These are from Paul Gronke, a political scientist at Reed College:
John asked in an earlier post about political science research into the incumbency advantage and the top two primary. The comments to John’s post and the debate in the NY Times (featuring a number of prominent scholars) lays out the issues fairly well. There simply isn’t a lot of empirical work into the topic because the top-two is an unusual primary system. These systems are hard to study in isolation because we lack a good comparative base: you can either try to study a “most similar” state, or study the same state which has offered, and then removed, the top two option. In either case, our scientific inferences are limited.
Many scholars have strong prior expectations about how these systems will operate strategically — code words in our discipline for formal theoretic expectations (see Matt Shugart’s comment and Barbara Sinclair’s comments at the NY Times roundtable. These are difficult to convey to lay audiences because they rely on the complex connections between party rules, candidate choices, and voter behavior. Put simply: the world ain’t simple. Full disclosure: many political scientists were involved in the debates over the top-two in Washington, Oregon, and California.
All that being said, there has been work on the impact of open primary systems,of which the top-two is a variant. None of the studies speak directly to the incumbency advantage. Eric McGhee provides a very good summary of the literature and new analyses in support of the California ballot measure. Elisabeth Gerber and Rebecca Morton find that more closed primary systems result in more moderate candidates in the general election. Michael Alvarez and Jonathan Nagler find little evidence of “crossover” voting in open vs. closed primaries and thus very little evidence of “strategic” voting by primary voters.