4.1. Laplacian regularization

We begin by describing how to incorporate network information in a choice model like MNL through Laplacian regularization (Ando & Zhang, Reference Ando and Zhang2007). Laplacian regularization encourages parameters corresponding to connected nodes to be similar through a loss term of the form $\lambda \boldsymbol{\alpha }^TL \boldsymbol{\alpha }$ , where $L$ is the graph Laplacian (as defined in Section 3), $\boldsymbol{\alpha }$ is the vector of parameter values for each node, and $\lambda$ is the scalar regularization strength. A famous identity is that $\boldsymbol{\alpha }^TL \boldsymbol{\alpha } = \sum _{(i, j) \in E} (\boldsymbol{\alpha }_i - \boldsymbol{\alpha }_j)^2$ , which more clearly shows the regularization of connected nodes’ parameters towards each other. This also shows that the Laplacian is positive semi-definite, since $\boldsymbol{\alpha }^TL \boldsymbol{\alpha } \ge 0$ , which will be useful to preserve the convexity of the multinomial logit’s (negative) log-likelihood.

The idea of using Laplacian regularization for discrete choice was proposed in (Zhang et al., Reference Zhang, Fountoulakis, Cao, Mahoney and Pozdnoukhov2017) (although they focused on regularizing intercept terms in binary logistic regression). We generalize the idea to be applicable to any logit-based choice model and show that it corresponds to Bayesian inference with a network correlation prior. We then specialize to the models we use in our experiments. Laplacian regularization is simple to implement, can be added to any logit-based choice model with per-chooser parameters, and only requires training one extra hyperparameter. Laplacian regularization also carries a number of advantages over another approach to accounting for structured preference heterogeneity, mixture modeling.

4.1.1. Theory of Laplacian-regularized choice models

Consider a general choice model, as in Equation (1). We split the parameters $\boldsymbol{\theta }$ into two sets $\theta _{\mathcal{A}}$ and $\theta _{G}$ , where parameters $\boldsymbol{\alpha } \in \theta _{\mathcal{A}}, \boldsymbol{\alpha }\in{\mathbb{R}}^n$ vary over choosers and parameters $\beta \in \theta _{G}, \beta \in{\mathbb{R}}$ are constant over choosers. The log-likelihood of a general choice model is

(2) \begin{equation} \ell (\boldsymbol{\theta };\;{\mathcal{D}}) = \sum _{(i, a, C) \in{\mathcal{D}}} \Big [\log (u_{\boldsymbol{\theta }}(i, C, a)) - \log \sum _{j \in C} \exp\!(u_{\boldsymbol{\theta }}(j, C, a))\Big ]. \end{equation}

The Laplacian- and $L_2$ -regularized log-likelihood (with $L_2$ regularization strength $\gamma$ ) is then

(3) \begin{align} \ell _L(\boldsymbol{\theta };\;{\mathcal{D}}) &= \ell (\boldsymbol{\theta };\;{\mathcal{D}}) - \frac{\lambda }{2}\sum _{\boldsymbol{\alpha } \in \theta _{\mathcal{A}}} \boldsymbol{\alpha }^T L \boldsymbol{\alpha } -\frac{\gamma }{2} \sum _{\boldsymbol{\alpha } \in \theta _{\mathcal{A}}} ||\boldsymbol{\alpha }||_2^2. \end{align}

We show that regularized maximum likelihood estimation of $\boldsymbol{\theta }$ corresponds to Bayesian inference with a prior on per-chooser parameters that encourages smoothness over the network. In contrast, existing results on priors for semi-supervised regression (Xu et al., Reference Xu, Dyer and Owen2010; Chin et al., Reference Chin, Chen, Altenburger and Ugander2019) typically split the nodes into observed and unobserved, fixing the observed values and only considering randomness over unobserved nodes. In choice modeling, observing choices at a node only updates our beliefs about their preferences, leaving some uncertainty. Our result also allows some parameters of the choice model to be chooser dependent and others to be constant across choosers, allowing it to be fully general over choice models. Finally, we note that $L_2$ regularization can also be applied to the global parameters $\beta$ , which as usual corresponds to a Gaussian prior on these parameters—however, we state the result with uniform priors to emphasize the Laplacian regularization on the per-chooser parameters $\boldsymbol \alpha$ .

Theorem. The maximizer $\boldsymbol{\theta }^*_{\text{MLE}}$ of the Laplacian- and $L_2$ -regularized log-likelihood $\ell _L(\boldsymbol{\theta };\;{\mathcal{D}})$ is the maximum a posteriori estimate $\boldsymbol{\theta }^*_{\text{MAP}}$ after observing $\mathcal{D}$ under the i.i.d. priors $\boldsymbol{\alpha } \sim \mathcal{N}(0, [\lambda L + \gamma I] ^{-1})$ for each $\boldsymbol{\alpha } \in \theta _{\mathcal{A}}$ and i.i.d. uniform priors for each $\beta \in \theta _{G}$ .

