Definition and General Propositions.

We develop our argument in a simple model where the outcome of policy depends on some underlying unobservable “state of the world.” Denote states of the world by points s in a set S of possible states. Each state of the world could be a fact about or aspect of how the world works; indeed it could be an entire “possible world.” In the case of monetary policy, for example, the two states can be “Keynesian” and “monetarist” worlds. ^e The state of the world is fixed and unaffected by policy, but different policies may result in different outcomes. ^f Actual policy outcomes are often multidimensional, complex, and not observed with much detail or accuracy. What can usually be observed is a summary indicator of the outcome, which is subject to random disturbances of measurement and estimation. For example, outcomes of monetary and fiscal policies include the effects on incomes and prices faced by millions of consumers and firms; what we observe is an index of inflation or unemployment constructed by the relevant bureau of statistics. Sometimes citizens are aware of only a binary indicator, such as an estimate of whether a policy is judged a success or failure. Denote the random disturbance affecting the observed magnitudes by u ∈ U. The product set Ω = S × U is then our sample space over which probabilities are defined, with typical element (sample point) denoted by ω = (s, u). ^g

Neither the true state of the world s nor the disturbance u is observable. However, each individual can observe the actual policy x that is being pursued and some (potentially noisy) indicator y of the policy outcome, which in turn depends on the policy and the true state of the world. Thus, there is a known functional relationship, y = Y(x, s, u). Each individual has his or her own prior probability distribution over S, the possible states of the world. These priors may be thought of as initial “world views” or beliefs about the “true nature” of the world we live in. Upon observing y, the individual updates his or her prior, using Bayes' rule to obtain a posterior concerning the state of the world. This is the individual's revised (or confirmed) world view or belief. ^h The individual cannot in general infer s from x and y, not only because of the random disturbance u but also because the mapping y = Y(x, s, u) from s to y, given x and u, need not be one-to-one. We will assume that the probability distribution of the random disturbance u is known, so no Bayesian revision of its probability distribution need be made, but even this can be generalized. We assume the random draws of s and u to be statistically independent.

Our concept of polarization of different individuals' probabilistic beliefs is illustrated in Figs. 1 and 2, where the horizontal axis represents a one-dimensional spectrum of states s of the world. Fig. 1 shows the prior and posterior cumulative distributions F, and Fig. 2 shows the corresponding probability density functions f, for two individuals identified by the colors red and blue. The priors are shown as solid curves, and the posteriors are shown as dashed curves. The prior of red is to the left of that of blue in the sense of first-order stochastic dominance. The posterior of red is even farther to the left than her prior, and the posterior of blue is even farther to the right than his prior. We call this polarization, and examine when it can arise.

It helps to start with a condition that rules out such polarization. Let g(y | x, s) be the probability density of the observable y conditioned on the policy x and state s. Polarization will be ruled out if, in response to a higher y, the posterior for s, calculated by using Bayes' Rule, always shifts in the same direction in the sense of first-order stochastic dominance. It is well known that a necessary and sufficient condition for this is that g(y | x, s) has the monotone likelihood ratio property (MLRP), namely that the likelihood ratio g(y | x, s ₁)/g(y | x, s ₂) for any policy x and states s ₁ > s ₂ is a monotone (increasing or decreasing) function of y. ⁱ

MLRP holds for many standard distributions, such as the normal, which is often used and has affected thinking about polarization. Suppose the observable y is a linear function of its arguments, with coefficients set equal to 1 by choice of units:

