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Driving to work this morning, I heard about the upset in the French primaries, and the announcer stated that “the French say that they do not vote for someone, but, rather, against someone”.

Of course, I am aware of tactical voting, but what if we took that literally?

If each person had a choice of voting +1 for a candidate, or -1 against another, would there be any significant change in outcome? Would things drift towards the centre, for instance, or is it a zero sum game?

Would there be marked difference in effect on different systems of voting, such as winner takes all and proportional representation?

I am sure that it would have ramifications which I cannot foresee, so thought that I would ask here.

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    When you can up-or down-vote every candidate independently, it's called approval voting. Are you suggesting this system or are you suggesting a system where each voter has one vote and needs to decide whether to use it to upvote a candidate or downvote a candidate? – Philipp♦ Nov 21 '16 at 13:48
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    one man, one vote :-) You can vote either up or down, but you only get a single vote – Mawg Nov 21 '16 at 15:50
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    In my Land (Hessen, Germany) we can strike out/delete candidates in communal elections. Technically it is a shortcut to upvoting many other candidates: we have as many votes as the parliament has seats and can distribute them among all candidates (panachage) - up to 3 upvotes to one candidate (cumulation). As that is a lot of work (ca. 100 votes in the bigger cities/Kreise) and thus error prone, there are shortcuts, e.g. you can still upvote a list/party. Deleting a candidate means upvotes for everyone in the list but the deleted candidate. The shotcuts are converted into upvotes for counting. – cbeleites supports Monica Nov 21 '16 at 18:27
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    The question is logically unanswerable because it is incomplete, since it doesn't specify how negative votes are counted with respect to the final winner, and other details of your proposed scheme. For instance, if you had a race between two unpopular candidates (e.g., the recent American election) and, let's say, no candidate ended up with a net positive vote count, is the winner the one with the smallest negative count? Would a candidate with no votes at all win over a candidate with lots of positive votes, but an equal number of negative notes? – user316117 Nov 21 '16 at 18:45
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For one thing, Duverger's law would likely be ineffective, meaning that there would no longer be a convergence to two parties in FPTP system.

This is because, typically, the way Duverger's law works is, by forcing people to excercise tactical voting by voting for whichever of the 2 main party candidates they find least objectionable. This is because voting for a minority 3rd party candidate they actually like - effectively counts as 1/2 vote for the major-two candidate they dislike more. So, Perot voters helped elect Bill Clinton (or so goes the popular narrative), and Nader voters helped elect G.W. Bush (also a popular narrative that, actually, was disproved if you look at the numbers). But the truthiness of the narrative is irrelevant - only the level of influence the narrative has on voting behavior.

However, if you can downvote a candidate, you can now downvote the major-two candidate you dislike more instead of upvoting the one you dislike less. This will not affect the balance between major candidates (since a downvote for A is same as upvote for B, in the balance between the two) - BUT, a downvote for major candidate A is also effectively an upvote for 3-d party candidate C!

As such, in elections where a large portion of electorate isn't terribly inspired by either candidates, and mostly votes for "lesser of two evils" in current FPTP, the two major party candidates just might accrue enough down-votes that a 3rd party candidate who isn't nearly as disliked will, on balance, win over both of them (or at the very least, acquire more than the abysmal 4% combined popular vote and 0 electoral vote like 2016 US presidential elections, despite 3rd party candidates combined likely being preferred by 40% of electorate, as a low bound).

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  • I'm not at all sure this weak version of approval voting alone is enough to kill Duverger's law, simply because the 'vote for one candidate only' restriction is so limiting. – origimbo Nov 21 '16 at 15:21
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    @origimbo - "kill", not sure. "significantly weaken" - likely. At least in USA, the election matrix at federal level tends to be simple - mainline R and D; and 2 "outliers", one roughly left one roughly right. In 2016, I wouldn't be surprised if Trump got a sum total of zero (or negative votes) from the ~50% of voters who voted for him; with 25% of voters from his downvoting Hillary; and 25% of Hillary's voters downvoting Trump; with the same happening on D side - which would actually let Gary Johnson win as he would get almost no downvotes. All hypothetical, of course. – user4012 Nov 21 '16 at 15:28
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    That was an election with one historically unfavourable candidate (who won) and a high percentile unfavourable candidate. It's certainly not unlikely that with a different voting system, primaries would work differently, and the Republican primary campaign in particular was a classic example of vote splitting among essentially identical candidates. – origimbo Nov 21 '16 at 15:38
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As with any change of voting system, it's hard to game the results without stating other parameters that have been assumed, but there are a few key cases that are fairly obvious. Proportional representation is difficult to arrange under your scoring scheme, since the sum of the candidates scores is no longer equal to the total number of votes cast and it's possible for a candidate to get a negative score (Indeed, it's possible for all candidate's scores to be negative). This would require modifications to the quotient system used to allocate seats. I can find proportional schemes for more general versions approval voting, but I don't think they work in the one candidate case.

Sticking with winner takes all, we can pick out a few key scenarios

A two party election

A.K.A US federal elections and most single member voting districts in parliamentary elections.

The result should remain unchanged, since a vote against party A is the same thing as a vote for party B.

One extremist and multiple identical moderates

For a sufficiently informed electoral, this voting scheme should make a plurality victory by an extremist less likely, since to guarantee a victory the candidate needs a larger proportion of positive voters to counteract negative voters who don't have strong views between the moderates. In other words this system is more resilient to vote splitting than straight first-past-the-post.

Multiple extremists, multiple moderates

At this point it's unclear that your proposed voting scheme is significantly likelier to guarantee a moderate plurality victory than regular first-past-the-post voting. In this case if your aim is to implement more regular approval voting. Note too that definitions of extremist will depend on the voters' political views.

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    While "The result should remain unchanged" may count towards 100% strict 2 party election, in practice, these would become rare (in US, at least) as such a system would immediately give a strong leg-up to 3rd party candidates, making sure that most elections wouldn't remaind 2-party affairs – user4012 Nov 21 '16 at 14:58
  • Ballparking from the numbers on FiveThirtyEight, this past election would have had an outcome of about net -2.5% Clinton, net -2.5% Trump, and we'd be preparing to welcome President Johnson, with about a net 3% of the vote. – Mark Nov 22 '16 at 0:57

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