chore: Release trueskill-tt version 0.4.0

`Game::ranked` and `Game::scored` validated `p_draw` and `score_sigma`
but never `convergence`. `ConvergenceOptions` has public fields and
`GameOptions` carries one, so a caller could hand the engine a set that
`HistoryBuilder`'s eager asserts never saw. Past that, the only guard
was a `debug_assert!`, which is gone in the profile users ship.

An `alpha` of zero is the bad case, and it fails silently rather than
loudly. Measured in release before the fix:

    likelihoods: [[Gaussian { pi: 0.0, tau: 0.0 }],
                  [Gaussian { pi: 0.0, tau: 0.0 }]]

Every EP update unapplied, every likelihood uninformative, inference
returning the priors it was given — and an `OwnedGame` that looks
entirely ordinary to the caller. `HistoryBuilder::convergence` already
documents exactly this hazard; the `Game` constructors just did not
share the check.

Adds `ConvergenceOptions::validate`, called by both constructors.
Rejects `alpha` outside `(0.0, 1.0]` and negative `epsilon`; NaN fails
both comparisons and is rejected too.

`tests/validation.rs` states the release-mode guarantee for the whole
public surface, not just this hole, and CI already runs the suite in
release. Probing the other conditions #18 lists found five of eight
already enforced — ties without a draw probability, per-event score
sigma, weight/team dimensions, draw-probability range, score-sigma
range — so this closes the remaining gap rather than the whole issue.
The engine keeps its `debug_assert!`s as invariant documentation.

Refs #18

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ

CHANGELOG.md

@@ -2,6 +2,29 @@
 
 All notable changes to this project will be documented in this file.
 
+## 0.4.0 - 2026-09-07
+
+### Breaking Changes
+
+- feat!: N-team outcome prediction with draw mass, replacing the 2-team panic
+- refactor!: close the remaining API gaps from #21
+- fix!: apply competitor configuration whenever it is supplied
+
+### Bug Fixes
+
+- fix(release): skip the changelog hook during a dry run
+- fix: stop destroying tail precision in evidence and truncation
+- fix: reject convergence options that silently disable inference
+
+### Documentation
+
+- docs: correct drifted documentation and compile the README in CI
+
+### Features
+
+- feat: add expected information gain for active matchup selection
+- feat: let observers be shared, boxed, or borrowed
+
 ## 0.3.0 - 2026-09-01
 
 ### Breaking Changes
@@ -25,6 +48,7 @@ All notable changes to this project will be documented in this file.
 ### Miscellaneous Tasks
 
 - chore: ignore proptest regression seed files
+- chore: Release trueskill-tt version 0.3.0
 
 ### Performance
 

Cargo.toml

@@ -1,6 +1,6 @@
 [package]
 name = "trueskill-tt"
-version = "0.3.0"
+version = "0.4.0"
 edition = "2024"
 rust-version = "1.85"
 description = "TrueSkill Through Time: Bayesian skill rating that tracks how skill evolves over time, via Gaussian message passing"

