"The additional cost of three elections over two is substantial." Political analyst Jim Engster says one of the reasons Louisiana used the old open primary system was to save money. Secretary of State Jay Dardenne says because of the closed primary system, Louisiana will spend an additional one million dollars on special elections in the spring and he says there is a very strong possibility the state will have to spend an additional five million dollars in the fall. Dardenne says money is not the reason for the change. "Under our old open primary system, we were potentially sending our congressmen later than other states who had completed their elections in November, whereas, we potentially had runoffs in December."
I do want to point out that it's not the closed nature of these primaries that is costing the extra money. These elections cost more because they're spread out over three stages, instead of two. A three-stage election would cost just as much even if it used open party primaries.
Instant runoff voting could allow us to compress the Congressional election cycle back down to two stages or less. If it were used in the party primaries, then the party nominees could be chosen in one round. Of course, in the general election IRV could provide an even greater benefit: the person elected would have to have the support of more than half of the voters casting ballots.
Alternatively, we could have just one general election in which all the candidates run, like the elections Louisianans are used to. However, we could use IRV in that election, thus allowing a candidate to be chosen by an absolute majority in one round. We could hold that election in November and our Congressmen would be chosen in November. It would solve the problem that prompted our electoral changes in the first place, but without adding the cost of extra elections and without resorting to plurality-voting.
If there was a concern that there would be too many candidates from one party in that general election, then the parties could devise and pay for some process (e.g., caucuses, conventions) to choose their own nominees.