I would not call it Playing chicken. I would call it respecting the space the other person would like to have in the R. Trust me, this creates more room for intimacy, not less. And it is not a game or manipulation. It is a way of being with others so that everyone has space to know what they want. Of course there is give and take in terms of who tries to change the R in different ways.
Look, you ask someone to go steady, they say no. Three options:
1) end the r 2) go steady by yourself 3) back off a step and see where things go
If you reject (1), (3) is the clear winner in terms of treating all involved well and creating opportunities for a deeper R.