Andy Murray has made it his goal to finish the season ranked third in the world, and he's well on his way to doing so. Roger Federer, presumably, cares less whether he is second, third or fourth. It's the No. 1 that matters for him.
Nevertheless, the gauntlet has been thrown, and here's what Murray needs to do to retain his current ranking, and what Federer needs to do to rob him off it at the very last minute.
Murray has 7,380 points, 710 ahead of Federer.
A player can earn up to 1,500 points in London if he goes undefeated through the tournament. Each round-robin victory earns the player 200 points. A semifinal win earns an additional 400 points and a final win another 500.
If Federer reaches the final, he'll have 600, 800 or 1,000 points, depending on how many round-robin matches he will have won. If Federer shall take the No. 3 ranking in this scenario, Murray will have to either win no matches at all, while Federer wins two in the round robins (800), or Murray wins one match in the round robins and Federer would then need a perfect run to the finals.
In any other scenario, save Federer winning the tournament, Murray gets to be year-end No. 3.
But what if Federer wins it?
Again, he can win the tournament with either one (1,100), two (1,300) or three (1,500) round-robin wins. In the first case, Murray would need to win two matches to stay ahead (by 10 points). In the second, he'll need to win all three (or one plus his semi) and in the third, he would need two round-robin wins plus a spot in the final.
If the two are to change positions, it appears Federer needs a damn good tournament. A spot in the final is not likely to do it, but it can if Murray under-performs grossly.
Murray can decide the outcome: Win two round-robin matches and his semifinal, and Federer can't touch him. But anything short of that and Federer might sneak back at No. 3 if he has a perfect or near-perfect tournament.