No, this is not about the 2016 U.S. election. Not just that, anyway.

First, I need to explain a weird glitch in our voting system. Let's say there's two candidates, Steven Square and Tracy Triangle , on a couple political axes. (for example, “left vs. right” and “globalist vs. nationalist”) Let's also say there's a voter who simply votes for whoever's political position is closest. What would that look like?

click & drag
the candidates and the voter:

It's a tough choice. Triangle's got some sharp points, but Square understands more sides! Alas, in the end, you can only vote for one.

Of course, there's more than just one voter in an election. Let's simulate what an election would look like with 100+ voters.

drag the candidates & voters around.
(to move voters, drag the middle of the crowd)
watch how that changes the election:

Now let's consider a different election. Say Tracy Triangle is already beating Steven Square in the polls, and a third candidate, Henry Hexagon , sees this. (Hexagon's supporters like how he tackles problems from more angles) Inspired by her success, Hexagon swoops in and takes a political position close to Triangle's.

Now, you'd think giving the voters more of what they want should result in a better choice, or at least, not result in a worse choice, right? Well...

at first, beats .
drag to just under ,
and see what happens:

That's right. Steven Square, our least popular candidate, now wins! This is because when you have two good candidates, they "steal" votes from each other, letting a bad third candidate win.

This is called the spoiler effect. The most famous real-world example of this was in 2000, when Ralph Nader "stole" votes from Al Gore, letting George Bush win. And though the spoiler effect didn't play a big role in 2016, its impact could still be felt.

In the Republican primary, one anti-establishment nominee, Trump, ran against sixteen GOP establishment nominees, who all "stole" votes from each other, letting Trump grab the nomination, easily. As for the Democratic primary, fear of splitting the vote prevented Sanders from running as independent. And to cap it all off, there was always the worry that other candidates like Johnson, Stein, and McMullin could spoil the election.

But again, this is not about the 2016 U.S. election.

This is about designing a democracy that people can trust.

Despite so much hoopla around the 2016 election, a full half of Americans did not vote. Even of those who voted for Clinton/Trump, 20% of them said their candidates were untrustworthy, and voted for them anyway. And around the world, people's trust in their governments – or the trustworthiness of their governments – has never been lower. It's more than America at stake. It's every democracy in the world. yeah, no pressure.

Rebuilding trust is a complex problem with no easy solutions. But I think there is an easy first step. It's a step that could get rid of our “lesser of two evils” problem, and give us citizens more choices, better choices. And yet, it won't be as daunting as fixing campaign finance or gerrymandering or lack of proportional representation, no, it'd just require changing a piece of paper, and how we count those pieces of paper.

This idea is not the most important issue. It won't solve everything. But as a first step? It'd give us the biggest bang-for-buck.

Let's talk about how to build a better ballot.

Now, some of you may have a couple objections!

First objection. Why would the people in power change the voting system that got them in power? Well, the spoiler effect has cost both Dems & Reps a major election before. Getting rid of that glitch would be a win-win for major and minor parties! Also, voting reform is already picking up steam. Just last month, Maine adopted Instant Runoff, and Justin Trudeau, Canada's Cutie-In-Chief, will be moving his nation towards a better voting system in 2017.

Second objection. Didn't some guy once prove that all voting systems will be unfair? Not quite. You're thinking of the infamous Impossibility Theorem by Kenneth Arrow, the mathematician in the 1950's who founded the whole study of voting systems.

Two answers to that: 1) some voting systems can still be more fair than others, even if none are perfect. And 2) Kenneth Arrow's proof doesn't apply to all voting systems! That's a misconception. It only applies to voting systems where you rank candidates. Later, we'll see some voting systems where you don't rank candidates – along with other alternatives to our current, glitchy voting system.

But first, let's take a closer look at the voting system we do have:


same as before. click & drag
the candidates and voter

How To Count: Simply add up the votes. Whoever gets the most votes, wins.

