If the number of constituencies in an assembly elected by FPTP was increased from 200 to 500 will that make it easier for third parties to win seats or does it not have any effect?
Assuming that the increase in seats is uniform and drawn fairly, you can generally expect all parties' seats to increase in proportion to the seats they had before.
Canada added 30 seats to its House of Commons ahead of the 2015 election. While this increase was not uniform (only 4 of 10 provinces gained seats), the increase of the third party (as all other parties) was still relatively proportional within those provinces.
Here is the official result of the 2011 election, along with Elections Canada's estimate of what the result would have been under the new districts:
| | 308 seats | 338 Seats | | Conservative | 166 | 188 (+22) | | NDP | 103 | 109 (+ 6) | | Liberal | 34 | 36 (+ 2) | | Bloc Québ. | 4 | 4 (± 0) | | Green | 1 | 1 (± 0) |
(Source: Elections Canada)
If fptp constituencies are geographic, increased seats ought to increase representation of geographically concentrated third parties, but not increase geographically dispersed third parties.
Consider the party for Newtown, which has great difficulty winning The Seat for Greater Sydney, but would have a better chance in The Seat for Newtown/Camperdown.
Consider the Party for Redheads which is unlikely to win the seat of Greater Sydney and similarly unlikely to win Newtown/Camperdown.
Duverger's law says that for a given district, first-past-the-post pushes to just two parties in that district. If there are two districts, then there could be four parties even under Duverger's law. That said, our experience has been that in a country like the United Kingdom, with only one type of district, multiple parties are competitive nationally, even if at most two are competitive in any one district.
In the United States, there is a national election (for president). Further, the Senate and House of Representatives have overlapping districts. The result has been that in the US, the system has only two national parties with representation. Only two Senators (and no Representatives) are not from those two parties. And one of those sought the Democratic nomination for president in 2016. Both caucus with the Democrats in the Senate.
Someone might argue that the Prime Minister is a national figure. But prime ministers (at least in the UK) are not directly elected by voters. They are elected by the legislature and may be replaced at any time, not just at elections. While the US president is technically picked by the electoral college, this is as a direct result of the popular vote and has elected the majority winner of the popular vote in every election where one existed except one (and there were serious allegations of fraud in that election). Several times a plurality winner has lost.
There is no evidence that number of seats matters with overlapping districts. It may make some difference in non-overlapping districts. But it can make a difference in either direction. A larger district may hide a minority that a smaller district would reveal. Or a smaller district may limit a minority to one district that would have still had one in a larger district, effectively reducing their percentage representation. And with either, careful selection of borders can limit minorities by not letting them have a majority in any district. Larger districts make that easier.