But since Chile has no previous experience of open primaries, pollsters are not so sure.
Both states held open primaries, which meant Republicans, independent voters and Democrats could cast ballots in the GOP contests.
The Conservative leader has previously spoken in favour of open primaries - based on the system used in some US states for selecting party candidates.
Fears about this radicalising effect of partisan primaries lay behind the victory on the same day of a ballot measure, Proposition 14, to change to open primaries.
It is not hard to see why open primaries might appeal to Mr Bloomberg, a media mogul who became a moderate Republican just before he ran for office.
And there is talk of open primaries in which people vote irrespective of their party affiliation, and then elect a candidate in a run-off between the top two vote-getters, whether from the same party or not.
David Cameron's party is also trying to get members involved in policy formation - a process that had effectively died out - and it has pioneered the use of "open primaries" to select Tory election candidates.
But during the 1990s several states threw open their primaries to independents.
The second is supporting Proposition 62, which would open up primaries.
For her part, Ms. Quinn has been walking a tightrope: Open opposition would upset the liberal voters who turn out for party primaries, while yielding would leave her open to criticism from the business community in November's general election.
For example, Tuesday's primaries in Illinois feature wide-open and hotly contested U.S. Senate races for both parties.
Ms. Quinn has been walking a tightrope: continued opposition would upset the liberal voters who turn out for party primaries, while yielding would leave her open to criticism from the business community in November's general election.
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