91Ó°ÊÓ

Consider the variant of Hotelling's model in which the candidates (like the citizens) care about the winner's position, and not at all about winning per se. There are two candidates. Each candidate has a favorite position; her dislike for other positions increases with their distance from her favorite position. Assume that the favorite position of one candidate is less than \(m\) and the favorite position of the other candidate is greater than \(m\). Assume also that if the candidates tie when they take the positions \(x_{1}\) and \(x_{2}\) then the outcome is the compromise policy \(\frac{1}{2}\left(x_{1}+x_{2}\right) .\) Find the set of Nash equilibria of the strategic game that models this situation. (First consider pairs \(\left(x_{1}, x_{2}\right)\) of positions for which either \(x_{1}m\) and \(x_{2}>m .\) Next consider pairs \(\left(x_{1}, x_{2}\right)\) for which either \(x_{1}

Step by step solution

01

Define the Problem

Identify the key elements of the situation: There are two candidates, each with a favorite position, one to the left of point \(m\) and the other to the right of point \(m\). The candidates' dislike increases with their distance from their favorite position. If positions \(x_1\) and \(x_2\) tie, the outcome is the compromise policy \(\frac{1}{2}(x_1 + x_2)\).

02

Examine Candidate Strategies for \(x_1 < m\) and \(x_2 < m\)

Both candidates are on the same side of \(m\), specifically to the left. In this scenario, both candidates can potentially move closer to \(m\) to capture more voters. However, both will eventually reach \(m\) because if one candidate deviates more towards \(m\) than the other, they can capture the median voter. Thus, this setup does not provide a stable equilibrium.

03

Examine Candidate Strategies for \(x_1 > m\) and \(x_2 > m\)

Both candidates are on the same side of \(m\), specifically to the right. The reasoning for this is similar to Step 2, where both candidates will eventually converge towards \(m\), otherwise they have an incentive to move closer to \(m\) to gain more voters. Hence, this scenario does not produce a Nash equilibrium either.

04

Examine Candidate Strategies for \(x_1 < m < x_2\) or \(x_2 < m < x_1\)

In these scenarios, one candidate is on the left and the other on the right of \(m\). They move toward each other until they may reach \(m\) from opposite directions. This setup achieves a stable Nash equilibrium where neither candidate can improve their position by moving unilaterally.

05

Examine Candidate Strategies for \(x_1 = m\) and \(x_2 eq m\) or \(x_1 eq m\) and \(x_2 = m\)

In this scenario, one candidate is exactly at \(m\), and the other is not. The candidate not at \(m\) has an incentive to move closer or exactly to \(m\) if they can capture the median voter, hence not a stable equilibrium.

06

Examine Candidate Strategies for \(x_1 = m\) and \(x_2 = m\)

Here, both candidates are at \(m\). If both candidates choose \(m\), neither one has an incentive to deviate because moving away will only distance them from the median voter. Thus, this is the Nash equilibrium.

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Key Concepts

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

Hotelling's model

Hotelling's model is a classic concept in game theory, often used to show how competitors (like businesses or political candidates) position themselves to capture the maximum market share or voter support.
This model originates from Harold Hotelling's 1929 paper, explaining how sellers of a homogeneous product (imagine ice cream vendors on a beach) choose their locations to optimize sales.
In our problem, think of the two candidates as the ice cream vendors. Each candidate has a favorite spot (position along a line), and their dislike increases the further they are from this spot.
However, unlike businesses, these candidates care about the final position chosen if they tie, which leads to what we call a 'compromise policy'.

Strategic Game

In game theory, a strategic game refers to a scenario where players make decisions to maximize their payoffs while considering others' actions.
Each candidate in our problem has their set of strategies, which are possible positions they can choose. Their goal is to minimize their dislike for the position, knowing how the other candidate might move.
Their choices create a strategic interplay - for each position chosen by one candidate, there's potentially a best response from the other, and vice versa.
This back-and-forth positioning continues until no player wants to change their choice, leading us to what we call a Nash equilibrium.

Candidate Strategies

Candidate strategies are different possible positions candidates can choose along the line. Let's break down the scenarios:

• Both candidates left of point m: They'll keep moving closer to m to capture more voters, eventually both reaching m. Not a stable equilibrium.
• Both candidates right of point m: Similar to the left-side scenario. They'll converge towards m, with no stable equilibrium.
• One candidate left, one right of m: Each moves toward m from opposite sides, stabilizing without shifting positions. This can be a Nash equilibrium.
• One candidate exactly at m, other not: The candidate not at m will try to move closer. Not stable.
• Both candidates at m: Everyone's happy. Neither has a reason to move away. This is the Nash equilibrium.

These strategies and outcomes show how each candidate adapts based on the other's position, leading to this game-theory equilibrium.

Compromise Policy

A compromise policy happens when both candidates tie at different positions. Instead of picking one winner's position, they compromise.
The tie leads to an average of their positions: \[\frac{1}{2}(x_{1} + x_{2})\]. For example, if one picks x1 and the other x2, they end up at the middle point between them.
This concept modifies the dynamics of how candidates position themselves. Knowing a tie leads to a compromise outcome means neither candidate wants to deviate too far from the median point m.
As a result, understanding this aspect helps in predicting where candidates might ultimately stabilize in their position choices to avoid unfavorable compromises.

Equilibrium Analysis

Equilibrium analysis in this context helps to find stable solutions where neither candidate can improve their position unilaterally.
We look for Nash equilibria, where each candidate's choice is optimal assuming the other candidate's choice stays fixed.

• For both on the same side of m: They can't stabilize without gaining by moving closer to m.
• One on each side: They stabilize opposite each other, centered around m.
• One exactly at m: The other will move closer, preventing stability.
• Both at m: No candidate can benefit by moving away, establishing a Nash equilibrium.

This analysis lets us predict that the stable outcome is both candidates choosing to position themselves exactly at m, unable to improve their satisfaction by changing their choice.