This chapter examines the effects of such conviction on candidates' equilibrium policy proposals in a model of one-dimensional spatial competition. In this model, the voter and the politician are uncertain about the competence of the politician and the appropriateness of her proposed policy.
Introduction
In the one-dimensional spatial model pioneered by Downs (1957) and Hotelling (1929), if voters have Euclidean preferences, the Median Voter Theorem states that the ideal point of a median voter is the Condorcet Winner (Black 1958). Thus, pairwise symmetry is ensured in the one-dimensional strategy space at the median ideal point.
Restricted Condorcet Winners
Preliminaries
While the policy space is two-dimensional, the restriction to one issue reduces the strategy space for each candidate to one dimension, and thus a pure strategic equilibrium exists in more general circumstances than in the classical spatial model. In Section 1.5 we show that these conditions are weaker than those in the classical spatial model, and that an equilibrium therefore exists in more general circumstances.
A Restricted Condorcet Winner
Guaranteed Supporter Sets and Median Hyperplanes
Hence, this voter is closer to a than b∗(θ), and thus a guaranteed support forum, if and only θ is above the equidistance hyperplane for aand (−ε,−ε). Note also that the median hyperplane for the r-direction coincides with the median hyperplane for the r-direction; so, M(r) =M(−r).
Characterizing Restricted Condorcet Winners
A policy a∈α is an RCW if and only if M(r) is weakly further from β than H(a, b(r)) for everyr∈ω(a, β). If a ∈ α is an RCW and there exists a hyperplane S(r) that uniquely supports a point x∗ ∈ GS(a, β), then it is necessary that M(r) weakly beyond β if S(r) is
Comparison to the Classic Spatial Model
Conditions for an Equilibrium
Because Lemmas 2 and 3 are valid even if the policy sets are not disjoint, it is necessary to be an RCW that M(r)⊂H(a, b(r))+. If β is convex, then at most half of all directions are vulnerable and is further an RCW if and only if for every vulnerable direction,M(r)⊂H(a, b(r))+.
Existence of an Equilibrium
Intuitively, this inequality says that if a candidate can get at least 2τ closer to the center of a yolk than her opponent, then there is an RCW in that candidate's policy. By a symmetric argument: if c lies below the leftmost dashed line, there is an RCW in β.
Robustness to Perturbations of the Distribution
Let M¯(r) be the median hyperplane for direction indicator when voter's ideal points are distributed according to f¯. If a is an RCW when the voter's ideal points are distributed according to f, then for any r∗ ∈ ω(a, β)+ and δ >0,a is an RCW when voters have ideal points distributed according to ¯ . Therefore, by Theorem 1.5, an RCW remains if any subset of the voters' ideal points is shifted by any distance in direction r∗.
Conclusion
If b instead has an arbitrarily small y-coordinate and anx-coordinate of −ε, thenrab is arbitrarily close to (0,1). A shift of the voter ideal points in the direction of r∗ implies that voters are moving toward more Republican policies on both gun rights and defense spending, at least weakly. Therefore, a shift in voter opinion toward more Republican policies does not decrease support for a given Republican policy.
Appendix
Since a convex set is the intersection of its supporting hyperplanes, we know that the supporting hyperplanes atx are equally spaced hyperplanes. By Lemma 1.5, the solutionx∗ must lie on the unique hyperplane S(r) to GS(a, β) in the directionr becauser·x≥r·x∗ for all x∈GS(a, β). Because the first term on the right is the integral of a strictly positive function on a positive-measured area, it is strictly positive.
Introduction
In my model, the uncertainty about the location of the central voter is not taken for granted. Other views in the literature also consider voters' inferences toward each other (McMurray 2018). However, none of these analyzes take into account the possibility that additional information depends on the content of the politician's policy proposals.
Setup
The Informational Environment
For simplicity, I assume that the voter can choose to observe the state perfectly at a cost of c > 0; otherwise, she must make her election decision based on her prior alone. This assumption also implies that voters are more likely to believe in states that are closer to their current ideal point. That a candidate has proposed a particular policy likely reflects the candidate's belief that there is evidence available that could persuade the voter to support it, and that may be somewhat informative; however, I assume this type of information is negligible.
Voter Preferences over Proposals
A voter may choose to pay attention to the information available during the campaign to learn about the country, but this is an expensive activity, as it requires time and effort to gather information about the country and update their beliefs. This assumption implies that other voters' ideal points, and thus their voting decisions, do not contain information about her own ideal point. Leaving aside any instrumental desire to learn about the state in order to influence outcomes, a voter may simply gain more utility from paying attention to an exciting, high-stakes race.
Voter Preferences over Candidates
Second, a representative voter is always decisive, and as a result she has a substantial interest in choosing the right policy. However, voters derive other benefits from researching an election, and their level of interest in an election is at least to some extent linked to their perception that something is at stake; that is, which of the two alternatives is chosen is likely to have material repercussions. For example, an investor may be very interested in the tax policies of two candidates, not because she expects to change the outcome of the election, but so that she can plan to move her assets accordingly.
The Decision to Pay Attention
When a = 1, the voter is better off when the state is ω = 1 or ω = 2 because she is able to get a policy closer to the true state. When the challenger takes a position of a= 2, the voter is significantly better off in the case that ω= 2 because of its quadratic loss utility function. If the challenger provides no reason for the voter to learn more about the race, the voter acts as if he has a known ideal point at 0.
