Except when you're not allowed to, which happens a lot if you're registered as an Independent...
> In a closed primary, only voters registered with a given party can vote in that party's primary. States with closed primaries include party affiliation in voter registration so that the state has an official record of what party each voter is registered as.
I don’t understand arguments for open primaries. If you’re not a member of a political party what business do you have in voting who they want to represent them in the presidential election?
What stops people voting for unpopular candidates in order to sabotage the party?
This has happened before. In the 2008 primaries, Rush Limbaugh ran an "Operation Chaos" campaign where he called on his listeners to vote in late, open primaries for the then-losing Hillary Clinton in order to extend the Democratic primary season. In 2016, even in some closed primaries like West Virginia, a majority of voters who supported Sanders in the primary, according to exit polls, still planned to vote for Trump in the general election.
In some states, like PA where I live, you have to be registered with one party or the other to vote in primaries. I changed my registration from independent recently due to this. There is talk of legislation to change it but I'm not sure of it's current status.
I tend to credit the two party system for there being a general lack of good candidates who make it into the general election with only two having more than a remote chance of winning.
Duverger's Law only applies to the general election, where each main party only fields one candidate. It doesn't work for primaries, where the party has lots of candidates running. There's nothing really stopping the parties from selecting better candidates in their primaries, since the parties have control over who they select anyway (they're not actually bound by the results of the election, and can rig it if they want, such as with "superdelegates").