Just briefly looking at the code, the first thing I spotted is 'post_type=game&posts_per_page=4&paged=$paged'.
Because you’re using single quotes, paged will never equal the current page.
You should use double quotes so the $paged variable is actually used in your query:
"post_type=game&posts_per_page=4&paged=$paged"
You can learn more about single quoted and double quoted strings in the PHP documentation.