Are you using ACF 4 or 5?
In either case you need to supply the correct post ID value. In ACF for this is "{$taonomy}_{$term_id}". In ACF 5 you can use the same value or you can use "term_{$term_id}"
In a tax archive template you can get the term ID like this
$queried_object = get_queried_object();
$taxonomy = $queried_object->taxonomy;
$term_id = $queried_object->term_id;
To get the field you should either use
// the_field() echos the value and you do not need to include your own echo statement
the_field('game_publisher', $taxonomy.'_'.$term_id);
or
echo get_field('game_publisher', $taxonomy.'_'.$term_id);
for more information see https://www.advancedcustomfields.com/resources/get_field/ Get a value from different objects
