Attentiveness in elections with impressionable voters

A tie is a measure zero event in our analysis, i.e., it occurs non-generically, and can be ignored. For completeness we can assume that ties are broken with equal probability.

See Sect. 4.1 for further discussion, and examples of the factors that determine voters’ attentiveness.

The two ways to define the “event” probabilities \(\Pr _{\theta }^{t}(.)\) are equivalent, and each can be derived from the other.

Strictly speaking, we would have to index the set K by subscript \(\theta\) for the group of voters, and superscript t for time. For simplicity of notation, however we drop these two indexes here and in analogous cases.

The parameter \(\lambda _{p}\) need not be the same across all voters. As we will prove in Proposition 1, the distribution of vote measures in a stationary distribution does not depend on \(\lambda _{p}\). This parameter can vary across e.g., groups of voters, without changing the stationary measures of votes.

An alternative method of updating aspirations in the literature is that of Cyert-March (Cyert & March, 1963) where aspirations are a linear mixture of previous period aspirations and an average of present period impressions. The analysis would be more difficult, but would yield qualitatively similar results.

We note that \(K_{\theta }^{\tau }\) is a random variable, with the distribution \(\{\Pr _{\theta }(K)\}_{K\subseteq N}\) as defined previously. We also note that if \(\sum _{\tau =1}^{t}|K_{\theta }^{\tau }|=0\), i.e., voters do not pay attention to any candidate up to period t, then the average of impressions from \(\tau =1\) to t is not well-defined. In this situation, we take, for example, \(a_{\theta }^{t+1}=a_{\theta }^{0}\) an arbitrary value.

If the events \(\{e_{\theta }(i)\}_{i\in N}\) are correlated, some results continue to apply, but others are more difficult to prove, and likely necessitate assumptions on the full distribution of the attentiveness vector \(e_{\theta }\). Specifically, the results concerning the impact of positive/negative impressions would not change, while the results concerning attentiveness would be limited.

In addition to the direct effect of the attention probabilities \(r_{\theta }(.)\) discussed, we note that attention probabilities also indirectly impact vote measures through the impression probabilities \(q_{\theta }(.)\). As aspirations are computed as average of impressions of candidates voters pay attention to, paying more attention to a candidate makes it more likely that voters’ impressions of that candidate have a larger weight on aspirations. In other words, attention probabilities affect the aspirations \(a_{\theta }\) in a stationary distribution. Furthermore, probabilities of positive impression \(q_{\theta }(.)\) depend on aspirations, yielding the indirect effect. However, this indirect effect is second-order, and not likely to dominate the direct effect we discuss above. For this reason, we set it aside.

In the case of full attentiveness, i.e., \(k=n\), which corresponds to the model of Andonie and Diermeier (2022), the process has a unique stationary distribution.

Under negative plurality, if attentiveness is endogenous then the distribution of the event vector \(e_{\theta }^{t}\) can never be stationary, and so Proposition 1 does not apply. Moreover, it is easy to see that the process cannot have distributions where the vote measures are stationary.

The formal condition that “the average \(v_{\theta }^{1}\) for candidate 1 is sufficiently larger than the same average for the other candidates” is complicated, and we omit stating it formally in the proposition. In general, the condition depends not only on the fixed components \(v_{\theta }^{i}\) and group shares \(s_{\theta }\), but also on other parameters, e.g, distribution F(.), etc. (see the proof to Proposition 3 in the appendix).

Recall that in our model aspirations are means of past impressions. Therefore, if past impressions are negative, then aspirations will be low.

Under negative plurality, we assume that \(r_{R}(3)\) is “sufficiently” close to 1. We assume this additional condition for the following reason. From previous propositions, under negative plurality, the vote measure of a candidate increases if voters pay less attention to that candidate. Therefore, even though all voters rank candidate 3 at the bottom, if the attention probability \(r_{R}(3)\) is low enough, the vote measure of candidate 3 can increase so that it surpasses that of candidate 1 or candidate 2. To shut down this effect, which we separately analysed previously, we make the assumption that \(r_{R}(3)\) is “sufficiently” large.

The cutoff \(v^{*}\) is characterized by the implicit equation: \((3-F( \frac{2v^{*}:-1}{4\alpha }))(8-F(\frac{2v^{*}-3}{4\alpha })-F(\frac{ 1+2v^{*}}{4\alpha }))=16\).

In the quasi-stationary distribution under negative plurality, each of the three candidates can win, but the probability that candidate 3 wins is smaller than the probability that candidate 1 or candidate 2 wins.

We can think of this specification as a simplified form of the following more detailed conception of news coverage. Suppose the distribution of impressions of candidate i among voters of type \(\theta\) is as follows:where \(\chi ^{i}\) is a binary random variable that can take values “\(+\nu ^{i}\)” and “\(-\nu ^{i}\)”, with probabilities \(\pi ^{i}\), and \(1-\pi ^{i}\) respectively; and \(\xi ^{i}\) is a continuous random variable with a symmetric, single-peaked, distribution around 0. The two additional random variables, \(\chi ^{i}\) and \(\xi ^{i}\), model the following two features. First, in some periods the news coverage of a candidate can be positive, i.e., when the realization of \(\chi ^{i}\) is \(+\nu ^{i}\), and in other periods news coverage can be negative, i.e., when the realization of \(\chi ^{i}\) is \(-\nu ^{i}\). Second, some voters may interpret some news as good, while others may interpret the same news as bad. This is captured by the second random variable \(\xi ^{i}\). This more detailed extension, however, is equivalent to the specification above, where \(\mu ^{i}\) corresponds to the expectation of \(\chi ^{i}\), and the uncertainty is combined into the random term \(\epsilon _{\theta }^{t}(i)\).

The effect of an increase of \(\mu ^{i}\) on the mean of impressions \(\pi _{\theta }(i)\) is larger than the effect on the aspirations \(a_{\theta }\), and so the probability of positive impression \(q_{\theta }(i)\) increases with \(\mu ^{i}\).

Here, we focus our analysis on plurality rule. The analysis under the two other rules is similar.

To see that, consider first the effect of an increase in the impression probability \(q_{\theta }(i)\) on the distribution of vote measures. An increase of \(q_{\theta }(i)\) (assuming \(q_{\theta }(l)\), for \(l\ne i\), are fixed) induces an increase in the vote measure of candidate i, \(S_{\theta }(i)\), and a decrease in the vote measure of each candidate \(j\ne i\), \(S_{\theta }(j)\). Consider now in addition the effect of a decrease in the impression probabilities \(q_{\theta }(l)\), for each candidate \(l\ne i\). First, \(S_{\theta }(i)\) further increases. Second, we note that the measure of abstentions \(S_{\theta }(A)\) increases as well, as voters become less likely to have positive impressions of candidates \(l\ne i\). This implies that, on average, the vote measure of a candidate \(j\ne i\) further decreases. The vote measures \(S_{\theta }(j)\), for \(j\ne i\), decrease on average, however, the vote measure of some candidates \(l\ne i\) may increase, depending on their ranks in the vector of fixed components \(u_{\theta }\), and the shape of the distribution function of the random component \(\epsilon\).

For technical reasons in the case of negative plurality we derive “quasi-stationary” distributions.