Plackett–Luce (Ranking MLE)
Bradley–Terry extended to full finishing orders: a placement is read as first being chosen from everyone, then second from the rest, and so on.
MLE over full placements: P(ordering) from skill values
Best at
Free-for-all groups who record complete placements and want the most rigorous estimate available.
Where it misleads
Expensive to compute over large fields, and as opaque as its pairwise sibling.
How it works
Plackett-Luce extends Bradley-Terry to handle full rankings in multi-player matches. It models the probability of any particular ordering of players, making it the gold standard for FFA statistical ranking.
How it works: The model treats a full placement as a sequence of "choices": 1st place is the most likely to win from all players, then 2nd from the remaining players, and so on. MLE finds the skill values that maximize the probability of the observed orderings.
When to use it: When you record full placements in FFA board games and want the most statistically rigorous skill estimates. It's essentially what TrueSkill approximates, but solved exactly.
Watch out for: Computationally expensive, especially with large fields or many matches. Requires iterative optimization. Only works with FFA placement data. Best for analytically-minded groups who appreciate statistical rigor.