change the 0 to a 1 or 2. Section numbers should not be used in short poster papers.

\subsubsection{Section Headings.}
Sections should be arranged and headed as follows: 

\subsubsection{Acknowledgments.}
The acknowledgments section, if included, appears after the main body of text and is headed ``Acknowledgments." This section includes acknowledgments of help from associates and colleagues, credits to sponsoring agencies, financial support, and permission to publish. Please acknowledge other contributors, grant support, and so forth, in this section. Do not put acknowledgments in a footnote on the first page. If your grant agency requires acknowledgment of the grant on page 1, limit the footnote to the required statement, and put the remaining acknowledgments at the back. Please try to limit acknowledgments to no more than three sentences. 

\subsubsection{Appendices.}
Any appendices follow the acknowledgments, if included, or after the main body of text if no acknowledgments appear. 

\subsubsection{References}
The references section should be labeled ``References" and should appear at the very end of the paper (don't end the paper with references, and then put a figure by itself on the last page). A sample list of references is given later on in these instructions. Please use a consistent format for references. Poorly prepared or sloppy references reflect badly on the quality of your paper and your research. Please prepare complete and accurate citations.

\subsection{Illustrations and Figures}
Figures, drawings, tables, and photographs should be placed throughout the paper near the place where they are first discussed. Do not group them together at the end of the paper. If placed at the top or bottom of the paper, illustrations may run across both columns. Figures must not invade the top, bottom, or side margin areas. Figures must be inserted using the \textbackslash usepackage\{graphicx\}. Number figures sequentially, for example, figure 1, and so on. 

The illustration number and caption should appear under the illustration. Labels, and other text in illustrations must be at least nine-point type. 

\subsubsection{Low-Resolution Bitmaps.}
You may not use low-resolution (such as 72 dpi) screen-dumps and GIF files---these files contain so few pixels that they are always blurry, and illegible when printed. If they are color, they will become an indecipherable mess when converted to black and white. This is always the case with gif files, which should never be used. The resolution of screen dumps can be increased by reducing the print size of the original file while retaining the same number of pixels. You can also enlarge files by manipulating them in software such as PhotoShop. Your figures should be a minimum of 266 dpi when incorporated into your document.

\subsubsection{\LaTeX{} Overflow.}
\LaTeX{} users please beware: \LaTeX{} will sometimes put portions of the figure or table or an equation in the margin. If this happens, you need to scale the figure or table down, or reformat the equation. Check your log file! You must fix any overflow into the margin (that means no overfull boxes in \LaTeX{}). If you don't, the overflow text will simply be eliminated. \textbf{Nothing is permitted to intrude into the margins.}

\subsubsection{Using Color.}
Your paper will be printed in black and white and grayscale. Consequently, because conversion to grayscale can cause undesirable effects (red changes to black, yellow can disappear, and so forth), we strongly suggest you avoid placing color figures in your document. Of course, any reference to color will be indecipherable to your reader. 

\subsubsection{Drawings.}
We suggest you use computer drawing software (such as Adobe Illustrator or, (if unavoidable), the drawing tools in Microsoft Word) to create your illustrations. Do not use Microsoft Publisher. These illustrations will look best if all line widths are uniform (half- to two-point in size), and you do not create labels over shaded areas. Shading should be 133 lines per inch if possible. Use Times Roman or Helvetica for all figure call-outs. \textbf{Do not use hairline width lines} --- be sure that the stroke width of all lines is at least .5 pt. Zero point lines will print on a laser printer, but will completely disappear on the high-resolution devices used by our printers.

\subsubsection{Photographs and Images.}
Photographs and other images should be in grayscale (color photographs will not reproduce well; for example, red tones will reproduce as black, yellow may turn to white, and so forth) and set to a minimum of 266 dpi. Do not prescreen images.

\subsubsection{Resizing Graphics.}
Resize your graphics \textbf{before} you include them with LaTeX. You may \textbf{not} use trim or clip options as part of your \textbackslash includgraphics command. Resize the media box of your PDF using a graphics program instead. 

\subsubsection{Fonts in Your Illustrations}
You must embed all fonts in your graphics before including them in your LaTeX document.

