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feat!: N-team outcome prediction with draw mass, replacing the 2-team panic

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`predict_outcome` asserted `teams.len() == 2` and returned `[p, 1 - p]`,
allocating no probability to a draw even with `p_draw > 0`. For a
draw-enabled model the numbers were simply wrong, at any team count.

It now returns `Result<Prediction, InferenceError>` and supports N teams.

Two algorithms, both deterministic:

- Who finishes first. Performances are independent Gaussians, so this
  separates into a one-dimensional integral per team rather than a
  multivariate orthant probability. Adaptive Gauss-Kronrod evaluates it
  to ~1e-15, matching the exact two-team closed form.
- A specific finishing order. The factor graph only constrains
  rank-adjacent teams, so a full order is a chain of local constraints,
  not a general orthant integral. That chain collapses into a sequential
  recursion over cumulative integrals: O(teams * grid) per order.

Fixed-node Gauss-Hermite is the obvious tool for the first and is a trap:
when a rival's sigma is small the CDF product becomes a step narrower
than the node spacing, and the nodes step over it. Measured 4.4e-4 off
the closed form on a mildly skewed matchup and 1.7e-2 on a small-sigma
one, while still returning something that looks like a probability.
Adaptive refinement is what makes that case safe, and
`win_probabilities_survive_a_rival_with_a_tiny_sigma` pins it down.

The acceptance test is an identity rather than a golden: the outcome
space is exhaustive and disjoint, so the probabilities sum to one. Any
drift is integration error and nothing else. Gauss-Hermite failed it at
4.4e-4; this holds to ~1e-9.

Also from #21: unknown keys are now reported rather than dropped, so a
team of strangers can no longer produce a confident-looking prediction.
`predict_quality` returns `Result` for the same reason.

BREAKING CHANGE: `predict_outcome` returns `Result<Prediction, _>`
instead of `Vec<f64>`; `predict_quality` returns `Result<f64, _>`.