Sounds logical enough. But as you saw earlier, it can lead to a weird glitch, where having two good candidates can make the election go to a third bad candidate. This is why some people vote "strategically", voting not for their actual honest favorite, but voting for the lesser of two evils. And strategic voting is fine – but! – ask yourself this: how can we expect our elected officials to be honest, when our voting system itself doesn't let us be honest?

So, to fix the spoiler effect, other voting systems have been suggested. Such as...


again, click & drag

How To Count: There's actually several different ways to count these kinds of ballots. Here, I'll just show you the top three:

Instant Runoff Voting (IRV): This one is the most popular alternative to First Past The Post (FPTP). Australia and Ireland use it in national elections. San Francisco, Minneapolis, and Portland, Maine use it in local elections. And Justin Trudeau, Prime Man-ister of Canada, is leaning towards Instant Runoff, too.

(Note: Instant Runoff Voting is also called “Ranked Choice Voting”, even though there's other ways to count ranked ballots. IRV is also often just called “Alternative Vote”, even though there's a flippin' dozen other voting methods. Such selfish naming! Sheesh!)

IRV is a bit more complicated than FPTP, but here's how it works:

  1. Count up the #1 choices.
  2. If someone has more than 50%, they win! END.
  3. If not, eliminate the last-place loser.
  4. Run a new "round" of the election, minus that loser.
  5. Repeat until someone has 50% or more.

If that seems like too much, there is a much simpler method of counting ranked ballots...

Borda Count: Simply add up the rank numbers. Like in golf, whoever has the lowest score, wins. Borda count is used in Slovenia and a bunch of tiny islands in Micronesia.

But if you want an even nerdier way of voting, you could try...

Condorcet Method: Run a simulated "election" between every pair of candidates, using the info on voters' ballots. IF there's a candidate who beats all other candidates in one-on-one "elections", that candidate wins the real election. However, that's a very big "IF". (as we'll see later...) The upside is, when this method does pick a winner, it's always the “theoretically best” candidate! Currently, this method is not being used by any governments, and is only being used by neeerrrrrds.

So, those are the voting systems where you rank candidates – the ones that Kenneth Arrow proved would always be unfair in some big way! But what of voting systems where you don't rank candidates? They're less well-known, but now, at least you'll know 'em:


yup, stiiiiill click & drag

How To Count: Simply add up the approvals. Whoever gets the most approvals, wins.

Wait, picking more than one candidate? Doesn't that violate the one-vote-per-person rule? I hear you ask. Well, your vote was never a single check mark, your vote was always the whole ballot. And on this ballot, you get to honestly express all the candidates you approve of, not just your favorite or strategic second-favorite.

But if you want a more expressive voting system, why not try...


you guessed it

How To Count: Simply add up the ratings. Whoever has the highest average score, wins. Kind of like Amazon reviews, but with democracy. (Note: this is not a ranking system, because two candidates can have the same score.)

So there's our top 6 voting systems: the one we use, and five popular alternatives. But how can we tell if these alternatives are actually better? What glitches might they have? And which voting system – if any – can we say is "the best"?

Like before, let's simulate 'em.

Remember that simulation of the spoiler effect from earlier? Well, here it is again, but now you can switch between the six different voting systems! Here's the "spoiler effect" simulation again. See how different voting systems deal with potential spoilers:

drag to just under to create a spoiler effect.
then compare the 6 different voting systems:
(note: in the rare cases there's a tie, i just randomly pick a winner)

As you could see, every voting system except First Past The Post is immune to the spoiler effect. So, that's it, right? Ding dong, the glitch is dead? Just pick any other alternative voting system and be done with it?

But, alas. In getting rid of one glitch, some of these alternative voting systems create other glitches – for some, the cure is even worse than the disease.

For example, here's a sim of Instant Runoff Voting. In the beginning, Tracy Triangle is already winning, and you're going to move the voters even closer to her. Obviously, if a candidate is already winning an election, and becomes even more popular, they should still win afterwards, right?

You can probably guess where this is going...

drag the voters slowly up towards :

What happened? Originally, is eliminated in the first round, so goes against a weaker , and wins. But when you move the voters closer to , the loser changes! So now, is eliminated in the first round, which means goes against a stronger , and loses.

