Federer, who has a combined 49-59 head-to-head record in all competitions against his fellow competitors in the 2019 ATP Finals, trails against two of the three players in his group (22-26 against Djokovic and 2-4 against Thiem).
In six matches against Djokovic and Thiem at the ATP Finals, Federer has three wins (two against Djokovic, one against Thiem) and three losses (all against Djokovic - 2012 final, 2013 RR and 2015 final).
It is to be noted that Federer's walkover against Djokovic in the final of the 2014 ATP Finals does not count as a loss in the player's head-to-head against the Serb.
Federer has the most match wins (57) in the history of the ATP Finals. The Swiss maestro's 41-win tally in the round-robin stage of the ATP Finals alone is more than the next highest tally of wins by any player (Ivan Lendl - 39) in the tournament.
Federer is making his record-extending 17th appearance at the ATP Finals. In his 16 previous appearances in the tournament, the six-time winner won all three group games on 10 occasions, doing so most recently in 2017, while enduring a group-stage exit only once (2008).
This is the third time Federer and Djokovic find themselves in the same group at the ATP Finals, with the pair splitting a pair of wins (2013-Djokovic, 2015-Federer) in the round robin stage of the competition. On both occasions, Djokovic went on to win the tournament.
If Federer gets through the group stage, he would meet one of Rafael Nadal, Sascha Zverev, Daniil Medvedev, and Stefanos Tsitsipas, for a place in a record-extending 11th ATP Finals final. The Swiss maestro is a combined 24-29 against the quartet in all competitions, with 24 of his losses coming against Nadal.
At the ATP Finals, Federer is 4-1 against Nadal (lost in the 2013 semis) and 1-1 against Zverev (lost in the 2018 semis). Medvedev and Tsitsipas are making their tournament debut at the 2019 ATP Finals.
Federer leads 4th-placed Medvedev by 485 points and 5th placed Thiem by 1165 points in the ATP rankings. The Swiss maestro needs two group-stage wins to avoid falling out of the top four, while a title run would ensure that Federer remains at No. 3 at the end of the tournament.
The Swiss maestro can also remain at No. 3 under the following additional scenarios.
Scenario 1: Federer reaches the semifinal, Medvedev does not
Theoretically speaking Federer needs at least one group stage win to make it to the last 4, which will give him 200 additional points and an overall lead of 685 points over Medvedev. Medvedev stands to earn a maximum of 600 points for three group-stage wins (3*200).
In the worst case scenario, if Federer makes the semis with just one group-stage win and Medvedev fails to get out of his group despite two group-stage wins, Federer will have a minimum lead of 285 (485 + 200 - 400) points over Medvedev .
Scenario 2: Federer beats Medvedev in the semis
The duo would meet in the semis if one wins the group and the other finishes runner-up.
In the worst case scenario for Federer, if he makes the semis with only one group-stage win (200 points) and Medvedev wins all three of his group games, Federer will have a lead of 85 points (485 + 200 - 600) over the Russian.
A win over Medvedev in the semis would thus keep Federer at No. 3 at the end of the tournament.
Scenario 3: Federer reaches the final, Medvedev does not
Federer would make a minimum of 600 points (200 points for a round robin win, 400 points for a semifinal win), for an overall lead of 1085 points over Medvedev, if the Swiss maestro reaches the final of the 2019 ATP Finals.
In that case, if Medvedev does not progress beyond the semis, he would only have a maximum of 600 points for his efforts (3*200 for three round robin wins), which would not be enough to overhaul his deficit to Federer.
Scenario 4: Federer wins the title
Any title run for Federer is guaranteed to give the Swiss a minimum of 1100 points (1*200 for one round robin win + 1*400 for a semifinal win + 1*500 for a final win), for an overall lead of 1585 points over Medvedev.
That would be enough to keep Federer at No. 3 regardless of what Medvedev accomplishes at the tournament.
Any scenario which sees Medvedev fail to earn more than 485 points than Federer at the 2019 ATP Finals would see the Swiss maestro end the season at World No. 3.