Why are U.S. Parties So Polarized?
A “Satisficing” Dynamical Model

We solve for the fixed points of the system both analytically and numerically with analytical results presented in section 4.2 and numerical results presented in Figs. 3 and 4. The results reveal that for sufficiently large inclusiveness parameters, $\sigma_{1,2}$, the two parties are attracted to a single stable equilibrium: $\mu_{1}=\mu_{2}=0$, at the center of the ideology space. However, for smaller $\sigma_{1,2}$, the two parties stabilize at a finite separation. Fig. 3A visualizes the dynamics of a two-party system by plotting the derivatives $d\mu_{1}/dt$ and $d\mu_{2}/dt$ in Eq. (3) as a vector field. In this example, the two parties stabilize at a finite separation. The fixed points and their stability are shown in Fig. 3B for the symmetric case, $\sigma_{1}=\sigma_{2}$. We note that Fig. 3B plots the positions of both parties on the same vertical axis. In the asymmetric case where $\sigma_{1}\neq\sigma_{2}$, the polarization, as measured by the distance between the two parties at steady state, is a function of both $\sigma_{1}$ and $\sigma_{2}$. The numerical results for this case are shown in Fig. 3C. The system equilibrates at a finite separation in most of the parameter space.

An intuitive understanding of this result is that when the inclusiveness parameters are large, few voters abstain. Thus, parties are attracted to the center of the ideology space through the same mechanism as in the classic Downsian model that predicts the convergence of both parties to the median voter [11]. However, as the inclusiveness parameters decrease, convergence to the center increases the number of voters at the tails of the ideology distribution abstaining from voting. In that case, the parties benefit from moving away from the center, with an equilibrium distance determined by the parameters. Although the model was presented in a one-dimensional ideology space, the model’s behavior is similar in two dimensions (see supplementary material section1).

The right hand side of the system specified by Eq. (3) can be computed analytically for the symmetrical case $\sigma_{1}=\sigma_{2}=\sigma$:

$$\frac{d\mu_{i}}{dt}=\frac{\sigma_{0}[\mu_{i}(\sigma^{2}+\sigma_{0}^{2})-\mu_{j}\sigma_{0}^{2}]}{2\sigma(\sigma^{2}+2\sigma_{0}^{2})^{3/2}}e^{-\frac{(\mu_{i}^{2}+\mu_{j}^{2})\sigma^{2}+(\mu_{i}-\mu_{j})^{2}\sigma_{0}^{2}}{2\sigma^{2}(\sigma^{2}+2\sigma_{0}^{2})}}-\frac{\mu_{i}\sigma_{0}\sigma e^{-\frac{\mu_{i}^{2}}{2(\sigma^{2}+\sigma_{0}^{2})}}}{(\sigma^{2}+\sigma_{0}^{2})^{3/2}}\;.$$

(4)

Solving for the equilibria $d\mu_{i}/dt=0$, and imposing symmetry in party positions $\mu_{2}=-\mu_{1}=\mu$ (this follows from the symmetry in $\sigma_{1}=\sigma_{2}$), we find

$$\mu\left[-2\sigma^{2}(\sigma^{2}+2\sigma_{0}^{2})^{1/2}e^{-\frac{\mu^{2}}{2(\sigma^{2}+\sigma_{0}^{2})}}+(\sigma^{2}+\sigma_{0}^{2})^{3/2}e^{-\frac{\mu^{2}}{\sigma^{2}}}\right]=0\;.$$

(5)

Clearly $\mu=0$ is a solution, and thus $\mu^{\star}=0$ is a fixed point of the system for all values of $\sigma$ and $\sigma_{0}$. Other possible fixed points are given by the solution of the equation $(\sigma^{2}+\sigma_{0}^{2})^{3/2}e^{-\frac{\mu^{2}}{\sigma^{2}}}-2\sigma^{2}(\sigma^{2}+2\sigma_{0}^{2})^{1/2}e^{-\frac{\mu^{2}}{2(\sigma^{2}+\sigma_{0}^{2})}}=0$, which can be solved for $\mu^{2}$ explicitly as

$$\left(\mu^{\star}\right)^{2}=\sigma^{2}\left(\frac{\sigma^{2}+\sigma_{0}^{2}}{\sigma^{2}+2\sigma_{0}^{2}}\right)\ln\left[\frac{\left(\sigma^{2}+\sigma_{0}^{2}\right)^{3}}{4\sigma^{4}\left(\sigma^{2}+2\sigma_{0}^{2}\right)}\right]\;;$$

(6)

the solid line in Fig. 3B is this curve. Note that this expression can be rewritten in terms of nondimensional parameters $\{\hat{\sigma}\equiv\sigma/\sigma_{0},~{}\hat{\mu}\equiv\mu/\sigma_{0}\}$ simply by setting $\sigma\mapsto\hat{\sigma}$ and $\sigma_{0}\mapsto 1$.