Proof. First, recall that $L$ is positive semi-definite, so $\lambda L + \gamma I$ (with $\gamma, \lambda \gt 0$ ) is positive definite and invertible. Now, using Bayes’ Theorem,

\begin{align*} \boldsymbol{\theta }^*_{\text{MAP}} &= \textrm{argmax}_{\boldsymbol{\theta }} p(\boldsymbol{\theta } \mid{\mathcal{D}})\\[5pt] &= \textrm{argmax}_{\boldsymbol{\theta }} \frac{\Pr ({\mathcal{D}} \mid \boldsymbol{\theta }) p(\boldsymbol{\theta })}{\Pr ({\mathcal{D}})}. \end{align*}

Since $\Pr ({\mathcal{D}})$ is independent of the parameters and $\log$ is monotonic and increasing,

\begin{align*} \boldsymbol{\theta }^*_{\text{MAP}} &= \textrm{argmax}_{\boldsymbol{\theta }} \!\left [\log \Pr ({\mathcal{D}} \mid \boldsymbol{\theta })+ \log p(\boldsymbol{\theta })\right ]. \end{align*}

Notice that the first term is exactly the log-likelihood $\ell (\boldsymbol{\theta };\;{\mathcal{D}})$ . Additionally, the priors of each parameter are independent, so

\begin{align*} \log p(\boldsymbol{\theta }) &= \sum _{\boldsymbol{\alpha } \in \theta _{\mathcal{A}}} \log p(\boldsymbol{\alpha }) + \sum _{\beta \in \theta _{G}} \log p(\beta ). \end{align*}

Since the priors $p(\beta )$ are uniform, they do not affect the maximizer:

\begin{align*} \boldsymbol{\theta }^*_{\text{MAP}} &= \textrm{argmax}_{\boldsymbol{\theta }} \!\left [\ell (\boldsymbol{\theta };\;{\mathcal{D}})+ \sum _{\boldsymbol{\alpha } \in \theta _{\mathcal{A}}} \log p(\boldsymbol{\alpha })\right ]. \end{align*}

Now consider the Gaussian priors $p(\boldsymbol{\alpha })$ :

\begin{align*} p(\boldsymbol{\alpha }) &= (2\pi )^{n/2} \det \![(\lambda L + \gamma I) ^{-1}]^{-1/2} \exp\! \left (-\frac{1}{2} \boldsymbol{\alpha }^T [(\lambda L + \gamma I) ^{-1}]^{-1}\boldsymbol{\alpha }\right ). \end{align*}

Simplifying the term in the $\exp$ reveals the two regularization terms:

\begin{align*} -\frac{1}{2} \boldsymbol{\alpha }^T [(\lambda L + \gamma I) ^{-1}]^{-1}\boldsymbol{\alpha }&=-\frac{1}{2} \boldsymbol{\alpha }^T (\lambda L+\gamma I) \boldsymbol{\alpha }\\[5pt] &= -\frac{\lambda }{2} \boldsymbol{\alpha }^T L\boldsymbol{\alpha } -\frac{\gamma }{2} \boldsymbol{\alpha }^T\boldsymbol{\alpha } \\[5pt] &= -\frac{\lambda }{2} \boldsymbol{\alpha }^T L\boldsymbol{\alpha } -\frac{\gamma }{2} ||\boldsymbol{\alpha }||_2^2. \end{align*}

We thus have for a constant $c$ independent of $\boldsymbol \alpha$ ,

\begin{align*} \log p(\boldsymbol \alpha ) &= \log \!\left ((2\pi )^{n/2} \det \![(\lambda L + \gamma I) ^{-1}]^{-1/2} \exp \!\left (-\frac{\lambda }{2} \boldsymbol{\alpha }^T L\boldsymbol{\alpha } -\frac{\gamma }{2} ||\boldsymbol{\alpha }||_2^2 \right )\right )\\[5pt] &= -\frac{\lambda }{2} \boldsymbol{\alpha }^T L\boldsymbol{\alpha } -\frac{\gamma }{2} ||\boldsymbol{\alpha }||_2^2 + c. \end{align*}

Plugging this is in and dropping the constants not affecting the maximizer yields

\begin{align*} \boldsymbol{\theta }^*_{\text{MAP}} &= \textrm{argmax}_{\boldsymbol{\theta }} \!\left [\ell (\boldsymbol{\theta };\;{\mathcal{D}}) - \frac{\lambda }{2} \sum _{\boldsymbol{\alpha } \in \theta _{\mathcal{A}}} \boldsymbol{\alpha }^T L \boldsymbol{\alpha } -\frac{\gamma }{2} \sum _{\boldsymbol{\alpha } \in \theta _{\mathcal{A}}} ||\boldsymbol{\alpha }||_2^2 \right ]\\[5pt] &= \textrm{argmax}_{\boldsymbol{\theta }} \ell _L(\boldsymbol{\theta };\;{\mathcal{D}})\\[5pt] &= \boldsymbol{\theta }^*_{\text{MLE}}. \end{align*}

