Two-round system
Based on Wikipedia: Two-round system
In 2002, France nearly elected a fascist president. Not because most French voters wanted one—they emphatically did not—but because the voting system they trusted to represent them had a critical flaw that almost nobody saw coming.
The story of how Jean-Marie Le Pen, leader of the far-right National Front, came within striking distance of the French presidency is really a story about mathematics, democracy, and the surprisingly difficult problem of figuring out what voters actually want.
The Promise of Two Chances
The two-round system sounds almost too sensible to criticize. Here's how it works: everyone votes once, and if no candidate wins more than half the votes, the top two finishers face off in a second election a few weeks later. Simple. The winner always has majority support. What could go wrong?
This system—sometimes called "ballotage" from its French origins, or "top-two runoff" in American parlance—emerged during France's July Monarchy in 1832. The government was grappling with a fundamental problem: what happens when nobody wins a clear majority? Their solution was elegant. Just hold another election, but this time with only two choices.
The French were onto something. Today, the two-round system is the most common way in the world to elect a single leader. It dominates presidential elections across South America, Eastern Europe, and Africa. It feels democratic. It feels fair.
But democracy is harder than it looks.
The 2002 Disaster
Going into the first round of France's 2002 presidential election, everyone knew what would happen. Jacques Chirac, the center-right incumbent, would face off against Lionel Jospin, the center-left Socialist prime minister. These were the leaders of France's two largest parties. Polls showed them as the clear frontrunners. The only question was which one would win the runoff.
There was just one problem: sixteen candidates were on the ballot.
The French left had fractured into competing camps. Two candidates from Jospin's own coalition—Jean-Pierre Chevènement and Christiane Taubira—refused to drop out. Together they pulled nearly eight percent of the vote away from the left. Jospin, confident in his lead, didn't push hard enough for them to withdraw.
That confidence was fatal.
When the first-round results came in, France went into shock. Chirac had 19.88 percent. Le Pen had 16.86 percent. Jospin had 16.18 percent.
Jospin was out. Le Pen was in.
The margin was tiny—less than one percentage point. But those rules about the top two advancing weren't guidelines. They were iron law. The second round would pit a mainstream conservative against a Holocaust-minimizing nationalist whose party had roots in Nazi collaboration.
The Scramble
What happened next was extraordinary. The French left, which had spent weeks attacking Chirac, immediately threw its support behind him. "Vote for the crook, not the fascist" became an unofficial slogan. Protesters filled the streets. Young people who had never voted conservative in their lives held their noses and marked their ballots for Chirac.
The result was a landslide: Chirac won 82.21 percent to Le Pen's 17.79 percent.
Democracy survived. But something had clearly gone wrong. Most French voters preferred Jospin to Le Pen. Most preferred Jospin to Chirac, for that matter, or at least enough did that it should have been a real contest. Instead, the voting system had eliminated the candidate who might have been the actual favorite before most voters ever got to weigh in on him.
Ironically, post-election analysis showed that Chirac was genuinely preferred to Jospin by a majority of French voters—the "right" candidate won. But the system had produced this result almost by accident, and there was no guarantee it would be so lucky next time.
Center Squeeze: The Hidden Weakness
What happened in France has a name among political scientists: center squeeze.
Here's the dynamic. Imagine three candidates: one on the left, one on the right, and one in the middle who might actually be acceptable to almost everyone. Under a two-round system, the centrist needs to finish in the top two to advance to the runoff. But centrists draw votes from both sides. Their support is wide but often shallow—many voters like them, but as a second choice.
Meanwhile, candidates at the extremes have intensely loyal bases. Their voters won't defect to the center; they're true believers. So in a crowded first round, the centrist can get squeezed out by candidates with smaller but more committed followings.
This is a variant of what's called the spoiler effect. A spoiler is a candidate who can't win themselves but whose presence changes who does win. In 2002, Chevènement and Taubira were spoilers—neither had any chance at the presidency, but together they spoiled Jospin's chances by splitting the left-wing vote.
The two-round system was supposed to fix the spoiler problem. After all, you get two chances to vote! If your favorite candidate loses in the first round, you can still vote for your second choice in the runoff.
But this protection only works if your preferred candidates make it to the second round. When they don't, you're stuck.
The American Jungle
Americans might think this is a peculiarly French problem. It's not.
Several American states use variations of the two-round system. Louisiana pioneered what's called the "jungle primary"—all candidates from all parties compete in a single first-round election, with the top two advancing to a runoff regardless of party affiliation. This means you can end up with two Democrats or two Republicans facing off in the final round.
California and Washington State adopted similar systems through ballot initiatives in 2010 and 2008 respectively, calling theirs the "nonpartisan blanket primary" or "top-two primary." Georgia uses a version where a runoff only happens if no candidate breaks fifty percent in the first round.
