In general English, we have the expression 'Finders keepers'. It means that whoever finds something has the right to keep it.
Expressions relating to children's games tend to be region-specific and dialectal. Where I grew up, kids said 'Bagsy' or 'Bags I', as in 'Bags I the first go on the swing!' This isn't something I'd expect to find in a dictionary, and I certainly wouldn't recommend you use it.
It would be interesting to hear what phrases other native speakers used when they were growing up.