Notice that the Gaussian in the theorem above has precision (i.e., inverse covariance) matrix $\lambda L + \gamma I$ . The partial correlation between the per-chooser parameters $\boldsymbol{\alpha }_i$ and $\boldsymbol{\alpha }_j$ , $i \ne j$ (controlling for all other nodes) is therefore

(4) \begin{equation} - \frac{\lambda L_{ij}}{\sqrt{(\lambda L_{ii} + \gamma ) (\lambda L_{jj} + \gamma )}} = \frac{\lambda A_{ij}}{\sqrt{(\lambda d_i + \gamma ) (\lambda d_j + \gamma )}} \end{equation}

using the standard Gaussian identity relating precision and partial correlation (Liang et al., Reference Liang, Song and Qiu2015) (where $d_i$ is the degree of $i$ ). If both $d_i, d_j \gt 0$ and $\gamma$ is small, then we can approximate

(5) \begin{equation} \frac{\lambda A_{ij}}{\sqrt{(\lambda d_i + \gamma ) (\lambda d_j + \gamma )}} \approx \frac{\lambda A_{ij}}{\sqrt{(\lambda d_i) (\lambda d_j)}} = \frac{A_{ij}}{\sqrt{d_i d_j}}. \end{equation}

This is easy to interpret: $\boldsymbol{\alpha }_i$ and $\boldsymbol{\alpha }_j$ have partial correlation 0 when $i$ and $j$ are unconnected ( $A_{ij} = 0$ ) and positive partial correlation when they are connected (larger when they have fewer other neighbors). That is, the Gaussian prior in the theorem assumes neighboring nodes have correlated preferences.

4.2. Graph neural networks

GNNs (Wu et al., Reference Wu, Pan, Chen, Long, Zhang and Philip2020) use a graph to structure the aggregations performed by a neural network, allowing parameters for neighboring nodes to influence each other. We test the canonical GNN, a graph convolutional network (GCN) (Kipf & Welling, Reference Kipf and Welling2017), where node embeddings are averaged across neighbors before each neural network layer. There are many other types of GNNs (see Wu et al. (Reference Wu, Pan, Chen, Long, Zhang and Philip2020) for a survey)—we emphasize that we do not claim this particular GCN approach to be optimal for discrete choice. Rather, we illustrate how GNNs can be applied to choice data and encourage further exploration.

In a depth- $d$ GCN, each layer performs the following operation, producing a sequence of embeddings $H^{(0)}, \dots, H^{(d)}$ :

(6) \begin{equation} H^{(i+1)} = \sigma (A'H^{(i)}W^{(i)}) \end{equation}

where $H^{(0)}$ is initialized using node features (if they are available—if not, $H^{(0)}$ is learned), $\sigma$ is an activation function, $W^{(i)}$ are parameters, and $A' = (D+2I)^{-{1}/{2}}(A+I)(D+2I)^{-{1}/{2}}$ is the degree-normalized adjacency matrix (with self-loops). Self-loops are added to $G$ to allow a node’s current embedding to influence its embedding in the next layer. We can either use $H^{(d)}$ as the final embeddings or concatenate each layer’s embedding into a final embedding $H$ . In our experiments, we use a two-layer GCN (both with output dimension 16) and concatenate the layer embeddings. For simplicity, we fix the dropout rate at 0.5.

To apply GCNs to discrete choice, we can treat the final node embeddings as chooser features and apply an MNL, modeling utilities as $u_\theta (i, C, a) = u_i + H_a^T\boldsymbol{\gamma}_{\boldsymbol{i}}$ , where $u_i$ and $\boldsymbol{\gamma}_{\boldsymbol{i}}$ are per-item parameters (the intercept and embedding coefficients, respectively). If item features are also available, we add the CL term $\boldsymbol{\theta }^T \boldsymbol{y}_{\boldsymbol{i}}$ . Thanks to automatic differentiation software such as PyTorch (Paszke et al., Reference Paszke, Gross, Massa, Lerer, Bradbury, Chanan and Chintala2019), we can train both the GCN and MNL/CML weights end-to-end. Again, any node representation learning method could be used for the embeddings $H$ —we use a GCN for simplicity.

In general, GNNs have the advantage of being highly flexible, able to capture complex interactions between the features of neighboring nodes. However, some recent research has indicated that nonlinearity is less helpful for classification in GNNs than in traditional neural networks tasks (Wu et al., Reference Wu, Souza, Zhang, Fifty, Yu and Weinberger2019). With the additional modeling power comes significant additional difficulty in training and hyperparameter selection (for embedding dimensions, depth, dropout rate, and activation function).

