Problem 6 - Advanced Macroeconomics

Question

The Condorcet paradox. Suppose there are three voters, \(1,2,\) and \(3,\) and three possible policies, \(A, B,\) and \(C .\) Voter 1 's preference ordering is \(A, B, C ;\) voter \(2^{\prime} \sin B, C, A ;\) and voter \(3^{\prime}\) s is \(C, A, B,\) Does any policy win a majority of votes in a two-way contest against each of the alternatives? Explain.

Step-by-Step Solution

Answer

No policy consistently wins against the others; it's a Condorcet paradox.

Step 1: Identify the voter preferences

First, let's list out the preferences of each voter clearly: Voter 1 prefers policy A over B over C, Voter 2 prefers policy B over C over A, and Voter 3 prefers policy C over A over B.

Step 2: Determine pairwise contests

We will conduct pairwise contests between each pair of policies: \(A\) vs \(B\), \(B\) vs \(C\), and \(C\) vs \(A\). In each contest, determine which policy is preferred by the majority of voters.

Step 3: Evaluate A vs B

Compare the preferences of all voters between policies A and B: - Voter 1 prefers A over B. - Voter 2 prefers B over A. - Voter 3 prefers C over A (and also B over A implicitly in their preference C, A, B), so they prefer B over A. Thus, 2 voters (2 and 3) prefer B over A; therefore, B wins the majority in this contest.

Step 4: Evaluate B vs C

Compare the preferences between policies B and C: - Voter 1 prefers A over B (and thus A over C through implicit preference), therefore implicitly prefers B over C. - Voter 2 prefers B over C. - Voter 3 prefers C over B. Thus, 2 voters (1 and 2) prefer B over C; therefore, B wins the majority in this contest.

Step 5: Evaluate C vs A

Compare policies C and A: - Voter 1 prefers A over C. - Voter 2 prefers C over A. - Voter 3 prefers C over A. Thus, 2 voters (2 and 3) prefer C over A; therefore, C wins the majority in this contest.

Step 6: Determine if there is a Condorcet winner

Analyze the results of the pairwise contests to see if any policy is a consistent winner: - Policy B wins over A but loses to C. - Policy C wins over A but hasn’t contested B previously. No policy wins against each of the other two policies in all contests, demonstrating the Condorcet paradox where there's a cycle with no single policy that is better than both other policies.

Key Concepts

Voter Preferences

In voting systems, understanding voter preferences is a vital foundation. Each voter has a list of options ordered according to their liking. In our example:

• Voter 1 likes policy A the most, followed by B, then C.


• Voter 2 favors policy B first, then C, and lastly A.


• Voter 3 prefers C over the others, with A next, and B last.

These preferences are often called a "ranking." Each voter ranks the available policies or candidates from most to least preferred. It is essential to clearly establish each voter's ranking since it directly influences the outcomes in further steps, like pairwise contests and understanding the overall group preference.

Pairwise Contests

In pairwise contests, each option is put against another in a direct comparison. The preferences of the voters ascertained earlier come into play here. By conducting these pairwise contests, we aim to determine which policy is preferred.

• First, we compare A against B. Most voters prefer B in this case.


• Next, B is compared to C, with B again being the more favorable choice.


• Finally, C is measured against A, and C takes the lead.

This process illustrates how each pair of options is weighed against each other according to the majority preferences of the voters. Pairwise contests are crucial because they reveal individual bouts among choices that are later used to find broader conclusions in collective decision making.

Majority Rule

Majority rule is a fundamental concept in democratic decision-making processes. It states that the choice, which gets more than half of the votes, wins. In pairwise contests:

• If voter 1 and voter 2 prefer policy B over A, then majority rule says that B wins in a contest between A and B.


• Similarly, when more voters prefer C over A, C wins in contests between C and A.

Majority rule finds the most favored option in individual comparisons, serving as a key method to identify a potential winner. However, as we'll see, it can still lead to paradoxes when applied across multiple options.

Voting Paradox

The voting paradox, also known as the Condorcet paradox, arises when individual rational preferences result in a collective irrational outcome. This happens when no candidate or option wins all pairwise contests. In our example:

• B defeats A in one contest.


• B then loses to C in another.


• Lastly, C wins against A.

Such cycles illustrate how it's possible for a group to lack a stable preference, even if each individual has consistent likes and dislikes. Despite each pair having a winner via majority rule, no single option is the overall winner. This highlights the complexity and sometimes the impracticality of simple majority-based systems when more than two choices are involved. The voting paradox showcases that even widely accepted voting methods can have inherent contradictions and may not always lead to a clear decision.