We are often hearing about two round elections, in Europe and elsewhere. It would seem they must be more expensive and certainly more time consuming. There must be some advantage to the notion. Perhaps the idea is worth looking into.
A good example of the merits might be the recent American election. No candidate received 50% of the vote. Many voters wanted to register their preference for third party candidates, some actually in protest. It is hard to believe that many Green party voters really wanted Donald Trump to win, but at the same time it seems likely that few Libertarian voters would have preferred Hillary Clinton to win. Each side cannot accept the other.
In other systems such as Canada and the United Kingdom the most powerful official is decided by locally elected members, often to the party that won only a plurality of voters. At least in the US. it is possible for one party to locally elect members of Congress while the president is of a different party which in itself is check on power abuse.
Many political parties adopt a system that assures only a candidate that obtains 50% +1 is declared the leader of that party. After the first round there is almost always some soul searching and attempts to build a coalition among like minded voters. In the end everyone can acknowledge that the winner is at least acceptable to the true majority and the minorities for the most part accept the decision. The same parties are often happy with the first past the post system and plan their strategies around that.
Not sure what happens when in a national election there are 4 or more candidates and they all receive similar results. From the party elections we know it is not always one from the top 2 of the first round that prevails, but if there are more than 2 rounds the expense and time consumption go up.
When the stark reality of only two candidates left with neither being totally satisfactory it becomes evident people are forced to give the matter more thought.
I still believe local issues need representation, but that when power is concentrated at the national level it is preferable to give voters a chance to focus on different concerns.
One pitfall is that some people vote strategically--by that I mean will vote for a third party that will have the impact of denying another party a final round or they forego the third party and vote for their second choice. It might even out and in any case you do get to choose between the two survivors.
An educated voter is crucial to a democracy. The system needs to support them as much as practical. Perhaps we in North America can take a closer look at how other jurisdictions handle elections. Elections are a vital part of democracy.