Behavior of the Challenger and the Three Equilibria
Equilibria
If the first condition of Corollary 2.1 is met, then when the voter is willing to pay attention to a= 1, the challenger prefers the position a= 1 to 0. If the second condition of Corollary 2.1 is met, then when the voter willing to pay attention to a= 2, the challenger prefers the position a= 2 toa= 0. If the voter is willing to pay attention only to a= 2, then the challenger is willing to gamble by the position a = 2 to offer.
Existence
Conclusion
Appendix
If ω = 2, then the voter's policy utility is greater because it is closer to ω than 0, but he sacrifices E(v), for a net benefit of 3−E(v). Further, because P(1,1)≥P(0, y) for these parameter values and P(1,1)> P(2,1) for any parameter value, a= 1 maximizes the candidate's probability of selection given the choice of the voter's attention. Therefore, a = 2 maximizes the candidate's probability of being elected given the voter's choice of attention, making it the unique equilibrium, given the assumption that the challenger does not choose a= 0 if she is indifferent.
The Public Vetting Model
Competence and Outcomes
In the event that the politician maintains the status quo, there is no change in the outcome, and ut = 0. Both the voter and the politician are uncertain about the outcome of the policy, if it is implemented, and the politician's competence. However, it is in reality possible that the politician has some supplementary information about his own competence.
Policy Vetting and Reputation
This signal is conditionally independent of the politician's competence, in the sense that P(σ|ut = 1) = P(σ|ut = 1, τ). This does not require my stronger assumption that policy proposal succeeds if and only if the politician is competent. As is standard in the literature, I refer to the probability that the politician is competent given the available information as her reputation.
Preferences
This assumption can be interpreted to mean that the signal only contains information about the benefits of the policy. Given the assumption that a policy is successful if and only if the politician is competent and the voter observes the probability that the policy is successful if it is implemented, the value of the received signal reputation, P(τ =H|σ), is simply P(u1 = 1|σ). For ease of notation, signals refer to the reputation they indicate. Based on his observation of the politician's action and its outcome, the voter updates the politician's reputation at the end of the period from one to r.
First Period Strategies with Public Vetting
- Implementation and Reputation
- The Incentive to Implement
- The Impact of Vetting
- Inefficient Implementation with Public Vetting
His information about the sitting president is the same as if the politician had consciously opted for the status quo. The politician who wants to be re-elected only wants to implement her policy if her signal is sufficiently low. Optimal experimentation requires the politician to implement her policy if and only if her signal is sufficiently high.
Private Vetting
- Setup
- The Voter’s Decision Rule
- The Politician’s Decision Rule
- Equilibrium Behavior
- The Impact of Private Vetting
- Private Vetting Can Improve Welfare in Both Periods
If σ remains less than σpri, then the politician's behavior is the same in both cases. If σ remains greater thanσpri, then the politician's behavior is the same in both cases. It is therefore extremely likely that when the politician chooses to implement her policy, her competence is revealed.
Conclusion
It is weighted by (1−γ), because this is the proportion of time that the outcome of their policy is observed in the public inquiry case. As γ increases, it is more likely that when an incumbent implements her policy, no outcome is observed. Hence, according to Bayes' rule, when no outcome is observed, it is more likely that the incumbent chose to implement her policy; that is, σ > σ∗.
Proofs of Results
If the politician were to implement his policy instead, then the second period's expected voter utility is σ[1]+(1−σ)[2rc−1]. Second, assume σ < rc so that the politician is not retained in the absence of additional information. If rc ≤ σ, then the politician will be retained in the choice ifx1 =s1, so integration of both sides of the inequality (3.10) establishes that.
Extensions
- Notation and Assumptions
- Generalization of the Public Vetting Equilibrium
- Generalization of the Private Vetting Equilibrium
- Partially Asymmetric Information
Consequently, there is a critical value σpub, so the politician prefers to implement her policy if and only ifσ <σpub. Ifσ(0)>0andσ(1)<1, there exists a unique pointσpub∈(0,1) such thatσ(σpub) =σ∗pub, and for allσ < σpub it holds thatσ < σ(σ) and the politician prefers for implemented its policy, while for allσ > σ∗pub, it holds that σ > σ(σ), and the politician prefers the status quo. Therefore, σ∗ = 0 is the only solution, and the policymaker prefers to implement her policy for every σ∈[0,1].
Policy sets for A and B in two party competition
The hyperplane of equidistance and its associated halfspaces
The GS set for a given policy a ∈ α
Policy a is strictly majority preferred to b
Policy a is majority preferred to b(r) and b(−r) if and only if M (r) lies between H(a, b(r)) and
The gray set is an example of β such that all directions are vulnerable at policy a
Possible locations for the center of a yolk
If there is a yolk with center c located above the upper dashed curve, then there exists a ∈ α
If a voter is moved the direction r ∗ , then the median hyperplane for r cannot move toward b(r) . 18
The set of directions ω(a, β) + for which voters can be moved while preserving a as an RCW
In region I, the voter does not observe the state for any position of the challenger. In region II,
The challenger’s preferences over positions given the values of π 1 and π 2
The equilibrium positions of the challenger given the values of π 1 and π 2
The sequence of events for the public information model
A politician with signal σ ˆ has a greater expected probability of re-election from implementation
The politician only has an incentive to implement her policy if σ is sufficiently low