\subsection{References} 
The aaai.sty file includes a set of definitions for use in formatting references with BibTeX. These definitions make the bibliography style fairly close to the one specified below. To use these definitions, you also need the BibTeX style file ``aaai.bst," available in the author kit on the AAAI web site. Then, at the end of your paper but before \textbackslash end{document}, you need to put the following lines:

\begin{quote}
\begin{small}
\textbackslash bibliographystyle\{aaai\}
\textbackslash bibliography\{bibfile1,bibfile2,...\}
\end{small}
\end{quote}

The list of files in the \textbackslash  bibliography command should be the names of your BibTeX source files (that is, the .bib files referenced in your paper).

The following commands are available for your use in citing references:
\begin{quote}
\begin{small}
\textbackslash cite: Cites the given reference(s) with a full citation. This appears as ``(Author Year)'' for one reference, or ``(Author Year; Author Year)'' for multiple references.\\
\textbackslash shortcite: Cites the given reference(s) with just the year. This appears as ``(Year)'' for one reference, or ``(Year; Year)'' for multiple references.\\
\textbackslash citeauthor: Cites the given reference(s) with just the author name(s) and no parentheses.\\
\textbackslash citeyear: Cites the given reference(s) with just the date(s) and no parentheses.
\end{small}
\end{quote}

\textbf{Warning:} The aaai.sty file is incompatible with the hyperref and natbib packages. If you use either, your references will be garbled.

Formatted bibliographies should look like the following examples.

\smallskip \noindent \textit{Book with Multiple Authors}\\
Engelmore, R., and Morgan, A. eds. 1986. \textit{Blackboard Systems.} Reading, Mass.: Addison-Wesley.

\smallskip \noindent \textit{Journal Article}\\
Robinson, A. L. 1980a. New Ways to Make Microcircuits Smaller. \textit{Science} 208: 1019--1026.

\smallskip \noindent \textit{Magazine Article}\\
Hasling, D. W.; Clancey, W. J.; and Rennels, G. R. 1983. Strategic Explanations in Consultation. \textit{The International Journal of Man-Machine Studies} 20(1): 3--19.

\smallskip \noindent \textit{Proceedings Paper Published by a Society}\\
Clancey, W. J. 1983b. Communication, Simulation, and Intelligent Agents: Implications of Personal Intelligent Machines for Medical Education. In Proceedings of the Eighth International Joint Conference on Artificial Intelligence, 556--560. Menlo Park, Calif.: International Joint Conferences on Artificial Intelligence, Inc.

\smallskip \noindent \textit{Proceedings Paper Published by a Press or Publisher}\\
Clancey, W. J. 1984. Classification Problem Solving. In \textit{Proceedings of the Fourth National Conference on Artificial Intelligence,} 49--54. Menlo Park, Calif.: AAAI Press. 

\smallskip \noindent \textit{University Technical Report}\\
Rice, J. 1986. Poligon: A System for Parallel Problem Solving, Technical Report, KSL-86-19, Dept. of Computer Science, Stanford Univ. 

\smallskip \noindent \textit{Dissertation or Thesis}\\
Clancey, W. J. 1979b. Transfer of Rule-Based Expertise through a Tutorial Dialogue. Ph.D. diss., Dept. of Computer Science, Stanford Univ., Stanford, Calif.

\smallskip \noindent \textit{Forthcoming Publication}\\
Clancey, W. J. 1986a. The Engineering of Qualitative Models. Forthcoming.



\section{Producing Reliable PDF\\Documents with \LaTeX{}}
Generally speaking, PDF files are platform independent and accessible to everyone. When creating a paper for a proceedings or publication in which many PDF documents must be merged and then printed on high-resolution PostScript RIPs, several requirements must be met that are not normally of concern. Thus to ensure that your paper will look like it does when printed on your own machine, you must take several precautions:
\begin{itemize}
\item Use type 1 fonts (not type 3 fonts)
\item Use only standard Times, Nimbus, and CMR font packages (not fonts like F3 or fonts with tildes in the names or fonts---other than Computer Modern---that are created for specific point sizes, like Times\~{}19) or fonts with strange combinations of numbers and letters
\item Embed all fonts when producing the PDF
\item Do not use the [T1]{fontenc} package (install the CM super fonts package instead)
\end{itemize}

