Illustrations of Electoral Failures

Many people find electoral systems abstract and esoteric. Party primary and plurality seem like obvious, fair systems, yet draw large amounts of attention for their unfair results and damage to our democracy; and the Electoral College gets attention when it disagrees with that same often-maligned popular vote system, ironically creating a call for a national popular vote. People aren’t generally experts in the economics of social decision making, and only recognize that there is absolutely something wrong here.

Fortunately, people are in general intelligent and capable of understanding complex concepts if someone explains them well enough. For complex electoral systems, we can use the United States 2016 Presidential Election as a frame of reference, and propose several different sets of votes. From these votes, we can calculate outcomes and show how each electoral system fails.

We must understand upfront the utmost importance of representation. Our elections must produce a representative result. That representation might not fit with what we as individuals or larger groups want; that’s not the point. The point is for us, as a majority, minority, or whatever portion, to have our say and to influence the election so that every vote counts—that means we must concede some things sometimes.

To emphasize this point, I’ve used a model in which there are more Republicans voting than Democrats in the 2016 election. In general, this means a Republican wins—but which Republican? That’s the key question.

In our first example, we have 24% ranking Rubio, then Trump, then Clinton: we’ve assumed the Republicans will always vote Republican first.

As seen above, Plurality selects only from first-ranked votes. More voters—a full 30%—voted Trump than anyone else, so Trump wins. We can see how flawed a system this is.

Party Primary is actually worse. In the real 2016 Republican primary, Trump won with 44% of the votes thanks to the same vote splitting seen above. The Party Primary system also faces non-mathematical flaws rooted in human economic behavior.

Party-line voters who don’t care who wins the primary are less-inclined to vote. In practice, you might see 75% of the Sanders voters turn out, and only 50% of the Clinton voters. Using our example here, with more Democrats preferring Clinton, Sanders would achieve a victory at 54% to 46% in the primary. Social media propaganda attacks capitalize heavily on this problem.

Our example also shows a top-two system, which just picks Trump and Clinton by plurality vote. Again, Trump wins.

Majority Runoff and its ranked choice voting version, Instant Runoff Voting, show the center squeeze effect: least-voted Bernie Sanders gets booted early, sending votes to Clinton. Clinton and Trump thus squeeze out Rubio, turning this into a Clinton-Trump contest, in which Clinton wins.

The last example is our Condorcet method. The graph on the right holds particular importance: it shows who defeats who, with the arrows pointing toward the losers in one-on-one contests.

Because Rubio defeats Trump, Clinton, and Sanders in the respective two-candidate elections, Rubio is the Condorcet winner. Systems like Tideman’s Alternative elect this winner, representing a consensus of the voters.

So okay, there were more Republicans, and Rubio wins under this system. Does that mean this is a better system, or is it just electing a candidate some of us personally dislike less than the winner?

Those aren’t mutually-exclusive propositions: 70% of voters preferred Rubio to Trump. A lot of us apparently personally disliked Rubio less than Trump, yet Trump got elected. Trump got elected despite a majority of voters not wanting to elect Trump.

Let’s fiddle with the numbers a little.

Hillary Clinton won the popular vote in 2016. A large body of Never-Trump Republicans had no alternative but to vote Clinton, and the above illustration supposes Clinton won the popular vote because of a great many Republicans ranking Rubio, then Clinton, then Trump. The Electoral College essentially stuffed these Republicans into strong Red jurisdictions, covering up their votes just like gerrymandering, so Trump won anyway.

Again, Trump wins Plurality; and again, Rubio remains the Condorcet candidate—so far, the correct winner, given the arguments above. What about the other systems?

Party Primary selects Clinton and Trump. This removes Rubio. The graph shows Rubio defeats Clinton, who defeats Trump; since Rubio has been removed, Clinton wins. Just as with Trump, 54% of the voters preferred Rubio to Clinton.

Top-Two again selects Clinton, and Rubio voters bail out on Trump. Runoff and Instant Runoff Voting also squeeze out Rubio, leading to a Clinton victory.

Even though the majority of the voters clearly prefer Rubio to Trump, Clinton, and Bernie, only the Condorcet system elects Rubio. The rest fail, this time electing Clinton instead of Trump.

A small, unimportant change to the votes has a big change to the outcome. How badly can this fail, anyway? Let’s find out what happens if fewer people vote Trump and more people vote Bernie Sanders.

