I feel a bit bad for Rin with the way her competition idea is being characterized by Toudou and Shirashiro. I think Rin's very sincerely shonen-brained about competition, very invested in the idea of being something that helps you understand yourself and others, and grow from that, with the part where you find out which of two people are better at something as nearly a tertiary aspect of it.
So like, when Toudou is saying that putting it up to a contest made her worry about how serious Rin was about their relationship, it feels wrong to me because I think it's specifically because Rin takes it seriously that she feels like she has to put it to a competition. And when Shirashiro is characterizing it as making Toudou just a prize for the victor it feels wrong, because I think to Rin it's not about creating a loser, a winner, and a prize, but a genuine belief that the way competition can make you grow and understand eachother makes it the best way to resolve this.
I don't mean to say the other girls had to be cool with the idea; that might be what competition means to Rin but that doesn't mean it has to be what it means to Toudou and Shirashiro. But I think it's kinda sad that they don't even seem to realize what this might mean to Rin.
I dunno, maybe I'm reading to much into it. I think competition is important to Rin on a deep level, but I think she also does like to win. Maybe the dating competition idea was more about the latter than the former and I'm just misunderstanding.