Let’s imagine a hypothetical 2020 presidential campaign between Elizabeth Warren, Donald Trump, and Michael Bloomberg. Let’s say that a month before the election, 40% of voters prefer Warren, 40% of voters prefer Trump, and 20% of voters prefer Bloomberg.
In a straight-up winner-take-all election, either Warren or Trump might narrowly win, but neither could claim a meaningful mandate. Whoever won, their opponent could delegitimize either as opposed by a majority of voters, hence the problem with winner-take-all elections. But under winner-take-all voting rules, Bloomberg would be less likely to run in the first place, and Warren would win 55% to 45%.
Let’s first treat this as a ranked-choice voting election. In such an election, Bloomberg can’t win with just 20% first-choice support, but he could exert considerable pull on both the candidates, since his supporters hold the pivotal balance of power based on who they name as their second-choice. If a majority of Bloomberg voters prefer Warren to Trump, Warren wins. If a majority of Bloomberg voters prefer Trump to Warren, Trump wins.
So, the strategy for both campaigns and voters is pretty straightforward. Both Warren and Trump should appeal to as many Bloomberg voters as possible, to win true majority support. Bloomberg’s strategy is to get enough first-choice support to not be eliminated in the first round, and then get enough second-choice preferences from whoever is in last place.
Let’s also assume that 75% of Bloomberg voters prefer Warren to Trump, so once Bloomberg is eliminated (as he would be once first preferences are counted), Warren picks up an additional 15%, which takes her over the top, thanks to Bloomberg supporters.
Things obviously get slightly more complicated with more candidates, but three candidates is enough to illustrate the basic dynamics. The logic is the same. Candidates try to build winning majority coalitions. Leading candidates compete for the support of trailing candidates, and trailing candidates get some influence by potentially directing support.
What about under approval voting? Now we also need to know how Warren and Trump voters feel about Bloomberg. Let’s say that 75% of Warren voters would also be okay with Bloomberg, and 50% of Trump voters would also be okay with Bloomberg.
Assuming everyone votes sincerely and not strategically (checking all the candidates they are okay with) Bloomberg would win. In addition to the 20% who like him best, you also have another 50% who are fine with him (75% of Warren’s 40%, plus 50% of Trump’s 40%, so 30% plus 20%), bringing Bloomberg’s total to 70%. By contrast, Warren would get 55% (her 40% plus 15% (75% of Bloomberg’s 20%), and Trump would get 45% (his 40% plus 5% (25% of Bloomberg’s 20%). So the final approval voting tally would be Bloomberg 70%, Warren 55%, and Trump 45%.
But hold on a second. Is this how voters would really vote?
Let’s view this election from the perspective of Warren’s supporters. Say you prefer Warren to Trump, but also Bloomberg to Trump. Should you vote for Warren and Bloomberg? Or just for Warren? Each vote for Bloomberg improves Bloomberg’s chances, and if everybody who was okay with Bloomberg voted for him, he’d win. So, if you really want Warren to win, you should just vote for Warren.
But, with polls tied 40-40, what if Trump wins? Wouldn’t it be better to support Bloomberg, too? And what if polls shifted slightly, and Trump was leading Warren 42-38? Then should Warren supporters all vote for Bloomberg? Or what if polls moved in the other direction, and Warren was leading Trump 42-38?
Approval voting strategies are highly contingent on polling, but as we know, polls bounce around and can be off by a few points. And without polling, voters would be even more hopeful about their candidates’ chances of winning, and less likely to approve of anyone else, since approving of a lesser choice essentially is giving up on your first choice winning. This is not a problem in ranked-choice voting. The strategies are the same regardless of small variations in polling, and voters can vote sincerely with or without available polling.
Let’s move onto Bloomberg supporters. What about Bloomberg supporters who prefer Warren to Trump? If you think Bloomberg has no chance of winning, then there’s no harm in voting for Warren, too, since each vote for Warren improves the chance she’ll defeat Trump.
But under approval voting, maybe there’s a chance Bloomberg could win, especially if it looks like Warren’s trailing Trump and her supporters are about to all approve of Bloomberg so Trump doesn’t win. If so, every additional vote for Warren reduces the chances that Bloomberg could win. Again, strategies are very different depending on small ripples in polling.
The Warren campaign would have an obvious campaign strategy: Tell supporters to “bullet vote” (vote only for Warren), and convince Bloomberg/Warren supporters that Bloomberg has no chance of winning, so his supporters should also support Warren. Even if Warren is down slightly, her campaign should tell supporters to only support her. As soon as she tells her supporters vote for Bloomberg, she is finished.
The Bloomberg campaign would have a different strategy: Try to convince Warren/Bloomberg supporters that Warren might lose, and so they should also support Bloomberg, while getting their own supporters to withhold support for Warren. If Warren and Bloomberg team up, they can defeat Trump. But if they team up, who wins? It depends which supporters withhold their votes. They are in a game of very high-stakes chicken.
Meanwhile, the Trump campaign would see its best shot at winning only if nobody votes for anybody else, so they should do everything they can to pit Warren and Bloomberg against each other, while making them both unacceptable to Trump supporters. Except, they’d also like to have the support of the Bloomberg / Trump supporters. But Trump voters would face the same dilemma as Warren voters — if Trump is down slightly, should they approve Bloomberg knowing that it would sacrifice Trump’s chances to defeat Warren.
If it sounds somewhat confusing and chaotic, that’s the problem. Factor in shifts in polling and campaign strategies, and approval winds up highly contingent on small differences in polling and difficult coordination. All potentially winning campaigns have strong incentives to go negative against other candidates to make sure their supporters don’t help competitors.
And so, in the end, the likeliest outcome of most approval voting is that the supporters of all potentially competitive candidates rank nobody but their own favored candidates. Following this logic, approval voting winds up looking like plurality voting, with the same kinds of negative campaigning. If you see genuine differences among the candidates, you should only vote for your favorite candidate, since all votes help other candidates. And all winning candidates have a strong incentive to encourage voters to see sharp differences, undermining the preference neutrality on which the whole system is premised.
Certainly, you can vary the dynamics of specific elections all you want, and if you really, really want to understand the differences between the two voting systems, by all means, spend an afternoon randomly generating different scenarios and their likely strategies and results.