Decomposition of Valence and Paradox of Primary Election

When does a political party decide to hold a primary election to choose a candidate for the next election, and when does an incumbent decide to announce his/her political retirement? Furthermore, if they make decisions interactively, how do their decisions affect the quality - "valence" - of the candidate who ends up running for the party? In this paper, we examine these questions by modeling a simple strategic game between a party and an incumbent in the context of Australia where political parties take important roles in selecting candidates and making policies. Specifically, by decomposing the valence into three factors, we show that primary elections may facilitate (potentially) high-quality incumbents to retire unexpectedly. We call it a "paradox of primary elect