DoctorJ The system you are advocating can be known by a number of names. It is often called the single-transferable vote, the alternative vote, or IIRC the preference vote. The former name is usually reserved for multi-seat election, however.
Ireland uses multi-seat STV for its parliament, and Australia uses it for its Senate. Australia uses single-member alternative voting for its House of Representatives.
Despite what Erik Raven implies, how many political parties function effectively in a voting system has nothing to do with how the executive is selected. No matter how many parties the U.S. had, under the current Constitution there would never be a coalition Presidency. The Italy/Israel/Austria examples are strawmen the anti-reformers always put for the to discredit multipartism.
I support a system where the President is directly elected with a majoritarian system. This system could either be the alternative vote (similar to the one advocated by Doctor J except you could number as many candidates as you wanted, starting with “1”), or with a second-ballot runoff. A lot of people seem to think that the runoff would be some terrible thing, and that the required threshold should be reduced from a voting majority to 40% of the vote. Runoffs, however, are not terrible things, and happen routinely in some countries, two or three weeks after the main election. Senate elections could be carried out similarly.
For U.S. House elections, I support proportional representation, with open list voting, D’Hondt’s highest-average system, and medium-sized constituencies with between 5 and 15 members. States entitled to more than 15 Representatives would be divided into two or more multi-member constituencies; states entitled to fewer than 5 could choose another system, or just use the default system.
The highest-average system means, in a given constituency, a seat would go to the party that would have the highest quotient of votes-per-seat if it won that seat. So the first seat would go to the largest party, the second seat would go to one of the two largest parties, etc.
Open list voting means you vote for one (or more, but I support just one) candidate on one party list. The party’s mandate is determined by votes of all its members; party seats are filled by its candidates in descending order of votes they received as individuals.
Any similarity in the above text to an English word or phrase is purely coincidental.