Sorry for the late reply, I've been away.
catacaustic quite capably explained your options, I just wanted to add more specificity related to template filenames.
If you choose not to use a custom post type, a page template can be named anything, the name shown in the dropdown template selection on the page editor screen comes from the header information. It can be based on page.php if you like, but you need to add something like Template Name: Games to the header.
OTOH, the filenames for templates for a custom post type is specific. They must be of the form single-{$post_type}.php and archive-{$post_type}.php, so if your CPT is "games", then the template filenames are single-games.php and archive-games.php.