your initiative in Matsuri, Survival Trial and Arena(s) are base on your overall initiative divided evenly between your positions then add treasure tools for each position so either that player has more overall initiative or a better position 1 treasure tool or both.

you'd be surprised how much initiative treasured tools put out. someone in my server has 80k bp less than me and I was @200k, he also had way less overall initiative and was still faster than me in survival trial, but in feeding all my resources into my position 1 treasured tool I ended up being faster in survival. just focus on your position 1 treasure tool is my advice (it should be a rope btw).