Refs #21, #39

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ
This commit is contained in:
logaritmisk
2026-09-07 14:55:11 +02:00
co-authored by Claude Opus 5
parent 87fca8dcca
commit bb2a845882
8 changed files with 1513 additions and 50 deletions
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src/predict.rs
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//! Outcome prediction: who wins, and how likely is a given finishing order.
//!
//! Prediction runs on *performances*, not skills. A competitor's skill is
//! inflated by their performance noise `beta` before any comparison, which is
//! what separates "how good are they" from "how will they do today".
//!
//! Two questions, two algorithms:
//!
//! - **Who finishes first.** Because performances are independent Gaussians,
//! the probability that team `i` beats every other team separates into a
//! *one-dimensional* integral — no multivariate orthant integral is
//! involved. [`quadrature::integrate`] evaluates it to near machine
//! precision for a few hundred `cdf` calls.
//! - **A specific finishing order.** The factor graph only ever constrains
//! rank-*adjacent* teams (see `Game::run_chain`), so the joint probability
//! of a full order is a chain of local constraints rather than a general
//! orthant probability. That chain collapses into a sequential recursion:
//! one cumulative integral per adjacent pair, `O(teams * grid)` overall.
//!
//! Both are deterministic. A sampler would have been easier to write and
//! would have made every `predict_*` call return a slightly different number,
//! which is not a property a rating library should have.
use crate::{Gaussian, quadrature};
/// Teams beyond this count make the outcome enumeration impractical.
///
/// Each realisation sorts into exactly one (permutation, tie-pattern) event,
/// so the space has `n! * 2^(n-1)` members: 24 at 3 teams, 192 at 4, 1_920 at
/// 5, 23_040 at 6. The jump to 322_560 at 7 is where enumerating stops being
/// a reasonable thing to do on a caller's behalf.
pub(crate) const MAX_TEAMS_FOR_DISTRIBUTION: usize = 6;
/// Relative tolerance for the first-place integrals.
///
/// Tightening past this buys nothing: the underlying `cdf` is a rational
/// approximation with fractional error ~1.2e-7, which contributes ~6e-9 to a
/// finished probability and dominates any further quadrature refinement.
const WIN_TOLERANCE: f64 = 1e-8;
/// Nodes for the ranking grid, and the floor below which a grid is pointless.
///
/// The recursion converges as O(h^2). Measured against the exact two-team
/// closed form, 2_048 nodes leave ~1.2e-6 of discretisation error while 8_192
/// reach ~1e-7 — at which point the residual is the `cdf` rational
/// approximation (~2.4e-8), not the grid, and refining further buys nothing.
const MIN_GRID_POINTS: usize = 8_192;
const MAX_GRID_POINTS: usize = 262_144;
/// How many standard deviations of support the grid and integrals cover.
///
/// The normal density is below 1e-18 of its peak past nine sigma, far under
/// the precision of everything else here.
const SUPPORT_SIGMAS: f64 = 9.0;
/// Standard normal CDF at `z`.
fn phi(z: f64) -> f64 {
crate::cdf(z, 0.0, 1.0)
}
/// Normal density of `x` under `g`.
fn density(g: Gaussian, x: f64) -> f64 {
let sigma = g.sigma();
let z = (x - g.mu()) / sigma;
(-0.5 * z * z).exp() / (sigma * (2.0 * std::f64::consts::PI).sqrt())
}
/// Per-pair draw margins.
///
/// The margin is *not* a single number for the whole game: inference derives
/// it per rank-adjacent pair from those two teams' betas (`Game::likelihoods`).
/// Prediction has to use the same per-pair values or it answers a question
/// about a different model than the one that will actually be fitted.
pub(crate) struct Margins {
n: usize,
values: Vec<f64>,
}
impl Margins {
/// Build from a per-pair margin function.
pub(crate) fn new<F: Fn(usize, usize) -> f64>(n: usize, f: F) -> Self {
let mut values = vec![0.0; n * n];
for i in 0..n {
for j in 0..n {
if i != j {
values[i * n + j] = f(i, j);
}
}
}
Self { n, values }
}
fn get(&self, i: usize, j: usize) -> f64 {
self.values[i * self.n + j]
}
/// True when no pair can draw, so every tie has probability zero.
fn all_zero(&self) -> bool {
self.values.iter().all(|&v| v == 0.0)
}
}
/// `P(team i finishes strictly first)` for every team.
///
/// Strictly means beating each rival by more than that pair's draw margin, so
/// with a non-zero margin these sum to less than one; the shortfall is the
/// probability that the top place is shared.
