How the math works

1 · Two kinds of votes

Ballots are your ordered top 5–15 at one position (quarterbacks, running backs, …). Duels are single head-to-head picks, usually across positions (“Mahomes or Garrett?”). Ballots decide who is best at a position; duels decide how positions compare and sharpen the very top.

Cadence: one ballot per position per day and up to 100 duels per day, per voter (an anonymous cookie). Every ballot counts, so a regular’s influence grows one day at a time — no single session can swamp the board.

2 · One strength per player

Every player gets a latent strength on one shared log-odds scale, built from three parts:

strength_i  =  μ[position]  +  δ_i  +  ε_i

μ   position baseline  — learned from cross-position duels, shared by everyone at the position
δ   ballot standing    — learned only from that position's ballots
ε   duel adjustment    — learned only from the player's own head-to-head record

Splitting it this way means a position with only a handful of duels still gets a well-estimated baseline (every duel involving any of its players informs μ), while an individual who keeps winning duels above his ballot standing gets lifted through ε.

3 · Ballots: Plackett–Luce over the whole position

A ballot b₁ > b₂ > … > b_k is read as k sequential choices: b₁ was chosen over all players at the position, b₂ over everyone except b₁, and so on. That makes an omission informative — if 500 voters leave a player off, the model hears that 500 times.

P(ballot)  =  ∏_j   exp(δ_{b_j})  /  Σ_{m still available}  exp(δ_m)

The scale is pairwise-interpretable: P(i over j) = σ(δ_i − δ_j), so ballot standing plugs straight into duel math.

4 · Duels: Bradley–Terry

P(i beats j)  =  σ(strength_i − strength_j)  =  1 / (1 + exp(strength_j − strength_i))

Every duel nudges the two position baselines and the two players’ ε. Within-position duels (“your #1 vs the crowd’s #1”) only move ε.

5 · Why two early votes can’t crown a #1

Every parameter has a Gaussian prior centred on “average” (δ ~ N(0, 2²), μ ~ N(0, 1.5²), ε ~ N(0, 0.75²)). With little evidence the estimate stays shrunk toward average and carries a wide error bar. And the model is refit from all votes every time — there is no running Elo score to carry momentum. Two early #1 votes versus 500 later ballots ranking the same player 18th is simply 2 against 500; the early votes get no seniority.

score_i  =  strength_i  −  z · se_i        (z = 1)

se_i  =  √( se(δ_i)²  +  se(μ)²  +  se(ε_i)² )     (Laplace approximation at the optimum)

The board is ordered by score, the lower edge of each player’s plausible range. Two players with the same point estimate are separated by how sure we are — thin evidence sinks, solid evidence floats. Position boards use the same idea on δ.

6 · Estimation

Maximum-a-posteriori fit by MM (minorize–maximize): Hunter’s minorizer for Plackett–Luce / Bradley–Terry with the prior kept exact, so every coordinate update is a tiny 1-D convex solve and every iteration provably raises the posterior. A full-league refit (2,188 players) takes well under a second, so it runs right after your submission.

7 · Which duels you get

Pairs are sampled from the crowd’s current top players plus your own picks, with probability proportional to expected information: p(1−p) · (Var[s_i] + Var[s_j]). Toss-ups between uncertain players are worth the most; lopsided, well-established match-ups are rarely asked. Your own picks get a bonus so every session feels personal.

8 · Everything is transparent

Every player page shows the raw evidence — how many ballots included him, where voters put him (1st…15th), first-place votes, mean consensus points, duel record, most common opponents — plus his rank history across refits, so you can literally watch an early #1 become an #18.

Data: nflverse snap counts (CC BY 4.0) — every player who took a snap in the 2025 regular season or playoffs. Voter identity is an anonymous cookie; one ballot per position per voter, always replaceable.