4.3. Choice fraction propagation

We also consider a baseline method that uses the graph to directly derive choice predictions, without a probabilistic model of choice. We extend label propagation (Zhou et al., Reference Zhou, Bousquet, Lal, Weston and Schölkopf2004; Jia & Benson, Reference Jia and Benson2021) to multialternative discrete choice. The three features distinguishing the choice setting from standard label propagation are that we can observe multiple “labels” (i.e., choices) per chooser, each observation may have had different available labels, and that not all labels are available at inference time. Given training data of observed choices of the form $(i, C, a)$ , where chooser $a \in{\mathcal{A}}$ chose item $i\in C \subseteq{\mathcal{U}}$ , we assign each chooser $a$ a vector $\boldsymbol{z}_{\boldsymbol{a}}$ of size $k = |U|$ with each item’s choice fraction. That is, the $i$ th entry of $\boldsymbol{z}_{\boldsymbol{a}}$ stores the fraction of times $a$ chose $i$ in the observed data out of all opportunities they had to do so (i.e., the number of times $i$ appeared in their choice set). We use choice fraction rather than counts to normalize by the number of observations for a chooser and not to count against an item instances when it was not available.

We then apply label propagation to the vectors $\boldsymbol{z}_{\boldsymbol{a}}$ over $G$ . Let $Z^{(0)}$ be the matrix whose rows are $\boldsymbol{z}_{\boldsymbol{a}}$ . As in standard label propagation, we iterate $ Z^{(i+1)} \gets (1-\rho ) Z^{(0)} + \rho D^{-{1}/{2}}AD^{-{1}/{2}} Z^{(i)}$ until convergence, where $\rho \in [0, 1]$ is a hyperparameter that controls the strength of the smoothing. Let $ Z^{(\infty )}$ denote the stationary point of the iterated map. For inference, we can use the $a$ th row of $Z^{(\infty )}$ (in practice, we will have a matrix arbitrarily close to $Z^{(\infty )}$ ), denoted $\boldsymbol{z}_{\boldsymbol{a}}^{(\boldsymbol\infty )}$ , to make predictions for chooser $a$ . Given a choice set $C$ , we predict $a$ will choose the argmax of $\boldsymbol{z}_{\boldsymbol{a}}^{(\boldsymbol\infty )}$ restricted to items appearing in $C$ . Note that in a semi-supervised setting, we do not observe any choices from the test choosers, so their entries of $Z^{(0)}$ will be zero. The term $(1-\rho )Z^{(0)}$ then acts as a uniform prior, regularizing the test chooser entries of $Z^{(\infty )}$ toward 0. Since choice fraction propagation does not use chooser or item features, it is best suited to scenarios where neither are available.

5.1. Friends and Family app data

The Friends and Family dataset (Aharony et al., Reference Aharony, Pan, Ip, Khayal and Pentland2011) follows over 400 residents of a young-family community in North America during 2010-2011. The dataset is remarkably rich, capturing many aspects of the participants’ lives. For instance, they were given Android phones with purpose-made logging software that captured app installation and usage as well as Bluetooth pings between participants’ phones. We use the installation and usage data to construct two separate choice datasets (app-install and app-usage) and use a network built from Bluetooth pings, as in (Aharony et al., Reference Aharony, Pan, Ip, Khayal and Pentland2011). We ignore uncommon and built-in apps (for instance, we ignore apps whose package names begin with com.android, com.motorola, com.htc, com.sec, and com.google), leaving a universe $\mathcal{U}$ of 127 apps in app-install and 121 in app-usage (e.g., Twitter, Facebook, and Myspace).

To construct app-install, we use scans that checked which apps were installed on each participant’s phone every 10 minutes—3 hours. Each time a new app $i$ appears in a scan for a participant $a$ , we consider that a choice from the set of apps $C$ that were not installed at the time of the last scan. We use a plain logit as the baseline model in app-install, since no item features are readily available. To construct app-usage, we use 30-second resolution scans of running apps. To separate usage into sessions, we select instances where a participant ran an app for the first time in the last hour. We consider such app runs to be a choice $i$ from the set of all apps $C$ installed on participant $a$ ’s phone at that time. Our discrete choice approach enables us to account for these differences in app availability. In app-usage, we use a CL with a single instance-specific item feature: recency, defined as $\log ^{-1}(\text{seconds since last use})$ or 0 if the user has not used the app. While it would be possible to construct more complex sets of features with additional effort (for instance categorizing different types of apps or tracking down their Android store ratings), a simple baseline suffices to demonstrate how social network structure can benefit choice modeling even in the absence of item and user features.

To form the social network $G$ over participants for both datasets, we use Bluetooth proximity hits—like the original study (Aharony et al., Reference Aharony, Pan, Ip, Khayal and Pentland2011), we only consider hits in the month of April between 7 am and midnight (to avoid coincidental hits from neighbors at night). For each participant $a$ , we form the link $(a, b)$ to each of their 10 most common interaction partners $b$ (we also tested thresholds 2–9, but our methods all performed very similarly). We perform this thresholding because the Bluetooth ping network is extremely dense and contains many edges that are likely not socially meaningful (for instance, nearby phones may ping each other when two strangers shop in the same store). Prior research on this data found that social contacts were useful in predicting app installations, but did not employ a discrete choice approach (Aharony et al., Reference Aharony, Pan, Ip, Khayal and Pentland2011; Pan et al., Reference Pan, Aharony and Pentland2011). Our discrete choice approach allows us to account for multi-hop social connections and the context of each installation (i.e., what apps were already installed).

