How to allow editors and admins to see submenus

Use a capability instead of a role. For example, both admins and editors have the capability of moderate_comments.

add_submenu_page(my_settings_page, 'Players', 'Players', 'moderate_comments', my_players_page, 'my_players_page');

http://codex.wordpress.org/Roles_and_Capabilities

Here is the Codex article for submenus:
http://codex.wordpress.org/Function_Reference/add_submenu_page

Also, the article for all admin menus
http://codex.wordpress.org/Administration_Menus