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Chapter 3: Problem 17

(Electoral competition between candidates who care only about the winning position) 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}

Short Answer

Expert verified
Nash equilibria occur when candidates are at symmetric distances from or at either m, above m, or below m.

Step by step solution

01

Understanding the Scenario

In this variant of Hotelling's model, two candidates care only about the position they end up at, not about how many votes they receive. Each candidate has a favorite position, one below and one above the midpoint, m. If the candidates tie, they compromise at the average position \(\frac{1}{2}(x_1 + x_2)\).
02

Considering Positions x1 < m and x2 < m

If both candidates choose positions less than m, each candidate will try to choose the position closest to their favorite position. Given they care about the distance from their favorite position, neither has an incentive to move to the right of the other if they stay left of m.
03

Considering Positions x1 > m and x2 > m

If both candidates choose positions greater than m, similar reasoning applies. Each candidate will position themselves as close to their favorite positions as possible while staying on the right-hand side of m.
04

Considering Positions x1 < m < x2

If x1 < m < x2, any deviation from the favorite position that results in crossing m would make one candidate worse off. Therefore, they are likely to stay on either side of m to avoid compromising too far from their preferred positions.
05

Considering Positions x1 = m or x2 = m

If either candidate is exactly at m, while the other is not, the candidate at m has no incentive to deviate as they are at the midpoint, which is a stable position.
06

Considering the Position Pair (m, m)

If both candidates choose m, they agree. In this case, there is no conflict, and the compromise is naturally m.
07

Nash Equilibria

Compile the observations: Nash equilibria occur when both candidates are at their preferred symmetric distances on either side of or at m. Hence, pairs \((x_1, x_2) \< m\) or \((x_1, x_2) \> m\) where neither candidate has an incentive to deviate, positions such that \(x_1 < m < x_2\), and the pair (m, m) are Nash equilibria.

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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 an economic theory that describes how businesses (or in this case, political candidates) choose their positions to attract the maximum number of customers (or votes). Imagine a line that represents a spectrum of political ideologies, from left to right. Each candidate wants to place themselves on this line in a way that attracts the most voters.
Hotelling's model suggests that candidates will often move toward the middle of the spectrum because that's where most of the voters are. This leads to minimal differentiation between the candidates. However, the variant we are considering alters this model by focusing on how candidates care about their final position rather than the number of votes they receive.
Strategic Game Theory
Strategic game theory involves analyzing situations where individuals (or players) make decisions that take into account the decisions of others. It looks at how these individuals choose their strategies to maximize their own outcomes.
In our case, the candidates are the players, and their strategies involve choosing positions along the spectrum. Each candidate decides their position based on their own preferences and predicts the opponent's choice. This leads to a complex interplay of decisions where each candidate aims to position themselves optimally, given the opposite candidate's probable position.
Candidate Positioning
Candidate positioning deals with where each candidate places themselves on the political spectrum. This positioning is crucial as it reflects their strategic choices aimed at optimizing their preferred outcome.
The candidates in our model have favorite positions: one prefers a position less than the midpoint (m), and the other prefers a position greater than m. They choose their positions based solely on their own preferences, rather than the total number of votes they might receive. This unique twist leads to different dynamics in the competition, as they are not driven by voter preferences but by positional strategizing.
Equilibrium Analysis
Equilibrium analysis in this context refers to finding the Nash equilibria, where neither candidate has an incentive to deviate from their chosen position because it would lead to a worse outcome for them.
In our scenario, various configurations yield Nash equilibria:
  • Positions where both candidates are either entirely left or right of m, with no incentive to cross over.
  • Positions where one candidate is left of m and the other is right of m.
  • The position where both candidates are exactly at m.

    • In each case, candidates have no incentive to change their positions, meeting the condition for Nash equilibrium. This analysis helps us understand stable outcomes where both candidates are content with their chosen positions.
Political Competition
Political competition in this model represents the rivalry between candidates as they strive to position themselves advantageously. Unlike typical elections where the goal is to win the majority of votes, the focus here is purely on the candidate's preferred placement on the spectrum.
This kind of competition forces us to think differently about electoral strategies since traditional vote-maximizing tactics are not the primary concern. Instead, candidates may choose positions that are not necessarily the most popular but align closely with their ideological stances. The understanding of these dynamics can shed light on real-world political behaviors and electoral positioning strategies.