The trivial solution $\mu^{\star}=0$ is the only fixed point for $\sigma>\sigma_{c}$ for some $\sigma_{c}$, and it is of interest to understand how $\sigma_{c}$ depends on system parameters²²2The result that the trivial solution $\mu^{\star}=0$ is the only fixed point for $\sigma>\sigma_{c}$, holds even for two parties with $\mu_{1}\neq\mu_{2}$, as can be seen with a series expansion for large $\sigma$ retaining only leading order terms.. This critical value can be understood by noting that the nontrivial fixed points for $\mu$ given in Eq. (6) cease to exist when the logarithm becomes negative, i.e., when the argument of the logarithm becomes less than one; $\sigma_{c}$ is the critical value at which the argument is exactly one. Expressing this condition in the nondimensionalized form gives

$$\frac{\left(\hat{\sigma}_{c}^{2}+1\right)^{3}}{4\hat{\sigma}_{c}^{4}\left(\hat{\sigma}_{c}^{2}+2\right)}=1\;,$$

(7)

and rearranging the terms leads to a cubic polynomial in $\hat{\sigma}_{c}^{2}$:

$$3\hat{\sigma}_{c}^{6}+5\hat{\sigma}_{c}^{4}-3\hat{\sigma}_{c}^{2}-1=0\;.$$

(8)

This equation has three real roots for $\hat{\sigma}^{2}_{c}$, at $\hat{\sigma}_{c}^{2}\approx\{-2.07,-0.247,0.652\}$, and only the last root allows for a real-valued solution $\hat{\sigma}_{c}\approx 0.807$. The system undergoes a subcritical pitchfork bifurcation at this point.

The exact shape of the fixed point curve is defined by Eq. (6), which cannot be solved explicitly for $\sigma$ (or $\hat{\sigma}$ in nondimensional form). A convenient approximation can be written by expanding the right-hand side to the leading order in $\hat{\sigma}$, yielding $\hat{\mu}^{2}\approx-\hat{\sigma}^{2}\ln(\sqrt{8}\hat{\sigma}^{2})$. Define $W\equiv-\hat{\mu}^{2}/\hat{\sigma}^{2}$, and rearrange the previous equation to get $-\sqrt{8}\hat{\mu}^{2}\approx W\exp(W)$. This equation is solved by $W=W(-\sqrt{8}\hat{\mu}^{2})$, where $W$ is now identified as the special function known as the Lambert $W$ function. Thus $\hat{\sigma}^{2}\approx-\hat{\mu}^{2}/W(-\sqrt{8}\hat{\mu}^{2})$.

The Lambert $W$ function is real-valued for negative arguments only when the argument has magnitude less than $1/e$. This sets a bound on the maximum $\hat{\mu}^{2}$ for which a solution exists since $\sqrt{8}\hat{\mu}_{\textrm{max}}^{2}\approx 1/e$ implies $\hat{\mu}_{\textrm{max}}\approx 2^{-3/4}e^{-1/2}\approx 0.361$. The corresponding value of $\hat{\sigma}$ is then immediate since $W(-1/e)=-1$, yielding $\hat{\sigma}_{\textrm{max}}\approx\hat{\mu}_{\textrm{max}}\approx 2^{-3/4}e^{-1/2}$. These approximations are useful in understanding the general shape of the bifurcation curve, though the critical points of the exact expression in Eq. (6) can be easily computed numerically.

