BY the old rules of Japanese politics, Prime Minister Kiichi Miyazawa yesterday was fully justified in declining to heed the advice of fellow politicians and the press that he should ''take responsibility'' for the ruling Liberal Democratic Party's (LDP) electoral debacle and immediately resign. The reason was simple. The LDP had lost its House of Representatives majority for the fourth time. It had failed to make up for the seats lost when 55 LDP members defected. But Mr Miyazawa himself had emerged from the election with the largest single LDP faction in the Lower House and with the second largest faction in both houses of parliament. By the old rules, this entitled him to a little respect. He might even have been allowed to complete his full term as LDP president and prime minister. Yesterday, few noticed this development. Factional logic now counts less than it once did. Very likely Mr Miyazawa himself was unaware of being slightly stronger. His great weakness as an LDP leader has always been his disdain for political machinations. The fact that the relative improvement in Mr Miyazawa's factional strength counted for nothing was one small sign among several that the LDP is slowly changing, almost despite itself. First, while the old factional line-up remains in place (see table), there are signs that factional discipline is not what it used to be. Perhaps this is the inevitable result of former finance minister Tsutomo Hata and former LDP secretary-general Ichiro Ozawa taking their faction out of the LDP, and setting up their own party, Shinseito, which has done extraordinarily well, garnering just over 10 per cent of the vote. Suddenly, the formerly tight world of mutual factional reliance, which was part of the essence of the LDP, looks more vulnerable. Secondly, in the desperate situation in which the party found itself after the passage of the no-confidence motion last month, the LDP had to bypass the factional system of seniority. Mr Miyazawa may have been prime minister but many candidates asked him to stay at home rather than campaign for them. Instead, the LDP pressed into service the so-called ''Three Arrows'' to wear the white gloves (signifying purity) and to bear the main oratorical burden of the campaign throughout Japan. These were former transport minister and controversial author Shintaro Ishihara, Chief Cabinet Secretary and former LDP dissident Yohei Kono, and former finance minister Ryutaro Hashimoto. Under the old factional system, Mr Ishihara was too much the maverick to get anywhere. Mr Kono still vies with former chief cabinet secretary Koichi Kato to succeed to the leadership of the Miyazawa faction. Mr Hashimoto is not even that close to leadingthe Obuchi faction one day. The ''Three Arrows'' may not be factional bosses, but they helped save the LDP in the sense that the party roughly held on to its pre-election seat strength, and kept its lost share of the total vote to 9.5 per cent, a better result than most polls predicted. Their performance inevitably makes it that much more difficult for the LDP to go back to its old system of factional leaders each having their turn at the prime ministership. Nevertheless, the old faction war horses are gearing themselves up for a run at the top job. Former deputy prime minister and foreign minister Michio Watanabe - despite severe illness - was yesterday urging Mr Miyazawa to step down in order that he couldstep up. Thirdly, former prime minister Toshiki Kaifu indicates another change as he implicitly suggests that policy preference should concern LDP politicians more than just personal, factional loyalties. Mr Kaifu, who unlike Mr Miyazawa at least tried to institute political reform, has set up an intra-factional LDP group, reported to number 109 members who favour political reform. It is for this reason that Mr Kaifu, who is still merely a lieutenant in the lowly Komoto faction, is being touted as a possible replacement for Mr Miyazawa. If the LDP is forced to accept political and electoral reform as the price for retaining office, Mr Kaifu will be useful as a front-man, and those 109 members will be unlikely to defect. But if the LDP decides to go on being its old self, resisting reform, Mr Kaifu and his allies could further erode LDP Diet strength. The LDP leaders might hasten their party's demise if they forget that Mr Kaifu has been and remains quite friendly with Mr Ozawa.