Let an observer's prior distribution of s (his or her “initial world view”) be normal with mean μ and precision τ_s (reciprocal of variance), and let the known distribution of u be normal with mean 0 and precision τ_u. When x and y are observed, and the distribution of s is updated according to Bayes' Rule, the posterior distribution for the state of the world (the observer's “revised world view”) is normal with mean ν given by where a convex combination of the prior μ and the “signal” y − x ≡ y′. The precision of the posterior is τ_s + τ_u. Thus, in this model, the posterior is more precise than the prior, and a higher (lower) realization of the observable indicator y leads to a higher (lower) mean of s under the posterior distribution, for any given prior μ. Suppose two observers have different prior means, say μ₁ and μ₂ and, possibly, different βs, β₁ and β₂. Choose the labels so that μ₁ > μ₂. When the two observe the same y and update their priors, let the posterior means be ν₁ and ν₂. Using the above formula for the posterior mean, we have the following three cases:

1. If y′ > μ₁, then μ₁ < ν₁ < y′, and μ₂ < ν₂ < y′, so both distributions shift to the right.

2. If y′ < μ₂, then y′ < ν₂ < μ₂, and y′ < ν₁ < μ₁, so both distributions shift to the left.

3. If μ₂ < y′ < μ₁, then μ₂ < ν₂ < y′, and y′ < ν₁ < μ₁, so the two distributions shift toward each other.

In no case can the distributions shift as in Figs. 1 and 2; polarization cannot occur. Users of the linear-normal model then have to explain the instances of belief polarization by invoking some biases of perception or learning to modify Bayesian updating. Gerber and Green (10) review this literature.

We now illustrate how the MLRP fails in some politicoeconomic situations where policies form a one-dimensional spectrum, such as left to right or dove to hawk. Think of a policy as a real number. In the example of monetary policy, this can be the rate of growth of the money supply, or the federal funds rate. If policy x is used when the state is s, this will generate a loss, L(x, s). The individual knows the actual policy, x, and observes an indicator of the ensuing loss. To contrast our approach with the usual approach in current political science and economics, where individuals hold the same prior but have distinct preferences and, hence, loss functions, we here assume that everyone is agreed about the loss function L; disagreement is limited to probabilities of the states of the world. We do not deny the realistic possibility that people also have distinct preferences. However, heterogeneity of beliefs about the nature of the world we live in is also realistic, and here we focus on its consequences by leaving out the other possibility. Preference differences can be an additional reason for polarization.

An optimal policy x*(s) in state s is one that minimizes the loss L(x, s). For the sake of simplicity, we assume that the optimal policy is unique in every state. Thus everyone is agreed about the function x* : S → ℝ that maps each state to its optimal policy, but people may disagree about what policy should be chosen to minimize the expected loss if they have distinct probabilistic beliefs about the true state of the world. For notational convenience, suppose that x*(s) ≡ s.

The loss is not directly observed, but only an indicator Y(x, s, u) is observed. We consider two cases of this. Suppose, first, that the function Y takes only two values; either 0, “success,” or 1, “failure,” and let X = S be a subset of the real line. Failure becomes more likely both when the policy is farther to the right of its optimum and when it is farther to the left of the optimum. Conversely, success becomes more likely when policy gets closer to its optimum from either direction. The individual knows the actual policy, x, and observes whether it succeeds or fails. So here, y simply is “success or failure” of policy x. For y ∈ {0, 1}, let g(y | x, s) denote the conditional probability of the event Y(x, s, u) = y, given the policy x, state s, and observation error u. For y = 1 (“failure”), this probability increases with the distance |x − s| between the actual and optimal policy, say g(1 | x, s) = ε + δ | x − s | for ε, δ > 0, such that g(1 | x, s) is always less than one over the ranges of the variables in the context. Consider any two states s ₁ and s ₂ > s ₁. Then the likelihood ratio for failure is and that for success (y = 0) is

For the MLRP to hold, one of these likelihood ratios, say the first, should always exceed the other. This is equivalent with the requirement that either x > (s ₁ + s ₂)/2 or x < (s ₁ + s ₂)/2 for all x and s. This not being the case, the posterior may just as easily shift in one direction as in the other, depending on the prior. Because all individuals know the policy and make the same observation as to its success or failure, their posteriors may shift in different directions depending on their priors concerning the state of the world.

