I've seen systems like this a few times. One was based in Scotland, it was a list of candidates with STV and an option to vote for and against. A candidate would need to have a positive on the for/against to get through to the STV round to actually be elected.
It's also normal on company AGMs to vote for/against election of directors, though they normally elect all candidates.
This is all assuming that there's a list of candidates and you get to vote for or against each of them.
The system is vulnerable to tactical voting. If you say they need a positive score, that means they have to get over 50% of the vote in a tactical situation (vote against everyone but the candidate you want to win). If you say they need only to have the highest net score, you could well end up electing people who have an overall negative vote, the least hated candidate.
In a two horse race, such as the US presidential election, it makes no difference. A vote for one candidate on a single-x two candidate election is equivalent to a vote against the other. Though you could require a re-run if neither candidate had a positive tally which is entirely possible.
It only starts to count when you have more horses in the race, but entering an extra candidate for your party risks splitting the vote and you always lose, better to keep investing everything in only one candidate. However it does mean that the field could open up to more parties rather than more candidates from the main parties. Someone with very few votes for, but even fewer against could walk away with the win.
Now assuming a system where you only get one vote, but it could be for or against any one candidate
There's a list of candidates, is it more important that you vote for your candidate or against the one you really don't want to get in? Is hate or desire the stronger emotion. That's going to be about how each candidate runs their campaign, if one rants against another, they may find that rather than voting for him, they're voting against the other, leaving them both losing out to a third candidate. It would lead to a greater tendency towards positive campaigning rather than attacking opponents as you need to encourage people to make the for vote.
The general theme of this is that the opportunity to downvote only makes a difference in a larger field of candidates.
The biggest difference to the outcomes of elections come from:
- Greater turnout of voters
- Changing to a system that opens the game to more players such as PR or STV
A few definitions for those who don't have to deal with this stuff all day every day.
Single-X: You mark an X/tick in the box of the candidate you choose, One person, one vote, for one candidate.
STV: Single Transferable Vote, known in the US as IRV, Instant Runoff Vote, you list your candidates in order of preference, if your favourite candidate is knocked out due to having the lowest number of votes, your vote is transferred to your next preference.
PR: Proportional Representation, used for large numbers of elected persons over considerable areas, selected from party lists, you vote for a party rather than a candidate and the number of delegates elected from each party is taken from the proportion of votes that party gets, normally requires a minimum of 5% or similar of the total vote for the first candidate from that party though the required proportion for further candidates may be lower than that.