\subsection{Creating Output Using PDF\LaTeX{} Is Required}
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\section{Related Work}
The widespread adoption of personalized recommender systems has led to diverse investigations of the potential societal consequences of these systems. One body of literature tackles bias in recommender systems \citep{chen2020survey,engin2013bias}, examining how they may 
systematically underrepresent minority views \citep{stoica2019hegemony}, 
serve predictions of uneven quality  across user groups \citep{yao2017parity}, or fail to recommend valuable items to certain users, e.g., showing job opportunities in STEM fields to fewer women than men \citep{lambrecht2019gender}.
Empirical observations, however, have often been mixed in nature, e.g., documenting how systems sometimes favor long-tail items \citep{fleder2009blockbuster,brynjolfsson2011goodbye} or (over-)favor popular items \citep{abdollahpouri2017popularity}. 

Another topic of societal concern is the possibility that personalized recommendations are pushing individuals into ``filter bubbles'' \citep{pariser2011bubble}. Social media users are known to selectively share content and connect to friends who agree with their existing opinions \citep{an2014selective,garimella2018bipartisan}, and there is some evidence that recommender systems exacerbate this dynamic of ideological segregation \citep{bakshy2015facebook,flaxman2016bubble}. 
Recommender systems seem to have a narrowing effect in other contexts as well: e.g., the sets of movies \citep{nguyen2014movies} and songs \citep{anderson2020spotify} recommended to users tend to be less diverse than the content that users find on their own. 
However, some studies have pointed out that the role of recommender systems is modest compared to the impact of user choice (e.g., whether to click on a recommended story) 
on narrowing consumption diversity \citep{bakshy2015facebook}. Other works have made the case that recommender systems actually \textit{increase} diversity in exposure \citep{flaxman2016bubble} and widen users' interests \citep{hosanagar2013tribes}. 


Part of the reason why it is so challenging to measure the true effects of recommender systems---and perhaps why empirical studies have not been able to reach a consensus---is that we typically only observe user outcomes under the recommender system, but we cannot assess user outcomes in the absence of these systems. Claims, e.g., that users are entering filter bubbles, must be understood relative to some counterfactual baseline. Thus, it is often helpful to design models that enable us to ``observe`` and compare these hypothetical realities. For example, \citet{dandekar2013polarization} show with a model of opinion formation that if users are biased in how they process evidence (drawing undue support for their initial opinion), recommender systems can cause a polarizing effect on users' opinions. \citet{stoica2019hegemony} show under their model that recommender systems accelerate hegemonic dynamics on social media (i.e., when a single viewpoint receives undue attention). 
\citet{perra2019opinion} model the impact of algorithmic personalization on user opinions, 
although the effect also depends on the structure of the users' social network, 
As a final example, \citet{chaney2018confounding} use models to analyze the consequences of feedback loops, when the recommender system is trained on data that was influenced by the system.

Using models to study the impact of recommender systems can be powerful when it is difficult to observe ``what-ifs'' in real life, e.g., what if we removed the feedback loop, what if recommendations were not personalized. 
In the present work, we take a model-based approach to ask arguably the most fundamental ``what-if'' question about recommender systems: what if there were no recommender system at all? 
We analyze our models using first-principles measures such as the expected item match for each user, their expected loss, and the variance over the population of matched items (Section \ref{sec:metrics}). These measures form the building blocks for many of the downstream phenomena of interest related to recommender systems, including polarization, filter bubbles, user satisfaction, user retention, and bias and fairness. Implicitly embedded in these more complex processes are the metrics we study, and thus, our work forms a foundation upon which future researchers can build. 


