In this section, we first define a general and abstract Markov chain model to mathematically capture empirically observed polarization trends (see Fig. 1). Our model describes a (macroscopic) set of different opinion classes. Empirically, these opinion classes correspond to mappings of a high-dimensional feature space (e.g., self-identification with a certain political party, views on certain political issues) to a single-number metric. We briefly describe the update dynamics and then focus on the characterization of the stationary distribution.

### 2.1 Definition of the ideology chain

We proceed in three steps to mathematically describe initiatives and the diffusion of individuals from one ideological position to adjacent ones, with step 1 developed in Sects. 2.1 and 2.2, step 2 in Sect. 2.3, and step 3 in Sect. 2.4.

In the first step, we consider a one-dimensional chain which consists of *N* different states denoted by *i* (\(i\in \{1,\dots,N\}\)). We use \(X_{i}\) to denote the fraction of society in state *i*, and hence \(\sum_{i=1}^{N} X_{i} = 1\). Next, we map the index *i* to an *ideological position* \(x\in [-1,1]\) according to \(x=2 (i-1)/(N-1)-1\). These positions represent the political spectrum in the following way: Very liberal individuals are located at the beginning of the chain (\(i=1\), \(x=-1\)), whereas strongly conservative individuals are found at the opposite side (\(i=N\), \(x=1\)). We next consider the evolution of a hypothetical society in discrete time. We interpret \(X_{i}^{n}\) as the fraction of voters of type *i* at time step \(n\in \mathbb{N}\). For every *n*, it holds that

$$ \sum_{i=1}^{N} X_{i}^{n}=1 $$

(1)

as a normalization condition. We employ a simple birth-death queue [28] of social interactions and assume that individuals may change their ideological position via interactions with their ideological neighbors. At the aggregate level, in a particular time step, we assume that transitions occur from state *i* to its nearest neighbors (\(i\rightarrow i+1\) and \(i\rightarrow i-1\)) with some probabilities \(p_{i}\in (0,1)\) and \(q_{i-1}\in (0,1-p_{i})\). For the moment, these transition probabilities are taken as given and will be estimated later. At the boundaries of the opinion chain, the probability of becoming more ideologically extreme is zero (\(q_{0}=p_{N}=0\)). The probability of staying at a certain ideological position *i* is given by \(r_{i}=1-p_{i}-q_{i-1}\). Together, these probabilities form the transition matrix *P*, with the following entries:

$$ P_{i i-1}=q_{i-1},\qquad P_{i i} = r_{i},\quad \text{and}\quad P_{i i+1}=p_{i}. $$

(2)

The probabilities in each row sum up to one, i.e. \(\sum_{j=0}^{3} P_{i i-1+j}=1\).

There exist different ways of motivating and microfounding our opinion-formation model. First, the mathematical structure of our model is similar to DeGroot’s [26] Markov-chain description of “social learning” in a group of communicating individuals. Second, the macroscopic distributions of opinions that we observe in our model can be recovered in random matching and bounded confidence models [29, 30] in which individuals adopt sufficiently close opinions through communication (see [31] for a comprehensive account on how such micro-level assumptions in a social network turn into macro-level implications and [32] for a general theory about such social interactions). After every meeting, individuals update their opinion and may switch to their partner’s opinion with some probability. Equivalently, individuals change their opinion if they meet a sufficient number of people with alternative opinions [17, 18, 33]. The probabilities \(p_{i}\) and \(q_{i}\) can then be interpreted as the resulting parameters at the aggregate level.^{Footnote 1} We show an example of an ideology chain with \(N=9\) states in Fig. 2. For the sake of clarity, we do not include self-loops described by \(r_{i}\) in this figure.

Next we focus on the dynamics of the model to account for the diffusion of individuals from one ideology to adjacent ones. The initial values of all states are given by \(X_{i}^{n=0}=X_{i}^{0}\). We use \(X^{0}= (X_{1}^{0},\dots,X_{N}^{0} )\) to denote the row vector of all initial states. The time evolution of the ideology distribution is then described by \(X^{0} P^{n} = X^{n}\).

### 2.2 Stationary distribution

To determine the stationary ideology distribution, we formulate the update rule of state \(X_{i}^{n}\) and find

$$ X_{i}^{n+1}=(1-p_{i}-q_{i-1})X_{i}^{n}+q_{i} X_{i+1}^{n}+p_{i-1} X_{i-1}^{n}. $$

(3)

We are not considering periodic boundaries, and thus find for \(i=1\),

$$ X_{1}^{n+1}=(1-p_{1})X_{1}^{n}+q_{1} X_{2}^{n}. $$

(4)

The Markov chain converges to a stationary distribution, which we denote by \(X_{i}\) with \(i\in \{1,\dots,N\}\). Based on Eq. (4), we obtain \(X_{2}= (p_{1}/q_{1} ) X_{1}\). Furthermore, using Eq. (3) we find by induction that [34]

$$ X_{i+1}= X_{1} \prod_{j=1}^{i} \frac{p_{j}}{q_{j}}. $$

(5)

To satisfy the normalization condition of Eq. (1), we set \(X_{1}=1\) and divide each state \(X_{i}\) by \(\sum_{i=1}^{N} X_{i}\). The stationary distribution \(X=(X_{1},\dots,X_{N})\) is unique since the transition matrix *P* is irreducible and aperiodic [35]. Irreducibility follows from the fact that any state in the Markov chain can be reaced from any other state, and aperiodicity is satisfied because of \(P_{ii}^{n} > 0\) for all \(n\in \mathbb{N}\) [35].