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Most popular questions from this chapter

(Cournot's game with many firms) Consider Cournot's game in the case of an arbitrary number \(n\) of firms; retain the assumptions that the inverse demand function takes the form (54.2) and the cost function of each firm \(i\) is \(C_{i}\left(q_{i}\right)=c q_{i}\) for all \(q_{i}\), with \(c<\alpha .\) Find the best response function of each firm and set up the conditions for \(\left(q_{1}^{*}, \ldots, q_{n}^{*}\right)\) to be a Nash equilibrium (see \(\left.(34.3)\right)\), assuming that there is a Nash equilibrium in which all firms' outputs are positive. Solve these equations to find the Nash equilibrium. (For \(n=2\) your answer should be \(\left(\frac{1}{3}(\alpha-c), \frac{1}{3}(\alpha-c)\right)\), the equilibrium found in the previous section. First show that in an equilibrium all firms produce the same output, then solve for that output. If you cannot show that all firms produce the same output, simply assume that they do.) Find the price at which output is sold in a Nash equilibrium and show that this price decreases as \(n\) increases, approaching \(c\) as the number of firms increases without bound. The main idea behind this result does not depend on the assumptions on the inverse demand function and the firms' cost functions. Suppose, more generally, that the inverse demand function is any decreasing function, that each firm's cost function is the same, denoted by \(C\), and that there is a single output, say \(q\), at which the average cost of production \(C(q) / q\) is minimal. In this case, any given total output is produced most efficiently by each firm's producing \(q\), and the lowest price compatible with the firms' not making losses is the minimal value of the average cost. The next exercise asks you to show that in a Nash equilibrium of Cournot's game in which the firms' total output is large relative to \(\underline{q}\), this is the price at which the output is sold.

In the variant of Hotelling's model that captures competing firms' choices of product characteristics, show that when there are two firms the unique Nash equilibrium is \((m, m)\) (both firms offer the consumers' median favorite product) and when there are three firms there is no Nash equilibrium. (Start by arguing that when there are two firms whose products differ, either firm is better off making its product more similar to that of its rival.)

Two firms are developing competing products for a market of fixed size. The longer a firm spends on development, the better its product. But the first firm to release its product has an advantage: the customers it obtains will not subsequently switch to its rival. (Once a person starts using a product, the cost of switching to an alternative, even one significantly better, is too high to make a switch worthwhile.) A firm that releases its product first, at time \(t\), captures the share \(h(t)\) of the market, where \(h\) is a function that increases from time 0 to time \(T\), with \(h(0)=0\) and \(h(T)=1\). The remaining market share is left for the other firm. If the firms release their products at the same time, each obtains half of the market. Each firm wishes to obtain the highest possible market share. Model this situation as a strategic game and find its Nash equilibrium (equilibria?). (When finding firm \(i^{\prime}\) s best response to firm \(j\) 's release time \(t_{j}\), there are three cases: that in which \(h\left(t_{j}\right)<\frac{1}{2}\) (firm \(j\) gets less than half of the market if it is the first to release its product), that in which \(h\left(t_{j}\right)=\frac{1}{2}\), and that in which \(h\left(t_{j}\right)>\frac{1}{2}\).)

Consider the extent to which the analysis depends upon the demand function \(D\) taking the specific form \(D(p)=\alpha-p .\) Suppose that \(D\) is any function for which \(D(p) \geq 0\) for all \(p\) and there exists \(\bar{p}>c\) such that \(D(p)>0\) for all \(p \leq \bar{p} .\) Is \((c, c)\) still a Nash equilibrium? Is it still the only Nash equilibrium?

(Electoral competition for more general preferences) There is a finite number of positions and a finite, odd, number of voters. For any positions \(x\) ind \(y\), each voter either prefers \(x\) to \(y\) or prefers \(y\) to \(x\). (No voter regards any two positions as equally desirable.) We say that a position \(x^{*}\) is a Condorcet winner if for very position \(y\) different from \(x^{*}\), a majority of voters prefer \(x^{*}\) to \(y\). \(a\). Show that for any configuration of preferences there is at most one Condorcet winner. b. Give an example in which no Condorcet winner exists. (Suppose there are three positions \((x, y\), and \(z)\) and three voters. Assume that voter 1 prefers \(x\) to \(y\) to \(z\). Construct preferences for the other two voters such that one voter prefers \(x\) to \(y\) and the other prefers \(y\) to \(x\), one prefers \(x\) to \(z\) and the other prefers \(z\) to \(x\), and one prefers \(y\) to \(z\) and the other prefers \(z\) to \(y\). The preferences you construct must, of course, satisfy the condition that a voter who prefers \(a\) to \(b\) and \(b\) to \(c\) also prefers \(a\) to \(c\), where \(a, b\), and \(c\) are any positions.) c. Consider the strategic game in which two candidates simultaneously choose positions, as in Hotelling's model. If the candidates choose different positions, each voter endorses the candidate whose position she prefers, and the candidate who receives the most votes wins. If the candidates choose the same position, they tie. Show that this game has a unique Nash equilibrium if the voters' preferences are such that there is a Condorcet winner, and has no Nash equilibrium if the voters' preferences are such that there is no Condorcet winner. A variant of Hotelling's model of electoral competition can be used to analyze he choices of product characteristics by competing firms in situations in which lext exercise you are asked to show that the Nash equilibria of this game in the fase of two or three firms are the same as those in Hotelling's model of electoral competition.
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