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This is the election system in France and Ukraine. In the first round anyone can run. If no candidate wins an absolute majority, then the top two candidates go to a second round where they face off against each other.
What are the relative disadvantages of this electoral system?
Related: What are the disadvantages of first-past-the-post electoral systems? which deals with the relative disadvantages of the other major electoral system.
In no particular order:
And, as with all democratic systems, the winner is based on popularity, and not on competence.
The main problem is that it uses a single-mark ballot, so that voters can only express an opinion about one candidate.
In a two-candidate general election, this is fine, but in a multi-candidate election, it results in vote-splitting and unrepresentative winners.
Imagine an election in which there are dozens of very good, representative candidates, and two unrepresentative extremists on opposite ends of the spectrum. The majority of voters prefer the representative candidates, but which one? Their votes are split between them, and each only gets a small fraction of the vote. The unrepresentative extremists, on the other hand, are unique, and have no nearby competitors in the ideological space, so they get solid pluralities of the vote, and advance to the next round.
In this round, one of those two remaining candidates will get "majority support", but this is an illusion created by eliminating everyone else. Any of the representative candidates would have received a majority of support over one of the two extremists, but they aren't in the race anymore because of vote-splitting.
Here is an illustration on a one-dimensional political spectrum:
Red, Blue, and Yellow are good representatives of the voters, but votes are split between them, leaving Orange and Green to get the most votes and proceed as finalists to the runoff. Green, an unrepresentative extremist, will win the election, despite Blue being preferred over Green by 70% of voters. (Here's a 2D version.)
This same vote-splitting problem also affects every other voting system that only counts first preferences in rounds, such as FPTP, exhaustive ballot, supplementary vote, contingent vote, and instant-runoff voting (called "ranked choice voting" in the US, or "alternative vote" in the UK).
A much better solution is an approval voting primary, like the one adopted in St Louis recently. Voters can select as many candidates as they approve of, which makes this vote-splitting much less likely.
Alternatively, a system that considers all voters' preferences simultaneously can be used to eliminate the primary altogether, such as a Condorcet ranked-choice system or STAR voting.
This method fails the monotonicity criterion. This disadvantage is somewhat technical in nature, and it's hard to prove that nonmonotonic behavior occurred if you have two rounds rather than an ordinal ballot, but it is a weird and arguably undesirable property.
Intuitively, this means that rating a candidate higher can cause that candidate to lose.
A voting method satisfies the monotonicity if the following property holds. In order to define it, I need a few other definitions first. Note that we can define this property without talking about agents/rational actors at all; it can be defined using just collections of ballots.
In order to define this property, I'll introduce the nonstandard notion of X-superiority.
Let A and B be ballots. Let X be a candidate. A is X-superior to B if and only if, for all pairs of distinct candidates (Y, Z), A and B have the same ranking of Y, Z when neither Y nor Z is X, and for any pairwise comparison involving X, A ranks X higher than B does or A and B have the same ranking.
Following Wikipedia's example, X > Z > Y is X-superior to Z > X > Y, but X > Y > Z is NOT X-superior to Z > X > Y because the order of Y and Z is reversed.
Let A and B be I-indexed sets of ballots. A is X-superior to B if and only if, for all i in I, it holds that A[i] is X-superior to B[i].
A voting method fails the monotonicity criterion if and only if, there exist I-indexed sets of ballots C and D such that the candidate X wins in C, the candidate X does not win in D and D is X-superior to C.
Here's the nonmonotonicity example from Wikipedia with an explanation.
C > B > A 28
C > A > B 5
A > B > C 30
A > C > B 5
B > A > C 16
B > C > A 16
In this election, B is eliminated first and A wins in the next round.
Next consider the election below
C > B > A 28
C > A > B 3
A > B > C 30
A > C > B 7
B > A > C 16
B > C > A 16
Now, C is eliminated first and B subsequently wins.
Two C > A > B were shifted to A > C > B ballots in this example, increasing candidate A's votes, and that caused candidate A to lose.
The main drawback has already been mentioned (but not emphasized enough in my opinion): It costs a lot to organise two rounds of voting for every election. That's also what you will find in guidance from international organisations.