As a warm-up data analysis, we show that people are more likely to install an app the more of their friends have it (but not if we randomize friendships). Let $n$ the total number of people, $n_i$ be the number of people who installed application $i$ , $f_a$ the number of friends of person $a$ , and $f_{ai}$ the number of friends of person $a$ who have app $i$ . Suppose app installations are independent of friendships. If we sample some person $a$ uniformly at random and check which of their friends have app $i$ , then the probability that $a$ also has app $i$ is $(n_i - f_{ai})/(n-f_a)$ (simply the remaining fraction of people who have the app, after observing the friends of $a$ ). However, if app installations correlate across friendships, the observed probability would be higher when $f_{ai}/ f_a$ is larger. We measure the empirical probability that a person has an app at different friend-installation fractions. Specifically, we measure

(7) \begin{equation} \frac{1}{nk} \sum _{i \in U, a\in{\mathcal{A}}} \left (\boldsymbol{1}_{ai} - \frac{n_i - f_{ai}}{n-f_a}\right ), \end{equation}

where $\boldsymbol{1}_{ai}$ is an indicator for whether person $a$ has app $i$ . Notice that if friendships are uncorrelated with app installations, the expectation of the summand is $0$ . Instead of taking the mean over all app pairs, we take the mean at each unique friend-installation fraction to see if having more friends with an app results in stronger deviations from uniform installations. This is exactly what we observe: when people have more friends with an app, they are more likely to install it (Figure 1). In contrast with two null models (a configuration model with the same degree distribution and an Erdős–Rényi graph with the same density), we see an increase in peoples’ installation probabilities as a larger fraction of their friends have an app. This is in line with findings that the probability an individuals joins a social network community increases with the number of their friends in the community (Backstrom et al., Reference Backstrom, Huttenlocher, Kleinberg and Lan2006). However, it is worth emphasizing that this finding is purely correlational—we have no way of knowing whether increased installation rates are due to homophily in the social network, word-of-mouth contagion, or other confounding factors.

Figure 1. Difference between actual and expected install rates (if friendships were irrelevant). The left subplot is with the real network, while the right two are two null models. Each dot is a (participant, app) pair. The black line marks the mean over five bins, with the shaded region showing the standard error of the mean.

5.2. County-level US presidential election data

US presidential election data is a common testbed for graph learning methods using a county-level adjacency network, but the typical approaches are to treat elections as binary classification or regression problems (predicting the vote shares of one party) (Jia & Benson, Reference Jia and Benson2020; Zhou et al., Reference Zhou, Dong, Wang, Lee, Hooi, Xu and Feng2020; Huang et al., Reference Huang, He, Singh, Lim and Benson2020). However, this ignores the fact that voters have more than two options—moreover, different candidates can be on the ballot in different states. The universe of items $\mathcal{U}$ in our 2016 election data contains no fewer than 31 different candidates (and a “none of these candidates” option in Nevada, which received nearly 4% of the votes in one county). While third-party candidates are unlikely to win in the US, they often receive a nontrivial (and quite possibly consequential) fraction of votes. For instance, in the 2016 election, independent candidate Evan McMullin received 21.5% of the vote in Utah, while Libertarian candidate Gary Johnson and Green Party candidate Jill Stein received 3.3% and 1.1% nationally (the gap between Clinton and Trump was only 2%). A discrete choice approach enables us to include third-party candidates and account for different ballots in different states. As a visual example, in Figure 2, we show the states in which McMullin appeared on the ballot as well as his per-county vote share. By accounting for ballot variation, we can make counterfactual predictions about what would happen if different candidates had been on the ballot, which is difficult without a discrete choice framework. For example, given McMullin’s regional support in Utah, it is possible that he would have fared better in Nevada (Utah’s western neighbor) than in an East Coast state like New York. Using the entire ballots also allows us to account for one possible reason why McMullin’s vote share appears not to spilled over into Colorado, while it did into Idaho: Colorado had fully 22 candidates on the ballot, while Idaho only had 8. A discrete choice approach handles this issue cleanly, while regression on vote shares does not. We note that, due to inherent limitations of observational data, we cannot be sure of the causes of the effects we observe (Tomlinson et al., Reference Tomlinson, Ugander and Benson2021)—nonetheless, a discrete choice approach enables more flexible modeling and can improve prediction performance regardless of the cause of preference correlations.

Figure 2. 2016 US presidential election vote shares for conservative independent Evan McMullin. Notice his regional popularity and the spillover from Utah to southeast Idaho. McMullin was not on the ballot in filled-in states. The lack of spillover into Colorado may be due to its crowded field (22 candidates) or because it is less conservative than Idaho.

