The model is Elo — the same family used for chess ratings — extended from 1v1 to 5v5. Each player has an MMR; each team's rating is a soft-max aggregate of its 5 players' MMRs; win probability between two teams follows the standard Elo logistic curve. The two free parameters T (soft-max temperature) and S (Elo scale) plus the 8 bucket MMRs were calibrated on 13,739 EUW solo/duo games across 120 players spanning the Emerald-to-Challenger ladder.
For the full calibration story, parameter derivations, and the human takeaways, see the linked write-up below.
Calibrated MMR per bucket (anchored to Master 0 LP = 0): HC +1125, LC +930, GM +793, HM +654, MM +398, LM +49, DM −34, EM −312.
Team rating = soft-max with T = 1317. Rteam = T · log(Σ exp(Ri / T)). T controls the carry-vs-team-play balance: at T = 1317, a player +1 T MMR above their teammates contributes ~2.7× the soft-max weight of each teammate — meaningful but not dominant. The soft-max aggregation approach (interpolating between mean and max) is supported by Dehpanah et al. (2021). The previous version used T = 400, which over-weighted solo carries; the data says they're rarer than that.
Win probability uses standard Elo with S = 510. P = 1 / (1 + 10^((Ropp − Rteam)/S)). A 510-point MMR gap makes the higher team 10× more likely to win. (Chess Elo uses S = 400; solo Q is noisier — 4 randos per side — so rank gaps convert to win-rate less steeply.)
The pool (Master+ mode only). 31,000 players: 80 HC (2500+ LP), 220 LC (2000–2500), 700 GM (1500–2000), 1,500 HM (1000–1500), 5,000 MM (500–1000), 23,500 LM (0–500). Used only for the "1 in X" occurrence numbers.
No occurrence rates in Diamond+/Emerald+ modes. Match-occurrence requires a defined "pool" with population counts. The Master+ pool is well-defined at hard-reset; the Diamond and Emerald populations are larger and far less constrained by ladder anchoring, and matchmaking targets balance rather than uniform random sampling, so static "1 in X" probabilities stop reflecting reality.
Match-occurrence probability (Master+ mode only). Assumes uniform random sampling of 10 distinct players from the 31,000-player pool. Real Riot matchmaking targets balance, so genuinely lopsided matchups are even rarer in practice. Meaningful only right after a hard reset; once MMR data accumulates, occurrence numbers stop reflecting reality.
Row counts. Master+: C(10,5) = 252 distinct compositions across 6 tiers, 252² = 63,504 ordered matchups. Diamond+: C(11,5) = 462 across 7 tiers, 462² = 213,444. Emerald+: C(12,5) = 792 across 8 tiers, 792² = 627,264.
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