/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 (Electoral competition with thre... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 15

(Electoral competition with three candidates) Consider a variant of Hotelling's model in which there are three candidates and each candidate has the option of staying out of the race, which she regards as better than losing and worse than tying for first place. Use the following arguments to show that the game has no Nash equilibrium. First, show that there is no Nash equilibrium in which a single candidate enters the race. Second, show that in any Nash equilibrium in which more than one candidate enters, all candidates that enter tie for first place. Third, show that there is no Nash equilibrium in which two candidates enter the race. Fourth, show that there is no Nash equilibrium in which all three candidates enter the race and choose the same position. Finally, show that there is no Nash equilibrium in which all three candidates enter the race, and do not all choose the same position.

Short Answer

Expert verified
No Nash equilibrium exists as each entering candidate can deviate to improve their outcome.

Step by step solution

01

No Single Candidate Enters the Race

If only one candidate enters the race, that candidate wins unopposed. Another candidate has an incentive to enter the race because they can either win or tie for first place. Hence, there is no Nash equilibrium in which a single candidate enters the race.
02

Tie for All Candidates in Equilibrium

Suppose there is a Nash equilibrium in which multiple candidates enter the race. For it to be an equilibrium, all candidates that choose to enter must tie for first place. If one candidate wins outright, the other candidates would prefer to change their strategy and enter a different position to at least tie for first place.
03

No Two-Candidate Entries to the Race

Assume two candidates enter the race. According to Step 2, they must tie for first place to be in equilibrium. However, if two candidates tie for first, the third candidate has an incentive to enter as well and share the prize, thus breaking the equilibrium condition of only two candidates entering the race.
04

No Three Candidates in Same Position

If all three candidates enter the race and choose the same position, they all tie for first place. However, any one candidate can deviate and choose a different position to potentially win outright or at least share the win with one other candidate, which is preferred over losing or tying with two others.
05

No Three Candidates in Different Positions

If three candidates enter and choose different positions, at least one candidate will not tie for first place. This candidate would prefer to switch to the position of one of the other candidates to at least share the win. Therefore, there is no Nash equilibrium where all three candidates enter the race and do not choose the same position.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Nash equilibrium
In the context of game theory, a **Nash equilibrium** occurs when players reach a point where no one can benefit by changing their strategy while the other players keep their strategies unchanged. In simple terms, everyone sticks to their game plan because changing wouldn't make any individual better off.
In the electoral competition problem, there is no Nash equilibrium, meaning no stable strategy where all candidates are satisfied with their choices. Any adjustment by one candidate would result in other candidates wanting to change their strategies as well.
Hotelling's model
Hotelling's model is a framework used to analyze competition in various fields, including electoral competition. The model often involves competitors positioning themselves along a spectrum to maximize their appeal. For example, candidates may position themselves politically to attract the most voters.
In the context of three candidates in an election, the model helps us understand how they choose their positions (strategies) to either win or tie for first place. Since the candidates have the uncertainty of other entrants adapting their positions, it remains a complex dynamic without a Nash equilibrium.
candidate strategy
In electoral competition, a **candidate strategy** involves choosing how to position oneself to either win the election or, at the minimum, tie for first place. Strategies are affected by the choices of other candidates, leading to continuous adjustments.
If a single candidate enters, others will enter to either win or tie. If two candidates enter and tie, a third might join to at least share first place. Hence, strategies are interdependent and not stable, ruling out a Nash equilibrium.
electoral competition
This term refers to the battle among candidates to secure the most votes and win the election. The goal of each candidate is to adopt a strategy that maximizes their chances of winning or, at worst, tying for first place.
Electoral competition involves various dynamics, including positioning, entry and exit decisions, and reaction to competitors' moves. In this problem, the lack of Nash equilibrium suggests that no set strategy satisfies all candidates simultaneously, leading to continuous repositioning and no stable outcome.
tie for first place
In this specific electoral competition problem, candidates will prefer to either win outright or tie for first place rather than finish second or third.
If all entering candidates tie for first, there's no incentive for any candidate to change positions. However, this situation never stabilizes due to the constant strategic movements 鈥� one candidate may change positions to win outright, prompting others to adjust as well. Therefore, the scenario keeps changing, and no stable tie for first place can be maintained in equilibrium.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(Bertrand's duopoly game with constant unit cost) 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?

(Multi-unit auctions) Two units of an object are available. There are \(n\) bidders. Bidder \(i\) values the first unit that she obtains at \(v_{i}\) and the second unit at \(w_{i}\), where \(v_{i}>w_{i}>0\). Each bidder submits two bids; the two highest bids win. Retain the tie-breaking rule in the text. Show that in discriminatory and uniform-price auctions, player \(i^{\prime}\) s action of bidding \(v_{i}\) and \(w_{i}\) does not dominate all her other actions, whereas in a Vickrey auction it does. (In the case of a Vickrey auction, consider separately the cases in which the other players' bids are such that player \(i\) wins no units, one unit, and two units when her bids are \(v_{i}\) and \(w_{i}\).) Goods for which the demand exceeds the supply at the going price are sometimes sold to the people who are willing to wait longest in line. We can model such situations as multi-unit auctions in which each person's bid is the amount of time she is willing to wait.

(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.

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.)

(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.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.