Page 163 says that in case of ties the initiave should be determined by lowest honor taking priority. I can understand where such an idea might come from, like that ruthlessness allows characters to act faster, but I think that is not a good approach for this setting, since looking at teh supreme military force in Rokugan the Lion and their high honor, one should realise that honor is not really a factor, since the Lion have usually so clear vissions that they also do not hesitate to do the right thing.
So, here my suggestion, instead of making the tie-breaker about honor, how about looking first at the strife, since the emotional state might could also be a good indicator of how eager one might be to act. And thus making a roll instead of reducing strife is even more tempting for characters that want to get stuff done! The game has the new mechanic, so why not use it?