The Setup
Imagine a game where each team enters with two players. A match is a game where two teams play against each other in the following way: one player from a team plays against a player from the opposing team and the other two players also play against each other but the positional advantage is reversed. Imagine each game position is pre-generated by the computer.
Example Game
Match 1: Game 1: Player A1 from Team A plays against Player B1 from Team B, and Player A2 from Team A plays against Player B2 from Team B but the position is reversed. (The specifics of the game is irrelevant. The point is that the game has a positional aspect. Naturally, the teams cannot complain about positional advantage or disadvantage.) Then, the score between A1 and B1 is compared against the score between A2 and B2. For example:
A1 v B1: +1 (which means +1 for Team A, thus -1 for Team B.)
B2 v A2: +1 (which means +1 for Team B, thus -1 for Team A.)
Net score of the match 0.
or, another match
A1 v B1: +1
B2 v A2: -3
Net score of the match +4 for Team A.
A Tournament
Now imagine a tournament where 8 teams compete playing this game. And the format of the tournbament is league-based; that in a total of 7 rounds each team will have played against each other and the team with the highest total points will be the winner.
Suppose that at each round, each of the 8 matches play the same given position. Suppose there are ten to the power of 64 possible starting positions (the number has no relevance), in each match the computer selects a random number. Say in round one the selected position number is 967. Thus:
Team A v Team B: A1 v B1, and B2 v A2
Team C v Team D: C1 v D1, and D2 v C2
Team E v Team F: E1 v F1, and F2 v E2
Team G v Team H: G1 v H1, and H2 v G2
Each 8 matches play the same position.
Individual Performance
After the tournament, to see how the individual players performed, individual performances are calculated. The idea is that just because Team A lost to Team B, it does not mean that both A1 and A2 played badly. It could be the case that A1 played an excellent game, but A2 played so badly that Team A lost the game. To put this in an objective manner, the player performances are calculated as follows. Imagine the following scores in a given round:
A1 v B1: +2
B2 v A2: +2
C1 v D1: +2
D2 v C2: +1
E1 v F1: -1
F2 v E2: +2
G1 v H1: +3
H2 v G2: +0
(We note that +2 is the common score for the first pair. (We can assume that going plus +2 for the first pair was not really an achivement; rather it was the "expected" normal outcome.) We further note that Player D2, E1, H1, and H2 performed badly. (E1 was the worst.))
The round average was 11/8. Thus, for that round, each player's individual score is calculated by his score minus the average score.
Imagine that at the end of the tournament, the individual scores are:
Individual Rankings
A1: 8
B2: 7
C2: 7
D1: 5
A2: 4
...
E2: 1
...
E1: -1
F1: -4
B1: -5
etc.
(We thus note that based on this tournament scores A1 is a strong player, B1 is a weak player, etc.)
Players obhject that above ranking is not purely objective
It may seem that the individual rankings reflect performances objectively. But player E1 complains that BASED ON FINAL INDIVIDUAL RANKINGS, and considering the 7 rounds of the tournament, he mostly played against stronger player of the opposing team. For example, in the match between Team B and Team E, suppose the pairings were: E1 v B2, and B1 v E2. E1 argues that had he played against B1 instead of B2, he could have a much higher score.
My Question
Though the setup was long, I think everything up to now is trivially simple. My question is: After the initial individual rankings, how can we adjust the scores so that we get an objective individual ranking.
Assume that the game involves no element of luck. Further assume that a player has no advantage with respect to his position in a match. For example:
A1 v B1 and B2 v A2
versus
B1 v A1 and A2 v B2
do not matter with respect how players perform. (Because a plus score does not automatically mean that a player earned that score because he was a good player; it could be that given the position of the game, that was the "expected" outcome.)
You could assign each player something like an Elo rating and each position a rating adjustment that gets added to the players' rating difference to reflect the positional advantage. In your example, that would be $16$ variables for the players and $7$ variables for the positions. You could determine these variables by minimizing the sum of squared errors between the expected and actual results. You can't use the Elo system one-to-one because you have outcomes other than win, lose and draw, but it shouldn't be too difficult to generalize. For instance, if there are upper and lower bounds to the scores, you could rescale them so that they lie in $[-1,+1]$ and consider them fractional wins/losses for purposes of the Elo system.
The ratings would only be defined up to an additive constant; you'd have to take that into account in the minimization.