README.md

@@ -164,6 +164,75 @@ h.event(1)
 h.converge().unwrap();
 ```
 
+## Prediction
+
+`predict_outcome` gives the full distribution over finishing orders. Each entry
+is a rank vector in the same shape `Outcome::ranking` takes — equal ranks mean a
+tie — so an outcome feeds straight back into inference.
+
+```rust
+use trueskill_tt::History;
+
+let mut h = History::builder().p_draw(0.1).build();
+h.record_winner(&"alice", &"bob", 1).unwrap();
+h.converge().unwrap();
+
+let p = h.predict_outcome(&[&[&"alice"], &[&"bob"]]).unwrap();
+
+// Probabilities are exhaustive and disjoint, so they sum to one.
+assert!((p.total() - 1.0).abs() < 1e-6);
+
+let (best, likelihood) = p.most_likely().unwrap();
+println!("most likely: {best:?} at {likelihood:.3}");
+println!("draw:        {:.3}", p.probability_of(&[0, 0]));
+```
+
+Supports any number of teams. Because the outcome space grows factorially, the
+full distribution is capped at `MAX_PREDICTED_TEAMS`; two cheaper entry points
+stay available at any size:
+
+- `predict_win_probabilities(teams)` — `P(team i finishes strictly first)`,
+  quadratic in team count.
+- `predict_ranking(teams, ranks)` — one specific finishing order.
+
+Unknown keys are an error, not a silent omission: a team the history has never
+seen cannot produce a confident-looking probability.
+
+## Which match to play next
+
+`quality()` measures whether a matchup is *fair*. That is not the same as
+whether it is *informative*, and the two only coincide for two evenly matched
+competitors. When each observation costs something, ask
+`expected_information_gain` instead — the outcome-weighted divergence between
+what you believe now and what you would believe afterwards.
+
+```rust
+use trueskill_tt::History;
+
+let mut h = History::builder().build();
+for t in 1..=10 {
+    h.record_winner(&"veteran", &"regular", t).unwrap();
+    h.record_winner(&"regular", &"veteran", t + 100).unwrap();
+}
+h.record_winner(&"veteran", &"newcomer", 500).unwrap();
+h.converge().unwrap();
+
+let settled = h.expected_information_gain(&[&[&"veteran"], &[&"regular"]]).unwrap();
+let unknown = h.expected_information_gain(&[&[&"veteran"], &[&"newcomer"]]).unwrap();
+
+// Playing the newcomer teaches you more than replaying a settled rivalry.
+assert!(unknown > settled);
+```
+
+The result is in nats, and is bounded by the entropy of the outcome: at most
+`ln 2 ≈ 0.693` for a two-way result, `ln 3` once draws are possible, `ln k` for
+`k` outcomes. A value near zero means you already know how it ends.
+
+This costs one full inference pass **per possible outcome**, so it is far more
+expensive than `quality()`. Scoring every pairing among `n` competitors is
+`O(n² × outcomes)` passes — shortlist with `quality()` or
+`predict_win_probabilities` first, then score only the shortlist.
+
 ## Todo
 
 - [x] Implement approx for Gaussian
@@ -172,6 +241,7 @@ h.converge().unwrap();
 - [x] Add examples (`examples/atp.rs`, `examples/scored.rs`)
 - [x] Add Observer (`Observer` / `NullObserver`)
 - [x] Benchmark the inference loop (`benches/batch.rs`, `benches/history_converge.rs`, `benches/ingest.rs`)
+- [x] N-team `predict_outcome` with draw mass, and `expected_information_gain`
 - [ ] Cross-check `quality()` against [sublee/trueskill](https://github.com/sublee/trueskill/tree/master) — N-group support works and is covered by invariants, but no reference values are asserted
 
 ## License

src/acquisition.rs

@@ -0,0 +1,352 @@
+//! Active learning: which comparison teaches you the most.
+//!
+//! [`quality`](crate::quality) answers "is this matchup *fair*". That is a
+//! different question from "is this matchup *informative*", and the two
+//! coincide only for two evenly matched competitors. When each observation
+//! costs something — a human click, a scheduled fixture — the question worth
+//! asking is the second one.
+//!
+//! The quantity here is expected information gain: the outcome-weighted
+//! divergence between what you believe now and what you would believe after
+//! seeing the result.
+//!
+//! ```text
+//! EIG(matchup) = SUM  P(outcome) * KL( posterior_after(outcome) || prior )
+//!              outcome
+//! ```
+//!
+//! It is the mutual information between the observed outcome and the skills,
+//! which is worth remembering because it pins the scale: information gain
+//! cannot exceed the entropy of the thing you are about to observe. A contest
+//! with `k` distinguishable outcomes can teach you at most `ln k` nats,
+//! whatever the ratings. That ceiling is the sharpest available test of an
+//! implementation — see [`expected_information_gain`].
+
+use crate::{
+    GameOptions, Gaussian, InferenceError, Outcome, Rating, drift::Drift, predict, time::Time,
+};
+
+/// Outcomes below this probability contribute nothing measurable and are not
+/// worth an inference pass.
+///
+/// The contribution of an outcome is `P * KL`, and `KL` is bounded in practice
+/// by tens of nats, so a probability this small moves the total by less than
+/// the quadrature error already present in `P` itself.
+const NEGLIGIBLE: f64 = 1e-12;
+
+/// `KL(q || p)` for two univariate Gaussians, in nats.
+///
+/// Both arguments are proper posteriors from inference, so the degenerate
+/// cases guarded here (zero or infinite variance) indicate that inference has
+/// broken down rather than anything a caller did.
+fn kl_divergence(q: Gaussian, p: Gaussian) -> f64 {
+    let (var_q, var_p) = (q.sigma().powi(2), p.sigma().powi(2));
+
+    if !(var_q.is_finite() && var_p.is_finite()) || var_q <= 0.0 || var_p <= 0.0 {
+        return 0.0;
+    }
+
+    let mean_gap = q.mu() - p.mu();
+    0.5 * ((var_p / var_q).ln() + (var_q + mean_gap * mean_gap) / var_p - 1.0)
+}
+
+/// Expected information gain of a hypothetical matchup, in nats.
+///
+/// Enumerates the outcomes this matchup could have, runs inference for each to
+/// get the belief it would produce, and weights the resulting divergence by
+/// that outcome's probability. A higher value means the result would teach you
+/// more.
+///
+/// # Interpreting the value
+///
+/// Nats. The upper bound is the entropy of the outcome variable: at most
+/// `ln 2 ≈ 0.693` for a two-way result, `ln 3 ≈ 1.099` once draws are
+/// possible, `ln k` for `k` outcomes. A value near the ceiling means the
+/// result is close to a coin flip *and* would move the posteriors a long way;
+/// a value near zero means you already know what will happen, or that the
+/// result would barely change your beliefs if you saw it.
+///
+/// This is not a monotone transform of [`quality`](crate::quality). A lopsided
+/// matchup between two uncertain competitors scores well on quality-times-
+/// variance heuristics and poorly here, because the near-certain outcome
+/// carries almost no information.
+///
+/// # Cost
+///
+/// One full inference pass per possible outcome, so this is far more expensive
+/// than `quality()` — which is one closed-form evaluation. The outcome count
+/// grows quickly with team count (3 outcomes for two teams that can draw, 13
+/// for three, 75 for four), and scoring every candidate pairing among `n`
+/// competitors is `O(n² × outcomes)` inference passes.
+///
+/// For a selector over many candidates, shortlist with the cheap
+/// [`quality`](crate::quality) or
+/// [`predict_win_probabilities`](crate::History::predict_win_probabilities)
+/// first and score only the shortlist here. The expected-variance-reduction
+/// proxy sometimes suggested as a cheaper alternative is *not* cheaper: it
+/// needs the same hypothetical posteriors, so it shares the dominant cost.
+///
+/// # Errors
+///
+/// - `NotEnoughTeams` if fewer than two teams are supplied.
+/// - `EmptyTeam` if any team has no members.
+/// - `TooManyTeams` if the outcome space is too large to enumerate; see
+///   [`MAX_PREDICTED_TEAMS`](crate::MAX_PREDICTED_TEAMS).
+/// - `InvalidProbability` if `options.p_draw` is outside `[0.0, 1.0)`.
+/// - Anything [`Game::ranked`](crate::Game::ranked) returns for a hypothetical
+///   outcome.
+pub fn expected_information_gain<T: Time, D: Drift<T>>(
+    teams: &[&[Rating<T, D>]],
+    options: &GameOptions,
+) -> Result<f64, InferenceError> {
+    if teams.len() < 2 {
+        return Err(InferenceError::NotEnoughTeams { got: teams.len() });
+    }
+    if teams.len() > crate::MAX_PREDICTED_TEAMS {
+        return Err(InferenceError::TooManyTeams {
+            got: teams.len(),
+            max: crate::MAX_PREDICTED_TEAMS,
+        });
+    }
+    if !(0.0..1.0).contains(&options.p_draw) {
+        return Err(InferenceError::InvalidProbability {
+            value: options.p_draw,
+        });
+    }
+    for (idx, team) in teams.iter().enumerate() {
+        if team.is_empty() {
+            return Err(InferenceError::EmptyTeam { team: idx });
+        }
+    }
+
+    // Prediction runs on performances: skill inflated by each member's beta.
+    let performances: Vec<Gaussian> = teams
+        .iter()
+        .map(|team| {
+            team.iter()
+                .fold(crate::N00, |acc, rating| acc + rating.performance())
+        })
+        .collect();
+
+    // Draw margins per pair, derived from the teams' betas exactly as
+    // inference derives them, so the outcomes weighted here are the outcomes
+    // that would actually be fitted.
+    let beta_sq: Vec<f64> = teams
+        .iter()
+        .map(|team| team.iter().map(|r| r.beta().powi(2)).sum())
+        .collect();
+    let p_draw = options.p_draw;
+    let margins = predict::Margins::new(teams.len(), |i, j| {
+        if p_draw == 0.0 {
+            0.0
+        } else {
+            crate::compute_margin(p_draw, (beta_sq[i] + beta_sq[j]).sqrt())
+        }
+    });
+
+    let mut gain = 0.0;
+
+    for (ranks, probability) in predict::outcome_distribution(&performances, &margins) {
+        if probability <= NEGLIGIBLE {
+            continue;
+        }
+
+        let game = crate::Game::ranked(teams, Outcome::ranking(ranks), options)?;
+        let posteriors = game.posteriors();
+
+        // Beliefs factorise across competitors, so the joint divergence is the
+        // sum of the per-competitor ones.
+        let divergence: f64 = teams
+            .iter()
+            .zip(&posteriors)
+            .flat_map(|(team, posterior)| team.iter().zip(posterior))
+            .map(|(rating, &after)| kl_divergence(after, rating.prior()))
+            .sum();
+
+        gain += probability * divergence;
+    }
+
+    Ok(gain)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::{BETA, ConstantDrift, GAMMA};
+
+    type R = Rating<i64, ConstantDrift>;
+
+    fn rating(mu: f64, sigma: f64) -> R {
+        R::new(Gaussian::from_ms(mu, sigma), BETA, ConstantDrift(GAMMA))
+    }
+
+    fn options(p_draw: f64) -> GameOptions {
+        GameOptions {
+            p_draw,
+            ..GameOptions::default()
+        }
+    }
+
+    fn eig(teams: &[&[R]], p_draw: f64) -> f64 {
+        expected_information_gain(teams, &options(p_draw)).unwrap()
+    }
+
+    /// The analytic ceiling. Information gain is the mutual information between
+    /// the outcome and the skills, so it cannot exceed the entropy of the
+    /// outcome variable — whatever the ratings. This is the check a subtly
+    /// wrong implementation fails while still returning plausible numbers: an
+    /// early prototype of this returned 4.77 nats from a sign error and passed
+    /// every monotonicity test.
+    #[test]
+    fn never_exceeds_the_entropy_of_the_outcome() {
+        let ceiling_two = std::f64::consts::LN_2;
+
+        for (a, b) in [
+            (rating(0.0, 6.0), rating(0.0, 6.0)),
+            (rating(0.0, 0.5), rating(0.0, 0.5)),
+            (rating(12.0, 6.0), rating(-12.0, 6.0)),
+            (rating(40.0, 1.0), rating(-40.0, 1.0)),
+            (rating(3.0, 6.0), rating(-2.0, 0.1)),
+            (rating(0.0, 25.0), rating(0.0, 25.0)),
+        ] {
+            let g = eig(&[&[a], &[b]], 0.0);
+            assert!(
+                g >= 0.0 && g <= ceiling_two,
+                "EIG {g} outside [0, ln 2] for mu=({}, {}) sigma=({}, {})",
+                a.prior().mu(),
+                b.prior().mu(),
+                a.prior().sigma(),
+                b.prior().sigma()
+            );
+        }
+    }
+
+    /// With draws enabled there are three outcomes, so the ceiling rises to
+    /// `ln 3` — and the two-outcome bound no longer applies.
+    #[test]
+    fn the_ceiling_follows_the_outcome_count() {
+        let ceiling_three = 3.0f64.ln();
+        for sigma in [0.5, 3.0, 6.0, 25.0] {
+            let g = eig(&[&[rating(0.0, sigma)], &[rating(0.0, sigma)]], 0.25);
+            assert!(
+                g >= 0.0 && g <= ceiling_three,
+                "EIG {g} outside [0, ln 3] at sigma {sigma}"
+            );
+        }
+    }
+
+    /// An even matchup between uncertain competitors is the informative one.
+    /// A hopelessly lopsided matchup teaches you almost nothing, because you
+    /// already know how it ends.
+    #[test]
+    fn an_even_matchup_beats_a_lopsided_one() {