Sanders is now preferred among Democrats; Rubio voters still prefer Clinton to Trump, which elected Clinton in our second example; and one percent of voters have moved from the Trump camp to the Never-Trump camp.

Plurality still elects Trump, who has the most votes—no surprise there. A Condorcet system still picks Rubio.

Party Primary now selects Bernie Sanders. Sanders can’t beat Trump, so Trump wins. Voting for Bernie instead of Clinton elected Trump.

Top-Two actually gets it right for once. Because more people voted for Rubio, the line-up is Rubio versus Trump, and Rubio beats everybody.

This outcome is precarious: if more than 1% of voters additionally switched from Clinton to Bernie, Top-Two would have selected a Bernie-Trump election, and Trump wins—once again, voting for Bernie Sanders elects Donald Trump. This should never happen, yet it keeps happening.

Runoff and Instant Runoff Voting again center-squeeze. This time it’s Bernie defeating Clinton, instead of Clinton defeating Bernie; and then Rubio goes, leaving a Bernie-Trump election once again. Instant Runoff Voting elects Trump—and once again, changing your vote from Clinton to Sanders elects Trump.

That’s all fine, but what if nobody’s on top?

Enter strategic bloc voting. The Trump voters are trying to eliminate Rubio by burying under Sanders—a known flaw that can manipulate Condorcet systems. Tideman’s Alternative carries a high degree of resistance.

In this example, Plurality elects Trump, Top-Two picks Rubio, and both Party Primary and run-off elect Clinton. Trump voters have identified Sanders as the weakest candidate, and so buried Rubio under Sanders.

The attack works fairly well: Sanders defeats Rubio and Clinton. Rubio defeats Clinton, who defeats Trump, who defeats Sanders; and Rubio defeats Trump, who defeats Sanders.

From any candidate, you can find a path of victories leading back to that candidate. Rubio defeats Clinton, who defeats Trump, who defeats Sanders, who defeats Rubio. Looks like we don’t have a Condorcet candidate, and every candidate except Sanders seems to win under these systems, so how do we decide who wins?

Enter Tideman’s Alternative.

Tideman’s Alternative identifies the top cycle—in this case the Smith Set, which is those candidates who are not defeated by any candidate not in the Smith Set.

To better illustrate, this example includes Gary Johnson, defeated by everybody. The green arrows show the Smith Set path. None of the candidates in the Smith Set are defeated by Johnson. You can trace a path of victories from any candidate in the Smith Set to any candidate in the election.

Tideman’s Alternative eliminates all non-Smith candidates—that being Johnson—and then performs one round of run-off (sometimes called “alternative vote”). This eliminates Sanders, leaving Rubio, Trump, and Clinton.

In round 2, the Smith set is Rubio. This makes Rubio the Condorcet winner, and so Tideman’s Alternative elects Rubio.

As shown here, Tideman’s Alternative can reliably find a consensus winner, and resists tactical manipulation. It does have failure modes, but those failures rely on a large amount of manipulation: Sanders defeated all non-Trump candidates in this example, so even if Rubio voters voted Rubio, then Trump, then Sanders in an attempt to bury Clinton, the outcome wouldn’t change.

The failure modes can also be unpleasant. Consider this attack if 4% more of the voter were Trump voters instead of Rubio voters.

Here we see more Trump voters and fewer Rubio voters. An honest vote, with Trump voters placing Rubio second, elects Rubio; but the Condorcet burying attack changes the outcome.

By attempting to corrupt our elections through tactical manipulation of their voting ranks—by ranking not by the order they favor, but rather in an attempt to mathematically distort the outcome—Trump voters succeed in electing Hillary Clinton.

This happens because a few Rubio voters are honestly opposed to Trump—these are the Never-Trump voters who voted Clinton in 2016. This attack also requires Trump to be the stronger candidate—and Trump won 44% of the Republican Primary, in which the more-moderate independent voters can’t even vote.

To make this clear: if just over 4% of Rubio voters defect to Clinton, Trump has fewer than 50% of the votes. Clinton has around 28%, and Sanders is eliminated. Sanders votes go to Clinton, granting over 50% of the votes, a Clinton victory. That mean this attack elects Clinton even if almost no Rubio voters reject Trump.

Our electoral system matters. Our elections need to find the consensus of voters, not the end result of manipulations and errors. We’re not here to elect a winner, but a representative.