\section{Simulations} \label{sec:simulations}
\begin{figure*}[tp]
    \centering
    \includegraphics[width=\linewidth]{images/finite_n_simulations.pdf}
        \vspace{-5mm}
    \caption{Simulation results over different values of $n$. As before, we set $\su = \sm = 1$, $\si = \sr = 0.5$, and $m = 300$. For the individual metrics, we run 5000 trials for each model and $n$, and evaluate the mean (left) and variance (middle) of the matched item position $\yki$ for a user positioned at $x_i = 0.75$. For the population metric, we run 500 trials for each setting and evaluate the mean variance over the matched item positions $\mathcal{Y}_k$ (right).
    }
    \label{fig:simulation_over_n}
\end{figure*}
Theorems 1--4 demonstrated that even the simplest versions of the organic and recommender processes result in vastly different matching behavior.
However, one may wonder how sensitive these results are to our assumptions (studying the asymptotic case, with single-dimensional, normally distributed user and item positions). In this section, we remove each of these assumptions, and show via simulation experiments that the key qualitative results we established in this section hold for much more realistic settings as well.

In Section \ref{sec:extensions}, we implement the single-dimensional Gaussian model described in Section \ref{sec:theorems}, and test how the individual-level and population-level metrics converge to their limits as we increase the number of available items. In Section \ref{sec:empirics}, we augment our process models to integrate multidimensional user and item embeddings learned from real ratings data.

\subsection{Exploring finite numbers of items} \label{sec:extensions}



First, in the single-dimensional Gaussian setting we analyze the behavior of our proposed metrics as the number of items $n$ grows, complementing our analytical results in Section \ref{sec:theorems}, where we focused on the asymptotic setting where $n \rightarrow \infty$. 
We consider both the MLE and MAP versions of the organic and recommender models and $n \in \{4, 6, \cdots, 200\}$. 
For the individual metrics, we simulate 5000 stochastic trials for each model and value of $n$, re-sampling per trial the positions of items and noisy samples, but fixing the location of the single hypothetical user to $x_i = 0.75$. For the population metric, we simulate 500 stochastic trials, since there is far less variance in this metric, and re-sample positions of users, items, and noisy samples per trial. 

As shown in Figure \ref{fig:simulation_over_n}, these simulations are consistent with and extend our earlier analyses. In the organic model, when the user employs the MLE, a biased match is formed and the average $\yki$ converges inward of $x_i = 0.75$; as expected, converging at $0.5$, since $\sm = 1$ and $\si$ = 0.5, and $\Eykiomle = (\sm/(\sm+\si)) x_i$ (Theorem 1). Meanwhile, the recommender MLE model forms an unbiased match, and the average $\yki$ converges to $0.75$ when the number of items $n$ reaches around 100. When MAP is used, the roles are reversed, as the organic model becomes unbiased and the recommender model becomes biased. Furthermore, every model converges to a different matched item variance: the recommender MLE model has the highest variance, followed by organic MAP, then organic MLE, then finally recommender MAP (following the order we would expect from Figure \ref{fig:population}, at $\su = \sm = 1$), and this ordering is observed when there are as few as 10 items. 

We also observe that as we increase the number of items $n$, the average $\yki$ and $\VYk$ converge to their limit from the ``inside'' (the side closer to 0). To provide some intuition for this behavior, at the core of our asymptotic analyses for $\Eyki$ is the understanding that when there are infinite items, the user will choose an estimate that lies exactly at $x_i$, regardless of the shape of the item distribution. However, with fewer (finite) items available, the more the item density influences $\Eyki$ and contracts it towards the item mean, which is 0. In the most extreme case, if there is only one item, then $\Eyki = 0$ for all users and $\VYk = 0$, because every user will be matched to the same item. Thus, as the number of items $n$ grows, $\Eyki$ and $\VYk$ converge from the side closer to 0.





\section{Model Analysis} \label{sec:theorems}
To develop a theoretical understanding of how these two models behave, we begin with simplified instances, where we assume that the user and item positions come from Gaussian distributions in a one-dimensional space. 
For now, we will assume that user and item positions are drawn independently from $N(0, \su)$ and $N(0, \sm)$, respectively (since $d=1$, we replace covariance matrices $\Sigma$ with scalar variances, $\sigma^2$). 
We also focus here on the asymptotic setting where the number of items $n$ approaches $\infty$ (in Section \ref{sec:extensions} we examine simulations with different finite values of $n$). Interestingly, we will see that even this simplified setting is sufficient to induce the phenomena we seek to understand, and that the results remain qualitatively similar when we explore non-Gaussian multidimensional user and item positions learned from data (Section \ref{sec:empirics}).