+        let even = eig(&[&[rating(0.0, 6.0)], &[rating(0.0, 6.0)]], 0.0);
+        let lopsided = eig(&[&[rating(12.0, 6.0)], &[rating(-12.0, 6.0)]], 0.0);
+        assert!(
+            even > lopsided,
+            "even {even} should beat lopsided {lopsided}"
+        );
+    }
+
+    /// Certainty is the thing information gain is measuring the absence of:
+    /// the less you know, the more there is to learn.
+    #[test]
+    fn gain_falls_as_certainty_rises() {
+        let mut previous = f64::INFINITY;
+        for sigma in [12.0, 6.0, 3.0, 1.0, 0.5, 0.1] {
+            let g = eig(&[&[rating(0.0, sigma)], &[rating(0.0, sigma)]], 0.0);
+            assert!(
+                g < previous,
+                "sigma {sigma}: {g} did not fall below {previous}"
+            );
+            previous = g;
+        }
+        assert!(previous >= 0.0);
+    }
+
+    /// The heuristic this replaces is `quality * sigma_a^2 * sigma_b^2`. It is
+    /// not a monotone transform of information gain — it ranks a lopsided
+    /// matchup above a confident even one, and EIG ranks them the other way.
+    /// Pinning the disagreement down is what stops a future "simplification"
+    /// from quietly reverting to the heuristic.
+    #[test]
+    fn disagrees_with_the_quality_times_variance_heuristic() {
+        let heuristic = |a: &R, b: &R| {
+            crate::quality(&[&[a.prior()], &[b.prior()]], BETA)
+                * a.prior().sigma().powi(2)
+                * b.prior().sigma().powi(2)
+        };
+
+        let (confident_a, confident_b) = (rating(0.0, 0.5), rating(0.0, 0.5));
+        let (lopsided_a, lopsided_b) = (rating(12.0, 6.0), rating(-12.0, 6.0));
+
+        assert!(
+            heuristic(&lopsided_a, &lopsided_b) > heuristic(&confident_a, &confident_b),
+            "the heuristic should prefer the lopsided matchup"
+        );
+        assert!(
+            eig(&[&[confident_a], &[confident_b]], 0.0) > eig(&[&[lopsided_a], &[lopsided_b]], 0.0),
+            "information gain should prefer the even matchup"
+        );
+    }
+
+    #[test]
+    fn supports_more_than_two_teams() {
+        let teams: Vec<Vec<R>> = vec![
+            vec![rating(0.0, 6.0)],
+            vec![rating(0.0, 6.0)],
+            vec![rating(0.0, 6.0)],
+        ];
+        let refs: Vec<&[R]> = teams.iter().map(Vec::as_slice).collect();
+        let g = expected_information_gain(&refs, &options(0.0)).unwrap();
+        // Six distinguishable orderings with no draws.
+        assert!(
+            g > 0.0 && g <= 6.0f64.ln(),
+            "three-team EIG {g} out of range"
+        );
+    }
+
+    #[test]
+    fn multi_member_teams_are_supported() {
+        let a = [rating(0.0, 6.0), rating(1.0, 4.0)];
+        let b = [rating(0.0, 6.0)];
+        let g = expected_information_gain(&[&a, &b], &options(0.0)).unwrap();
+        assert!(g > 0.0 && g <= std::f64::consts::LN_2, "{g}");
+    }
+
+    #[test]
+    fn degenerate_shapes_are_errors() {
+        let a = [rating(0.0, 6.0)];
+        assert!(matches!(
+            expected_information_gain(&[&a], &options(0.0)),
+            Err(InferenceError::NotEnoughTeams { got: 1 })
+        ));
+        let empty: [R; 0] = [];
+        assert!(matches!(
+            expected_information_gain(&[&a, &empty], &options(0.0)),
+            Err(InferenceError::EmptyTeam { team: 1 })
+        ));
+        assert!(matches!(
+            expected_information_gain(&[&a, &a], &options(1.5)),
+            Err(InferenceError::InvalidProbability { .. })
+        ));
+    }
+
+    #[test]
+    fn kl_divergence_is_zero_for_identical_beliefs() {
+        let g = Gaussian::from_ms(3.0, 2.0);
+        assert!(kl_divergence(g, g).abs() < 1e-15);
+    }
+
+    #[test]
+    fn kl_divergence_is_non_negative_and_grows_with_separation() {
+        let prior = Gaussian::from_ms(0.0, 3.0);
+        let mut previous = 0.0;
+        for mu in [0.0, 0.5, 1.0, 2.0, 4.0] {
+            let d = kl_divergence(Gaussian::from_ms(mu, 3.0), prior);
+            assert!(d >= 0.0, "negative divergence at mu {mu}: {d}");
+            assert!(d >= previous, "not increasing at mu {mu}");
+            previous = d;
+        }
+    }
+}

src/convergence.rs

@@ -20,6 +20,37 @@ pub struct ConvergenceOptions {
     pub alpha: f64,
 }
 
+impl ConvergenceOptions {
+    /// Reject values that would make inference silently meaningless.
+    ///
+    /// `HistoryBuilder::convergence` asserts these eagerly, but the fields are
+    /// public and `GameOptions` carries a `ConvergenceOptions` — so a caller
+    /// can hand `Game::ranked` a set the builder never saw. In release the
+    /// engine's `debug_assert!`s are gone, and an `alpha` of zero leaves every
+    /// EP update unapplied: inference returns the priors, with every likelihood
+    /// uninformative and nothing to indicate anything went wrong.
+    ///
+    /// # Errors
+    ///
+    /// `InvalidParameter` if `alpha` is outside `(0.0, 1.0]` or `epsilon` is
+    /// negative. NaN fails both comparisons and is rejected.
+    pub(crate) fn validate(&self) -> Result<(), crate::InferenceError> {
+        if !(self.alpha > 0.0 && self.alpha <= 1.0) {
+            return Err(crate::InferenceError::InvalidParameter {
+                name: "alpha",
+                value: self.alpha,
+            });
+        }
+        if self.epsilon.is_nan() || self.epsilon < 0.0 {
+            return Err(crate::InferenceError::InvalidParameter {
+                name: "epsilon",
+                value: self.epsilon,
+            });
+        }
+        Ok(())
+    }
+}
+
 impl Default for ConvergenceOptions {
     fn default() -> Self {
         Self {

src/error.rs

@@ -25,11 +25,6 @@ pub enum InferenceError {
     /// result has no representable likelihood. Configure a positive `p_draw`
     /// (via `HistoryBuilder::p_draw` or `GameOptions::p_draw`) to admit ties.
     TieWithoutDrawProbability { teams: (usize, usize) },
-    /// Convergence exceeded `max_iter` without falling below `epsilon`.
-    ConvergenceFailed {
-        last_step: (f64, f64),
-        iterations: usize,
-    },
     /// Inference produced a non-finite value (NaN or infinity).
     ///
     /// Indicates numerical breakdown; the resulting skills are meaningless
@@ -38,8 +33,19 @@ pub enum InferenceError {
         context: &'static str,
         step: (f64, f64),
     },
-    /// Negative precision: a Gaussian with `pi < 0` slipped into an API call.
+    /// One batch declared two different values for the same competitor's
-    NegativePrecision { pi: f64 },
+    /// configuration.
+    ///
+    /// `prior` and `drift_scale` configure a competitor, not an event, so a
+    /// batch that sets one of them twice with different values has no
+    /// well-defined meaning: events within a batch are not ordered, so
+    /// "last one wins" would make the result depend on iteration order.
+    /// Declaring the same value repeatedly is fine and is the expected shape
+    /// when a competitor's configuration is a property of the domain.
+    ConflictingCompetitorConfig {
+        competitor: usize,
+        field: &'static str,
+    },
     /// A prediction referenced a key the history has no skill for.
     ///
     /// Reported rather than skipped: dropping unknown keys turns a team of
@@ -96,18 +102,12 @@ impl fmt::Display for InferenceError {
             Self::InvalidParameter { name, value } => {
                 write!(f, "{name} is invalid: {value}")
             }
-            Self::ConvergenceFailed {
+            Self::ConflictingCompetitorConfig { competitor, field } => {
-                last_step,
-                iterations,
-            } => {
                 write!(
                     f,
-                    "convergence failed after {iterations} iterations; last step = {last_step:?}"
+                    "competitor {competitor}: this batch sets {field} to two different values"
                 )
             }
-            Self::NegativePrecision { pi } => {
-                write!(f, "precision must be non-negative; got {pi}")
-            }
             Self::UnknownKey { team, member } => {
                 write!(
                     f,

src/event.rs

@@ -50,9 +50,17 @@ impl<K> Default for Team<K> {
 /// `weight` applies per event and defaults to 1.0.
 ///
 /// `prior` and `drift_scale` are **competitor configuration**, not per-event
-/// values: both are captured when the competitor is first created and ignored
+/// values. Setting either applies to the competitor for the whole history, not
-/// on every later appearance. Setting either on a key the history already knows
+/// just to this event, and applies whenever it is supplied — including on a key
-/// has no effect.
+/// the history already knows. Because configuration lives on the competitor and
+/// `converge` refits from competitor state, configuring one late still refits
+/// the whole history rather than taking effect only from that event onward.
+///
+/// Repeating the same value is inert, which is the expected shape when the
+/// configuration is a property of the domain. Supplying two *different* values
+/// for one competitor within a single batch is
+/// `InferenceError::ConflictingCompetitorConfig`: events in a batch have no
+/// order, so there would be no well-defined winner.
 #[derive(Clone, Debug)]
 pub struct Member<K> {
     pub key: K,

src/factor/trunc.rs

@@ -2,6 +2,7 @@ use crate::{
     N_INF, approx, cdf,
     factor::{Factor, VarId, VarStore},
     gaussian::Gaussian,
+    sf,
 };
 
 /// EP truncation factor on a diff variable.
@@ -74,16 +75,29 @@ impl Factor for TruncFactor {
 
 /// P(diff > margin) for non-tie, P(|diff| < margin) for tie.
 ///
-/// Clamped to a positive floor: for a near-certain outcome the tail rounds to
+/// Both branches pick whichever tail keeps their terms *small*, because the
-/// exactly 0.0, and the `erfc` approximation used by `cdf` carries ~1e-7 error
+/// alternative is subtracting two numbers that both approach 1. That
-/// so it can even return slightly more than 1.0, making the difference
+/// subtraction is not a rounding detail: it loses every digit of an unlikely
-/// negative. Either would send `log_evidence` to `-inf` or NaN and poison the
+/// outcome's evidence, and an unlikely outcome is precisely the one worth
-/// sum across the whole history.
+/// scoring. `1 - cdf` returned exactly zero past ~8.3 sigma, where the true
+/// probability is 1e-19; clamped, that reached `log_evidence` as -708 instead
+/// of -43.
+///
+/// The clamp remains as a guard rather than a workaround: `erfc` carries ~1e-7
+/// relative error, so a probability of exactly 1 can still come back a hair
+/// above it, and `ln` of a negative would poison the sum for the whole history.
 fn cavity_evidence(diff: Gaussian, margin: f64, tie: bool) -> f64 {
+    let (mu, sigma) = (diff.mu(), diff.sigma());
+
     let raw = if tie {
-        cdf(margin, diff.mu(), diff.sigma()) - cdf(-margin, diff.mu(), diff.sigma())
+        if mu < -margin {
+            // Both CDFs sit against 1 here; both survival terms are small.
+            sf(-margin, mu, sigma) - sf(margin, mu, sigma)
         } else {
-        1.0 - cdf(margin, diff.mu(), diff.sigma())
+            cdf(margin, mu, sigma) - cdf(-margin, mu, sigma)
+        }
+    } else {
+        sf(margin, mu, sigma)
     };
 
     raw.clamp(f64::MIN_POSITIVE, 1.0)
@@ -132,6 +146,52 @@ mod tests {
         assert_eq!(f.evidence_cached.unwrap(), first);
     }
 
+    /// The defect this guards: `1 - cdf` collapsed to zero for a surprising
+    /// result, the clamp turned that into `f64::MIN_POSITIVE`, and
+    /// `log_evidence` reported ln of *that* — about -708 whatever the truth
+    /// was. An upset is the observation a model-comparison score exists to
+    /// notice, so it was wrong exactly where it mattered.
+    #[test]
+    fn evidence_of_an_upset_is_not_flattened_to_the_clamp_floor() {
+        // diff ~ N(-9, 1) with margin 0: the favoured side lost by nine sigma.
+        let evidence = cavity_evidence(Gaussian::from_ms(-9.0, 1.0), 0.0, false);
+
+        assert!(
+            evidence > f64::MIN_POSITIVE,
+            "evidence collapsed onto the clamp floor: {evidence}"
+        );
+        // P(X > 0) for X ~ N(-9, 1) is the standard normal tail at 9 sigma.
+        assert!(
+            (evidence - 1.128_588e-19).abs() / 1.128_588e-19 < 1e-6,
+            "expected ~1.13e-19, got {evidence}"
+        );
+        assert!(
+            (evidence.ln() + 43.628).abs() < 1e-2,
+            "log evidence {} should be about -43.6, not -708",
+            evidence.ln()
+        );
+    }
+
+    /// Evidence must stay finite and positive however extreme the mismatch,
+    /// since `log_evidence` sums across the whole history and one `-inf` or
+    /// `NaN` poisons all of it.
+    #[test]
+    fn evidence_stays_positive_and_finite_at_any_separation() {
+        for mu in [-300.0f64, -50.0, -9.0, 0.0, 9.0, 50.0, 300.0] {
+            for tie in [false, true] {
