Fix PredictWin/PredictDraw formulas and documentation

[thesprockee]

Summary

Fixes three bugs in PredictWin and PredictDraw prediction formulas, verified against the Weng & Lin (2011) paper and the Python openskill.py reference implementation.

Bug 1: Pairwise denominator (PredictWin + PredictDraw)

TeamSigmaSquared (already σ²) was squared again, producing σ⁴. Additionally, the beta coefficient used n (number of teams) instead of 2.

Per Weng & Lin (2011) eq. (49): c_iq = sqrt(σ²_i + σ²_q + 2β²) — always 2β², and sigma values appear directly, not squared.

Before: sqrt(n*β² + σ²_A*σ²_A + σ²_B*σ²_B)
After: sqrt(2*β² + σ²_A + σ²_B)

Bug 2: PredictDraw iteration and draw probability base

• Used 1/n (team count) instead of 1/total_player_count for draw probability
• Iterated ordered permutations instead of unordered combinations
• Inconsistent normalization between 2-team and n-team cases

Rewritten to match openskill.py's itertools.combinations approach.

Bug 3: Incorrect documentation comments

Every default value comment in OpenSkillOptions was wrong (Z said 3.29 not 3, Epsilon said 0.000001 not 0.0001, Beta said 0.5 not sigma/2, Model said Glicko2 not PlackettLuce, Tau said 0.5 not nil).

Impact

The inflated denominator (~75% too large) compressed all predictions toward 0.5, making skill differences nearly invisible to any system using PredictWin or PredictDraw.

Using real player ratings from a production EchoVR matchmaking server (mu ≈ 1–14, sigma ≈ 3–5):

Matchup	Old P(win)	New P(win)	Error
μ=1.2 vs μ=9.3	0.38 vs 0.62	0.10 vs 0.90	+0.28
μ=1.2 vs μ=13.9	0.35 vs 0.65	0.04 vs 0.96	+0.31
μ=9.3 vs μ=13.9	0.43 vs 0.57	0.24 vs 0.76	+0.19

4v4 team match example:

• Old: Blue 54.5% win probability (looks balanced)
• New: Blue 82.4% win probability (clearly lopsided — 11 mu advantage)

The old predictions were telling matchmakers the match was basically a coin flip when one team had an overwhelming skill advantage. Any system relying on PredictWin/PredictDraw for match quality assessment was operating nearly blind to skill gaps.

References

• Weng, R. C., & Lin, C.-J. (2011). A Bayesian approximation method for online ranking. JMLR, 12, 267–300. Equations (5), (49).
• Cross-validated against openskill.py v6.2.0

Test plan

• All existing tests updated and passing
• Test values cross-validated against Python openskill.py output
• Formulas verified against Weng & Lin (2011) equations (5), (49)

🤖 Generated with Claude Code

[Copilot]

Pull request overview

This PR corrects the win/draw prediction math in rating to align with the intended Weng & Lin (2011) pairwise formulation, and updates related option documentation and tests to reflect the corrected behavior.

Changes:

• Fix PredictWin pairwise denominator to use 2*beta^2 + sigmaSqA + sigmaSqB (instead of effectively using n*beta^2 and squaring variances again).
• Rewrite PredictDraw to average unordered pairwise draw probabilities and base draw margin on total player count.
• Update prediction test expectations and refresh OpenSkillOptions field comments.

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 3 comments.

File	Description
types/types.go	Updates OpenSkillOptions documentation comments to match current default behaviors.
rating/predict.go	Fixes pairwise denominator in PredictWin and rewrites PredictDraw to use combinations + corrected constants.
rating/predict_test.go	Updates expected probabilities to reflect the corrected prediction formulas.

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