This is a real life problem. A group of people meet once a week to play a game between two teams. Each round 2 people are randomly appointed captains. Each captain takes turns picking people to be on their team. So far 17 people have turned up to these games. They are each represented by a different letter in the alphabet from A to Q. The players have different capabilities. The teams are usually 5 on 5 but it can vary if an odd number of people turn up for the game of if the captains agree that 2 weak players are equal to 1 strong player.
In the 7 rounds so far the scores have been:
ROUND1: TEAM 1 = AEHJK; TEAM 2 = BCDFIM; TEAM 1 SCORE = 10; TEAM 2 SCORE = 2; ROUND2: TEAM 1 = ABDFJ; TEAM 2 = CEGHI; TEAM 1 SCORE = 10; TEAM 2 SCORE = 3; ROUND3: TEAM 1 = ACEFK; TEAM 2 = BDGNO; TEAM 1 SCORE = 5; TEAM 2 SCORE = 1; ROUND4: TEAM 1 = ACFGI; TEAM 2 = BDJKPQ; TEAM 1 SCORE = 1; TEAM 2 SCORE = 4; ROUND5: TEAM 1 = ABIJ; TEAM 2 = CDFKOP; TEAM 1 SCORE = 3; TEAM 2 SCORE = 4; ROUND6: TEAM 1 = ABCFH; TEAM 2 = DEIKL; TEAM 1 SCORE = 10; TEAM 2 SCORE = 1; ROUND7: TEAM 1 = ACDIJ; TEAM 2 = BEFHK; TEAM 1 SCORE = 9; TEAM 2 SCORE = 0;
The organizers want to understand the strengths of each of the players so that they can organize more even match ups. If a teams strength is the sum of the strengths of each of the players then what is the strength of each player? It's possible that some combinations of players lead to a better result than would be predicted by a simple additive model. Is there a better way to accurately predict the results of the teams combinations? What is it?
Microsoft have developed a ranking algorithm for just this problem that they use in their Xbox Live service.
(Xbox Live is an online gaming platform players meet to compete in games like football and Quake-like games). Their algorithm is called TrueSkill. They use it to measure player's skill and pair them up with players of similar skill on servers, so that everyone has a good time.
Here's the technical paper. http://research.microsoft.com/pubs/67956/NIPS2006_0688.pdf
There's also the classic ELO rating system which is widely used. I don't think it works for individual skill determination in team games though. http://en.wikipedia.org/wiki/Elo_rating_system