Secondly, consider an example where a suboptimal policy generates a loss on a continuum scale. Common forms in the political science and economics literatures for the loss function are where x is the actual policy and x*(s) the optimal policy in state s. Suppose that the loss is observed only with noise: y = Y(x, s, u) = L(x, s) + u, where u is an observation error. Suppose the actual policy x is known, but x*(s) and u are not known or observed. Then, the observer will infer that, conditional on y and x, the optimal policy in the current state satisfies in the respective cases. Whether the plus or the minus part gets more posterior probability weight, after x and y have been observed and, therefore, whether the person's preferred policy shifts to the left or the right, depends on the person's prior. Thus, the usual locational or spatial spectrum model of policy creates an automatic bimodality in revisions of beliefs about the optimal policy.

Here are some concrete examples of bimodality. (i) Suppose country A intervenes militarily in country B, and the result is violence and ethnic conflict in B. This could be happening either because each ethnic group in B supports its own militants to resist A's forces, and the militants then turn on each other or because A's forces are not strong enough to maintain law and order. That is, the ideal policy could be either no intervention or a much stronger intervention, and the actual policy may be failing in the respective cases because it is too much or because it is too little. (ii) Suppose a country is experiencing high unemployment. Those who take a Keynesian view of the world may think this is because monetary policy is too tight, whereas those who take a monetarist view may think that the policy is too loose and that businesses are not hiring because they think that the loose policy will lead to inflation and then to much higher interest rates.

The kind of polarization we find does not last forever. Under mild technical conditions, the difference between the posteriors eventually goes to zero when these are successively updated after a sequence of observations; see Blackwell and Dubins (11) ^j . But there is no general guarantee that the convergence occurs monotonically. Our examples show how divergences can temporarily increase and, thereby, help us understand the process of polarization in greater detail. Moreover, these examples show that political polarization can arise quite naturally and consistently with Bayesian updating, without any need to invoke selective perception or biased learning. Of course these things exist in reality and can further aggravate the polarization.

Binary Indicator of Success.

We now develop these two examples of polarization of individual prior beliefs into fuller models of political polarization, where the policy is chosen by majority rule applied to the votes of the same diverse individuals. Our modeling of politics is admittedly special but does bring out some useful intuitions. Specifically, we assume that all voters vote and that they vote sincerely, that is, each voter in each election votes for his or her currently most preferred alternative according to his or her current belief. We do not consider more sophisticated strategic, forward-looking behavior. The context we have in mind is that of an election with numerous voters, where each has a negligible probability of being pivotal to the outcome, and therefore no one has the ability to manipulate the outcome strategically. ^k

Suppose there are five states of the world, s ∈ S = {1, 2, 3, 4, 5}, and equally many voter types, θ ∈ θ = {1, 2, 3, 4, 5}, with equally many voters of each type. Each voter type holds a distinct subjective prior about the true state of the world. Table 1 shows the prior probabilities at the start of period 1, with voter types as rows and states of the world as columns.

The general motivation behind these specific numbers is as follows. (i) Each type of voter assigns high positive probability to the state of the world corresponding to his type, a significant probability for it being one position away from his type, but very small probabilities for it being farther away ^l . (ii) Extremist individuals attach higher probabilities to their own type being the “right” type. (iii) The voter of type 3 has a slightly higher prior probability of the state being 2 than of it being 4.

There are five policies: Far Left (FL), Left (L), Center (C), Right (R) and Far Right (FR), which we label x = 1, 2,…,5. Formally, X = S. The loss function equals 1 if the policy fails and 0 if it succeeds. Therefore, minimization of expected loss is equivalent to minimizing the probability of failure. Then x = 1 (FL) is the (unique) optimal policy in state 1, x = 2 (L) the optimal policy in state 2, etc. Formally: x* (s) = s for all s ∈ S. We develop the example assuming that the true state of the world is 4, so x = 4 (R) is the optimal policy.