In this section, we organize our results into a series of theorems. In Theorems 1--2, we will analyze how the models differ in terms of individual metrics, by deriving 3 quantities for each model: (1) $\Eyki$, the expected item match; (2) $\Vyki$, the variance in the match; (3) $\mathbb{E}[l_i]$, the expected loss; all for any user $i$ at position $x_i$. In Theorems 3--4, we will then study how the models differ at the population level, showing that even though shrinkage can reduce MSE for both processes, a key difference is that shrinkage increases the variance of matched items under the organic model but decreases this variance under the recommender model.




\paragraph{Theorem 1.1.} In the organic MLE model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykiomle \rightarrow \LimEykiomle$
    \item $\Vykiomle \rightarrow \BigLimVykio$
    \item $\Elomle \rightarrow \left(\frac{\sm}{\sm + \si}-1\right)^2 x_i^2 + \BigLimVykio$.
\end{enumerate}
\begin{proof}
In the organic MLE model, the user makes an estimate of each item position, $\hijmle = \zij$, where $\zij$ is a sample drawn from $N(y_j, \si)$. Since there are an infinite number of items and the support of the item and noise distributions is $\mathbb{R}$, then there must be at least one item estimate located at any given point in $\mathbb{R}$. Thus, when the user chooses the estimate that is closest to her own position, $x_i$, we can expect that the chosen estimate lies at $x_i$. Given the knowledge that the user's chosen item produced a sample at $x_i$, and that the chosen item's position $\ykiomle$ was drawn from $N(0, \sm)$, we can derive a closed form solution for the posterior mean and variance of $\ykiomle$ (since both distributions are Gaussian, forming a conjugate pair), forming the expressions above for $\Eykiomle$ and $\Vykiomle$. 

Finally, recall that in estimation, mean squared error can be decomposed into the sum of the squared bias and the variance of the estimator. If we view the position of the matched item, $\yki$, as an estimate for user $i$'s true position, $x_i$, then the expected squared distance between the two is simply $(\Eykiomle - x_i)^2 + \Vykiomle$, thus yielding $\Elomle$.
\end{proof}

\paragraph{Theorem 1.2.} In the organic MAP model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykiomap \rightarrow x_i$
    \item $\Vykiomap \rightarrow \BigLimVykio$
    \item $\Elomap \rightarrow \BigLimVykio$.
\end{enumerate}
\begin{proof}
When the user uses MAP, her estimate of item $j$'s position becomes $\hijmap = (\sm/(\sm+\si)) \zij$. Since there are an infinite number of items, she will still choose an estimate that is equal to $x_i$; however, this means that the chosen item must have produced a sample at $((\sm + \si)/\sm)x_i$. Again, we can derive a closed form solution for the posterior distribution of the chosen item's position $\ykiomap$, knowing that it was also drawn from $N(0, \sm)$. We find that the terms cancel out, giving us $\Eykio = x_i$, and the variance does not change compared to the MLE version of the model. Since $\Eykio$ is now unbiased, the user's expected loss becomes only the variance.
\end{proof}

\paragraph{Theorem 2.1.} In the recommender MLE model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykirmle \rightarrow x_i$
    \item $\Vykirmle \rightarrow \sr$
    \item $\Elrmle \rightarrow \sr$.
\end{enumerate}
\begin{proof}
In the recommender MLE model, the recommender system makes an estimate of the user's true position, $\hrimle = \zri$, where $\zri$ is a sample drawn from  $N(x_i, \sr)$. Then, the system finds the item whose position is closest to $\hrimle$. Echoing the proof in Theorem 1, if there are an infinite number of items, then there must be at least one item located at any given point in $\mathbb{R}$. Thus, the chosen item position, $\ykirmle$, must be equal to $\hrimle$, and so the distribution of $\ykirmle$ reduces to the distribution of $\hrimle$, which has an expected value $x_i$ and variance $\sr$. Since $\ykirmle$ forms an unbiased estimate for $x_i$, the expected loss for the user is again just the variance.
\end{proof}

