Learning model

An approach that highlights the interaction between candidates and voters might suppose that candidates send certain information about themselves and voters process

8A few further assumptions about the unobserved factors in V_ik are necessary to get to a standard probit representation. As already discussed, if researchers observe that a voter chooses candidate 1 over candidate 2, the expected utility plus unobserved other factors Vij=_E[U_ij] +_eikhas to be higher than for the second candidate V_i1>V_i2. Making the assumption that the additional unobserved factors follow a normal distribution with an arbitrary error variance of .5, thus: e_ik∼N(0, 0.5), allows to derive a probit model with difference in policy distance. The choice of the error variance is arbitrary and made for the purpose of simplicity. Supposing a more general variance σ would effect the importance of policy distance in the probit model, by a deviation of two times σ: _2σ^β. This would not change the general implications in any meaningful way.

this to learn what they can expect from the candidates. This learning model focuses on how the communicated policy platforms alter difference in expected utility. The argument based on this is relatively straightforward: For voters with inconsistent policy preferences, information contained in platform signals is less informative, compared to voters with consistent preference. This holds because the information contained in the candidate signal depends on the degree of uncertainty a voter possesses about her ideal platform. Expected utility is less strongly affected by policy distance for voters with a strong policy belief variance who are uncertain about their ideal policy-platform, when compared to those with a low belief variance.

In order to cover this idea more formally, a model might suppose that a priori, a voter possesses certain information about the utility difference between candidate 1 and candidate 2 that is normal distributed around zero with some variance τ⁹. In this modeling approach, prior utility difference is depicted by a distribution P U_ij−U_ik
.

P U_ij−U_ik

∼ N (0, τ) (4.7)

In the next step candidates signal their policy platform C_j, C_k. As the primary focus of this analysis is on inconsistency, we might suppose that these signals are perfectly observed by all voters. The information contained in these signals about the utility difference between the candidates U_i1−U_i2, follows the difference in policy distance as outlined by 4.3. If candidate 1 signals that her policy platform is the same as the voter’s mean belief, and candidate 2 signals a more extreme platform, the information for the voter is that candidate 1 will yield considerable higher utility. However, although the candidates’ signals are perfect, the information about utility is not, because voters themselves are uncertain about their ideal policy outcome.

The information in candidate signals can more formally be denoted as the likelihood.

This function, essentially, depicts how likely the signals are given varying differences in utility between the candidates. I obtain the likelihood of the signals by plugging in voters’ policy beliefs (as outlined model by 4.3) in a utility function that is affected by the difference in quadratic loss between two candidates platforms. Again, simplifying

9I suppose that the prior utility is centered around zero to simplify the posteriori expression.

4.2. THEORY

the expression by dividing policy beliefs in mean belief x_i and random belief variance θ_i ∼N(_{0, σΘ}²

i)results in the following¹⁰:

P(C₁, C₂|U_i1−U_i2) = −β

(x_i+θ_i−C₁)²− (x_i+θ_i−C₂)^2 (4.8)

∼ N −β

(C₁−x_i)²− (C₂−x_i)²_{ , 4β}²(C₂−C₁)²σ_Θ²_i
The likelihood is normally distributed around the difference in quadratic loss of the candidates signaled positions (C1and C2) to a voter’s mean policy belief x_i. Of special concern is the variance term: 4β²(C₂−C₁)²σ_Θ²

i. What can be seen from this expression is that the variance depends on a voter’s belief variance. For voters with consistent policy preferences, the variance is smaller, which makes the signals more informative.

For voter with a wider belief variance, the information in the signal is weaker.

Voters can use the information contained in the signals to update their prior utility difference, addressing the question of what the utility difference is given the platform signals P(U_i1−U_i2|C₁, C₂). Supposing that voters form this expression employing Bayesian updating implies that the posterior beliefs are proportional to the prior and the likelihood P(U_i1−U_i2|C₁, C₂)_{∝ P}(U_i1−U_i2)P(C₁, C₂|U_i1−U_i2). The posteriori is then the mixture between two normal distributions. What matters for vote choice (because voters employ expected utility) is the expectation of this posterior. As a mixture, the expectation will lie between the expectation of the prior and the expectation of the likelihood. Because I suppose that the prior is centered around zero, the posterior expectation will lie between zero and expectation of the likelihood. Depending on degree of information in the signals, which is affected by the variance term, this will be closer to zero, or closer to the expectation from the likelihood. The expectation of the posterior can be formulated in the following way¹¹:

10Intermediate steps to get to this expression. I first rearrange terms

P(C1, C2|U_i1−U_i2) = −β

Because θ_iis a random variable the whole expression is a random variable as well. The expectation of this expression is equal to the first part−β

(C1−xi)²− (C2−xi)², because the expectation of θiis zero.

The variance of the expression is constructed from the second part which is normal (because θ_iis normal) with standard deviation of 4β²(C₂−C₁)²σ_Θ²

11 i

The complete posterior distribution is normal, according to:

P(U_i1−U_i2|C₁, C₂) ∼N −β τ

Inconsistency in policy preferences

Marginal effect of policy distance

low high

0.5 1.0 1.5 2.0

Expected utility model Learning model

Figure 4.1: Contrasting observable implications of two models how voters form expected utility for an illustrative example

E[P(U_i1−U_i2|C₁, C₂)] = −w_iβ

(x_i−C₁)²− (x_i−C₂)^2 (4.9) where the weight w_i = ^τ

4β²(C1−C2)²σ_Θi² +τ depends on a voter’s variance in policy beliefs σ_Θ²_i and thereby on the inconsistency in policy preference. If inconsistency increases, the weight on policy distance decreases. This is contrary to the expectation from the expected utility approach, because in addition to the homogeneous effect parameter β, voters put different weights on policy aspects. Following a similar voting model to the one in 4.3.1, this shows that the voting probabilities are less strongly affected when comparing voters with same mean positions but different inconsistencies. Formally, the marginal effect of distance on expected utility w_iβdecreases with higher levels of inconsistency. For voters with consistent beliefs (lim_σ2

Θi→0w_iβ = β), the effect is the same as in the expected utility model, but for voters with inconsistent beliefs policy distance less strongly impacts utility. In the most extreme case, policy distance has no effect at all (lim_σ2

Θi→+_∞w_iβ=0), because the platform signals are not informative to these voters. This analysis permits to derive observable implications that shall be outlined in the next subsection.