These systems all share the same vulnerability. In a crowded primary field, the two candidates who advance might not be the two that most voters actually prefer. They're just the two who happened to consolidate support most effectively among their particular factions.
Most other American states use a different approach: partisan primaries that effectively function as a two-round system. First, each party holds its own election to choose a nominee. Then the nominees face off in the general election. The difference is that voters can only participate in one party's primary—you have to choose your team before you can vote.
This creates its own problems. Primary electorates tend to be smaller and more ideologically extreme than the general electorate. Candidates who would appeal to the middle often lose primaries to candidates who fire up the base. The general election then offers voters a choice between two options that neither fully represents them.
The Search for Something Better
Voting theorists—yes, this is a real academic field—have spent decades trying to design systems that avoid these pitfalls. The mathematics of voting turns out to be surprisingly deep, touching on game theory, economics, and some genuinely difficult proofs about what's possible and what isn't.
One alternative is instant-runoff voting, known in the United States as ranked-choice voting and in Australia as preferential voting. Instead of voting twice, you vote once but rank the candidates in order of preference. If no candidate wins a majority, the candidate with the fewest first-choice votes is eliminated, and their supporters' votes transfer to their second choices. This continues until someone has a majority.
Alaska and Maine now use this system for most elections. It solves the logistical headache of making people show up twice to vote. But mathematically, it has many of the same problems as the two-round system—it can still squeeze out centrist candidates and produce winners who aren't actually the most broadly popular.
Ireland uses a related system called the single transferable vote for its presidential elections. The distinction matters because when applied to multi-member districts—where you're electing several representatives rather than just one—single transferable vote becomes a proportional system rather than a majoritarian one. Candidates don't need an outright majority; they just need to reach a quota based on the number of seats available.
Another family of alternatives goes by the name of Condorcet methods, after the eighteenth-century French mathematician and philosopher who first analyzed them. The Marquis de Condorcet proposed a simple criterion: the winner should be whichever candidate would beat every other candidate in head-to-head matchups. If Alice would beat Bob one-on-one, and Alice would beat Carol one-on-one, then Alice should win—regardless of how the votes split in any particular round.
Condorcet methods have elegant theoretical properties, but they come with their own complications. Sometimes there is no Condorcet winner—candidate A beats B, B beats C, but C beats A, creating an endless loop. What then?
Still other reformers advocate for approval voting, where you simply vote for as many candidates as you find acceptable. No ranking, no runoffs—whoever gets the most approval wins. It's simple, it's hard to game, and it tends to elect broadly acceptable candidates. But it requires voters to make a different kind of judgment than they're used to, treating candidates as either acceptable or unacceptable rather than ranking them.
Strategic Games
Every voting system creates incentives, and people respond to incentives.
Under the two-round system, sophisticated voters sometimes engage in what's called "pushover" tactics. The idea is to vote for a weak opposing candidate in the first round, helping them advance to the runoff where they'll be easy to beat. It's risky—if too many people try it, your preferred candidate might not make the runoff at all—but it happens.
Candidates and parties play strategic games too. A political faction that runs multiple candidates risks splitting its own vote and getting shut out of the runoff entirely. This creates pressure to consolidate behind a single candidate, which is why political parties exist in the first place. But coordination is hard. Ambitious politicians don't always want to step aside, as Chevènement and Taubira demonstrated in 2002.
The flip side is that a well-funded faction might deliberately sponsor extra candidates on the opposing side, hoping to split their opponents' votes. If the left is going to run five candidates, why not quietly encourage a sixth?
France has developed an interesting adaptation to deal with some of these problems. In legislative elections—as opposed to the presidential contest—French rules allow more than two candidates to advance to the second round if enough of them clear certain vote thresholds. This leads to "triangular" elections with three candidates in the runoff. But it also creates its own strategic dynamic: candidates who advance often withdraw from the second round to avoid spoiling a similar candidate's chances. The 2024 French legislative elections saw widespread strategic withdrawals of this kind.
The Exhaustive Alternative
If two rounds are good, would more rounds be better?
The exhaustive ballot takes this logic to its conclusion. Instead of eliminating all but two candidates after the first round, you eliminate only the last-place finisher. Then you vote again. Keep going until someone has a majority.
This sounds tedious, and it is—which is why the exhaustive ballot isn't used for large public elections. But it's common in smaller settings where getting the decision right matters more than efficiency. The Conservative Party in the United Kingdom uses it to select parliamentary candidates. The International Olympic Committee uses it to choose host cities. FIFA uses it to award World Cups.
