Self-join unique pairs

Question

I have a table (players) containing a list of players, I would like to pair up these players in a unique way, so that every player's opponent is unique every round (ie. every time the SELECT query is called).

The player_opponent_log column is an integer[] which contains the player_ids of players who have played with that player in a previous round (and is used to help pick out unique players). This column is populated afterwards using the results of the SELECT query - and is outside the scope of this question.

The table for instance would have the following data;

 player_id | player_opponent_log 
-----------+---------------------
         1 | {2,3}
         2 | {1}
         3 | {1}
         4 | {}
         5 | {}
         6 | {}
         7 | {8}
         8 | {7}

What I am trying to achieve would be something along the following lines:

 player_id1 | player_id2 
------------+------------
          1 |          4
          2 |          3
          5 |          7
          6 |          8

I have tried countless different approaches, including GROUP BY, DISTINCT ON (), self JOINS, but haven't been able to come to a working solution.

The following result is what I am currently getting and am trying to avoid:

 player_id1 | player_id2 
------------+------------
          1 |          4
          2 |          3
          3 |          4
          4 |          1
          5 |          1
          6 |          1
          7 |          1
          8 |          1

Where I'm getting stuck is how to eliminate a single player being allocated for a round more than once per SELECT query.

Any ideas on how to solve this would be highly appreciated.