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“Thad Cochran got 766,000 votes in his last reelection,” Mr. Barbour said. “There are not 766,000 Republicans in Mississippi, but there are that many in the whole state electorate who will vote for a strong GOP candidate like Thad. We welcome all those voters, but understand they won’t all become Republicans.”

At primary election time, however, that would seem exactly what is wrong with the open-primary system, in the views of members such as Bruce Ash, Bill Armistead and Ed Martin.

The nuances may differ, but the attitudes of most Republicans is summed up by Michigan RNC member Dave Agema.

“A closed primary is necessary to prevent [Democrats] from nominating liberal candidates within our party.” Mr. Agema said. “When I see candidates in Michigan advertising that Democrats can vote for them and cross over, it indicates that that candidate is weak and desires his re-election more than the principles of the RNC platform and the will of the Republicans in that district.”