To examine changes in real-world party positions over time, we employ ideological positions for Democratic and Republican legislators calculated from their congressional roll call voting records using the Dynamic, Weighted, Nominal Three-Step Estimation (DW-NOMINATE) method [7]. DW-NOMINATE is a multi-dimensional scaling method that first calculates a pairwise distance for every two members of the U.S. House of Representatives based on similarities in their roll call voting records. It then projects the resulting high-dimensional network of representatives to a low dimension while preserving the pairwise distance relation as much as possible. The representatives’ relative positions in this low-dimensional space are referred to as their ideology scores.³³3Note that this measure is relative in nature—it reflects the similarities and differences among members’ voting records across a large number of bills, not their positions on any specific issue. The same method can be used for scaling the positions of U.S. Senators, and in what follows we use a combined dataset of both representatives and senators (see supplementary material section 2 for data source).

As an example, Fig. 4A displays a histogram of Democratic and Republican legislators’ ideological positions during the 2013-15 term, the most recent in the data.

For purposes of comparing our model with data, we take the mean of each party’s distribution to represent its position, $\mu_{i}$, in the model and the standard deviation of each party’s distribution to represent its inclusiveness parameter, $\sigma_{i}$. As members of Congress move farther from the party mean, we infer that the party is more inclusive—and may thus appeal more to voters farther away from its center. Repeating that procedure for all available Congresses leads to a series of party positions over time since 1861, shown as the solid curves in Fig. 4B. We note that two distinct polarization metrics—the distance between party means (used in Fig. 4) and the standard deviation of the legislators’ ideology distribution (used in Fig. 1)—consistently demonstrate the striking increase in party polarization since the 1960s.

Importantly, Fig. 4B also compares the model’s predictions for the two major parties’ positions to historical data. We set the two parties’ positions in 1861 as their initial conditions and numerically simulate the dynamical system (3) to predict their positions in subsequent years. In the simulations, we update the inclusiveness parameter according to the data every Congress, and the model outputs the positions of both parties according to Eq. (3). Parameters $\sigma_{0}$, $k$, and a proportionality constant associating the standard deviation in the DW-NOMINATE score with the inclusiveness parameter are determined from the fit to the data. An animated visualization of Figs. 4A and 4B is included as a supplementary video.

Our theory shows good agreement with the data (Pearson correlation 0.75; see Figs. 4C and 4D). Notable deviations are observed around the times of the First and Second World Wars and following the recent rightward move by the Republican Party, as shown in Fig. 4B.

A central prediction of our model is that lower inclusiveness (more homogeneity within parties) will lead to higher party polarization. We find support for this prediction in empirical data, as shown in Fig. 4D. This finding suggests that it is not necessary for the voting population to become polarized in order for increased political party polarization to occur. Indeed, the electorate need not change at all; it is held constant in our simulations. We also perform robustness checks using an alternative measure of party polarization as well as an independent data source for the inclusiveness parameter and reach the same qualitative conclusions (see supplementary material section 3).

When determining the parameters from the data, we relate the units of the DW-NOMINATE ideology scores to units for variables in our model by assuming a linear scaling. For example, the party inclusiveness parameter, $\sigma_{i}$, is assumed to be linearly related to the DW-NOMINATE data through the scaling $\sigma_{i}=b\;\sigma_{\text{data},i}$. We fitted the parameters in the model by minimizing the 1-norm for differences between the time series predicted by the model and the time series from data. Three parameters are fitted: (1) $\sigma_{0}$, the standard deviation of the population ideology distribution; (2) $b$, the proportionality constant relating the standard deviation in the DW-NOMINATE data with the party inclusiveness parameter in the model, $\sigma_{i}$; and (3) $k$, the time scale constant. We set the initial conditions for the two parties as given by the data. The best fitting parameters are: $\sigma_{0}=0.93$, $b=3.73$, and $k=2.54$.

The satisficing assumption is essential to our model’s behavior. Here, we compare the results of our satisficing model with a maximizing model in the same framework (assuming two parties) and show that the predictions from the two models are fundamentally different. We consider the maximizing voters to be defined in the Downsian sense [11]—they maximize their utility function by voting for the ideologically closest party. With maximizing voters, the number of votes each party receives is

$$V^{(m)}_{l}=\int_{-\infty}^{(\mu_{l}+\mu_{r})/2}\rho(x)dx,\;\text{and}\;V^{(m)}_{r}=\int_{(\mu_{l}+\mu_{r})/2}^{\infty}\rho(x)dx\;,$$

(9)

where $(l,r)=(1,2)$ if $\mu_{1}<\mu_{2}$, and $(l,r)=(2,1)$ if $\mu_{1}>\mu_{2}$. If $\mu_{1}=\mu_{2}$, both parties receive the same number of votes,

$$V^{(m)}_{1}=V^{(m)}_{2}=\frac{1}{2}\int_{-\infty}^{\infty}\rho(x)dx\;.$$

(10)

We then solve for the fixed points of the system described by Eq. (3) with the two types of vote calculations and find their stabilities.