At any time, the voter can observe what policy is actually being followed and can observe a binary indicator of the outcome (success or failure). Table 2 shows the probabilities of failure for each policy in each state of the world (with policies as rows and states of the world as columns).

The policy x = 1 is optimal in state 1 and fails with probability only 0.01, but it is farther from optimal in states 2, 3, 4, and 5 and fails with higher probabilities and similarly for other policies. Thus, there is a small probability that even the optimal policy will fail, ^m and failure probabilities rise when the actual policy is farther away from the optimal. Note that policy 3 is optimal in state 3; its failure probability is higher when the true state is 2 or 4 and higher still when the true state is 1 or 5. This is the natural bimodality that arises in the spatial model and drives our results.

We assume that each type of voter knows the table of failure rates. In period 1, given their priors, each type's sincere vote is for the policy that coincides with his type. Preferences are single-peaked, and the outcome is the median voter's preferred policy, which, in this case, is 3. So, policy x = 3 is adopted. Suppose, however, that policy 3 leads to failure. The voters now revise their priors using Bayes' Rule. The posterior from period 1 become the priors at the start of period 2. For each voter type, the posterior probability that the true state is j is proportional to

The actual probabilities are found by normalizing these products. Table 3 shows these products, with voter types as rows and states as columns. Then Table 4 shows (to four significant digits) the resulting posteriors, found by dividing each entry by the sum of all of the cells in its row (still with voter types as rows and states as columns).

Now, consider period 2 voting, given these as the new priors. Voter types 1, 2, 4, and 5 obviously vote for their corresponding policies 1, 2, 4, and 5, respectively. But voter type 3's prior has become bimodal because the observed failure of policy 3 causes him to revise that probability drastically. So we must be more careful and calculate voter type 3's estimate of the probability of failure for all five policies to determine his most preferred policy, given his new prior. The failure probability of each policy i ∈ X is Table 5 shows these numbers.

So voter 3's preferences are still single-peaked, and his best choice is 2; this is his new most preferred policy. Therefore, the median voter is at 2, and policy 2 is adopted (policy shifts in the “wrong” direction).

Suppose policy x = 2 also leads to failure. Now the Bayesian revision yields posteriors shown in Table 6.

So voter type 3 has an epiphany: his probability distribution switches drastically to state 4. Hence, his most preferred policy is now 4. And the others become polarized: even type 4's most preferred policy now switches to 5.

This example serves to make three points: (i) Even slight asymmetries in initial beliefs can build into substantial differences. (ii) Polarization can occur in a way that even voters who are moderately biased in one direction come to favor the extreme policy in that direction. (iii) The outcome of an election can be determined by the switching of a very small number of the centrist type 3 voters, but everyone else is polarized to favor extreme policies; therefore the outcome is likely to cause a lot of dispute and acrimony.

Observable Continuous Loss.

Our second example allows policies to range over a continuum. Here, we focus on the possibility that polarization can arise because the function Y(x, s, u) is not monotonic in x; therefore, we assume that there is no error term u.

Voters are denoted by an index θ, ranging over the unit interval, and, for simplicity of exposition, they are assumed to be uniformly distributed over this range. Each voter has a continuous prior distribution about the true state of the world s, where s ranges over S = ℝ, the real line. The prior probability density function of voter θ is f _θ, that is, θ assigns probability f _θ(s) ds to the event that the true state of the world lies in the interval (s, s + ds). Let X = S.

The loss associated with an outcome is equal to the absolute value of the difference between the actual policy and state. The optimal policy in any state s is thus x = s, that is, x*(s) ≡ s. Writing x for the actual policy, the observable is therefore given by

Voter θ likes best the policy x that minimizes the expected loss, 𝔼_θ[L(x, s) | x], calculated under his prior. This is θ's most preferred policy, given his or her prior. We write this loss as:

The first-order condition for its minimization is where F _θ is the cumulative distribution function for voter θ's prior. And so the second-order condition is globally satisfied. Therefore, the optimum is given by F _θ = 1/2. Hence, voter θ's most preferred policy x _θ is the median of his prior distribution over the states of the world. Moreover, each voter's preferences are single-peaked around his most preferred policy.

