Check that I have the highest score for each user submission

Question

This is a hypothetical scenario where:

1. We have a list of games
2. For each game, users can submit their scores
3. I, as the 'Admin', have to make a submission with a higher score for EACH user
   • It doesn't matter if I have the highest score for that game, I HAVE to create a submission for each user

For example, here is a list of scores for Game 1

id  game_id     user_id     opponent_id     score
1   1           1           2               1003
2   1           2           null            1002
3   1           3           null            1001

Submissions where opponent_id is NULL are the ones that I have to beat - but I don't have to beat the same user twice, as long as I have a submission for EACH user.

So, in the above example, although I have a higher score in general, I only have a submission against opponent_id = 2, so that means I need another one for opponent_id = 3

How do I get a list of all the games where I'm not the best player? That means all games where a user has a submission and I don't have one that is higher than his specifically.

What I need to do is:

1. Get the highest score for each user where opponent_id = NULL
2. Check that I have score for each of those users that is higher than their highest score

I can get step 1, but I'm stumped by step 2

The output would be a list of all games where a player has put a score I need to beat.

A chess game analogy would be that each game is a chess game and I can play each game with multiple players. What I want to see is if there's a game where it's (in essence) my turn. Hope this makes sense.

The wording is a bit weird but here is a fiddle that might make it more clear : https://dbfiddle.uk/xUxfj4XQ

Is this even possible with this database structure? Is there something that can be changed to make it easier/more performant?

EDIT: Based on @user1191247 submission, this is what I got

SELECT *
FROM games
WHERE id IN (SELECT game_id
             FROM (SELECT game_id, user_id, MAX(score) AS max_score
                   FROM scores
                   WHERE opponent_id IS NULL
                   GROUP BY game_id, user_id) g
             WHERE NOT EXISTS (SELECT 1
                               FROM scores
                               WHERE game_id = g.game_id
                                 AND opponent_id = g.user_id
                                 AND score > g.max_score)
             )

Answer 1

I think this meets the criteria set out in your two steps, but I am not sure about the apparent contradiction mentioned in the comments above.

SELECT *
FROM (
  SELECT game_id, user_id, MAX(score) AS max_score
  FROM scores
  WHERE opponent_id IS NULL
  AND user_id <> 1
  GROUP BY game_id, user_id
) g
WHERE NOT EXISTS (
  SELECT 1 FROM scores
  WHERE game_id = g.game_id
  AND user_id = 1
  AND opponent_id = g.user_id
  AND score > g.max_score
);

For the data in your fiddle this outputs:

game_id	user_id	max_score
1	3	1001
3	3	1001
4	2	1001
4	3	1002

Here's an updated db<>fiddle.