The familiar argument in brief is that when voters are free to express their opinions, either unfavorable or favorable (or neither), regarding any or all of the candidates, then (for example) partisan Republicans will vote support for the Republican candidate but also vote in opposition to the Democratic candidate. Likewise, but with reversed roles, for partisan Democrats. The result will be that the partisan Democrats and Republicans will mostly cancel each other's votes. In effect, the polarized duopoly will defeat itself; well almost (in that mostly falls a bit short of totally). Margins in elections, even now, are often quite small in which case it takes only minor changes in voting patterns to alter the outcome. BAV does not change that fact. It does no more than to trim back the quite significant advantages of heading a ticket for one side of the duopoly. Can the duopoly survive such a clipping? I think not, but that question can be answered with full confidence by a real-world experience in elections.
It is easy to jump to a false conclusion that with BAV, only third-party candidates could ever win, but that is far from the case. BAV merely removes the bias that so many other voting systems gift to the largest political parties and occasionally even to other famous individuals. That bias, favoring the very famous, exists whenever a voting system that ignores popular opposition is used.
By way of analogy, in evaluating someone's wealth would, would it seem sensible to consider only at their assets and pay no attention to their debts? Risking money on this basis probably seems ridiculous even to people who are quite comfortable with a similar approach when selecting our political leaders. There is always strong resistance to breaking a well-ingrained habit, no matter how foolish the habit.
The natural tendency of a voter who has no opinion about a candidate, at least when possible, to be truthful and express neither support nor opposition. Such an option to abstain is provided to BAV voters and it seems likely that abstention is what most would do. It follows that both support and opposition votes for a candidate must be drawn largely from the pool of voters who actually do know about the candidate. The size of that pool will vary from one candidate to another and it will be particularly large for famous candidates. It follows that to count only support and disregard opposition gives a clear and significant advantage the most famous candidates. Perhaps this, more than merit, helps explain why we have such slow turnover in our elected political leaders who become relatively famous simply by virtue of their initial election. A relatively unknown challenger, given that massive a handicap, surely finds it close to impossible to compete.
Adopting BAV would be a simple way to level the playing field (or at least make it a much closer approximation to level). It would make elections more democratic but in no way would it block Republicans or Democrats from winning. Candidates from the largest political parties would retain advantages such as access to more money and perhaps more favorable treatment by media and it is quite possible they would continue winning often. Nonetheless, adopting BAV would open a door for smaller parties to compete more fairly and at least on occasion to win elections.
Margins in our elections tend to be small, at least when minor parties are ignored, so we have to delve into some details. So let us consider an example election soon after BAV is adopted. Old habits dying hard, perhaps 92% of voters retain their loyalty to either the Democratic or the Republican party. This 92% can be expected to vote both for their favored party's candidate and since with BAV it is now possible, also against the other big-party's candidate. And let us assume that the Green party has nominated the only minor-party or independent candidate and that the remaining 8% of voters support the Green candidate. Does this mean the Green candidate will win? To be sure, that is one possibility, but there are other factors to consider. Perhaps a large number of Green supporters would decide to hedge their bets and vote support for both the Green and Democratic candidates (the motivation for Green voters take that precaution would surely decline in time, but just after BAV comes into use, even Green voters may not believe their candidate has a ghost of a chance). That alone could hand the election to the Democrat, no doubt to the great discomfort of many Green voters.
But also, we have to take into account the votes coming from the 92% that may not abstain regarding the Green candidate. It seems quite possible that a substantial number of Republican voters would vote opposition to the Green candidate. But likewise there would be Democratic voters who would choose to vote support for the Green candidate; would these cancel each other or might a disparity change the election outcome? And if it changed the election outcome, is it clear in whose favor would it be?
Both major parties may include voters who ideologically favor the two-party duopoly and they would vote against any third party. It would not take a huge number of such voters to perpetuate that ideology if that were the only factor. But on the flip side there could be Republicans as well as Democrats who oppose the two-party duopoly and so would be inclined to support nearly any third-party candidate. It is hard to say how the election would come out. None of the candidates are certain of a win but none are certain of losing. Is that not how elections should be, with aggregate voter opinion controlling the outcome?
The important point to notice, is that losing is no longer such a certainty for the Green Party or for any other party. Voting becomes more democratic when BAV is adopted; the outcome will be what the voters, through their ballots, say they want. And with BAV, voters would have much more freedom of expression so they could say more clearly what they want. With improved ability to express themselves, the interpretation of the voters' wishes can correspond more closely with the voters' actual wishes.
The significant change is that the election is, with BAV, brought into play for third-party and independent candidates. In time as minor parties have success, that success would encourage voters to no longer dismiss these other candidates but rather give them more of the consideration they are due. More minor parties would likely form to promote their views. The decline of the duopoly would come, but probably not suddenly, if only because BAV adoption can surely come only one state at a time. BAV in no way threatens any sudden major disruption of our political system. It is something we could try in one city or state and eventually other states may choose to do the same. It is probably more accurate to think of it as an opportunity for an evolutionary improvement than as a revolution that that could bring on a sudden dramatic change.