The other objections are mostly theoretical and hard to support empirically. For example, it does not seem obvious at all that countries with a two-round system have a much higher level of voter fatigue or non-voting than others, in general.
The advantages of a two-round system over a simple first-past-the-post system should not be underestimated. Being able to vote for the least bad choice while keeping the system transparent and simple to understand for the voters is a feature not a bug. Ranked voting systems fail in that respect and even if their proponents tend to dismiss this as a non-issue, it is in fact an important trade-off.
This is a variation of STV, except instead of Instant Runoff, it's Separate Election Runoff, and votes gets transferred only once. As such, many of the advantages and disadvantages are similar to that of STV. Perhaps the most infamous case of this was when Louisiana had a runoff between Edwin Edwards, who was widely considered corrupt, and David Duke, a former KKK grand wizard, leading to the bumper sticker slogan "Vote for the crook: It's important."
As James K said, inclusion on the next round depends only on first-place preferences, and ignores lower preferences. So you could theoretically have a candidate that is ranked as first or second place preference by all the voters, but doesn't make it to the runoff. Thus, candidates who appeal to a broad base, but only moderately, can be at a disadvantage to candidates who are strongly supported by a minority. For instance, suppose one candidate is running on a platform of tax breaks for Republicans, one on tax breaks for Democrats, and one on no partisan-based tax breaks. If there's broad support for no tax breaks, but everyone mildly supports tax breaks for their side over no tax breaks, while strongly opposing tax breaks for the other side, then the runoff will be between the two tax breaks candidates, even though almost everyone supports the "no tax breaks for anyone" compromise.
And with a large field, a candidate needs only a few people to have that candidate as their top choice to get into the runoff, and voting blocs getting their preferred candidate largely comes down to which ones can build coalitions and coalesce around single candidates.
Another issue is that this can result in both candidates being from the same party. This can be an advantage or disadvantage depending on one's point of view. On the one hand, if most people support one party, then arguably it makes more sense for the voters to choose between candidates in that party than to have an election with a foregone conclusion. On the other hand, this can make people in the minority party feel disenfranchised. Moreover, depending on how the votes are split, it's possible for both of the final candidates to be from the minority party.
James K gave a good answer, but I would add one more issue:
The system is subject to gaming by organized factions or parties using stalking-horse candidates.
It is possible, under this system, for a clever and organized group to introduce candidates who appear to be members of ethnic or class voting blocs, or who campaign on slightly altered variations of a given ideology, in order to split the vote from those groups and influence who the top two final candidates are. In first-past-the-post systems, you can't afford to do this, because it might lead to a first-place finisher from outside your faction winning the race outright with a very small percentage of the vote. But in multi-round voting this can be an effective strategy.
@Acccumulation's last paragraph approaches what is IMO the most important flaw, but understates the problem.
Imagine a country which is 45% blue-ish and 55% reddish, but where the blue parties are more united so there's two blue candidates and five red candidates. Unless one of the red candidates does much better than the others, the runoff will be between the blues so blue would win. If you instead had a full runoff system (instant, or as many rounds of voting as it took) you would expect most of the red votes to accumulate on the remaining red candidate (some light reds might prefer light blue to dark red), so the final round would be red versus blue and red would win.
Sure, it's better than FPTP, but the extra cost and time would be way more than enough to do IRV.
(You could say "why don't the red parties unite?" but that just gets back to the main problem of FPTP, and, bringing it down to real politics, there's a lot more room for disagreement among those who want the state to do more than there is among those who want it to do less.)
Here is an example: in a 3-candidate election there is a centrist candidate that everybody could live with but is not a popular first choice (as it is often the case). Not having many first choices the centrist candidate gets eliminated on the first round. The run-off is among two extremists, both supported by a minority, so more than half of the population gets stuck with an extremist they really dislike.
From a theoretical point of view, IRV has all problems that STV has, only worse. This question has an answer that is a good analysis that focuses on a three-candidate election, which is a case where IRV and STV are in theory equivalent (you don't call to the polls twice in IRV and eliminated candidates don't have the time to endorse the runner-ups but mathematically it's the same).
IRV is coarser than STV, because it discards all 1st round results except who are the first two candidates. As such, it suffers more from random-like effects on the performance of the losing candidates and is more prone to the spoiler effect.
