You can search subparts of your site by using a combination of site:
operator and webSearchQueryAddition
parameter on gcse
element.
webSearchQueryAddition appends additional search term to your user's query. If for each of the "games" you change the webSearchQueryAddition to point to the "game" base url, the search results will be matching that url. You can inject that parameter programmatically with e.g. javascript, for each of the "games".
Documentation is here: https://developers.google.com/custom-search/docs/element#supported_attributes
And here is working example: http://jsfiddle.net/t2s5M/