+                let e = cavity_evidence(Gaussian::from_ms(mu, 1.0), 1.0, tie);
+                assert!(
+                    e.is_finite() && e > 0.0 && e <= 1.0,
+                    "mu={mu} tie={tie}: evidence {e} is not a probability"
+                );
+                assert!(
+                    e.ln().is_finite(),
+                    "mu={mu} tie={tie}: ln evidence is not finite"
+                );
+            }
+        }
+    }
+
     #[test]
     fn tie_evidence_uses_two_sided() {
         let mut vars = VarStore::new();

src/game.rs

@@ -433,6 +433,9 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
 impl<T: Time, D: Drift<T>> Game<'_, T, D> {
     /// # Errors
     ///
+    /// - `InvalidParameter` if `options.convergence` is out of range — an
+    ///   `alpha` of zero would leave every EP update unapplied and silently
+    ///   return the priors.
     /// - `InvalidProbability` if `options.p_draw` is outside `[0.0, 1.0)`.
     /// - `MismatchedShape` if the outcome's rank count differs from `teams.len()`.
     /// - `WrongOutcomeKind` if `outcome` is not `Outcome::Ranked`.
@@ -444,6 +447,7 @@ impl<T: Time, D: Drift<T>> Game<'_, T, D> {
         outcome: crate::Outcome,
         options: &GameOptions,
     ) -> Result<OwnedGame<T, D>, crate::InferenceError> {
+        options.convergence.validate()?;
         if !(0.0..1.0).contains(&options.p_draw) {
             return Err(crate::InferenceError::InvalidProbability {
                 value: options.p_draw,
@@ -491,8 +495,8 @@ impl<T: Time, D: Drift<T>> Game<'_, T, D> {
 
     /// # Errors
     ///
-    /// - `InvalidParameter` if `options.score_sigma` is not strictly positive,
+    /// - `InvalidParameter` if `options.score_sigma` is not strictly positive
-    ///   or is NaN.
+    ///   or is NaN, or if `options.convergence` is out of range.
     /// - `MismatchedShape` if the outcome's score count differs from `teams.len()`.
     /// - `WrongOutcomeKind` if `outcome` is not `Outcome::Scored`.
     pub fn scored(
@@ -500,6 +504,7 @@ impl<T: Time, D: Drift<T>> Game<'_, T, D> {
         outcome: crate::Outcome,
         options: &GameOptions,
     ) -> Result<OwnedGame<T, D>, crate::InferenceError> {
+        options.convergence.validate()?;
         if options.score_sigma <= 0.0 || options.score_sigma.is_nan() {
             return Err(crate::InferenceError::InvalidParameter {
                 name: "score_sigma",

src/history.rs

@@ -172,6 +172,23 @@ impl Default for HistoryBuilder<i64, ConstantDrift, NullObserver, &'static str>
     }
 }
 
+/// Configuration a caller attached to a competitor via `Member`.
+///
+/// Carries *what was explicitly set* rather than a merged `Rating`, so a member
+/// that sets only `drift_scale` does not also assert the default prior — which
+/// would spuriously conflict with a prior seeded on an earlier event.
+#[derive(Clone, Copy, Default)]
+pub(crate) struct CompetitorConfig {
+    prior: Option<Gaussian>,
+    drift_scale: Option<f64>,
+}
+
+impl CompetitorConfig {
+    fn is_empty(self) -> bool {
+        self.prior.is_none() && self.drift_scale.is_none()
+    }
+}
+
 pub struct History<
     T: Time = i64,
     D: Drift<T> = ConstantDrift,
@@ -538,10 +555,27 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
             .sum()
     }
 
-    /// Each team's performance Gaussian, and its member count.
+    /// The configured observer.
     ///
-    /// Performance is skill inflated by `beta`: the question a prediction
+    /// `History` takes its observer by value, so this is how a caller inspects
-    /// answers is "how will they do today", not "how good are they".
+    /// one it did not keep a handle to. For an observer that accumulates
+    /// state, prefer passing an `Arc` and keeping a clone — see the
+    /// [`Observer`] docs.
+    #[must_use]
+    pub fn observer(&self) -> &O {
+        &self.observer
+    }
+
+    /// Consume the history and return its observer.
+    ///
+    /// Useful for reclaiming a non-shared observer's accumulated state after
+    /// `converge` without needing interior mutability.
+    #[must_use]
+    pub fn into_observer(self) -> O {
+        self.observer
+    }
+
+    /// Every team's member skills, validated.
     ///
     /// # Errors
     ///
@@ -549,39 +583,60 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
     /// reported rather than dropped — silently skipping them would turn a team
     /// of strangers into a confident-looking prediction about nobody, which is
     /// the failure this replaced.
-    fn performances(&self, teams: &[&[&K]]) -> Result<(Vec<Gaussian>, Vec<usize>), InferenceError> {
+    fn member_skills(&self, teams: &[&[&K]]) -> Result<Vec<Vec<Gaussian>>, InferenceError> {
         if teams.len() < 2 {
             return Err(InferenceError::NotEnoughTeams { got: teams.len() });
         }
 
-        let mut performances = Vec::with_capacity(teams.len());
+        let mut gathered = Vec::with_capacity(teams.len());
-        let mut sizes = Vec::with_capacity(teams.len());
 
         for (team_idx, team) in teams.iter().enumerate() {
             if team.is_empty() {
                 return Err(InferenceError::EmptyTeam { team: team_idx });
             }
 
-            let mut total = crate::N00;
+            let mut members = Vec::with_capacity(team.len());
             for (member_idx, key) in team.iter().enumerate() {
                 let unknown = InferenceError::UnknownKey {
                     team: team_idx,
                     member: member_idx,
                 };
                 let index = self.keys.get(*key).ok_or(unknown.clone())?;
-                let skill = self
+                members.push(
-                    .time_slices
+                    self.time_slices
                         .iter()
                         .rev()
                         .find_map(|ts| ts.skills.get(index).map(|s| s.posterior()))
-                    .ok_or(unknown)?;
+                        .ok_or(unknown)?,
-                total = total + skill.forget(self.beta.powi(2));
+                );
             }
 
-            performances.push(total);
+            gathered.push(members);
-            sizes.push(team.len());
         }
 
+        Ok(gathered)
+    }
+
+    /// Each team's performance Gaussian, and its member count.
+    ///
+    /// Performance is skill inflated by `beta`: the question a prediction
+    /// answers is "how will they do today", not "how good are they".
+    ///
+    /// # Errors
+    ///
+    /// As [`History::member_skills`].
+    fn performances(&self, teams: &[&[&K]]) -> Result<(Vec<Gaussian>, Vec<usize>), InferenceError> {
+        let skills = self.member_skills(teams)?;
+
+        let performances = skills
+            .iter()
+            .map(|team| {
+                team.iter()
+                    .fold(crate::N00, |acc, s| acc + s.forget(self.beta.powi(2)))
+            })
+            .collect();
+        let sizes = skills.iter().map(Vec::len).collect();
+
         Ok((performances, sizes))
     }
 
@@ -618,38 +673,55 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
     ///
     /// `NotEnoughTeams`, `EmptyTeam`, or `UnknownKey`.
     pub fn predict_quality(&self, teams: &[&[&K]]) -> Result<f64, InferenceError> {
-        let mut groups: Vec<Vec<Gaussian>> = Vec::with_capacity(teams.len());
+        let groups = self.member_skills(teams)?;
-
-        for (team_idx, team) in teams.iter().enumerate() {
-            if team.is_empty() {
-                return Err(InferenceError::EmptyTeam { team: team_idx });
-            }
-            let mut members = Vec::with_capacity(team.len());
-            for (member_idx, key) in team.iter().enumerate() {
-                let unknown = InferenceError::UnknownKey {
-                    team: team_idx,
-                    member: member_idx,
-                };
-                let index = self.keys.get(*key).ok_or(unknown.clone())?;
-                members.push(
-                    self.time_slices
-                        .iter()
-                        .rev()
-                        .find_map(|ts| ts.skills.get(index).map(|s| s.posterior()))
-                        .ok_or(unknown)?,
-                );
-            }
-            groups.push(members);
-        }
-
-        if groups.len() < 2 {
-            return Err(InferenceError::NotEnoughTeams { got: groups.len() });
-        }
-
         let group_refs: Vec<&[Gaussian]> = groups.iter().map(Vec::as_slice).collect();
         Ok(crate::quality(&group_refs, self.beta))
     }
 
+    /// Expected information gain of running this matchup, in nats.
+    ///
+    /// Answers "which comparison should I run next" rather than "who will
+    /// win": the outcome-weighted divergence between current beliefs and the
+    /// beliefs each possible result would produce. Higher means the result
+    /// would teach you more.
+    ///
+    /// Uses each competitor's current skill as the prior, and the history's
+    /// own `beta`, `drift` and `p_draw`, so the outcomes weighted here are the
+    /// ones that would actually be fitted if the matchup were played and
+    /// recorded.
+    ///
+    /// Distinct from [`History::predict_quality`], which measures *fairness*.
+    /// The two coincide for two evenly matched competitors and diverge
+    /// elsewhere. See [`expected_information_gain`](crate::expected_information_gain)
+    /// for the scale, the analytic `ln k` ceiling, and the cost.
+    ///
+    /// # Errors
+    ///
+    /// As [`History::member_skills`], plus `TooManyTeams` and anything
+    /// inference returns for a hypothetical outcome.
+    pub fn expected_information_gain(&self, teams: &[&[&K]]) -> Result<f64, InferenceError> {
+        let skills = self.member_skills(teams)?;
+
+        let ratings: Vec<Vec<Rating<T, D>>> = skills
+            .iter()
+            .map(|team| {
+                team.iter()
+                    .map(|&skill| Rating::new(skill, self.beta, self.drift))
+                    .collect()
+            })
+            .collect();
+        let team_refs: Vec<&[Rating<T, D>]> = ratings.iter().map(Vec::as_slice).collect();
+
+        crate::expected_information_gain(
+            &team_refs,
+            &crate::GameOptions {
+                p_draw: self.p_draw,
+                score_sigma: self.score_sigma,
+                convergence: self.convergence,
+            },
+        )
+    }
+
     /// `P(team i finishes strictly first)`, for every team.
     ///
     /// Supports any number of teams. Because performances are independent
@@ -821,7 +893,7 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
         times: Vec<T>,
         mut weights: Option<Vec<Vec<Vec<f64>>>>,
         kinds: Vec<EventKind>,
-        mut priors: HashMap<Index, Rating<T, D>>,
+        priors: HashMap<Index, CompetitorConfig>,
     ) -> Result<(), InferenceError> {
         if results
             .as_ref()
@@ -888,17 +960,60 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
 
             this_agent.push(*agent);
 
-            if !self.agents.contains(*agent) {
+            let config = priors.get(agent).copied().unwrap_or_default();
-                self.agents.insert(
+
-                    *agent,
+            if self.agents.contains(*agent) {
-                    Competitor {
+                // Seeding a competitor the history already knows. This used to
-                        rating: priors.remove(agent).unwrap_or_else(|| {
+                // be dropped on the floor: `remove` was only reached on the
-                            Rating::new(
+                // create path, so a prior applied on a competitor's very first
+                // event and was silently ignored ever after.
+                if config.is_empty() {
+                    continue;
+                }
+
+                let rating = &mut self.agents.get_mut(*agent).unwrap().rating;
+                if let Some(prior) = config.prior {
+                    rating.prior = prior;
+                }
+                if let Some(scale) = config.drift_scale {
+                    rating.drift_scale = scale;
+                }
+                let seeded = rating.prior;
+
+                if config.prior.is_some() {
+                    // The prior is not re-derived every pass the way drift is.
+                    // A competitor's earliest slice has its forward message set
+                    // to the prior once, at ingestion, and `iteration` refreshes
+                    // only slices after the first — so without this, a late
+                    // prior would reach the drift terms and nothing else, which
+                    // is a subtler version of the silent drop this replaced.
+                    //
+                    // `clean` has just nulled every message, so the earliest
+                    // slice's forward is exactly the prior.
+                    for slice in &mut self.time_slices {
+                        if let Some(skill) = slice.skills.get_mut(*agent) {
+                            skill.forward = seeded;
