Condorcet Voting Paradox: Videos & Practice Problems

The Condorcet voting paradox illustrates how majority voting can yield inconsistent results when multiple options are present. For instance, if A beats B, B beats C, but C beats A, the transitive property fails. By manipulating the voting agenda, policymakers can influence outcomes; for example, eliminating C first allows A to win against B. This highlights that majority voting alone may not reflect societal preferences, as it can be strategically manipulated, leading to potential market failures and inefficiencies in decision-making.

Condorcet Voting Paradox

Alright. Now let's discuss another issue related to voting, the Condorcet voting paradox. The Condorcet voting paradox shows that majority voting can result in inconsistent results. We might get inconsistent results when we do a majority vote, right? The majority vote is where whoever is in the majority wins the vote. So let's think of this situation where we've got 3 groups faced with 3 choices. The first group has these preferences: 1st choice, 2nd choice, 3rd choice, they pick A over B over C. The second group picks B, C, A, and the third group picks C, A, B.

So let's see what happens if we send these choices to a vote and they have to go up against each other. The first one would be if we sent A versus B to a vote. So, the first group, would they pick A or B if that was the vote? Well, their first choice is A, so they would pick A. What about the second group? The second group's first choice is B, so they would pick B. And how about the third group? The third group's first choice is C, but that's not available. This is a vote between A and B, so they're going to vote for A and A is going to win this vote.

Now what happens if we put the vote between B and C? Well, what is group 1 going to vote for? They can't get their first choice of A, so they will take their second choice of B. What about group 2? They get their first choice, B, so they're going to vote for B. And group 3, their first choice is C which is, oh it is in the vote, so that is what they're going to vote for. Group 3 votes for C, in this case B wins, right? B comes out over C.

And guess what's going to happen in this third scenario? If you can't guess, let's go ahead and analyze it. We've got C versus A in this case. What is group 1 going to pick? Group 1 is going to pick their first choice of A and they'll vote for A. Group 2, well their first choice of B is not available, so they're going to vote for C, and group 3 is going to vote for their first choice of C up here. Alright? So, group 3 is going to vote for C and we're going to see that C wins, right?

This is the Condorcet voting paradox showing us that depending on the pairing, A beats B, B beats C, but then C beats A. This is an example where the transitive property, which we remember from algebra, that if A equals B and B equals C, then A must equal C, does not hold. A was able to beat B and B was able to beat C, so we would imagine that A could beat C, but then C beat A. Crazy, right? This is the paradox here. Alright, so let's pause here and then in the next video, let's see how we can use this knowledge of the agenda and the order of the vote to force certain policies to win. Let's pause here and then we'll continue with that example.

Order of the Vote

Alright, so now let's see how the order of the voting agenda is going to affect the outcomes of the vote, okay? So if we're policyholders or we're the policymakers, we might have certain agendas, right? Maybe we want A to win. Personally, we want A to win so we can manipulate the order of the vote to force A to come out on top. So let's see what happens here. If we want A to win, well the first thing we're going to want to do is we're going to want to get rid of C, right? Because if we look above, C beats A, right? C versus A, C won the vote. So the first thing we want to do is we want to vote between B versus C, right? And as we saw above, when B went against C, well B was the agenda that came out on top, right? So B wins in that case. So now that B won, C is off the table. C is no longer being voted for because B was preferred. So now we set a vote of A versus B, right? And now if we go A versus B, look what happens up above. We already figured this out and we saw that A wins, right? So A wins and that knocks out B and A becomes the policy that gets instated, right? A wins the overall vote and becomes the policy. Alright, let me get out of the way here and let's see what we would do if we want C to win, alright? Let's see what we want to do if C wins. So, notice if we go B versus C up here, if we're doing the B versus C, B is going to beat C. So we need to get rid of B first. So the first vote we want to do is A versus B. If we do A versus B, well, who's going to win in A versus B? A wins, right? So B is off the table, B is no longer being voted on and now we can do the vote of A versus C or as we have it written above, C versus A and the outcome there is that C wins, okay? So we can use this knowledge or at least politicians can use this knowledge to their advantage when they're setting the voting agenda to get their policies to be the ones that go through, alright? So another conclusion we see here is that majority voting by itself does not necessarily provide the outcomes that society wants, right? It's not just going to create the right outcomes. Majority voting can be manipulated just as we saw above. Alright? So let's go ahead and pause here and move on to the next video.

Which of the following is true regarding the Condorcet voting paradox?

• A. Majority voting is the best method for allocating society's resources
• B. Voters will not vote if their first choice is not available
• C. Only unanimous voting systems are efficient
• D. Different outcomes can occur based on the order of the voting agenda

Answer: (D)

Here’s what students ask on this topic:

What is the Condorcet voting paradox?

The Condorcet voting paradox occurs when majority voting among three or more options results in inconsistent outcomes. For example, if option A beats option B, and option B beats option C, one might expect option A to beat option C. However, in the paradox, option C can beat option A, violating the transitive property. This paradox highlights the potential for majority voting to produce cyclical preferences, where no single option is the overall winner, leading to inconsistencies in decision-making.

How does the Condorcet voting paradox illustrate the failure of the transitive property?

The Condorcet voting paradox illustrates the failure of the transitive property in majority voting. The transitive property states that if A > B and B > C, then A > C. However, in the paradox, if A beats B, B beats C, but C beats A, the transitive property fails. This inconsistency shows that majority preferences can be cyclical and not straightforward, leading to paradoxical outcomes where no single option is the clear winner.

How can policymakers manipulate voting outcomes using the Condorcet voting paradox?

Policymakers can manipulate voting outcomes by strategically setting the voting agenda. For instance, if they want option A to win, they can first eliminate option C by arranging a vote between B and C, where B wins. Then, they set a vote between A and B, where A wins. By controlling the order of votes, policymakers can ensure their preferred option comes out on top, demonstrating how majority voting can be manipulated to achieve specific outcomes.

What are the implications of the Condorcet voting paradox for majority voting systems?

The Condorcet voting paradox implies that majority voting systems can produce inconsistent and cyclical outcomes, which may not reflect true societal preferences. This paradox shows that majority voting can be strategically manipulated, leading to potential inefficiencies and market failures in decision-making. It highlights the need for alternative voting methods or additional safeguards to ensure fair and representative outcomes in democratic processes.

Can the Condorcet voting paradox occur with more than three options?

Yes, the Condorcet voting paradox can occur with more than three options. As the number of options increases, the likelihood of encountering cyclical preferences and inconsistent outcomes also increases. This makes it even more challenging to determine a clear winner using majority voting, further complicating decision-making processes and highlighting the limitations of majority rule in complex voting scenarios.