The first candidate to be put forward, almost as soon as it became clear that Mr Hosokawa was going, was Tsutomu Hata, the Foreign Minister and co-leader of Shinseito, the Japan Renewal Party. Mr Hata was one of the front-runners for the prime ministership when the coalition took office last year, but hostility towards him from some other members of the alliance was one of the main reasons why Mr Hosokawa got the job instead.
Many of the coalition's leading figures spent years as senior members of the Liberal Democratic Party (LDP), whose 38-year grip on power was loosened when Japan voted for change in last year's election. The new government threw them together with recent opponents in the Socialist Party, which was also seen by the voters as outdated, but which still won enough seats to be the biggest force in the coalition. While Mr Hosokawa was given credit for having quit the LDP when it was still firmly established, Mr Hata and others who made a later exit are seen by many left- wingers as opportunists whose ideas remain unchanged.
If this was considered to rule out Mr Hata, however, the same is true of his main rival, Masayoshi Takemura. Like the Foreign Minister and Mr Hosokawa, he broke away from the LDP at the head of a personal following, but his Sakigake party won only 13 seats to 55 for Shinseito and 35 for the outgoing Prime Minister's Japan New Party. Mr Takemura became chief cabinet secretary - a powerful position in Japanese governments - but has been accused of seeking the top job from the outset. On several occasions he undermined Mr Hosokawa by seeming to contradict him.
Then there is Ichiro Ozawa, who was once the protege of the arch-fixer in Japanese politics, Shin Kanemaru. Instead of taking over from his former mentor in the LDP, Mr Ozawa is using the methods he learnt from the now-disgraced Mr Kanemaru to pull the strings of the coalition. Political observers have speculated that if a successor cannot be found, Mr Ozawa might allow the alliance to dissolve and seek to form a new one with the help of further defections from the LDP. This would almost certainly exclude the Socialists, who reject his ideas of Japan playing a bigger part in the world, unhindered by the pacificism of the post-1945 constitution.