+                            break;
+                        }
+                    }
+                }
+            } else {
+                let mut rating = Rating::new(
                     Gaussian::from_ms(self.mu, self.sigma),
                     self.beta,
                     self.drift,
-                            )
+                );
-                        }),
+                if let Some(prior) = config.prior {
+                    rating.prior = prior;
+                }
+                if let Some(scale) = config.drift_scale {
+                    rating.drift_scale = scale;
+                }
+
+                self.agents.insert(
+                    *agent,
+                    Competitor {
+                        rating,
                         message: None,
                         last_time: None,
                     },
@@ -1115,7 +1230,7 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
         let mut times: Vec<T> = Vec::with_capacity(events.len());
         let mut weights: Vec<Vec<Vec<f64>>> = Vec::with_capacity(events.len());
         let mut kinds: Vec<EventKind> = Vec::with_capacity(events.len());
-        let mut priors: HashMap<Index, Rating<T, D>> = HashMap::new();
+        let mut priors: HashMap<Index, CompetitorConfig> = HashMap::new();
 
         for ev in events {
             if ev.outcome.team_count() != ev.teams.len() {
@@ -1149,23 +1264,37 @@ impl<T: Time, D: Drift<T>, O: Observer<T>, K: Eq + Hash + Clone> History<T, D, O
                         }
                     }
 
-                    // `prior` and `drift_scale` are competitor configuration,
+                    // `prior` and `drift_scale` configure the competitor, not
-                    // captured here and consumed at competitor creation. Both
+                    // the event. Both land in the same entry so a member may
-                    // land in the same entry so a member may set either alone.
+                    // set either alone.
+                    //
+                    // Events within a batch are not ordered, so a batch that
+                    // sets one field twice with different values has no
+                    // well-defined result — "last one wins" would depend on
+                    // iteration order, which `tests/ingestion_equivalence.rs`
+                    // exists to rule out. Repeating the *same* value is fine,
+                    // and is the expected shape when the configuration is a
+                    // property of the domain rather than of one event.
                     if member.prior.is_some() || member.drift_scale.is_some() {
-                        let rating = priors.entry(idx).or_insert_with(|| {
+                        let entry = priors.entry(idx).or_default();
-                            Rating::new(
-                                Gaussian::from_ms(self.mu, self.sigma),
-                                self.beta,
-                                self.drift,
-                            )
-                        });
 
                         if let Some(prior) = member.prior {
-                            rating.prior = prior;
+                            if entry.prior.is_some_and(|held| held != prior) {
+                                return Err(InferenceError::ConflictingCompetitorConfig {
+                                    competitor: idx.get(),
+                                    field: "prior",
+                                });
+                            }
+                            entry.prior = Some(prior);
                         }
                         if let Some(scale) = member.drift_scale {
-                            rating.drift_scale = scale;
+                            if entry.drift_scale.is_some_and(|held| held != scale) {
+                                return Err(InferenceError::ConflictingCompetitorConfig {
+                                    competitor: idx.get(),
+                                    field: "drift_scale",
+                                });
+                            }
+                            entry.drift_scale = Some(scale);
                         }
                     }
                 }

src/lib.rs

@@ -110,6 +110,7 @@ pub(crate) mod arena;
 mod time;
 mod time_slice;
 pub use time_slice::{EventKind, TimeSlice};
+mod acquisition;
 mod color_group;
 mod competitor;
 mod convergence;
@@ -132,6 +133,7 @@ mod rating;
 pub(crate) mod schedule;
 pub mod storage;
 
+pub use acquisition::expected_information_gain;
 pub use competitor::Competitor;
 pub use convergence::{ConvergenceOptions, ConvergenceReport};
 pub use drift::{ConstantDrift, Drift};
@@ -166,6 +168,21 @@ pub const ITERATIONS: usize = 30;
 pub const MAX_PREDICTED_TEAMS: usize = predict::MAX_TEAMS_FOR_DISTRIBUTION;
 
 const SQRT_TAU: f64 = 2.5066282746310002;
+/// `1 / sqrt(pi)`, the leading factor of the `erfcx` continued fraction.
+const FRAC_1_SQRT_PI: f64 = 0.564_189_583_547_756_3;
+/// `sqrt(2 / pi)`, the numerator of the inverse Mills ratio in scaled form.
+const SQRT_2_OVER_PI: f64 = 0.797_884_560_802_865_4;
+/// How many window widths into the tail before a tie window is treated as a
+/// half-line. Beyond this the truncated mass is concentrated within `1/alpha`
+/// of the near edge, so the far edge contributes nothing measurable.
+const HALF_LINE_WINDOW: f64 = 10.0;
+/// Where `v - alpha` switches from subtraction to its asymptotic series.
+///
+/// The subtraction loses roughly `eps * alpha^2` of relative precision, and the
+/// four-term series is good to ~1e-10 by here, so the two are at their closest
+/// agreement around this point. Below it the subtraction is exact; above it the
+/// series is.
+const ASYMPTOTIC_MILLS_ALPHA: f64 = 100.0;
 
 pub const N01: Gaussian = Gaussian::from_ms(0.0, 1.0);
 pub const N00: Gaussian = Gaussian::from_ms(0.0, 0.0);
@@ -257,6 +274,50 @@ pub(crate) fn cdf(x: f64, mu: f64, sigma: f64) -> f64 {
     0.5 * erfc(z)
 }
 
+/// `P(X > x)` for `X ~ N(mu, sigma^2)`.
+///
+/// The survival function, computed directly rather than as `1 - cdf(..)`.
+///
+/// The two are algebraically identical and numerically are not. `cdf` returns
+/// a value approaching 1 for an upper tail, so subtracting it from 1 cancels
+/// away every significant digit the tail had: measured against this function,
+/// `1 - cdf` carries 7% error by four sigma past the mean and returns exactly
+/// zero beyond about 8.3 sigma — where the true value is still 1e-19 and
+/// perfectly representable. `erfc` itself holds ~1e-7 *relative* accuracy down
+/// to 1e-296, so the precision is there to keep; only the subtraction threw it
+/// away.
+///
+/// This matters most where evidence is smallest, which is exactly where an
+/// upset makes it interesting: `ln` of a clamped zero is -708 regardless of
+/// whether the truth was -43 or -600.
+pub(crate) fn sf(x: f64, mu: f64, sigma: f64) -> f64 {
+    0.5 * erfc((x - mu) / (sigma * SQRT_2))
+}
+
+/// `e^(x^2) * erfc(x)`, the scaled complementary error function, for `x >= 0`.
+///
+/// Exists so the exponential factor common to a Gaussian density and its tail
+/// integral can be cancelled *analytically* instead of being computed twice
+/// and divided. Both underflow to zero past about 26 sigma, and their ratio is
+/// then `0/0` — finite in the limit, `NaN` in floating point.
+fn erfcx(x: f64) -> f64 {
+    if x < 2.0 {
+        // Below the crossover neither factor is extreme: erfc is O(1) and
+        // exp(x^2) is at most e^4, so the direct product is exact enough and
+        // cheaper than the continued fraction.
+        (x * x).exp() * erfc(x)
+    } else {
+        // erfcx(x) = 1/sqrt(pi) * 1/(x + (1/2)/(x + 1/(x + (3/2)/(x + ...)))),
+        // evaluated by backward recurrence. Converges quickly for x >= 2 and,
+        // unlike the product form, never touches an exponential.
+        let mut f = 0.0;
+        for n in (1..=60u32).rev() {
+            f = (f64::from(n) * 0.5) / (x + f);
+        }
+        FRAC_1_SQRT_PI / (x + f)
+    }
+}
+
 fn pdf(x: f64, mu: f64, sigma: f64) -> f64 {
     let normalizer = (SQRT_TAU * sigma).powi(-1);
     let functional = (-((x - mu).powi(2)) / (2.0 * sigma.powi(2))).exp();
@@ -264,25 +325,100 @@ fn pdf(x: f64, mu: f64, sigma: f64) -> f64 {
     normalizer * functional
 }
 
+/// Truncated-Gaussian correction terms `(v, w)`.
+///
+/// `v` shifts the mean and `w` shrinks the variance. Both are ratios whose
+/// numerator and denominator underflow together in the tails, so both are
+/// computed in scaled form there: the shared `exp(-alpha^2 / 2)` is cancelled
+/// analytically rather than evaluated and divided out. Without that, a
+/// truncation point beyond about 39 sigma produced `0 / 0` and put `NaN`
+/// straight into the posterior.
+/// Truncation terms for a boundary `alpha` standard deviations into the upper
+/// tail, from the asymptotic expansion of the inverse Mills ratio.
+///
+/// `v` tends to `alpha` out here, so the gap between them cannot be obtained by
+/// subtracting one from the other — the series computes the gap directly, and
+/// `w = v * gap` then never forms the difference of two large near-equal
+/// numbers. A far-tail *window* behaves like a half-line once it is more than a
+/// few multiples of its own width from the mean, so the tie branch shares this.
+fn half_line_truncation(alpha: f64) -> (f64, f64) {
+    let inv = alpha.recip();
+    let inv_sq = inv * inv;
+    let gap = inv * (1.0 - inv_sq * (2.0 - inv_sq * (10.0 - 74.0 * inv_sq)));
+    let v = alpha + gap;
+
+    (v, v * gap)
+}
+
 fn v_w(mu: f64, sigma: f64, margin: f64, tie: bool) -> (f64, f64) {
     if !tie {
         let alpha = (margin - mu) / sigma;
 
-        let v = pdf(-alpha, 0.0, 1.0) / cdf(-alpha, 0.0, 1.0);
+        // v is the inverse Mills ratio, phi(alpha) / Phi(-alpha), and w needs
-        let w = v * (v + (-alpha));
+        // the gap `v - alpha` as well as v itself. Far into the tail v tends to
+        // alpha, so that gap is a subtraction of two nearly equal numbers and
+        // loses every digit it has: at alpha = 1e6 it drove w above 1 and made
+        // `sqrt(1 - w)` NaN. Past the crossover the gap comes from its
+        // asymptotic series instead, which has no subtraction in it.
+        if alpha >= ASYMPTOTIC_MILLS_ALPHA {
+            return half_line_truncation(alpha);
+        }
 
-        (v, w)
+        let (v, gap) = if alpha > 0.0 {
+            // Both terms carry exp(-alpha^2 / 2); in scaled form it cancels
+            // and the result stays exact however far into the tail alpha sits.
+            let v = SQRT_2_OVER_PI / erfcx(alpha / SQRT_2);
+            (v, v - alpha)
         } else {
+            // Phi(-alpha) >= 1/2 here, so the direct ratio loses nothing.
+            let v = pdf(-alpha, 0.0, 1.0) / cdf(-alpha, 0.0, 1.0);
+            (v, v - alpha)
+        };
+
+        (v, v * gap)
+    } else {
+        // v is odd in mu and w is even, so fold to mu <= 0. Both truncation
+        // points then sit in the upper tail, where the scaled form applies.
+        let flipped = mu > 0.0;
+        let mu = if flipped { -mu } else { mu };
+
         let alpha = (-margin - mu) / sigma;
         let beta = (margin - mu) / sigma;
 
-        let v = (pdf(alpha, 0.0, 1.0) - pdf(beta, 0.0, 1.0))
+        // `w` comes out of `v * v - u`, and both terms grow as alpha^2 while
-            / (cdf(beta, 0.0, 1.0) - cdf(alpha, 0.0, 1.0));
+        // their difference stays O(1) — at alpha = 1e9 that subtraction had no
-        let u = (alpha * pdf(alpha, 0.0, 1.0) - beta * pdf(beta, 0.0, 1.0))
+        // digits left and returned w = -128, making `sqrt(1 - w)` nonsense.
-            / (cdf(beta, 0.0, 1.0) - cdf(alpha, 0.0, 1.0));
+        // Once the window sits many of its own widths into the tail it is
+        // indistinguishable from a half-line, so the asymptotic covers it with
+        // no subtraction at all.
+        if alpha >= ASYMPTOTIC_MILLS_ALPHA && alpha * (beta - alpha) >= HALF_LINE_WINDOW {
+            let (v, w) = half_line_truncation(alpha);
+            return (if flipped { -v } else { v }, w);
+        }
+
+        let (v, u) = if alpha > 0.0 {
+            // beta > alpha > 0, so this ratio of exponentials is at most 1 and
+            // cannot overflow.
+            let scale = (0.5 * (alpha * alpha - beta * beta)).exp();
+            let denominator = 0.5 * (erfcx(alpha / SQRT_2) - scale * erfcx(beta / SQRT_2));
+
+            (
+                (1.0 - scale) / SQRT_TAU / denominator,
+                (alpha - beta * scale) / SQRT_TAU / denominator,
+            )
+        } else {
+            // The interval straddles the mean, so nothing here is small.
+            let denominator = cdf(beta, 0.0, 1.0) - cdf(alpha, 0.0, 1.0);
+
+            (
+                (pdf(alpha, 0.0, 1.0) - pdf(beta, 0.0, 1.0)) / denominator,
+                (alpha * pdf(alpha, 0.0, 1.0) - beta * pdf(beta, 0.0, 1.0)) / denominator,
+            )
+        };
+