The data that we show in Fig. 1 suggests that the ideological overlap between Democrats and Republicans was larger in the 1990s and early 2000s compared to the last 15 years. To capture the ideology distributions of Democratic- and Republican-leaning segments of the U.S. public with our model, we consider two opinion chains *A* and *B* in the subsequent sections, and then account for the impact of influential actors and their initiatives.

### 2.3 Two populations

In the second step, we introduce two populations in which members influence each other regarding their ideological position. This allows us to examine how the distribution of ideologies among Democrats and Republicans evolves over time. Specifically, we consider two populations, *A* and *B*, with the corresponding stationary ideology distributions given by Eq. (5):

$$\begin{aligned} X^{A}_{i+1}=X^{A}_{1} \prod _{j=1}^{i} \frac{p^{A}_{j}}{q^{A}_{j}} \end{aligned}$$

(6)

and

$$\begin{aligned} X^{B}_{i+1}=X^{B}_{1} \prod _{j=1}^{i} \frac{p^{B}_{j}}{q^{B}_{j}}. \end{aligned}$$

(7)

### 2.4 Influential actors

In the third step, we account for influential actors of both parties that inject political/cultural concepts—simply called *initiatives* in our paper—into one of the populations. The literature has identified the importance of such influential actors and their initiatives (see e.g. [8, 25]). Typically, initiatives increase the attractiveness of coalescing around a particular ideological position. They can also increase the cohesion within each population and the identity value of belonging to a population.

Mathematically, we describe the impact of influential actors on the two populations in terms of rescaling the transition probabilities of Eq. (2) by \(\lambda _{A}\) and \(\lambda _{B}\) at a particular point in time according to

$$ p_{i}^{A}\rightarrow p_{i}^{A}/\sqrt{ \lambda _{A}} \quad\text{and}\quad q_{i}^{A} \rightarrow q_{i}^{A} \sqrt{\lambda _{A}}. $$

(8)

For opinion group *B* (e.g., Republicans), the rates are modified as follows:

$$ p_{i}^{B}\rightarrow p_{i}^{B} \sqrt{ \lambda _{B}} \quad\text{and}\quad q_{i}^{B} \rightarrow q_{i}^{B} /\sqrt{\lambda _{B}}. $$

(9)

The interpretation of the scaling factor is as follows: A value of \(\lambda _{A}>1\) and \(\lambda _{B}>1\) means that an initiative in the populations A and B is introduced, which attracts individuals towards the left and right ends of the underlying ideology chains, respectively. The larger \(\lambda _{A}\) and \(\lambda _{B}\), the greater is the attractiveness of these initiatives. Based on Eqs. (8) and (9), we obtain the following modified stationary states:

$$ X^{A}_{i+1}=\lambda _{A}^{-i} X^{A}_{1} \prod_{j=1}^{i} \frac{p^{A}_{j}}{q^{A}_{j}} $$

(10)

and

$$ X^{B}_{i+1}=\lambda _{B}^{i} X^{B}_{1} \prod_{j=1}^{i} \frac{p^{B}_{j}}{q^{B}_{j}}. $$

(11)

The outlined multiplicative rescaling of the transition probabilities leads to a directly-interpretable modification of the stationary opinion distribution. If \(\lambda _{B} > 1\), we obtain a stationary opinion distribution whose mean is shifted to the right compared to the case where \(\lambda _{B} = 1\). Similarly, if \(\lambda _{A} > 1\), the stationary opinion distribution moves towards the left. We thus refer to *λ* as the *initiative impact*.

When we move to the empirical application, we have to recognize that a particular value of \(\lambda _{B}\) or \(\lambda _{A}\), which we infer from the data, is open to different interpretations. On the one hand, it could represent changes in the way individuals communicate and influence each other, resulting in a shift regarding political/cultural views. On the other hand, it could represent the impact of new ideas (political or cultural) that affect the attractiveness of different ideological positions as we have outlined in our model.

While it is difficult to disentangle these two interpretations, we will interpret our findings in light of the second interpretation. We do this for two reasons: First, changes in communication, e.g. due to increased Internet use [3] cannot explain the rise in polarization, and [36] recently summarized that the evidence about whether social media increase political polarization is not conclusive. Second, there exists a series of legal acts and policy proposals that have been at the center of communication in the community leaning towards the Democratic party. Such legal acts include the Dodd–Frank Financial Reform Act and the Affordable Care Act. Examples of policy proposals are universal-health care programs, higher taxes on wealth, tighter gun control, and initiatives to slow down climate change. Opposition against tighter gun control, higher taxes, and same-sex marriages are examples of major ideas in the communities leaning towards the Republican party. These examples suggest that major new initiatives have been mainly introduced in the communities leaning towards the Democratic party. We examine whether the data are consistent with this interpretation. We also note that some of the initiatives such as same-sex marriage and abortion rights are culture-dependent since they concern norms and beliefs about how people should be able to live in families, groups and communities.

In the following sections, we show that the outlined rescaling approach is able to model empirically-observed opinion polarization in the U.S. public. In principle, we could also consider values of \(\lambda _{A}\) and \(\lambda _{B}\) that depend on the position in ideology space. It is, however, possible to capture a substantial part of the polarization effects with a constant value of \(\lambda _{A}\) and \(\lambda _{B}\), as shown in Sect. 3.2.