The exhaustive ballot gives voters more information at each stage. You can see exactly where support stands before casting your next vote, which lets you adjust your strategy in real time. Whether this produces better outcomes or just more opportunities for manipulation depends on your perspective.
What We Can Prove
Mathematicians have spent considerable effort trying to determine which voting systems are "best." The answer is frustrating: it depends on what you're trying to achieve, and some desirable properties are mathematically incompatible with others.
Kenneth Arrow won a Nobel Prize partly for proving that no ranked voting system can satisfy a certain set of seemingly reasonable criteria simultaneously. This result, Arrow's impossibility theorem, is sometimes overstated as proving that all voting systems are equally flawed. That's not quite right—some systems are clearly worse than others—but it does show that every system requires tradeoffs.
For the two-round system specifically, we can prove some things. If voters have fixed preferences and vote sincerely according to them, the system is mathematically equivalent to the contingent vote—a variant of instant-runoff voting that only does two rounds of counting rather than eliminating candidates one by one.
But voters don't always have fixed preferences, and they don't always vote sincerely. When game theory enters the picture, things get complicated. In some equilibrium models where all voters strategize perfectly, the two-round system will elect the Condorcet winner whenever one exists—at least in races with three or fewer serious candidates. That's a meaningful advantage over pure single-round plurality voting.
The catch is that these equilibrium models assume perfect information. Every voter knows every other voter's preferences. In a real election with millions of voters, that's clearly not the case. Small elections might approximate these conditions better than large ones, but presidential elections definitely don't.
The Fundamental Tension
At its heart, the two-round system embodies a particular theory of democracy: that legitimacy requires majority support. A leader should be able to claim that more than half the voters chose them over the alternative. This is an appealing idea. It feels right.
The problem is that "the alternative" is doing a lot of work in that sentence. In the French 2002 election, Chirac won 82 percent of the vote in the second round. But this massive majority was against Le Pen, not for Chirac. Against Jospin, Chirac's support might have been narrower—or he might have lost entirely. The "majority" that elected Chirac was an artifact of which candidates happened to be available in the final round.
Critics call this an "artificial majority," and they have a point. The two-round system doesn't so much discover majority preferences as construct them. By limiting the final choice to two options, it guarantees a majority result—but the options on offer have already been filtered through a process that might have eliminated the most popular candidate.
Defenders argue that this is still better than plurality systems where someone can win with thirty percent of the vote. At least runoff winners have to appeal to more than just their base. There's something to this. The need to win a majority in the second round does pull candidates toward the center, at least in the final stage.
Living With Imperfection
No voting system is perfect. The two-round system is vulnerable to center squeeze, strategic nomination, and the chaos of crowded first rounds. Instant-runoff voting has similar problems. Condorcet methods can produce weird results when there's no clear Condorcet winner. Approval voting requires voters to make artificial binary judgments about candidates they might feel ambivalent about.
What matters is understanding the tradeoffs and designing systems appropriate to their contexts. The two-round system works reasonably well when there are genuinely only two major factions competing. It becomes more dangerous as the political landscape fragments.
France has learned this the hard way. After 2002, there were calls for electoral reform. After Marine Le Pen—Jean-Marie's daughter—made the runoff again in 2017 and 2022, those calls grew louder. Social choice theorists have proposed alternatives like rated voting systems. Whether France will actually change its system remains an open question.
The United States is slowly experimenting. Alaska and Maine have adopted ranked-choice voting. Other states and cities are watching to see how it goes. The jungle primary systems in California, Washington, and Louisiana continue to produce occasional surprises when the top-two finishers aren't who everyone expected.
Democracy is an ongoing experiment. We're still learning how to count votes.
The Deeper Problem
Perhaps the most important lesson from all this analysis is that "what voters want" isn't as simple as it sounds.
When we ask who should win an election, we're really asking a mathematical question about how to aggregate millions of individual preferences into a single collective choice. Different voting systems are different answers to that question. They weight different aspects of those preferences differently—intensity versus breadth of support, first choices versus compromise acceptability, strategic sophistication versus sincere expression.
There is no obviously correct answer. Arrow's theorem tells us there can't be. What we can do is think carefully about which properties matter most in which contexts, and design systems that reflect those priorities.
The two-round system prioritizes a particular kind of legitimacy—the ability to claim majority support in a head-to-head contest. It sacrifices some other properties to achieve this. Whether that tradeoff is worth it depends on what you think democracy is for.
In the meantime, voters in two-round systems should understand how their system works and what its failure modes are. If you live in France, be wary of crowded first rounds. If you live in California, pay attention to the jungle primary. And if you're ever tempted to vote for a "pushover" candidate to manipulate the outcome—maybe think twice. These systems are complicated enough without adding more game theory to the mix.