         let w = -(u - v.powi(2));
 
-        (v, w)
+        (if flipped { -v } else { v }, w)
     }
 }
 
@@ -465,6 +601,151 @@ mod tests {
         assert_eq!(sort_time(&[0i64, 1, 2, 0], true), vec![2, 1, 0, 3]);
     }
 
+    /// Upper-tail values of the standard normal, from published tables. The
+    /// point is not the digits — `erfc` only carries ~1e-7 relative — but that
+    /// a number comes back at all: `1 - cdf` returned exactly zero for every
+    /// one of these.
+    #[test]
+    fn survival_function_survives_the_far_tail() {
+        for (z, expected) in [
+            (9.0f64, 1.128_588e-19),
+            (12.0, 1.776_482e-33),
+            (20.0, 2.753_624e-89),
+            (37.0, 5.725_571e-300),
+        ] {
+            let got = sf(z, 0.0, 1.0);
+            assert!(got > 0.0, "sf({z}) collapsed to zero");
+            assert!(
+                (got - expected).abs() / expected < 1e-6,
+                "sf({z}) = {got}, expected ~{expected}"
+            );
+            assert_eq!(
+                1.0 - cdf(z, 0.0, 1.0),
+                0.0,
+                "the naive form should still be zero here"
+            );
+        }
+    }
+
+    /// Where no cancellation happens the two forms must agree exactly enough
+    /// that nothing else in the crate shifts.
+    #[test]
+    fn survival_function_matches_the_naive_form_where_that_form_works() {
+        for z in [-4.0f64, -1.0, 0.0, 0.5, 1.0, 2.0, 3.0, 4.0] {
+            let naive = 1.0 - cdf(z, 0.0, 1.0);
+            let direct = sf(z, 0.0, 1.0);
+            // Bounded by `erfc`'s own ~1e-7 relative error, not by the
+            // subtraction: the two forms evaluate `erfc` at different points
+            // and the approximation is not exactly antisymmetric.
+            assert!(
+                (naive - direct).abs() < 1e-6,
+                "z={z}: naive {naive} vs direct {direct}"
+            );
+        }
+    }
+
+    #[test]
+    fn survival_and_cdf_partition_the_mass() {
+        for z in [-3.0f64, -0.5, 0.0, 1.0, 2.5] {
+            let total = sf(z, 1.0, 2.0) + cdf(z, 1.0, 2.0);
+            // `erfc(z) + erfc(-z) == 2` only to the accuracy of the
+            // approximation, which is ~1e-7 relative.
+            assert!((total - 1.0).abs() < 1e-6, "z={z}: {total}");
+        }
+    }
+
+    /// `erfcx` switches formulation at x = 2; the two sides must meet.
+    #[test]
+    fn erfcx_is_continuous_across_its_crossover() {
+        for x in [1.90f64, 1.99, 1.999, 2.0, 2.001, 2.01, 2.10] {
+            let direct = (x * x).exp() * erfc(x);
+            let scaled = erfcx(x);
+            assert!(
+                (direct - scaled).abs() / scaled < 1e-6,
+                "x={x}: direct {direct} vs erfcx {scaled}"
+            );
+        }
+    }
+
+    /// The whole reason `erfcx` exists: it stays finite and O(1/x) exactly
+    /// where `exp(x^2)` overflows and `erfc(x)` underflows.
+    #[test]
+    fn erfcx_stays_finite_where_its_factors_do_not() {
+        for x in [27.0f64, 50.0, 1.0e3, 1.0e8] {
+            let scaled = erfcx(x);
+            assert!(scaled.is_finite() && scaled > 0.0, "erfcx({x}) = {scaled}");
+            // Asymptotically erfcx(x) -> 1 / (x * sqrt(pi)).
+            let asymptote = 1.0 / (x * std::f64::consts::PI.sqrt());
+            assert!(
+                (scaled - asymptote).abs() / asymptote < 1e-2,
+                "erfcx({x}) = {scaled} strays from its asymptote {asymptote}"
+            );
+            assert!(
+                (x * x).exp().is_infinite(),
+                "x={x} should overflow the direct form"
+            );
+        }
+    }
+
+    /// Truncation must never produce a non-finite posterior. Before the scaled
+    /// formulation these returned NaN from `0 / 0` past about 39 sigma.
+    #[test]
+    fn truncation_stays_finite_arbitrarily_far_into_the_tail() {
+        for alpha in [0.0f64, 8.0, 38.0, 40.0, 100.0, 1.0e3, 1.0e6, 1.0e9, 1.0e15] {
+            for tie in [false, true] {
+                let (v, w) = v_w(-alpha, 1.0, if tie { 1.0 } else { 0.0 }, tie);
+                assert!(v.is_finite(), "alpha={alpha} tie={tie}: v = {v}");
+                assert!(w.is_finite(), "alpha={alpha} tie={tie}: w = {w}");
+                // sigma_trunc = sigma * sqrt(1 - w) must stay real.
+                assert!(
+                    (0.0..=1.0).contains(&w),
+                    "alpha={alpha} tie={tie}: w = {w} leaves sqrt(1 - w) imaginary"
+                );
+
+                let (mu_t, sigma_t) = trunc(-alpha, 1.0, if tie { 1.0 } else { 0.0 }, tie);
+                assert!(
+                    mu_t.is_finite() && sigma_t.is_finite(),
+                    "alpha={alpha} tie={tie}: trunc = ({mu_t}, {sigma_t})"
+                );
+            }
+        }
+    }
+
+    /// The Mills gap switches from subtraction to series at alpha = 100. Both
+    /// are supposed to be right there; if they disagree, the crossover is in
+    /// the wrong place.
+    #[test]
+    fn the_mills_gap_series_meets_the_scaled_form() {
+        for alpha in [50.0f64, 99.0, 100.0, 101.0, 200.0] {
+            let scaled = SQRT_2_OVER_PI / erfcx(alpha / SQRT_2) - alpha;
+            let inv = alpha.recip();
+            let inv_sq = inv * inv;
+            let series = inv * (1.0 - inv_sq * (2.0 - inv_sq * (10.0 - 74.0 * inv_sq)));
+            assert!(
+                (scaled - series).abs() / series < 1e-9,
+                "alpha={alpha}: scaled {scaled} vs series {series}"
+            );
+        }
+    }
+
+    /// Folding the tie branch to `mu <= 0` is only valid if v is odd in mu and
+    /// w is even. Assert the symmetry the implementation relies on.
+    #[test]
+    fn tie_truncation_is_odd_in_v_and_even_in_w() {
+        for mu in [0.5f64, 3.0, 20.0, 40.0, 100.0, 1.0e3] {
+            let (v_pos, w_pos) = v_w(mu, 1.0, 1.0, true);
+            let (v_neg, w_neg) = v_w(-mu, 1.0, 1.0, true);
+            assert!(
+                (v_pos + v_neg).abs() < 1e-9,
+                "mu={mu}: v should be odd, got {v_pos} and {v_neg}"
+            );
+            assert!(
+                (w_pos - w_neg).abs() < 1e-9,
+                "mu={mu}: w should be even, got {w_pos} and {w_neg}"
+            );
+        }
+    }
+
     #[test]
     fn test_quality() {
         let a = Gaussian::from_ms(25.0, 3.0);

src/observer.rs

@@ -26,6 +26,83 @@ pub trait Observer<T: Time>: Send + Sync {
     fn on_converged(&self, _iters: usize, _final_step: (f64, f64), _converged: bool) {}
 }
 
+/// Shared and boxed observers forward to what they point at.
+///
+/// `History` takes its observer by value, so a caller who wants to *read* what
+/// an observer recorded has to keep a handle to it. Without these impls the
+/// natural spelling does not compile:
+///
+/// ```
+/// # use std::sync::{Arc, Mutex};
+/// # use trueskill_tt::{History, Observer};
+/// #[derive(Default)]
+/// struct Recorder {
+///     iterations: Mutex<Vec<usize>>,
+/// }
+///
+/// impl Observer<i64> for Recorder {
+///     fn on_iteration_end(&self, iter: usize, _step: (f64, f64)) {
+///         self.iterations.lock().unwrap().push(iter);
+///     }
+/// }
+///
+/// let recorder = Arc::new(Recorder::default());
+/// let mut h = History::builder().observer(Arc::clone(&recorder)).build();
+/// h.record_winner(&"a", &"b", 1).unwrap();
+/// h.converge().unwrap();
+///
+/// // The caller's handle sees what the history's copy recorded.
+/// assert!(!recorder.iterations.lock().unwrap().is_empty());
+/// ```
+///
+/// The alternative was for every observer to wrap each of its own fields in an
+/// `Arc` and derive `Clone` — one allocation and one lock per field, and a
+/// pattern each implementor had to rediscover.
+///
+/// `?Sized` is deliberate: it makes `Arc<dyn Observer<T>>` and
+/// `Box<dyn Observer<T>>` work, so observers can be chosen at runtime.
+impl<T: Time, O: Observer<T> + ?Sized> Observer<T> for std::sync::Arc<O> {
+    fn on_iteration_end(&self, iter: usize, max_step: (f64, f64)) {
+        (**self).on_iteration_end(iter, max_step);
+    }
+
+    fn on_slice_processed(&self, time: &T, slice_idx: usize, n_events: usize) {
+        (**self).on_slice_processed(time, slice_idx, n_events);
+    }
+
+    fn on_converged(&self, iters: usize, final_step: (f64, f64), converged: bool) {
+        (**self).on_converged(iters, final_step, converged);
+    }
+}
+
+impl<T: Time, O: Observer<T> + ?Sized> Observer<T> for Box<O> {
+    fn on_iteration_end(&self, iter: usize, max_step: (f64, f64)) {
+        (**self).on_iteration_end(iter, max_step);
+    }
+
+    fn on_slice_processed(&self, time: &T, slice_idx: usize, n_events: usize) {
+        (**self).on_slice_processed(time, slice_idx, n_events);
+    }
+
+    fn on_converged(&self, iters: usize, final_step: (f64, f64), converged: bool) {
+        (**self).on_converged(iters, final_step, converged);
+    }
+}
+
+impl<T: Time, O: Observer<T> + ?Sized> Observer<T> for &O {
+    fn on_iteration_end(&self, iter: usize, max_step: (f64, f64)) {
+        (**self).on_iteration_end(iter, max_step);
+    }
+
+    fn on_slice_processed(&self, time: &T, slice_idx: usize, n_events: usize) {
+        (**self).on_slice_processed(time, slice_idx, n_events);
+    }
+
+    fn on_converged(&self, iters: usize, final_step: (f64, f64), converged: bool) {
+        (**self).on_converged(iters, final_step, converged);
+    }
+}
+
 /// ZST no-op observer; the default when none is configured.
 #[derive(Copy, Clone, Debug, Default)]
 pub struct NullObserver;

tests/competitor_config.rs

@@ -0,0 +1,222 @@
+//! `Member::with_prior` / `with_drift_scale` — competitor configuration.
+//!
+//! Both were previously consumed only on the branch that *creates* a
+//! competitor, so configuration supplied for a key the history already knew was
+//! dropped with no error. `with_prior` had no coverage in this directory at
+//! all, which is how that survived.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConvergenceOptions, Event, Gaussian, History, InferenceError, Member, Outcome, Team,
+};
+
+const CONVERGENCE: ConvergenceOptions = ConvergenceOptions {
+    max_iter: 2_000,
+    epsilon: 1e-12,
+    alpha: 1.0,
+};
+
+fn history() -> History {
+    History::builder()
+        .mu(25.0)
+        .sigma(25.0 / 3.0)
+        .beta(25.0 / 6.0)
+        .p_draw(0.0)
+        .convergence(CONVERGENCE)
+        .build()
+}
+
+/// One event, optionally configuring `a`.
+fn bout(
+    a: &'static str,
+    b: &'static str,
+    time: i64,
+    prior: Option<Gaussian>,
+    scale: Option<f64>,
+) -> Event<i64, &'static str> {
+    let mut member = Member::new(a);
+    if let Some(p) = prior {
+        member = member.with_prior(p);
+    }
+    if let Some(s) = scale {
+        member = member.with_drift_scale(s);
+    }
+
+    Event {
+        time,
+        teams: smallvec![
+            Team::with_members([member]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }
+}
+
+fn skill_of(h: &History, key: &str) -> Gaussian {
+    h.current_skill(&key).expect("key in history")
+}
+
+/// Baseline: the mechanism works at all on a competitor's first appearance.
+#[test]
+fn a_prior_applies_to_a_new_competitor() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut with = history();
+    with.add_events(vec![bout("a", "b", 0, Some(seeded), None)])
+        .unwrap();
+    with.converge().unwrap();
+
+    let mut without = history();
+    without
+        .add_events(vec![bout("a", "b", 0, None, None)])
+        .unwrap();
+    without.converge().unwrap();
+
+    assert!(
+        (skill_of(&with, "a").mu() - skill_of(&without, "a").mu()).abs() > 1.0,
+        "a seeded prior should move the fit"
+    );
+}
+
+/// The defect in #10: a prior supplied for a competitor the history already
+/// knows was silently discarded, and the caller got output computed from the
+/// default prior with no indication anything had been dropped.
+#[test]
+fn a_prior_applies_to_a_competitor_the_history_already_knows() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut late = history();
+    late.add_events(vec![bout("a", "b", 0, None, None)])
+        .unwrap();
+    // "a" now exists. Configuring it here used to do nothing whatsoever.
+    late.add_events(vec![bout("a", "b", 1, Some(seeded), None)])
+        .unwrap();
+    late.converge().unwrap();
+
+    let mut never = history();
+    never
+        .add_events(vec![
+            bout("a", "b", 0, None, None),
+            bout("a", "b", 1, None, None),
+        ])
+        .unwrap();
+    never.converge().unwrap();
+
+    assert!(
+        (skill_of(&late, "a").mu() - skill_of(&never, "a").mu()).abs() > 1.0,
+        "a late prior must not be silently dropped: {} vs {}",
+        skill_of(&late, "a").mu(),
+        skill_of(&never, "a").mu()
+    );
+}
+
+/// Configuration is competitor-scoped, not event-scoped, and `converge` refits
+/// from competitor state — so seeding late reaches the same fit as seeding from
+/// the start. This is the documented scope, asserted rather than assumed.
+#[test]