Figure 5 shows the stable fixed points for party positions as a function of $\sigma_{1}=\sigma_{2}=\sigma$, for normalized variables $\hat{\sigma}=\sigma/\sigma_{0}$ and $\hat{\mu}=\mu/\sigma_{0}$. With satisficing voters, the two parties stabilize at a finite separation for a range of parameters (as shown above). However, with maximizing voters, the position $(0,0)$ is the only stable fixed point, recovering Downs’ result [11], a classic finding in political science.⁴⁴4This result can be immediately seen for the symmetric case $\mu_{1}=-\mu_{2}$. In that case, each party is drawn toward the center to increase votes, despite having split the number of votes equally with the opposing party. This difference indicates that the voters’ satisficing decision making is indeed a key ingredient in producing the model’s results.

Our model offers new contributions to the literature. First, we present empirical evidence of and an explanation for the relationship between party homogeneity and polarization, showing that simply changing the shape of parties’ satisficing functions (via changes to inclusiveness) is sufficient to lead to divergence in their positions. In particular, even though the impact of intra-party heterogeneity on a party’s competitiveness has been discussed before [18], the mechanism linking increasing ideological homogeneity with diverging party positions has remained unclear. Second, the model also shows why appealing to an extreme segment of the electorate can be a winning political strategy in times of greater intra-party ideological homogeneity (i.e., decreasing party inclusiveness), which may be especially relevant for interpreting current trends in U.S. politics. Ideological homogeneity itself may be driven by other factors, such as partisan redistricting [8] or media echo chambers [19], and explicitly modeling how these factors influence intra-party cohesion remains an open area of research. Third, our approach offers a new quantitative framework that incorporates satisficing behavior into a voting model.

In addressing the call for combining interdisciplinary methods to study human social behavior [15], this article offers insight into the complex process of political elections and democratic responsiveness through a parsimonious model, and suggests several directions for future work. For example, do all individuals engage in satisficing, and if so, do they do so in the same way? When and how can minor parties gain traction? And, how might outcomes change if parties could control both their position and their level of inclusiveness? For simplicity, we left out a number of electoral variables, such as party primaries, campaign financing, and Southern realignment, but it will be important for future research to understand how these factors interact with vote satisficing. We hope our work will spur further development of quantitative frameworks to incorporate human bounded rationality into mathematical models.

A copy of the DW-NOMINATE dataset used is included as a supplementary file. The computer code for numerically simulating the model is available in the following GitHub repository: https://github.com/vc-yang/satisficing_election_model.

In the main text, we assumed the public’s ideology distribution to be unimodal and approximated it as Gaussian. Here we use empirical data to justify this assumption.

The American National Elections Studies (ANES) conducted yearly surveys from 1972 to 2012 asking people’s self-identification on a 7-point scale between extremely liberal and extremely conservative. The self identification shows a unimodal distribution centered at “moderate.” The data were downloaded in November 2015 from http://www.electionstudies.org/studypages/anes_timeseries_cdf/anes_timeseries_cdf.htm.
The histogram for the U.S. public self-identified ideology is shown in Fig. S1. Here, the distribution remains peaked at the moderate position for all years, despite fluctuations, and the Gaussian distribution is a good approximation of this distribution.

In the main text, we are motivated by the reality that political idealogy in the U.S. is largely one-dimensional. The model can be generalized to higher dimensions by replacing the scalar variables $x$, $\mu_{i}$, and $\sigma_{i}$ with vectors. Figure S2 shows some example trajectories for the two-party case in a two-dimensional ideology space. In these simulations, for ease of numerical computation, we used discrete time steps and restricted party movements in each period to either vertical or horizontal directions, but the steady states reached are equivalent to those found by integrating the differential equations. The behavior of the system is analogous to that of the one-dimensional problem. For large values of the parameters $\sigma_{i}$, both parties converge to the center. For small-to-moderate $\sigma_{i}$, however, the parties stabilize at a finite separation on opposite sides of a circle (Fig. S2).