Under Instant Runoff, it's possible for a winning candidate to lose, by becoming more popular. What a glitch!

How often does this actually happen in real life? There's a couple confirmed examples, and mathematicians estimate this glitch would happen about 14.5% of the time. But sadly, we can't know for sure, because governments usually don't release enough info about the ballots to reconstruct an IRV election & double-check the results.

So, not only is Instant Runoff's glitch as undemocratic as First Past The Post's glitch, it's possibly worse – because while FPTP's counting method is simple and transparent, Instant Runoff is anything but. And a lack of transparency is an even deadlier sin nowadays, when our trust in government is already so low.

(But wait! We'll be talking about the risk of strategic voting later. Can IRV can make a comeback? Stay tuned...)

So much for the most popular alternative. What about the second-most popular, Borda Count? In this next simulation, you move a losing candidate closer to another losing candidate. Under FPTP, the spoiler effect would split their votes, making both of them lose even more. But watch what happens under Borda Count instead...

drag to just slightly left of :

Yup. Borda Count has a reverse spoiler effect. Instead of one good candidate hurting another good candidate by moving closer, with Borda Count, one bad candidate can help another bad candidate by moving closer.

Here's what happened: at first, some voters ranked >>, but when you moved closer to , those voters then swung to ranking >>, hurting enough to make her lose to .

Still, Borda's not the worst, and at least it's simpler and more transparent than Instant Runoff. But how does Condorcet Method compare? When Condorcet picks a winner, it's always the “theoretically best” winner – but that's when it picks a winner.

So far, I've just been simulating voters as a single group, with a center and some spread. But seeing how polarized politics is nowadays, one could imagine several groups of voters, with totally different centers. Now, Condorcet tries to pick the candidate who beats all other candidates in one-on-one races. But with polarized voters, you could end up with a Rock-Paper-Scissors-like loop, where a majority of voters prefer A to B, B to C, and C to A.

In certain situations, the other voting systems just had glitches. In Condorcet, the voting system crashes. Try it out for yourself:

create your own “condorcet cycle”!
move the voters in such a way that NOBODY wins:

Now, in actual practice – not that any government actually uses this voting system – when Condorcet fails to find a winner, the election falls back to another method like Borda Count. But if you do that, it'll get the glitches of its backup method. So it goes.

First Past The Post. Instant Runoff. Borda Count. Condorcet Method. Those were all the voting systems that use ranking – the ones that our math boy, Kenneth Arrow, proved would always be unfair or glitchy in some big way. What about the voting systems that don't use ranking, like Approval & Score voting? Well...

...I couldn't come up with a simulation to show their flaws. Because, in theory, they don't have many big flaws.

But that's a really, really, really big “in theory!” It may be that, in practice, strategic voters use Approval & Score Voting exactly like First Past The Post – only approving or giving 5 stars to their top candidate, and disapproving or giving 1 star to all others, even if they actually like the others. (See FairVote's critique of Approval Voting, and defense of Instant Runoff)

Then again, even if Approval & Score Voting disincentivize you from expressing an honest second choice, FPTP and IRV punish you for expressing an honest first choice. Besides, if Approval can be "gamed", then that goes double for IRV. (See this mathematician's critique of FairVote's critique, and defense of Approval) So, in the end... [confused shrugging sounds]

We're gonna need a hecka lot more simulations.

So, below is a chart (source), showing the results of 2.2 million simulations. A huge variety of scenarios were tested. All-honest voters. All-strategic voters. Half-honest, half-strategic. Voters who know each others' preferences. Voters who don't know each others' preferences. Voters who only sorta-know each others' preferences. And so on. You can tell that a real mathematician made this chart, because it's makin' my eyes bleed:

Each voting method's results is shown as an ugly-blue bar. The further to the right a voting method is, the more it "maximizes happiness" for the voters. The higher up a voting method is, the simpler it is. And a bar's width shows the range of a voting system's performance, given different ratios of honest-to-strategic voters.

