This paper addresses the issue of gridlocks in the approval of prime ministers in semi-presidential countries and presidential countries with prime ministers, focusing on the case of South Korea. A game-based scheme for prime minister approval specifically designed for presidential systems like Korea's is then proposed, along with two supporting provisions. The scheme has an adjustable parameter tentatively set at 3/5 to ensure a balanced system, and it also introduces a new form of cohabitation — hard cohabitation — where the prime minister is not nominated by the president. The supporting provisions aim to cope with two challenges likely to arise from hard cohabitation: potential cabinet instability and difficulties in cooperation between the president and the prime minister. By adopting a different transition model than in version 1.0, this version 2.0 resolves several issues identified, and introduces a mechanism absent in version 1.0, which is for the parliament to replace the prime minister following a midterm parliamentary election. The reform package proposed here is thus believed to be smooth, fair, and effective, and is strongly hoped to be adopted by South Korea or other similar countries in the future.