\paragraph{Theorem 2.2.} In the recommender MAP model as $n \rightarrow \infty$,
\begin{enumerate}
    \item $\Eykirmap \rightarrow \frac{\su}{\su + \sr} x_i$
    \item $\Vykirmap \rightarrow \left(\frac{\su}{\su + \sr}\right)^2 \sr$
    \item $\Elrmap \rightarrow \left(\frac{\su}{\su + \sr} - 1\right)^2 x_i^2 + \left(\frac{\su}{\su+\sr}\right)^2 \sr$.
\end{enumerate}
\begin{proof}
In the MAP setting, the recommender model's estimate of the user's position becomes $\hrimap = (\su/(\su + \sr)) \zri$. Since there are an infinite number of items, it is still true that the chosen item's position, $\ykirmap$, will match $\hrimap$ perfectly, and so the distribution of $\ykirmap$ reduces to the distribution of $\hrimap$. Since $\hrimap$ is simply a sample drawn from $N(x_i, \sr)$ scaled by $\su/(\su + \sr)$, then $\hrimap$ must be normally distributed with mean $(\su/(\su + \sr))x_i$ and variance $(\su/(\su + \sr))^2 \sr$. This provides the expressions above for $\Eykirmap$ and $\Vykirmap$, and the expected squared loss is the sum of the squared bias, $(\Eykirmap - x_i)^2$, and the variance $\Vykirmap$.
\end{proof}

\begin{figure}[]
    \centering
    \includegraphics[width=0.8\linewidth]{images/expected_user_loss.pdf}
    \caption{Expected user loss as a function of user position (Theorems 1--2); $\sm = 1$ and $\si = \sr = 0.5$.}
    \label{fig:individual_mse}
\end{figure}

Theorems 1 and 2 imply that if we treat the matched item position $\yki$ as an estimate of the user position $x_i$, then the organic MLE model forms biased matches for users, while the recommender MLE model does not. In particular, the organic MLE model is biased in that, relative to $x_i$, the expected match for user $i$ is contracted towards the center of the item distribution (Theorem 1.1). Furthermore, since the bias term grows with $x_i^2$, this means that users closer to the center will be less impacted by the bias than users who are farther out. This is a fundamental difference between the MLE versions of the processes: the organic model favors users who are closer to the center of the item distribution, while the recommender model is agnostic to user position, as we show in Figure \ref{fig:individual_mse}. 
Interestingly, the trend reverses when we switch from MLE to MAP: the organic model becomes  unbiased, and the recommender model becomes favorable to central users. Some of this effect is particular to our form of shrinkage here (namely, that the user perfectly erases the bias in the organic model by taking the MAP based on the true prior distribution of items), but in general, shrinkage will always increase bias for the recommender model, and, if the item distribution is heaviest around its mean, shrinkage will also reduce bias for the organic model.

From Theorems 1 and 2, we can also establish that switching from MLE to MAP will \textit{always} reduce mean user loss (i.e., MSE) in the organic model, and that it can sometimes reduce MSE in the recommender model. First, Theorem 1 shows that in the organic model, expected loss strictly decreases for every user (with non-zero $x_i$) from MLE to MAP; thus, the mean loss must fall as well. From Theorem 2.2, we can imagine the conditions under which MAP will achieve a lower MSE in the recommender model: if the user variance $\su$ is small, then the bias terms will be smaller because all of the $x_i^2$'s will be closer to $0$, and the relative reduction that MAP achieves on the variance, scaling by a factor of $(\frac{\su}{\su + \sr})^2$, will be larger. More generally, we can conclude that there are certainly reasons to switch from MLE to MAP for both models, which makes the consequences of the switch all the more interesting -- the individual-level consequences which we have analyzed, in terms of biased matches and favored users, and now population-level consequences, which we analyze below.