We gathered county-level 2016 presidential voting data from (Kearney, Reference Kearney2018) and county data from (Jia & Benson, Reference Jia and Benson2020),Footnote ³ which includes a county adjacency network, county-level demographic data (e.g., education, income, birth rates, USDA economic typology,Footnote ⁴ and unemployment rates), and the Social Connectedness Index (SCI) (Bailey et al., Reference Bailey, Cao, Kuchler, Stroebel and Wong2018) measuring the relative frequency of Facebook friendships between each pair of counties. We aggregate all votes at the county level, treating each county as a chooser $a$ and using county features as $\boldsymbol{x}_a$ [modeling voting choices in aggregate is standard practice (Alvarez & Nagler, Reference Alvarez and Nagler1998)]. For the graph $G$ , we tested using both the geographic adjacency network and a network formed by connecting each county to the 10 others with which it has the highest SCI. We found almost identical results with both networks, so we only discuss the results using the SCI network. We refer to the resulting dataset as us-election-2016.

5.3. California precinct-level election data

The presidential election data is particularly interesting because different ballots have different candidates, all running in the same election. For instance, this is analogous to having different regional availability of goods within a category in an online shopping service. In our next two datasets, ca-election-2016 and ca-election-2020, we highlight a different scenario: when ballots in different locations may have different elections. Extending the online shopping analogy, this emulates the case where different users view different recommended categories of items. Although it is beyond the scope of the present work, a discrete choice approach would enable measuring cross-election effects, such as coattail effects (Hogan, Reference Hogan2005; Ferejohn & Calvert, Reference Ferejohn and Calvert1984) where higher-office elections increase excitement for down-ballot races.

To construct these datasets, we used data from the 2016 and 2020 California general elections from the Statewide Database.Footnote ⁵ This includes per-precinct registration and voting data as well as shapefiles describing the geographic boundaries of each precinct (California has over 20,000 voting precincts). The registration data contain precinct-level demographics (counts for party affiliation, sex, ethnicity, and age ranges), although such data were not available for all precincts. We restrict the data to the precincts for which all three data types were available: voting, registration, and shapefile (99.8% of votes cast are included in our processed 2016 data and 99.0% in our 2020 data). Again, we treat each precinct as a chooser $a$ with demographic features $\boldsymbol{x}_a$ .

Our processed California data include elections for the US Senate, US House of Representatives, California State Senate, and California ballot propositions. We set aside presidential votes due to overlap with the previous dataset and state assembly votes to keep the data size manageable. Due to California’s nonpartisan top-two primary system,Footnote ⁶ there are two candidates running for each office—however, each voter has a different set of elections on their ballot due to differences in US congress and California state senate districts (the state has 53 congressional districts and 40 state senate districts). A discrete choice approach enables us to train a single model accounting for preferences over all types of candidates. We use the precinct adjacency network $G$ (since SCI is not available at the finer-grained precinct level), which we constructed from the Statewide Database shapefiles using QGIS (https://qgis.org).

6.1. Improved sample complexity with Laplacian regularization

By leveraging correlations between node preferences through Laplacian regularization, we need fewer samples per node in order to achieve the same inference quality. When preferences are smooth over the network, an observation of a choice by one node gives us information about the preferences of its neighbors (and its neighbors’ neighbors, etc.), effectively increasing the usefulness of each observation. In Figure 3, we show the sample complexity benefit of Laplacian regularization in synthetic data with 100-node Erdős–Rényi graphs ( $p=0.1$ ) and preferences over 20 items generated according to the prior from Section 4.1.1. In each of 8 trials, we generate the graph, sample utilities, and then simulate a varying number of choices by each chooser. We repeat this for different homophily strengths $\lambda$ . For each simulated choice, we first draw a choice set size uniformly between 2 and 20, then pick a uniformly random choice set of that size. We then measure the mean-squared error in inferred utilities of observed items (fixing the utility of the first item to 0 for identification). When applying Laplacian regularization, we use the corresponding value of $\lambda$ used to generate the data (in real-world data, this needs to be selected through cross-validation). We train the models for 100 epochs.

Figure 3. Estimation error of item utilities with (left) and without (right) Laplacian regularization on synthetic data generated according to the priors in Section 4.1.1, with varying homophily strength $\lambda$ . Error bars (most are tiny) show standard error over 8 trials. Using Laplacian regularization can improve sample complexity by orders of magnitude.

In this best-case scenario, we need orders of magnitude fewer samples per chooser if we take advantage of preference correlations: with Laplacian regularization, estimation error with only one sample per chooser is lower than the estimation error with no regularization and 1000 samples per chooser. The stronger the homophily, the fewer observations are needed to achieve optimal performance, since a node’s neighbor’s choices are more informative.

6.2. Prediction performance comparison

We now evaluate our approaches on real-world choice data. In the style of semi-supervised learning, we use a subset of choosers for training and held-out choosers for validation and testing. This emulates a scenario where it is too expensive to gather data from everyone in the network or existing data is not available for all nodes (e.g., perhaps not all individuals have consented to choice data collection). We vary the fraction of training choosers from 0.1 to 0.8 in increments of 0.1, using half of the remaining choosers for validation and half for testing. We perform 8 independent sampling trials at each fraction in the election datasets and 64 in the smaller Friends and Family datasets.

