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Chapter 3: Problem 18

(Citizen-candidates) Consider a game in which the players are the citizens. Any citizen may, at some cost \(c>0\), become a candidate. Assume that the only position a citizen can espouse is her favorite position, so that a citizen's only decision is whether to stand as a candidate. After all citizens have (simultaneously) decided whether to become candidates, each citizen votes for her favorite candidate, as in Hotelling's model. Citizens care about the position of the winning candidate; a citizen whose favorite position is \(x\) loses \(\left|x-x^{*}\right|\) if the winning candidate's position is \(x^{*}\). (For any number \(z,|z|\) denotes the absolute value of \(z:|z|=z\) if \(z>0\) and \(|z|=-z\) if \(z<0 .\) ) Winning confers the benefit \(b\). Thus a citizen who becomes a candidate and ties with \(k-1\) other candidates for first place obtains the payoff \(b / k-c\); a citizen with favorite position \(x\) who becomes a candidate and is not one of the candidates tied for first place obtains the payoff \(-\left|x-x^{*}\right|-c\), where \(x^{*}\) is the winner's position; and a citizen with favorite position \(x\) who does not become a candidate obtains the payoff \(-\left|x-x^{*}\right|\), where \(x^{*}\) is the winner's position. Assume that for every position \(x\) there is a citizen for whom \(x\) is the favorite position. Show that if \(b \leq 2 c\) then the game has a Nash equilibrium in which one citizen becomes a candidate. Is there an equilibrium (for any values of \(b\) and \(c\) ) in which two citizens, each with favorite position \(m\), become candidates? Is there an equilibrium in which two citizens with favorite positions different from \(m\) become candidates? Hotelling's model assumes a basic agreement among the voters about the ordering of the positions. For example, if one voter prefers \(x\) to \(y\) to \(z\) and another voter prefers \(y\) to \(z\) to \(x\), no voter prefers \(z\) to \(x\) to \(y\). The next exercise asks you to study a model that does not so restrict the voters' preferences.

Short Answer

Expert verified
If b 鈮� 2c, a Nash equilibrium exists with one candidate. An equilibrium with two candidates is unlikely under most conditions.

Step by step solution

01

Introduction

Understand the main elements of the game: citizens decide whether to become candidates, vote for their favorite candidate, and base their payoffs on the winning candidate's position, the cost to become a candidate, and the benefit of winning.
02

- Examine Payoffs

Examine the payoffs for different scenarios:1. A citizen who becomes a candidate and ties or wins.2. A citizen who becomes a candidate and loses.3. A citizen who does not become a candidate. Define the critical thresholds where these decisions change.
03

- Single Candidate Equilibrium

To show a Nash equilibrium with one candidate:1. Calculate the payoff for a citizen who decides to be the only candidate.2. Compare this payoff to the payoff for not becoming a candidate.If the single candidate's payoff exceeds the non-candidate's payoff when another citizen would like to step in, this establishes an equilibrium when one citizen becomes the candidate.
04

- Condition for Single Candidate Equilibrium

Establish that if the benefit for winning, b, is less than or equal to twice the cost of becoming a candidate, c, then indeed, there is a Nash Equilibrium with one candidate:If another citizen enters, the cost is shared, and the overall benefit becomes less attractive.
05

- Equilibrium with Two Candidates at Same Position

Determine if there can be an equilibrium when two citizens with the same favorite position both become candidates:Calculate and compare payoffs, taking into account the potential benefit and costs divided, to see if either would deviate from being a candidate.
06

- Equilibrium with Two Candidates at Different Positions

Analyze the scenario where two citizens with different positions both become candidates:1. Calculate the potential payoff for each with their respective positions.2. Discuss the likelihood and the influence of differing positions on each other's decisions to become candidates or not.
07

- Conclusion

Summarize the Nash equilibrium findings:1. With b 鈮� 2c, there is an equilibrium with a single candidate.2. Analyze under which conditions, if any, two citizens at the same position or different positions could sustain an equilibrium.

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Key Concepts

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

Nash Equilibrium
In game theory, a Nash equilibrium is a scenario where no player can benefit by changing their strategy while the other players keep their strategies unchanged. In the context of this Citizen-Candidates game, a Nash equilibrium occurs when citizens choose whether or not to become candidates in such a way that none of them can improve their payoff by changing their decision.
In this game, a single citizen becoming a candidate can form a Nash equilibrium if their payoff from winning exceeds the cost and potential losses from others entering the race. If the benefit of winning, denoted as \(b\), satisfies \(b \leq 2c\), where \(c\) is the cost to become a candidate, it is less likely that multiple citizens will enter, thus maintaining equilibrium.
Hotelling's Model
Hotelling's model is originally an economics model that describes how businesses or political candidates choose their positions based on consumer or voter preferences. In the Citizen-Candidates game, this model supports the idea that candidates locate themselves at positions that maximize their share of the vote.
Assume voters have an ordered preference. If a candidate's position closely matches a voter's preference, that candidate is more likely to receive that voter's vote. Therefore, strategic positioning is crucial. The model stipulates that candidates will ideally position themselves in a way that best attracts their largest share of the voter base.
Payoff Calculation
Calculating payoffs under different scenarios is essential to understanding the game dynamics. For the Citizen-Candidates game, the payoffs depend on the following:
  • A citizen who does not become a candidate has a payoff of \(-\left| x - x^* \right| \), where \(x^*\) is the winning candidate's position.
  • A citizen who becomes a candidate and wins has a payoff of \(b - c\), where \(b\) is the benefit of winning, and \(c\) is the cost of becoming a candidate.
  • A citizen who becomes a candidate and ties for the win has a payoff of \(\frac{b}{k} - c\), where \(k\) is the number of candidates who tie.
  • A citizen who becomes a candidate but loses has a payoff of \(-\left| x - x^* \right| - c\).
Understanding these payoffs helps determine the strategic choices of whether citizens should become candidates based on expected outcomes.
Voter Preferences
Voter preferences in this game align closely with Hotelling's model assumptions. The game assumes that for each position \(x\), there is exactly one citizen who considers \(x\) as their favorite position. Voters will always vote for the candidate closest to their own preference.
Preferences create a competitive dynamic where citizens need to evaluate positions and payoffs accurately to decide whether to run for candidacy. For example, a citizen whose favorite position is \(m\) will vote for the candidate whose platform is closest to \(m\), ensuring minimal loss. Therefore, knowing voter preferences is crucial for candidates to decide strategically on their positions.
Strategic Decision-Making
Strategic decision-making in the Citizen-Candidates game involves evaluating the costs and benefits of becoming a candidate, considering the positions of other potential candidates, and the preferences of the electorate. Candidates must decide:
  • Whether the benefit \(b\) of winning outweighs the cost \(c\) of becoming a candidate.
  • If entering the race would split the vote in a way that makes winning less likely, or makes losing more costly.
  • Where to position themselves to attract the maximum number of votes, taking into account voter preferences and opponents' positions.
Understanding these strategic considerations helps explain why a single-candidate outcome may more likely occur under certain conditions (like when \(b \leq 2c\)), and why multiple-candidate scenarios may be less stable as equilibria.

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Most popular questions from this chapter

There is a finite number of positions and a finite, odd, number of voters. For any positions \(x\) and \(y\), 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 every 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.

(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.

(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.

Consider Cournot's game in the case of an arbitrary number \(n\) of firms; retain the assumptions that the in-verse demand function takes the form (54.2) and the cost function of each firm \(i\) is \(\mathrm{C}_{i}\left(q_{i}\right)=c q_{i}\) 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 \(q_{\prime}\) this is the price at which the output is sold.

(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.
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