\paragraph{Theorem 3.} In the organic model as $n \rightarrow \infty$, 
\begin{enumerate}
    \item $\VYkomle \rightarrow \frac{m-1}{m}\left[(\frac{\sm}{\sm + \si})^2 \su + \LimVykio\right]$
    \item $\VYkomap \rightarrow \frac{m-1}{m}\left[\su + \LimVykio\right]$.
\end{enumerate}
\begin{proof}
In general, we can think of each chosen item position $\yki \in \Yk$ as the sum of its expected value conditioned on $x_i$, $\Eyki$, and some independent random noise $\epsilon$ with variance $\Vyki$ (which does not depend on $x_i$ in any of our models). 

Using this approach, the process of generating terms in $\Ykomle$ in the organic MLE model can be seen as first drawing a random user position $x$ from $N(0, \su)$, scaling it by $\sm/(\sm + \si)$, then adding this noise term $\epsilon$. We scale $x$ because $\Eykiomle = \sm/(\sm + \si)x_i$ (Theorem 1.1). This output becomes the sum of two random variables, where the first term is the scaled $x$ and the second is $\epsilon$. The variances of these terms are $(\sm/(\sm + \si))^2 \su$ and $\Vykio = \LimVykio$, respectively, so the expected variance of $\Ykomle$ is the sum of these expressions multiplied by $(m-1)/m$, because there are $m$ elements in $\Ykomle$.

In the organic MAP case, the process of generating terms in $\Ykomap$ can also be seen as first drawing a random user position $x$ from $N(0, \su)$, then adding a noise term $\epsilon$. We do not scale $x$ in this case, since $\Eykiomap = x_i$ (Theorem 1.2). The output is again the sum of two random variables, where the first term is $x$ and the second is $\epsilon$. Their variances are $\su$ and $\LimVykio$, respectively, so the expected variance of $\Ykomap$ is their sum scaled by $(m-1)/m$. 
\end{proof}

\paragraph{Theorem 4.} In the recommender model as $n \rightarrow \infty$, 
\begin{enumerate}
    \item $\VYkrmle \rightarrow \frac{m-1}{m}(\su + \sr)$
    \item $\VYkrmap \rightarrow \frac{m-1}{m}\left[\frac{\su}{\su+\sr} \su \right]$.
\end{enumerate}
\begin{proof}
We can use the same approach from Theorem 3 to derive $\VYk$ here. In the recommender MLE model, the process of generating terms in $\Ykrmle$ can be seen as drawing a random user position $x$ from $N(0, \su)$ then adding a noise term $\epsilon$ that has variance $\Vykirmle = \sr$ (Theorem 2.1). We do not scale $x$ here either because $\Eykirmle = x_i$. The sum of these terms' variances is $\su + \sr$, and again we scale this by $(m-1)/m$. 

In the recommender MAP model, to generate terms in $\Ykrmap$, we draw a random user position $x$ from $N(0, \su)$, scale it by $\su/(\su + \sr)$, then add a noise term $\epsilon$ (Theorem 2.2). The variance of the scaled $x$ is $(\su/(\su + \sr))^2 \su$ and the variance of $\epsilon$ is $\Vykirmap = (\su/(\su + \sr))^2 \sr$. 
Adding these terms together, simplifying, and multiplying by $(m-1)/m$ yields expression for $\VYkrmap$.
\end{proof}

\begin{figure}[]
    \centering
    \includegraphics[width=\linewidth]{images/varYk.pdf}
    \caption{Variance of matched items $\VYk$ (Theorems 3--4); $\si = \sr = 0.5$ and $m=300$. On the left, we fix $\sm = 1$ and vary $\su$; on the right, we fix $\su = 1$ and vary $\sm$.}
    \label{fig:population}
\end{figure}

From Theorems 3 and 4, we can see that the variance of matched items will \textit{always} increase in the organic model when switching from MLE to MAP, but it will \textit{always} decrease in the recommender model when making the same switch. 
We demonstrate these two effects in Figure \ref{fig:population}: across different values for the user variance $\su$ and item variance $\sm$, the matched item variance for the recommender MLE model remains well above that of the recommender MAP, and the matched item variance for the organic MLE model is always below that of the organic MAP. This quantifies our earlier intuition from Figure \ref{fig:model} that shrinkage acts as a diversifying force in the organic model (increasing variance), but a homogenizing force in the recommender model (decreasing variance), and we have proven that this is always true in this setting.


