FWIW, our constitution is mostly based on systems from France and England. After a few quick searches: The English parliament party with the most members generally has their pick named PM, by the monarch, aka Queen now. The French have multiple parties, but 2 major ones. First vote open to all accepted candidates. If no on gets a pure majority, there is a runoff between the top 2. Of course they only have 67 mil people. I cannot see a second general election date working in the US.