As a baseline, we use standard logit models with no network information. For the election datasets, we use an MNL that uses county/precinct features to predict votes. This approach to modeling elections is common in political science (Dow & Endersby, Reference Dow and Endersby2004). For app-install, we use a simple logit. For app-usage, we use a CL with recency (as defined in Section 5.1). We then compare the three graph-based methods we propose to the baseline choice model: a GCN-augmented MNL (or CML), a Laplacian-regularized logit (or CL/MNL) with per-chooser utilities, and choice fraction propagation. Aside from propagation, we train the other methods with batch Rprop (Riedmiller & Braun, Reference Riedmiller and Braun1993), as implemented in PyTorch (Paszke et al., Reference Paszke, Gross, Massa, Lerer, Bradbury, Chanan and Chintala2019). For each dataset–model pair, we select the hyperparameters that result in the lowest validation loss in a grid search; we tested learning rates $10^{-3}, 10^{-2}, 10^{-1}$ and $L_2$ regularization strengths $10^{-5}$ , $10^{-4}$ , $10^{-3}$ , $10^{-2}$ , $10^{-1}$ (we also tested no $L_2$ regularization in the two app datasets). We similarly select Laplacian $\lambda$ using validation data from $10^{-5}, 10^{-4}, 10^{-3}, 10^{-2}$ in the election datasets (in addition to these, we also test $10^0, 10^{-1}, 10^{-6}, 10^{-7}$ in the app datasets) and propagation $\rho$ from $0.1, 0.25, 0.5, 0.75, 1$ . The smaller hyperparameter ranges in the election datasets were used due to runtime constraints. We train the likelihood-based models for 100 epochs, or until the squared gradient magnitude falls below $10^{-8}$ . For propagation, we perform 256 iterations, breaking if the sum of squared differences between consecutive iterates falls below $10^{-8}$ . We note that we did not aggressively fine-tune the GCN beyond learning rate and $L_2$ regularization strength, since it has many more hyperparameters than our other approaches and is more expensive to train. Our GCN results should therefore be interpreted as the performance a discrete choice practitioner should expect to achieve in a reasonable amount of time using the model, which we believe is an important metric.

In Figure 4, we show results of all four approaches on all five datasets. We evaluate the three likelihood-based methods using their test set NLL and use MRR (Tomlinson & Benson, Reference Tomlinson and Benson2021) to evaluate propagation. For one sample, MRR is defined as the relative position of the actual choice in the list of predictions in decreasing confidence order (where 0 is the beginning of the list and and 1 is the end). We then report the mean MRR over the test set. In app-install, both Laplacian regularization and propagation improve prediction performance over the baseline logit model, and the advantage increases with the fraction of participants used for training (up to $6.8\%$ better MRR). However, the GCN performs worse than logit in terms of likelihood and the same or worse in terms of MRR. In contrast, graph-based methods do not outperform a CL in app-install. In the three election datasets, Laplacian-regularized MNL consistently outperforms MNL (with up to $2.6\%$ better MRR in us-election-2016; the margin in the California data is small but outside errorbars), while the GCN performs on par with MNL in us-election-2016 and worse in the California datasets.

Figure 4. Test negative log likelihoods (NLL; top row; lower is better) and mean relative ranks (MRR; bottom row; lower is better) on the two Friends and Family datasets and three election datasets (error bars show standard error over chooser sampling). “Logit” signifies plain logit in app-install, CL in app-usage, and MNL in the election datasets. Laplacian regularization improves performance in app-install, while no method improves on CL in app-usage. In the election data, Laplacian MNL, but not GCN, outperforms MNL across train fractions. Propagation performs well on app-install, but very poorly on app-usage, as it does not utilize recency. Despite not using county/precinct features, propagation can be competitive in the election data.

These results yield insight into the role networks play in different choice behaviors. In app-usage, we find no benefit from using social network structure using any method. Instead, the recency feature appears to dominate, with propagation (which has no access to item features) performing much worse than the three models that do incorporate recency. This indicates that app usage is driven by individual habit rather than by external social factors. On the other hand, our results show that app installation has a strong social component: even simple Bluetooth proximity between friends provides a signal that they will install (but not necessarily use) similar apps. This finding highlights how combining a discrete choice approach with network data can illuminate the role social networks play in different choice behaviors. In the election data, especially ca-election-2016, even simple choice propagation performs remarkably well, despite entirely ignoring demographic features. This reveals that many of the important predictive demographic features (such as party affiliation, age, and ethnicity) are so strongly correlated over the adjacency network that we do not need to know information about you to predict your vote: it suffices to know about your neighbors or your neighbors’ neighbors.

We also compare the runtime of each method. To measure runtime, each model was run on a 50-25-25 train-validation-test split of each dataset four times. Since the hyperparameters are not crucial for runtime measurements (especially because Rprop is not sensitive to initial learning rate as an adaptive method), we fixed the learning rate at 0.01, $L_2$ regularization strength at 0.001, Laplace regularization strength at 0.0001, and propagation $\rho$ at 0.5. For each trial, we trained and tested each model once, shuffling the order of models to avoid systematic bias due to caching. Laplacian regularization has very low overhead over CL/MNL, while GCN is up to $4\times$ slower in the smaller datasets (see Table 2). In the larger datasets, PyTorch’s built-in parallelism reduces this relative gap. Propagation is more than $10\times$ faster than the choice models in every dataset.