Consider the first election (Period 1) under this setup. Under majority rule, the median of the most preferred policies becomes the chosen policy. To keep the notation simple, suppose this is the point 0 on the policy spectrum. Suppose the optimal policy, in the true state of the world, is different from this; for the sake of definiteness, suppose the true state is s = 1 and, hence, x* = 1.

The actual policy and the loss are, by assumption, observable without error. These observations enable people to infer that the true state must be either s ⁺ = x + L(x, s), or s⁻ = x − L(x, s). Thus, the continuous prior is updated to a two-point posterior. To keep the notation simple, again suppose L(x, s) = 1. With x = 0, the posteriors then become concentrated on 1 and −1. From Bayes' Rule, the posterior probabilities for voter θ are and Pr [s = −1] = 1 − Pr [s = 1].

Suppose there is a number z > 1/2, such that the priors of voters in the range 0 < θ < z satisfy f _θ(−1) > f _θ (1), and the priors of voters in the range z < θ < 1 satisfy f _θ(−1) < f _θ(1). Fig. 3 below illustrates this, with red curves showing the prior densities of voters θ < z and the blue curves showing those of voters θ > z.

Then the voters in (0, z) have posteriors with Pr(s = −1) > 1/2 > Pr(s = 1), and those in (z, 1) have posteriors with Pr(s = −1) < 1/2 < Pr(s = 1). These posteriors become the new priors in the next election (Period 2). Therefore, in that election, more than half of the voters vote for the policy x = −1, their new most preferred policy, and fewer than half vote for the policy x = 1, their new most preferred policy. Whereas the election in Period 1 was a contest with a continuum of opinions leading to a moderate policy (albeit not the optimal, given the state), the election in Period 2 is polarized between two quite distinct positions, and the choice shifts away from the optimal policy. This can happen even if z is very close to one half.

With the optimal policy at 1 and the actual policy at −1, the outcome in Period 2 will be a loss equal to 2. With the priors concentrated on 1 and −1 and the actual policy at −1, this loss can arise only if the optimal policy is 1. Therefore, Bayesian updating will lead to a convergence of opinions at the optimal policy, and that policy will be adopted unanimously in the election of Period 3.

In this example, once again, quite small differences among voters can create polarization and nonmonotonic shifts in priors. However, the special structure with no error term in the loss function leads to quick reversal of the polarization and convergence to the optimal policy. An error term with a suitably large dispersion can slow down this process. We omit the details because the algebra gets complicated.

Concluding Comments.

We have seen how an electorate can become polarized, and policies can shift away from the optimal when the observable indicators of policy outcomes are not monotonic in the policy choice and how such polarization is perfectly consistent with voters agreeing on values and using Bayesian updating and voting for the conditionally optimal policy, given their information.

Political polarization entails quite serious risks; political debates get bitter, and the very existence of a civil society may be threatened. Current examples are policies concerning discrimination, immigration, gender, religion, welfare state, human rights, terrorism, civil wars, national sovereignty, and nuclear armament. One way to reduce these risks, therefore, is to attempt to create observable indicators that are not bimodal like the ones above, and satisfy the MLRP. Of course, that can still leave untouched the additional problems caused by biased perception and learning. Moreover, such indicators may be hard to identify. However, our argument unambiguously supports the case for searching out and publicizing such indicators, under the here maintained hypothesis that people broadly agree on values but may have differing beliefs about the world. Contrary to the current tendency in many countries to avoid highlighting socially and politically controversial and pressing issues, our simple examples suggest that political polarization may be reduced rather than increased if, instead, more information about the factual current situation and the effect of used policies is made available in the public debate, even when the issues at hand are controversial.