+fn a_prior_is_whole_history_scoped_not_per_event() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut late = history();
+    late.add_events(vec![bout("a", "b", 0, None, None)])
+        .unwrap();
+    late.add_events(vec![bout("a", "b", 1, Some(seeded), None)])
+        .unwrap();
+    late.converge().unwrap();
+
+    let mut early = history();
+    early
+        .add_events(vec![
+            bout("a", "b", 0, Some(seeded), None),
+            bout("a", "b", 1, Some(seeded), None),
+        ])
+        .unwrap();
+    early.converge().unwrap();
+
+    let (l, e) = (skill_of(&late, "a"), skill_of(&early, "a"));
+    assert!(
+        (l.mu() - e.mu()).abs() < 1e-9 && (l.sigma() - e.sigma()).abs() < 1e-9,
+        "late seeding should refit the whole history: {l:?} vs {e:?}"
+    );
+}
+
+#[test]
+fn repeating_the_same_prior_is_inert() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut once = history();
+    once.add_events(vec![
+        bout("a", "b", 0, Some(seeded), None),
+        bout("a", "b", 1, None, None),
+    ])
+    .unwrap();
+    once.converge().unwrap();
+
+    let mut every_time = history();
+    every_time
+        .add_events(vec![
+            bout("a", "b", 0, Some(seeded), None),
+            bout("a", "b", 1, Some(seeded), None),
+        ])
+        .unwrap();
+    every_time.converge().unwrap();
+
+    let (o, e) = (skill_of(&once, "a"), skill_of(&every_time, "a"));
+    assert!(
+        (o.mu() - e.mu()).abs() < 1e-12 && (o.sigma() - e.sigma()).abs() < 1e-12,
+        "declaring the same prior repeatedly changed the fit: {o:?} vs {e:?}"
+    );
+}
+
+/// Events within a batch have no order, so two different values for one
+/// competitor have no well-defined winner. Rejecting is what keeps the answer
+/// independent of iteration order.
+#[test]
+fn a_batch_declaring_two_different_priors_is_rejected() {
+    let mut h = history();
+    let err = h
+        .add_events(vec![
+            bout("a", "b", 0, Some(Gaussian::from_ms(40.0, 1.0)), None),
+            bout("a", "b", 1, Some(Gaussian::from_ms(10.0, 1.0)), None),
+        ])
+        .expect_err("two different priors for one competitor in one batch");
+
+    assert!(
+        matches!(
+            err,
+            InferenceError::ConflictingCompetitorConfig { field: "prior", .. }
+        ),
+        "got {err:?}"
+    );
+}
+
+/// A member setting only `drift_scale` must not also assert the default prior,
+/// or it would silently undo a prior seeded earlier. This is why the collected
+/// configuration tracks each field separately rather than a merged `Rating`.
+#[test]
+fn setting_one_field_late_leaves_the_other_alone() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut h = history();
+    h.add_events(vec![bout("a", "b", 0, Some(seeded), None)])
+        .unwrap();
+    // Only the scale this time — the prior above must survive.
+    h.add_events(vec![bout("a", "b", 1, None, Some(0.5))])
+        .unwrap();
+    h.converge().unwrap();
+
+    let mut both_upfront = history();
+    both_upfront
+        .add_events(vec![
+            bout("a", "b", 0, Some(seeded), Some(0.5)),
+            bout("a", "b", 1, None, None),
+        ])
+        .unwrap();
+    both_upfront.converge().unwrap();
+
+    let (a, b) = (skill_of(&h, "a"), skill_of(&both_upfront, "a"));
+    assert!(
+        (a.mu() - b.mu()).abs() < 1e-9 && (a.sigma() - b.sigma()).abs() < 1e-9,
+        "setting drift_scale late clobbered the earlier prior: {a:?} vs {b:?}"
+    );
+}

tests/drift_scale.rs

@@ -341,13 +341,20 @@ fn zero_scale_pins_a_competitor_in_the_filtered_pass() {
     );
 }
 
-/// `drift_scale` is competitor configuration captured at first appearance, the
+/// `drift_scale` is competitor configuration, and configuration supplied for a
-/// same as `prior` — a later `with_drift_scale` on a key the history already
+/// competitor the history already knows is now *applied* rather than dropped.
-/// knows is ignored. This guards that decision rather than driving it: the
+///
-/// behaviour falls out of where the capture happens, and the point of the test
+/// This test previously asserted the opposite. It was written as a deliberate
-/// is that moving the capture would be a visible break, not a silent one.
+/// change-detector — "moving the capture would be a visible break, not a silent
+/// one" — and that is exactly what happened: the capture moved, and the
+/// assertion inverted rather than being deleted.
+///
+/// Because configuration lives on the competitor and `converge` refits from
+/// competitor state, a late pin applies to the *whole* history, not just to
+/// events after it. So a scale set on the second batch must reach the same fit
+/// as one set from the very first event.
 #[test]
-fn drift_scale_is_ignored_after_first_appearance() {
+fn drift_scale_applies_when_set_after_first_appearance() {
     let mut late = History::builder()
         .mu(25.0)
         .sigma(25.0 / 3.0)
@@ -368,7 +375,7 @@ fn drift_scale_is_ignored_after_first_appearance() {
     }])
     .unwrap();
 
-    // Second batch asks for a pin. Too late: the competitor already exists.
+    // Second batch asks for a pin. No longer too late.
     late.add_events(vec![Event {
         time: 1000,
         teams: smallvec![
@@ -380,23 +387,113 @@ fn drift_scale_is_ignored_after_first_appearance() {
     .unwrap();
     late.converge().unwrap();
 
-    let ignored = curve(&late, "anchor");
+    let applied = curve(&late, "anchor");
-    let drifting = curve(&fit(distant_pair(None), 25.0 / 300.0), "anchor");
+    let pinned_from_the_start = curve(&fit(distant_pair(Some(0.0)), 25.0 / 300.0), "anchor");
+    let never_pinned = curve(&fit(distant_pair(None), 25.0 / 300.0), "anchor");
 
-    for ((t_l, g_l), (t_r, g_r)) in ignored.iter().zip(drifting.iter()) {
+    for ((t_l, g_l), (t_r, g_r)) in applied.iter().zip(pinned_from_the_start.iter()) {
         assert_eq!(t_l, t_r);
         assert!(
             (g_l.sigma() - g_r.sigma()).abs() < 1e-9,
-            "a scale set after first appearance must be ignored, leaving the fit \
+            "a late pin should refit the whole history: t={t_l}, {} vs {}",
-             identical to one that never set it: t={t_l}, {} vs {}",
             g_l.sigma(),
             g_r.sigma()
         );
     }
 
-    let pinned = curve(&fit(distant_pair(Some(0.0)), 25.0 / 300.0), "anchor");
+    // And it must actually have done something.
     assert!(
-        (ignored[1].1.sigma() - pinned[1].1.sigma()).abs() > 1e-6,
+        applied
-        "sanity: the pinned fit must actually differ, or the assertion above is vacuous"
+            .iter()
+            .zip(never_pinned.iter())
+            .any(|((_, a), (_, b))| (a.sigma() - b.sigma()).abs() > 1e-9),
+        "the pin had no effect at all — the silent drop is back"
+    );
+}
+
+/// Re-declaring the same configuration must be inert. This is the shape a
+/// caller gets when the configuration is a property of the domain — "layouts
+/// are static" — so every ingestion path repeats it on every event.
+///
+/// Both histories see exactly the same events; only how many times the scale
+/// is declared differs.
+#[test]
+fn repeating_the_same_configuration_changes_nothing() {
+    let events = |declare_every_time: bool| {
+        let anchor = |first: bool| {
+            if first || declare_every_time {
+                Member::new("anchor").with_drift_scale(0.0)
+            } else {
+                Member::new("anchor")
+            }
+        };
+        vec![
+            Event {
+                time: 0,
+                teams: smallvec![
+                    Team::with_members([anchor(true)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(0, 2),
+            },
+            Event {
+                time: 1000,
+                teams: smallvec![
+                    Team::with_members([anchor(false)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(1, 2),
+            },
+        ]
+    };
+
+    let once = curve(&fit(events(false), 25.0 / 300.0), "anchor");
+    let every_time = curve(&fit(events(true), 25.0 / 300.0), "anchor");
+
+    for ((t_l, a), (t_r, b)) in once.iter().zip(every_time.iter()) {
+        assert_eq!(t_l, t_r);
+        assert!(
+            (a.sigma() - b.sigma()).abs() < 1e-12,
+            "t={t_l}: declaring the same scale repeatedly changed the fit, {} vs {}",
+            a.sigma(),
+            b.sigma()
+        );
+    }
+}
+
+#[test]
+fn a_batch_that_contradicts_itself_is_rejected() {
+    let mut h = History::builder().convergence(CONVERGENCE).build();
+
+    let err = h
+        .add_events(vec![
+            Event {
+                time: 0,
+                teams: smallvec![
+                    Team::with_members([Member::new("anchor").with_drift_scale(0.0)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(0, 2),
+            },
+            Event {
+                time: 1,
+                teams: smallvec![
+                    Team::with_members([Member::new("anchor").with_drift_scale(1.0)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(0, 2),
+            },
+        ])
+        .expect_err("two different scales for one competitor in one batch");
+
+    assert!(
+        matches!(
+            err,
+            InferenceError::ConflictingCompetitorConfig {
+                field: "drift_scale",
+                ..
+            }
+        ),
+        "got {err:?}"
     );
 }

tests/ingestion_equivalence.rs

@@ -30,6 +30,22 @@ fn event(a: &str, b: &str, time: i64) -> Event<i64, String> {
     }
 }
 
+/// Like [`event`], but `a` carries competitor configuration.
+///
+/// `prior` and `drift_scale` configure the competitor rather than the event, so
+/// they are the part of ingestion most exposed to order: they are consumed once,
+/// where the competitor's state is written.
+fn configured_event(a: &str, b: &str, time: i64, scale: f64) -> Event<i64, String> {
+    Event {
+        time,
+        teams: smallvec![
+            Team::with_members([Member::new(a.to_string()).with_drift_scale(scale)]),
+            Team::with_members([Member::new(b.to_string())]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }
+}
+
 fn converged_skills(events: Vec<Event<i64, String>>, batched: bool) -> Vec<(String, Gaussian)> {
     let mut h: History<i64, _, _, String> =
         History::builder_with_key().convergence(tight()).build();
@@ -145,3 +161,65 @@ fn back_dated_event_matches_batched() {
     let incremental = converged_skills(events, false);
     assert_same(&batched, &incremental, "back-dated event");
 }
+
+/// The invariant this file protects was only ever checked for *unconfigured*
+/// competitors — every helper above built members with `Member::new`.
+///
+/// Configuration is the part most exposed to ordering, because it is consumed
+/// once at the point the competitor's state is written rather than replayed per
+/// event. These cover it.
+#[test]
+fn configured_competitors_are_order_independent() {
+    let events = vec![
+        configured_event("a", "b", 0, 0.0),
+        configured_event("a", "c", 1, 0.0),
+        configured_event("a", "b", 2, 0.0),
+        event("b", "c", 3),
+    ];
+
+    assert_same(
+        &converged_skills(events.clone(), true),
+        &converged_skills(events, false),
+        "configuration repeated on every appearance",
+    );
+}
+
+/// Configuration supplied only on a *later* event is the case that used to be
+/// silently dropped. It must now reach the same fit either way it is ingested.
+#[test]
+fn late_configuration_is_order_independent() {
+    let events = vec![
+        event("a", "b", 0),
+        configured_event("a", "c", 1, 0.0),
+        event("a", "b", 2),
+    ];
+
+    assert_same(
+        &converged_skills(events.clone(), true),
+        &converged_skills(events, false),
+        "configuration supplied after first appearance",
+    );
+}
+
+/// And it must actually be doing something — an implementation that dropped
+/// configuration entirely would pass both tests above.
+#[test]
+fn configuration_changes_the_fit_however_it_is_ingested() {
+    let configured = vec![
+        event("a", "b", 0),
+        configured_event("a", "c", 1, 0.0),
+        event("a", "b", 2),
+    ];
+    let plain = vec![event("a", "b", 0), event("a", "c", 1), event("a", "b", 2)];
+
+    for batched in [true, false] {
+        let with = converged_skills(configured.clone(), batched);
+        let without = converged_skills(plain.clone(), batched);
+        assert!(
+            with.iter()
+                .zip(&without)
+                .any(|((_, x), (_, y))| (x.sigma() - y.sigma()).abs() > 1e-9),
+            "batched={batched}: configuration had no effect, so the order tests are vacuous"
+        );
+    }
+}

tests/observer.rs

@@ -8,14 +8,13 @@ use std::sync::{Arc, Mutex};
 
 use trueskill_tt::{History, Observer};
 
-/// `History` takes its observer by value and never hands it back, so a test
+/// Plain fields. `Arc<O>` implements `Observer`, so the caller shares the
-/// that wants to read what was recorded shares the storage rather than the