Table 2. Runtime in seconds to train and test each model, with standard err over 4 trials

6.3. Facebook and Myspace communities in app-install

Given that we observed significant improvement in prediction performance in app-install, we take a closer look at the patterns learned by the Laplacian-regularized logit compared to the plain logit. In particular, the Facebook and Myspace apps were in the top 20 most-preferred apps under both models. Given that these were competitor apps at the time,Footnote ⁷ we hypothesized that they might be popular among different groups of participants. This is exactly what we observe in the learned parameters of the Laplacian-regularized logit. Facebook and Myspace are in the top 10 highest-utility apps for 70 and 27 participants, respectively (out of 139 total; we refer to these sets as $F$ and $M$ ). Intriguingly, the overlap between $F$ and $M$ is only 3. Moreover, looking at the Bluetooth interaction network, we find the edge densities are more than twice as high within each of $F$ and $M$ than between them (Table 3), indicating they are true communities in the social network. In short, the Laplacian-regularized logit learns about two separate subcommunities, one in which Facebook is popular and one in which Myspace is popular.

Table 3. Edge densities within/between the groups preferring Facebook ( $|F| = 70$ ) and Myspace ( $|M| = 27$ ) in app-install. Left: including the 3 choosers in $F\cap M$ . Right: excluding $F\cap M$

6.4. Counterfactuals in the 2016 US election

One of the powerful uses of discrete choice models is applying them to counterfactual scenarios to predict what might happen under different choice sets [e.g., in assortment optimization (Rusmevichientong et al., Reference Rusmevichientong, Shen and Shmoys2010)]. For instance, we can use our models to make predictions about election outcomes if different candidates had been on ballots in 2016. However, we begin this exploration with a warning: making predictions from observational data is subject to confounders, unobserved factors that affected both who was on which ballot and how the states voted. For example, only Nevadans had the option to vote for “None of these options,” and Nevada is an outlier in a number of ways that are likely to impact voting, including its reliance on tourism, high level of diversity, and lack of income tax. This makes it less likely that the preferences of Nevadans for “None of these options” will neatly generalize to voters in other states. There are causal inference methods of managing confounding in discrete choice models; for instance, our county covariates act as regression controls (Tomlinson et al., Reference Tomlinson, Ugander and Benson2021). If those covariates fully described variation in county voting preferences, then the resulting choice models would be unbiased, even with confounding (Tomlinson et al., Reference Tomlinson, Ugander and Benson2021). However, we do not believe the covariates fully describe voting, since we can improve prediction by using regional or social correlations not captured by the county features. Nonetheless, examining our model’s counterfactual predictions is still instructive, demonstrating an application of choice models, providing insight into the model’s behavior, and motivating randomized experiments to test predictions about the effect of ballot changes. We note that the MNL we use obeys IIA, preventing relative preferences for candidates changing within a particular county when choice sets change. However, since states contain many counties, they are mixtures of MNLs (which can violate IIA), so their outcomes can change under the model.

Table 4. Maximum likelihood 2016 election outcomes under our model under the three scenarios in Section 6.4. We show mean vote shares (with 95% confidence interval over trials) for the top three predicted candidates and differences in state outcomes between the counterfactual prediction and reality. C: Clinton, T: Trump, Outcome: Electoral College votes. T $\rightarrow$ C denotes that a state won by Trump goes for Clinton under the model. States abbreviated by postal code

$^*$ Maine allocates Electoral College votes proportionally—we assume a 3-1 split.

A widespread narrative of the 2016 election is that third-party candidates cost Clinton the election by disproportionately taking votes from her (Chalabi, Reference Chalabi2016; Rothenberg, Reference Rothenberg2019). To test this hypothesis, we examine three counterfactual scenarios: (Scenario 1) all ballots have five options: Clinton, Trump, Johnson, Stein, and McMullin; (Scenario 2) ballots only list Clinton and Trump, and (Scenario 3) ballots are as they were in 2016, but “None of these candidates” is added to every ballot. For each scenario, we take the best (validation-selected) Laplacian-regularized MNL trained on 80% of counties from each of the 8 county sampling trials and average their vote count predictions. Maximum-likelihood outcomes under the model are shown in Table 4. We find no evidence to support the claim that third-party candidates hurt Clinton more than Trump. None of the scenarios changed the two major measures of outcome: Clinton maintained the popular vote advantage, while Trump carried the Electoral College. A few swing states change hands in the predictions. The model places more weight on “None of these candidates” than seems realistic (for instance, predicting it to be the plurality winner in Rhode Island), likely because training data is only available for this option in a single state, leading to confounding. We also note that under the true choice sets, the model’s maximum likelihood state outcomes are the same as in Scenarios 1 and 2. A more complete analysis would examine the full distribution of Electoral College outcomes rather than just the maximum likelihood outcome, but we leave such analysis for future work as it is not our main focus.