When will we break away from the two party system?

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Most new parties in america originate as machines to run for president with.

The trouble is not only FPTP (India, Canada and the UK all use that stupid system, and have multi party systems although the lack of PR does cause weird results) although that certainly doesn't help, but the peculiarity of the American party system (a political party in Australia, Japan, Norway or Brazil works very differently to the dems and reps do); the extreme presidentialism of the politics and the majoritarian structure of how the legislature works (for example, in the senate you basically have to cacus with the majority or the minority, or you risk being essentially useless).

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I suppose, what are realistic scenarios for the breakup of the Democratic vs. Republican duopoly on political power? Surely this party system can't last forever, can it?

In the near future: Trump and the republican party fallout, he purges the White House of GOP allies and he announced he is starting a new party. that's my most realistic scenario, and I don't consider it very realistic. The democrats will survive, and change to fit the time (honestly the party must have the weirdest history in the western world).

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I think eventually you can see a third party called the "Moderate Party" which would consist off Centrists and Establishment types or you could see a "Populist Party" rise up as a third party.

No