The first thing to note is that strategic voting makes voters less happy than honest voting – in all voting systems! I was very surprised when I first learnt that. (But it makes sense, if you think about, say, a crowded room full of people trying to talk. Any one person can be "strategic" by shouting over others, but if everybody is "strategic", nobody can hear anybody, and all you're left with is sore throats and sad peeps.)

The other thing to note is which voting systems make people the happiest. If you have mostly honest voters, Score Voting is best. (with Borda Count a close second) And if you have mostly strategic voters, then both Approval & Score Voting are best. (and with strategic voters, IRV does just as bad as FPTP)

However, those are still computer simulations. How would these different voting systems play out in real life? Well, we can't just get the DeLorean up to 88, go back in time before the 2016 election, change the voting system, and see what would happen...

...or can we?!

No, no we can't. But last month, researchers did something close enough. A polling study asked 1,000+ U.S. registered voters to rank & rate the six presidential candidates, to simulate who would've won the (popular) vote under different voting systems! (But keep in mind that if we had a different voting system in the primaries, we'd have different candidates entirely. So take this study with a pillar of salt.) The results: under Instant Runoff, Condorcet, and Approval Voting, the winner would've been Hillary Clinton. But under Score Voting, the winner would've been Donald Trump. And under Borda Count, the winner would've been... uh... Gary Johnson?


a guesstimated model of the 2016 US election?...
how Clinton wins IRV, Trump wins Score, and Johnson wins Borda??


Before we wrap all this up – remember Kenneth Arrow? The infamous mathematician who founded the study of voting systems in the 1950's? Well, in an interview 60 years later, Kenneth Arrow had this to say, about which voting method he likes most now:

“Well, I’m a little inclined to think that score systems [like Approval & Score Voting] where you categorize in maybe three or four classes [so, giving a score out of 3 or 4, not 10 or 100] probably – in spite of what I said about manipulation [strategic voting] – is probably the best.”

That's as strong an endorsement as you'll ever squeeze out of a math-head.


(and everyone else around the world pushing for voting reform)

Thank you for taking this small but powerful first step! We've known for way too long that our current voting system – First Past The Post – forces voters to be dishonest, creates a polarizing "lesser of two evils" scenario, and screws over both major and minor candidates.

However, you're probably only considering Instant Runoff Voting. Which, to be fair, is better than than First Past The Post, and if it's a choice between just those two, definitely go for Instant Runoff. But IRV still has a glitch as undemocratic as FPTP's – and worse, in our age of distrust, Instant Runoff's lack of transparency may be deadly for democracy. Yes, sure, IRV was the best voting system we could come up with... in 1870. And since then, IRV has dominated the conversation, unwittingly framing the whole voting reform debate as “simple vs expressive”.

But that is a false choice. Thanks to computer simulations, real-life studies, and a bunch of math nerds, we now know of voting systems that are both simple and expressive.

Personally, I'm leaning towards Score Voting. It's simple, very expressive, and already familiar to anyone who's seen Amazon's or Yelp's “five star” review system. But that's just my humble opinion. You could also make the case that Approval Voting is more practical, because it's even simpler, and would already work with existing voting machines! All you'd need to do is change the instructions from “vote for the candidate you like” to “vote for the candidates you like”.

Or maybe I'm completely wrong about Instant Runoff Voting, and it's actually pretty okay. Heck, you could even go for Borda Count, as a hilarious prank.

I won't claim to know which voting system is The Best™. I shall keep open this discussion, just as long as we have this discussion. For three reasons:

1) If I claim one voting system is the best, end of story, all the social-choice-theory nerds will be on my butt, yelling, BUT NICKY WHAT ABOUT QUADRATIC VOTE BUYING

2) We still need to test these alternative voting systems with actual experience, not just annoying internet flame wars between IRV advocates and Score Voting advocates theory. All the more reason for small towns, local states, and nations like Canada to be pioneers, to bravely experiment!

3) Keeping the discussion going is what democracy is.

A recent study found that in many Western countries – from Sweden to Australia to the United States – support for democracy has plummeted over the last several generations. In 2011, almost a full quarter of young Americans said democracy was a "bad" or "very bad" way to run a country. And today, one in six Americans say it'd be "good" or "very good" to be under actual military rule.

