91Ó°ÊÓ

(Electoral competition with asymmetric voters' preferences) Consider a variant of Hotelling's model in which voters's preferences are asymmetric. Specifically, suppose that each voter cares twice as much about policy differences to the left of her favorite position than about policy differences to the right of her favorite position. How does this affect the Nash equilibrium? In the model considered so far, no candidate has the option of staying out of the race. Suppose that we give each candidate this option; assume that it is better than losing and worse than tying for first place. Then the Nash equilibrium remains as before: both players enter the race and choose the position \(m\). The direct argument differs from the one before only in that in addition we need to check that there is no equilibrium in which one or both of the candidates stays out of the race. If one candidate stays out then, given the other candidate's position, she can enter and either win outright or tie for first place. If both candidates stay out, then either candidate can enter and win outright. The next exercise asks you to consider the Nash equilibria of this variant of the model when there are three candidates.

Step by step solution

01

- Understand the Asymmetric Preferences

Recognize that each voter cares twice as much about policy differences to the left of their favorite position compared to the right. This implies that the utility loss is higher for deviations to the left. Let's denote the favorite position of a voter by \( p_v \). If a policy is implemented at position \(P\), then the utility loss for the voter is \(2 \cdot |P-p_v| \) if \(P < p_v\) and \( |P-p_v| \) if \(P > p_v\).

02

- Set Up Modified Hotelling Model

Consider a linear spectrum where policy positions can be placed. In the basic Hotelling model, voters are uniformly distributed along this line, and candidates choose a position to maximize votes. With asymmetric preferences, positions to the left of a voter's favorite position cause twice as much disutility.

03

- Define Candidates' Strategies

In the traditional two-candidate model, the Nash equilibrium is for both candidates to choose the median voter's position \(m\). Now, introducing three candidates, the aim is to see how they choose their positions, given voters' asymmetrical preferences.

04

- Analyze Three-Player Dynamic

Consider the positions chosen by the first two candidates. The third candidate will position herself relative to these positions to maximize her share of votes, considering asymmetrical voter preferences. Since no candidate can choose a position that gives them fewer votes than staying out of the race, each candidate must strategically position themselves such that moving slightly left or right would gain more votes than it costs.

05

- Identify Nash Equilibrium

With asymmetrical preferences, the Nash equilibrium may shift from the median position as candidates try to take advantage of the fact that small leftward deviations result in a greater loss of utility for voters. Candidates are likely to position themselves closer around a point that reflects voters' lessened sensitivity to rightward deviations, but still accounting for centralizing near the median.

06

- Verify No-One Stays Out

Finally, confirm that none of the three candidates prefer to stay out of the race. Even if they would lose or tie, remaining in the race offers better payoffs than the outside option due to the possibility of strategic positioning.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Asymmetric Voter Preferences

In this scenario, voters' preferences are not uniform. Each voter places greater importance on policy differences to the left of their favorite position, meaning deviations to the left cause twice as much disutility.

Let's break this down further: If a voter's favorite position is denoted by \( p_v \), and a policy position is \( P \), the disutility (unhappiness) they experience is given by:

• \( 2 \cdot |P - p_v| \) if \( P < p_v \)
• \( |P - p_v| \) if \( P > p_v \)

This setup means that candidates need to be very careful about policies that are more leftward. Such policies will have a more significant negative impact on voters whose favorite positions are more rightward.

In conclusion, understanding and acknowledging asymmetric voter preferences is crucial for candidates. Deviations to the left have more weight, compelling candidates to strategize around this asymmetry.

Hotelling's Model

Hotelling's model is all about spatial competition applied to politics, where candidates are like businesses competing for market share in a linear space. In this model, candidates select positions along this spectrum to maximize their votes.

In the simplest form, voters are uniformly distributed along a line, and two competing candidates will both position themselves at the median voter's position \( m \).

When asymmetric preferences enter the picture, candidates need to consider the increased disutility on one side of voters' favorite positions.

This means:

• They may choose positions slightly right of center to avoid the heavy disutility penalties of leftward deviations.
• They must balance their desire to be near the median with the asymmetrical dislike voters have for leftward policies.

This adjustment changes the traditional approach in Hotelling's model, bringing in new strategic elements as candidates try to capture the maximum number of votes while accounting for voter asymmetry.

Three-Candidate Elections

Adding a third candidate complicates the dynamic. Each candidate must position themselves in such a way that moving left or right yields more votes without causing a significant loss.

Here's how it breaks down:

• Any extreme positioning will leave space for others to capture the center votes, especially with asymmetric preferences.
• The third candidate analyzes the first two's positions and adjusts to maximize their vote share.

The simplest equilibrium, where each candidate earns a third of the votes, might shift due to the asymmetric response to policies.

All three candidates need to consider not just the vote count but the resulting disutility for voters on the asymmetrical side. This balancing act typically sees:

• Positions clustering around central but slightly rightward points to leverage less disutility from rightward deviations.
• None of the candidates choosing positions that cause them to drop out of the race because strategic placement offers better incentives than staying out.

Thus, understanding the multi-candidate scenario under the lens of asymmetric preferences showcases a more intricate strategic interaction among candidates.