+/// observer itself rather than wrapping each field in its own `Arc`.
-/// observer: the handles are cloned, the buffers are not.
+#[derive(Default)]
-#[derive(Clone, Default)]
 struct Recorder {
-    iterations: Arc<Mutex<Vec<usize>>>,
+    iterations: Mutex<Vec<usize>>,
-    slices: Arc<Mutex<Vec<(i64, usize, usize)>>>,
+    slices: Mutex<Vec<(i64, usize, usize)>>,
-    converged: Arc<Mutex<Vec<(usize, bool)>>>,
+    converged: Mutex<Vec<(usize, bool)>>,
 }
 
 impl Observer<i64> for Recorder {
@@ -37,8 +36,8 @@ impl Observer<i64> for Recorder {
 
 #[test]
 fn every_observer_callback_fires() {
-    let recorder = Recorder::default();
+    let recorder = Arc::new(Recorder::default());
-    let mut h = History::builder().observer(recorder.clone()).build();
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
 
     h.record_winner(&"a", &"b", 1).unwrap();
     h.record_winner(&"b", &"c", 2).unwrap();
@@ -61,8 +60,8 @@ fn every_observer_callback_fires() {
 
 #[test]
 fn slice_callbacks_report_the_slice_they_swept() {
-    let recorder = Recorder::default();
+    let recorder = Arc::new(Recorder::default());
-    let mut h = History::builder().observer(recorder.clone()).build();
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
 
     h.record_winner(&"a", &"b", 10).unwrap();
     h.record_winner(&"a", &"b", 20).unwrap();
@@ -90,8 +89,8 @@ fn slice_callbacks_report_the_slice_they_swept() {
 
 #[test]
 fn a_single_slice_history_still_reports_its_sweep() {
-    let recorder = Recorder::default();
+    let recorder = Arc::new(Recorder::default());
-    let mut h = History::builder().observer(recorder.clone()).build();
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
 
     h.record_winner(&"a", &"b", 1).unwrap();
     h.converge().unwrap();
@@ -103,3 +102,60 @@ fn a_single_slice_history_still_reports_its_sweep() {
     );
     assert!(slices.iter().all(|&(t, idx, _)| t == 1 && idx == 0));
 }
+
+/// The gap #40 closed: without `impl Observer for Arc<O>`, an observer that
+/// accumulates anything had to wrap every field in its own `Arc` and derive
+/// `Clone`, because `History` consumes the observer and never hands it back.
+#[test]
+fn a_shared_observer_reaches_the_callers_handle() {
+    let recorder = Arc::new(Recorder::default());
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
+
+    h.record_winner(&"a", &"b", 1).unwrap();
+    h.converge().unwrap();
+
+    assert!(!recorder.iterations.lock().unwrap().is_empty());
+    assert!(!recorder.slices.lock().unwrap().is_empty());
+    assert!(!recorder.converged.lock().unwrap().is_empty());
+}
+
+/// `?Sized` on the blanket impls means the observer can be chosen at runtime.
+#[test]
+fn a_trait_object_observer_works() {
+    let boxed: Box<dyn Observer<i64>> = Box::new(Recorder::default());
+    let mut h = History::builder().observer(boxed).build();
+    h.record_winner(&"a", &"b", 1).unwrap();
+    h.converge().unwrap();
+
+    let shared: Arc<dyn Observer<i64>> = Arc::new(Recorder::default());
+    let mut h = History::builder().observer(Arc::clone(&shared)).build();
+    h.record_winner(&"a", &"b", 1).unwrap();
+    h.converge().unwrap();
+}
+
+/// A non-shared observer can be reclaimed after convergence instead.
+#[test]
+fn into_observer_returns_the_accumulated_state() {
+    let mut h = History::builder().observer(Recorder::default()).build();
+    h.record_winner(&"a", &"b", 1).unwrap();
+    h.converge().unwrap();
+
+    // Readable in place...
+    assert!(!h.observer().iterations.lock().unwrap().is_empty());
+
+    // ...and reclaimable by value.
+    let recorder = h.into_observer();
+    assert!(!recorder.slices.lock().unwrap().is_empty());
+}
+
+/// Borrowing works too, for an observer that outlives the history.
+#[test]
+fn a_borrowed_observer_works() {
+    let recorder = Recorder::default();
+    {
+        let mut h = History::builder().observer(&recorder).build();
+        h.record_winner(&"a", &"b", 1).unwrap();
+        h.converge().unwrap();
+    }
+    assert!(!recorder.iterations.lock().unwrap().is_empty());
+}

tests/prediction.rs

@@ -212,3 +212,80 @@ fn team_size_affects_the_prediction() {
     assert!((p.total() - 1.0).abs() < 1e-6, "total = {}", p.total());
     assert!(p.probability_of(&[0, 0]) > 0.0);
 }
+
+// ---------------------------------------------------------------------------
+// Expected information gain
+// ---------------------------------------------------------------------------
+
+/// The whole point of #39: "which comparison should I run next?" is a
+/// different question from "who will win?" or "is this fair?".
+#[test]
+fn information_gain_prefers_the_uncertain_pairing() {
+    let mut h = History::builder().build();
+
+    // "known" and "rival" have played a lot; "newcomer" has played once.
+    for t in 1..=15 {
+        h.record_winner(&"known", &"rival", t).unwrap();
+        h.record_winner(&"rival", &"known", t + 100).unwrap();
+    }
+    h.record_winner(&"known", &"newcomer", 500).unwrap();
+    h.converge().unwrap();
+
+    let settled = h
+        .expected_information_gain(&[&[&"known"], &[&"rival"]])
+        .unwrap();
+    let unknown = h
+        .expected_information_gain(&[&[&"known"], &[&"newcomer"]])
+        .unwrap();
+
+    assert!(
+        unknown > settled,
+        "pairing against the newcomer should teach more: {unknown} vs {settled}"
+    );
+}
+
+/// The analytic ceiling, through the `History` entry point rather than the
+/// standalone one.
+#[test]
+fn information_gain_respects_the_entropy_ceiling() {
+    let h = history_with(&["a", "b", "c"], 0.0);
+
+    let two = h.expected_information_gain(&[&[&"a"], &[&"b"]]).unwrap();
+    assert!(
+        (0.0..=std::f64::consts::LN_2).contains(&two),
+        "two-team EIG {two} outside [0, ln 2]"
+    );
+
+    let three = h
+        .expected_information_gain(&[&[&"a"], &[&"b"], &[&"c"]])
+        .unwrap();
+    assert!(
+        (0.0..=6.0f64.ln()).contains(&three),
+        "three-team EIG {three} outside [0, ln 6]"
+    );
+}
+
+#[test]
+fn information_gain_reports_unknown_keys() {
+    let h = history_with(&["a", "b"], 0.0);
+    assert_eq!(
+        h.expected_information_gain(&[&[&"a"], &[&"ghost"]])
+            .unwrap_err(),
+        InferenceError::UnknownKey { team: 1, member: 0 }
+    );
+}
+
+/// A draw-enabled history has three outcomes to weigh rather than two, so the
+/// draw branch must actually be reachable through this path.
+#[test]
+fn information_gain_accounts_for_draws() {
+    let with_draws = history_with(&["a", "b"], 0.25);
+    let g = with_draws
+        .expected_information_gain(&[&[&"a"], &[&"b"]])
+        .unwrap();
+    assert!(g > 0.0 && g <= 3.0f64.ln(), "{g}");
+
+    // The draw outcome carries mass, so it is genuinely being weighed.
+    let dist = with_draws.predict_outcome(&[&[&"a"], &[&"b"]]).unwrap();
+    assert!(dist.probability_of(&[0, 0]) > 0.0);
+}

tests/validation.rs

@@ -0,0 +1,186 @@
+//! Input validation must hold in **release**, where `debug_assert!` is gone.
+//!
+//! The engine guards itself with `debug_assert!`, which documents invariants
+//! but vanishes in the profile users actually ship. Anything reachable from the
+//! public API has to be rejected with an `InferenceError` instead, at the
+//! boundary, rather than becoming NaN or an out-of-bounds panic deep inside
+//! `run_chain`.
+//!
+//! `GameOptions` and `ConvergenceOptions` both have public fields, so the
+//! eager asserts on `HistoryBuilder` do not cover the `Game` constructors —
+//! a caller can build the options struct directly.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, Game, GameOptions, Gaussian, History, InferenceError,
+    Member, Outcome, Rating, Team,
+};
+
+type R = Rating<i64, ConstantDrift>;
+
+fn rating() -> R {
+    R::new(
+        Gaussian::from_ms(25.0, 25.0 / 3.0),
+        25.0 / 6.0,
+        ConstantDrift(0.0),
+    )
+}
+
+fn options_with_alpha(alpha: f64) -> GameOptions {
+    GameOptions {
+        convergence: ConvergenceOptions {
+            alpha,
+            ..ConvergenceOptions::default()
+        },
+        ..GameOptions::default()
+    }
+}
+
+/// `alpha == 0.0` leaves every EP update unapplied, so inference silently
+/// returns the priors — the worst possible failure, since the output looks
+/// entirely reasonable.
+#[test]
+fn ranked_rejects_a_zero_damping_factor() {
+    let (a, b) = (rating(), rating());
+    let err = Game::<i64, _>::ranked(
+        &[&[a], &[b]],
+        Outcome::winner(0, 2),
+        &options_with_alpha(0.0),
+    )
+    .expect_err("alpha = 0 must be rejected");
+    assert!(
+        matches!(err, InferenceError::InvalidParameter { name: "alpha", .. }),
+        "got {err:?}"
+    );
+}
+
+#[test]
+fn ranked_rejects_an_out_of_range_damping_factor() {
+    let (a, b) = (rating(), rating());
+    for alpha in [-0.5, 1.5, f64::NAN] {
+        let err = Game::<i64, _>::ranked(
+            &[&[a], &[b]],
+            Outcome::winner(0, 2),
+            &options_with_alpha(alpha),
+        )
+        .expect_err("alpha out of (0, 1] must be rejected");
+        assert!(
+            matches!(err, InferenceError::InvalidParameter { name: "alpha", .. }),
+            "alpha={alpha}: got {err:?}"
+        );
+    }
+}
+
+#[test]
+fn scored_rejects_a_bad_damping_factor() {
+    let (a, b) = (rating(), rating());
+    let err = Game::<i64, _>::scored(
+        &[&[a], &[b]],
+        Outcome::scores([21.0, 9.0]),
+        &options_with_alpha(0.0),
+    )
+    .expect_err("alpha = 0 must be rejected");
+    assert!(
+        matches!(err, InferenceError::InvalidParameter { name: "alpha", .. }),
+        "got {err:?}"
+    );
+}
+
+/// Already covered by `Game::ranked`, asserted here so the release-mode
+/// guarantee is stated in one place.
+#[test]
+fn ranked_rejects_an_out_of_range_draw_probability() {
+    let (a, b) = (rating(), rating());
+    for p_draw in [-0.5, 1.0, 1.5] {
+        let options = GameOptions {
+            p_draw,
+            ..GameOptions::default()
+        };
+        assert!(
+            Game::<i64, _>::ranked(&[&[a], &[b]], Outcome::winner(0, 2), &options).is_err(),
+            "p_draw={p_draw} must be rejected"
+        );
+    }
+}
+
+#[test]
+fn scored_rejects_a_non_positive_noise() {
+    let (a, b) = (rating(), rating());
+    for score_sigma in [0.0, -1.0, f64::NAN] {
+        let options = GameOptions {
+            score_sigma,
+            ..GameOptions::default()
+        };
+        assert!(
+            Game::<i64, _>::scored(&[&[a], &[b]], Outcome::scores([21.0, 9.0]), &options).is_err(),
+            "score_sigma={score_sigma} must be rejected"
+        );
+    }
+}
+
+/// A tie with no draw probability makes the truncation margin zero and the
+/// two-sided update evaluate 0/0. Ingestion must refuse it.
+#[test]
+fn ingestion_rejects_a_tie_without_a_draw_probability() {
+    let mut h = History::builder().p_draw(0.0).build();
+    let err = h
+        .add_events(vec![Event {
+            time: 0,
+            teams: smallvec![
+                Team::with_members([Member::new("a")]),
+                Team::with_members([Member::new("b")]),
+            ],
+            outcome: Outcome::draw(2),
+        }])
+        .expect_err("a tie with p_draw = 0 must be rejected");
+    assert!(
+        matches!(err, InferenceError::TieWithoutDrawProbability { .. }),
+        "got {err:?}"
+    );
+}
+
+/// `Outcome::scores_with_sigma` documents that a non-positive sigma is
+/// accepted at construction and rejected at ingestion.
+#[test]
+fn ingestion_rejects_a_non_positive_per_event_score_sigma() {
+    for sigma in [0.0, -1.0, f64::NAN] {
+        let mut h = History::builder().build();
+        let err = h
+            .add_events(vec![Event {
+                time: 0,
+                teams: smallvec![
+                    Team::with_members([Member::new("a")]),
+                    Team::with_members([Member::new("b")]),
+                ],
+                outcome: Outcome::scores_with_sigma([21.0, 9.0], sigma),
+            }])
+            .expect_err("a non-positive per-event sigma must be rejected");
+        assert!(
+            matches!(err, InferenceError::InvalidParameter { .. }),
+            "sigma={sigma}: got {err:?}"
+        );
+    }
+}
+
+/// Per-team weights must match that team's membership. The top-level length
+/// checks in ingestion do not cover the inner dimension.
+#[test]
+fn ingestion_rejects_weights_that_do_not_match_their_team() {
+    let mut h = History::builder().build();
+    let mut team = Team::with_members([Member::new("a"), Member::new("b")]);
+    team.members[0].weight = 1.0;
+
+    let err = h
+        .event(0)
+        .team(["a", "b"])
+        .team(["c"])
+        // Three weights for a two-member team.
+        .weights([1.0, 1.0, 1.0])
+        .winner(0)
+        .commit()
+        .expect_err("a weight/member length mismatch must be rejected");
+    assert!(
+        matches!(err, InferenceError::MismatchedShape { .. }),
+        "got {err:?}"
+    );
+}