Our age of distrust goes a lot deeper than the technical details of a voting system. There isn't gonna be One Weird Trick to fix democracy. But as a first step, a low-hanging fruit, a way to show that, yes, you will make the system respond to the needs and wants and pains and hopes and dreams of your people – well, fixing our voting system's a good start as any.

Because, this isn't just about trying to build a better ballot.

This is about trying to build a better democracy.

~ Nicky Case

P.S: Since you've read & played this all the way, here, have a bonus! A “Sandbox Mode” of the election simulator, with up to five candidates. You can also save & share your very own custom election scenario with others. Happy simulating!

SANDBOX MODE! (link to just this)

One hope for Sandbox Mode is that readers can debate with me and each other using this tool! Not just telling me I'm wrong, but showing me I'm wrong. For example – here's a model I made in Sandbox Mode, showing an interesting argument against Approval & Score Voting. Granted, this tool is very limited – it doesn't handle strategic voting or imperfect information – but I think it's a start, and may help improve our Democratic Discourse™

Zero rights reserved. I'm giving away all my art/code/words, so that you teachers, mathematicians, hobbyists, activists, and policy wonks can use them however you like! This is for you. Get my source code on GitHub!

For citizens: Remember, think global, but act local. Change from the bottom-up lasts longer. If you're in the US, find your representative and badger 'em. If you're in Canada, find your Member of Parliament and badger 'em. Also if you're Canadian, fill out the survey before the end of 2016! This survey has a few questions specifically about voting reform! (sadly, the question is still framed as "simple vs expressive". that is why i've been so gung-ho about Approval & Score, and maybe a bit too mean towards IRV)

For learners: Watch CGP Grey's Politics in the Animal Kingdom series! It's charming, and covers more ground than I did here – it explains gerrymandering, proportional representation, and more. Also, read Gaming The Vote by William Poundstone. It's a thrilling read, with dramatic human stories of crooks & conmen trying to game our glitchy voting systems – and sometimes, succeeding.

For teachers: This entire "explorable explanation" is public domain, copyright-free, meaning you already have permission to use this freely in your classes! You can even use the Sandbox Mode to create your own material, or as a tool for students to make something on their own.

For coders: This is all open source! So you can get my code on GitHub, and remix it to your heart's content. (sorry in advance for my messy code)

Check out these organizations: Though they may differ on what voting system they like best, they all have a common goal: to reform the one we have. Electology likes Approval Voting most, FairVote likes Instant Runoff most, and likes Score Voting most.


This "explorable explanation" was directly inspired by these two projects:

Voting Sim Visualization by Ka-Ping Yee (2005) was a real eye-opener. (hat tip to Bret Victor for sharing it with me!) I've heard lots of written debate over FPTP vs IRV vs Condorcet vs Approval vs blah blah blah, but I'd never seen their difference visualized so clearly! It gave me instant insight. And it actually changed my mind – I used to think IRV was pretty good, but after seeing IRV's messiness (as shown above), I realized it's actually kinda stinky cheese.

However, even this brilliant visualization was still too abstract. And since it wasn't interactive, I couldn't test the many questions & scenarios that came to mind. So that's why my second inspiration was...

Up and Down the Ladder of Abstraction by Bret Victor (2011). It's one of the web's earliest "explorable explanations" (also a term Bret coined) and it is gorgeous. Obviously, I borrowed the format of mixing words & "games" to explain things, but I also followed the formula of starting concrete – one voter – then moving up to the more abstract – a whole election.

You can learn more about Explorable Explanations here.

And last but not least, thank you to all the math & policy nerds who spent way too much time thinking about all this.


Every once in a while, I'll fall into an endless rabbithole – like this one on voting systems – and slowly crawl my way out, bloodied and bruised, with a new interactive thing for you! If you wanna find out when I finally get around to making new shtuff, you can...

And if you wanna see more of my past projects, check out my wobsite!

See you again soon! Have a Happy New Year 2017, or try to, anyway.

aimee jarboe
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matt hughes
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ben fei
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