What POSSIBLE reason do you have for using GROUP BY in that query?
That's the culprit.
Any other DB wouldn't allow you to use it like that, but MySQL not only allows it but makes it have some very weird side effects.
Incidentally, your use of RIGHT JOIN there is incorrect. Or, rather, because of the rest of the query your RIGHT JOIN is silently being converted into an INNER JOIN.
You can *NOT* use any condition on the DEPENDENT table in an outer join in the WHERE clause. Such conditions *MUST* be made in the ON clause. (Sole exceptions: tests for NULL or NOT NULL.)
SELECT team_member, stage, points
RIGHT JOIN team_members
ON team_members.team_id = team_member_scores.team_id
AND team_member_scores.team_id = '".$team_id."'
WHERE game_id = '".$game_id_info."'
ORDER BY points DESC LIMIT 5
Though I would bet a latte that you do not need or want the apostrophes around '$team_id' or '$game_id_info'. You should never put apostrophes around NUMBERS if you are comparing with or assigning to numeric fields.
May I *strongly* suggest that before you use a query in your PHP code you first check it out using some DB query tool? MySQLWorkbench, perhaps. Or even ugly MySQLPHPAdmin or whatever.