tion into their choice, and this changes the magnitude of their expected payoff from the competency dimension (see Lemma 1). In Figure 2.5 the competency component of utility changes as a function of S + 1, the number of signals that each voter has access to. In Figure 2.5 S ranges from 0 to 25 and the value of γ = .10 is fixed.
The expected benefit of competency to each voter is calculated (and thus to the more competent candidate). That is, as the number is signals increases, the voter becomes increasingly more certain about which candidate is relatively more competent. This certainty adds additional weight to the component of their utility that is based upon competency, which thus allows the candidates to move increasingly further away from the median voter’s policy platform preferences. Note that this effect is similar to the comparative statics in p as well. As the number of signals increases the voter’s ex- pected benefit from competency converges to γ, as expected, since as the number of signals grows the beliefs about competency converge to the truth. Thus any so- cial structure which produces fewer signals reduces the voter’s expected benefit of competency, and any structure which produces more signals will increase the voter’s expected benefit of competency. When voters truthfully reveal their signals in the communication phase of the game then increasing the number of available signals adds additional weight to competency, but when voters have the option of distorting the message, however, more signals do not necessarily increase the amount of information
— and as a consequence, the voters’ expected utility.
Number of signals Expected Candidate Benefit
Figure 2.5: Expected Candidate Benefit of Competency as a Function of Number of Signals.
Note: Fixed γ =.1.
2.7.2 Truth Telling
The following sections analyze candidate location outcomes under a different assump- tion of voter behavior. Here voters will be assumed to communicate their sincere pref- erence in the communication phase, as opposed to communicating their competency signal. There is a large literature which suggests that voters will indeed communicate based upon partisan bias (Bartels 2002, Gerber and Green 1998, Achen 1992, Achen 1975, Zaller and Feldman 1992) and this behavioral assumption — that voters will communicate their candidate preference and not their competency signal — aligns much more closely with observed voter behavior. Here again voters will have payoffs which are determined based upon for whom they vote. This ensures that voters will not condition their vote upon their pivotality in the election.5
Note first that all voters observe each other’s ideological preferences, so that a voter’s ideology is common knowledge. Note that then voteriobservespA, pB, si (her private competency signal), andxj, the ideology of voter j, for allj ∈N. Voter ican state a message mij ∈ {A, B} to voter j. That is, she can claim that she observed either si =A orsi =B when communicating with j.
Recall λ is defined as the marginal expected benefit of voting for the more com- petent candidate. The voters are sorted into three groups. A voter is defined as an ideological voter if ||xi −pA| − |xi −pB|| ≥ λ. A voter is defined as a competency voter if ||xi−pA| − |xi−pB||< λ. Finally, a voter is defined as anindifferent voter if ||xi−pA| − |xi−pB||=λ.
Lemma 3: Voterjwill update her beliefs about competency if voteriis a competency voter.
Proof: i sends j a message mi which reveals her most-preferred candidate. j has observed the ideology ofi,xi, the candidate locationspAand pB and knows the value
5Suppose that voter payoffs were based upon the majority winner. This clearly affects their desire to tell other voters the truth when they reveal their private competency signal. With a large electorate, however, it is unlikely that voters will communicate strategically to others within their social context. Additionally, if voters are assumed to communicate sincerely, then even if a voter conditions upon her pivotality (consistent with the setup of Austen-Smith and Banks 1996) when casting her ballot under majority rule she will continue to vote sincerely. Note here that all voters observe each other’s ideological preferences and the common prior on the competency of either candidate is 1/2.
of λ. Thus, it is possible forj to calculate which groupiwill belong to by examining the value of||xi−pA| − |xi−pB||and λ. Ifi is an ideological voter, then the message mi will be based upon i’s ideological preference. If i is an indifferent voter, then the messagemi is equally likely to be based upon i’s ideological preference as it is to be based upon her information. Thus, the only informative message which ican send to j is if i is a competency voter, in which case her most preferred candidate is based upon her own signal.
2.7.3 Candidate Polarization with Most-Preferred Commu- nication
Recall that since voters randomly draw their ideological preferences from the uniform distribution, there is no correlation, ex ante, between their location on the circle and their preferences. Given the cases outlined above, communication which consists of the voters revealing their most-preferred candidate has the consequence of dramati- cally reducing the amount of information available for the competency voters, as now the only voters from whom they can receive useful signals are other competency vot- ers, and being a competency voter is dependent upon ideology. This implies that each competency voter will have less information (so the expected utility of competency is smaller) implying that λ will in fact be smaller than in the situation in which all voters are required to tell the truth. Referring to earlier work with respect to the comparative statics of λ and candidate locations in equilibrium, note that as λ is reduced less candidate polarization is observed.
2.7.4 Information Distribution Changes
Next the distribution of voter ideology changes. Here there is now strong correlation between voter ideology and their geographic location. Assume that after each voter draws their ideology, they sort themselves by ideology, and within ideology, by groups
— thus competency voters have competency neighbors, indifferent voters have indif- ferent neighbors, and ideology voters have ideology neighbors. This produces a circle
with the extremists together and the moderates together.
By considering this special distribution, the aim is to capture the notion that voters may separate themselves into groups in a way that depends on their ideology.
In other words, people who live in close proximity to each other, or who have close relationships with each other, are more likely to hold similar political views. Under this extreme form of correlation, for example, every voter on the left half of the circle is a Democrat while every voter on the right half of the circle is a Republican, and within parties, the voters are again sorted by group. This situation is referred to as
“voter polarization”.
The voters know the polarization setup (by allowing them, as before, to observe each other’s types). Now the competency voters, because they are neighbors, will be able to receive a great deal more information via competency signals. They will again have more certainty about which candidate is more competent. Here it is again possible refer to earlier work with respect to the comparative statics of λ and candidate locations. With an increased number of signals λ will increase relative to the uncorrelated case, which will produce an increase in candidate polarization.