For a robustness check of our findings, we employed an alternative estimate of the satisficing function parameter $\sigma$ using a dataset that is independent of the DW-NOMINATE scores shown in the main text.

The data used is the American National Elections Studies (ANES) 1948-2012 time series dataset, as described in section 1 above. We used three variables from the survey: the self-reported liberal-conservative scale (VCF0803), the feeling thermometer for liberals (VCF0211), and the feeling thermometer for conservatives (VCF0212). Data on these variables are available for 18 years between 1972 and 2012, for 55,674 individuals in total.

The feeling thermometer questions ask participants to report their feeling towards a group on a 0-100 scale. Participants are told that 50–100 means they feel favorably towards the group, 0–50 means they do not feel favorably towards the group, and 50 means they do not feel particularly warm or cold.

We assume that thermometer $\geq 50$ means satisfied with the liberal/conservative position. The self-reported liberal-conservative scale is 1 to 7, where 1 represents most liberal, and 7 most conservative. Fig. S1 shows the aggregated distribution of the data from 1972 to 2012. Given the definition of the 7-point scale, we interpret the target group (liberal and conservative) as positioned at 1 and 7 of this scale, respectively. Then, we calculate the proportion of the population satisfied with the target group as a function of their distance to the group. We fit a Gaussian to this curve and get the parameter $\sigma$ from the fit, which corresponds to the width of the satisficing function. Fig. S3 shows, as an example, the behavior of the data in 2012. The two satisficing curves for conservatives and liberals collapse, and are well approximated by a Gaussian function.

We repeat the analysis for all 18 years of data available. Fig. S4 plots the best fitting $\sigma$ value to the ANES data of each year (horizontal axis) against the $\sigma$ value estimated from the DW-NOMINATE data of the Congress that includes the same year. The Pearson correlation is $0.51$ ($p=0.03$). The $\sigma$ value estimated from ANES correlates with distance between parties (i.e., party polarization) in the DW-NOMINATE data (Congress data) with Pearson correlation $-0.58$ ($p=0.01$).

This analysis serves as a robustness check for the negative relationship between party separation (i.e., political polarization) and the narrowing of the satisficing function (i.e., increasing intra-party ideological homogeneity or reducing inclusiveness).

The DW-
NOMINATE score has been criticized at times for its computational complexity and for the difficulty of interpreting its axes. Here, we present a simpler and easier-to-understand metric and show that a negative relationship between polarization and inclusiveness is still observed.

We use data from the U.S. House of Representatives’ roll call records for the period 1941–2015. The data were downloaded from https://www.govtrack.us/data/ in March 2016.

We calculate two scores for each representative. For each representative in each Congress, we calculate a Republican alignment score ($S_{R}$). This score measures the proportion of times she or he voted in agreement with the Republican majority position. For example, if 60% of Republican representatives supported a bill, then a representative voting in favor of that bill would be counted as voting in agreement with the Republican majority. More explicitly,

$$S_{R}=\frac{N_{R}}{N},$$

(S1)

where $S_{R}$ is the Republican alignment score, $N_{R}$ is the number of times voting with the Republican majority, and $N$ is the total number of votes. Similarly, we also calculate a Democratic alignment score for each representative,

$$S_{D}=\frac{N_{D}}{N},$$

(S2)

where $N_{D}$ is the number of times a representative votes with the Democratic majority.

Finally, we combine the two scores into one alternative metric:

$$S=\frac{1}{2}(S_{R}-S_{D}).$$

(S3)

Results of this analysis are shown in Fig. S5. The polarization trend is observable even using this simple metric (shown in the second column of Fig. S5). The negative relationship between the interparty distance (i.e., party polarization) and party inclusiveness is still present. The Pearson correlation in the DW-NOMINATE data is $-0.77$ ($p=2\times 10^{-16}$), and the Pearson correlation in the alternative metric is $-0.59$ ($p=2\times 10^{-8}$). Note that the Republican and Democratic scores (shown in the third and fourth columns of Fig. S5) exhibit asymmetry due to a number of unanimous votes.