Let's say my main page is website.com. What are the effects of creating games.website.com? Would google consider it a separate site and return a beginner's pagerank?
Google will not consider it for separate site. But PR relates to individual page. For example your games.website.com initial PR is 0. Afterwords it may gain PR2 even though your main page PR is 1. So PR relates to individual page rank according to say traffic, umber of links to that page.
Thank you for the clear explanation. If I wanted to release an in-browser game for my website, would you advise using website.com/game instead?
Well my friend as john said above all page have page rank 0 at the beginning. Even if you consider to do website.com/game it is new page for google and will have page rank 0 at beginning.
Well I'm interested in games.website.com because of the inclusion of "games.", which I assume would greatly help with upping my spot in search results. I would only want to do website.com/games if it helped increase the PR of website.com (which I just found out it doesn't). So, based on the advice, I think the correct move is to stick with games.website.com?