pub(crate) fn win_probabilities(perf: &[Gaussian], margins: &Margins) -> Vec<f64> {
(0..perf.len())
.map(|i| {
let (mu, sigma) = (perf[i].mu(), perf[i].sigma());
let (lo, hi) = (mu - SUPPORT_SIGMAS * sigma, mu + SUPPORT_SIGMAS * sigma);
// Each rival's CDF turns over near its own mean plus the margin.
// Seeding there is what keeps a rival with a tiny sigma — a step
// function in disguise — from being stepped over.
let mut seeds = Vec::with_capacity(3 * perf.len());
for (j, rival) in perf.iter().enumerate().filter(|&(j, _)| j != i) {
let centre = rival.mu() + margins.get(i, j);
seeds.extend_from_slice(&[centre - rival.sigma(), centre, centre + rival.sigma()]);
}
quadrature::integrate(
|x| {
let d = density(perf[i], x);
if d == 0.0 {
return 0.0;
}
let beaten: f64 = (0..perf.len())
.filter(|&j| j != i)
.map(|j| phi((x - margins.get(i, j) - perf[j].mu()) / perf[j].sigma()))
.product();
d * beaten
},
lo,
hi,
&seeds,
WIN_TOLERANCE,
)
})
.collect()
}
/// Grid bounds and resolution covering every team's support.
///
/// Resolution is set by the *smallest* feature in play — the narrowest sigma,
/// or a draw margin narrower still — because that is what the recursion has to
/// resolve. A grid sized off the widest team would step over the narrow one.
fn grid_shape(perf: &[Gaussian], margins: &Margins) -> (f64, f64, usize) {
let lo = perf
.iter()
.map(|g| g.mu() - SUPPORT_SIGMAS * g.sigma())
.fold(f64::INFINITY, f64::min);
let hi = perf
.iter()
.map(|g| g.mu() + SUPPORT_SIGMAS * g.sigma())
.fold(f64::NEG_INFINITY, f64::max);
let narrowest = perf
.iter()
.map(Gaussian::sigma)
.fold(f64::INFINITY, f64::min);
let smallest_margin = margins
.values
.iter()
.copied()
.filter(|&m| m > 0.0)
.fold(f64::INFINITY, f64::min);
let feature = narrowest.min(smallest_margin);
let wanted = if feature.is_finite() && feature > 0.0 {
((hi - lo) / (feature / 12.0)).ceil()
} else {
MIN_GRID_POINTS as f64
};
let points = if wanted.is_finite() {
(wanted as usize).clamp(MIN_GRID_POINTS, MAX_GRID_POINTS)
} else {
MIN_GRID_POINTS
};
(lo, hi, points)
}
/// Densities of each team sampled on the shared grid.
struct Sampled {
lo: f64,
step: f64,
points: usize,
density: Vec<Vec<f64>>,
}
impl Sampled {
fn new(perf: &[Gaussian], margins: &Margins) -> Self {
let (lo, hi, points) = grid_shape(perf, margins);
let step = (hi - lo) / (points - 1) as f64;
let density = perf
.iter()
.map(|&g| {
(0..points)
.map(|i| density(g, lo + i as f64 * step))
.collect()
})
.collect();
Self {
lo,
step,
points,
density,
}
}
fn node(&self, i: usize) -> f64 {
self.lo + i as f64 * self.step
}
}
/// `P(order[0] >= order[1] >= ... )` with the given adjacency pattern.
///
/// `tied[k]` says whether `order[k]` and `order[k + 1]` finish within that
/// pair's draw margin. The recursion runs bottom-up: `carry` holds, for each
/// grid node, the probability that everything *below* the current team holds
/// given that team landed on that node. A strict gap reads a cumulative
/// integral; a tie reads a window. Both are O(1) against one prefix array,
/// so each level costs O(grid) and the whole order costs O(teams * grid).
fn order_probability(margins: &Margins, sampled: &Sampled, order: &[usize], tied: &[bool]) -> f64 {
let mut carry = vec![1.0; sampled.points];
for k in (0..order.len() - 1).rev() {
let below = order[k + 1];
let above = order[k];
let margin = margins.get(above, below);
let integrand: Vec<f64> = (0..sampled.points)
.map(|i| sampled.density[below][i] * carry[i])
.collect();
let cumulative = quadrature::Grid::from_values(sampled.lo, sampled.step, integrand);
carry = (0..sampled.points)
.map(|i| {
let x = sampled.node(i);
if tied[k] {
// Sorted order already implies `below <= above`, so the
// tie window is one-sided: [x - margin, x].
cumulative.integral_between(x - margin, x)
} else {
cumulative.integral_to(x - margin)
}
})
.collect();
}
let top = order[0];
let integrand: Vec<f64> = (0..sampled.points)
.map(|i| sampled.density[top][i] * carry[i])
.collect();
quadrature::Grid::from_values(sampled.lo, sampled.step, integrand).total()
}
/// Dense ranks implied by a sorted order and its tie pattern.
fn ranks_of(order: &[usize], tied: &[bool], n: usize) -> Vec<u32> {
let mut ranks = vec![0u32; n];
let mut rank = 0u32;
ranks[order[0]] = 0;
for k in 0..order.len() - 1 {
if !tied[k] {
rank += 1;
}
ranks[order[k + 1]] = rank;
}
ranks
}
/// Every (order, tie-pattern) event, or only the strict ones when no pair can
/// draw — a tie then has probability exactly zero and is not worth integrating.
fn events(n: usize, strict_only: bool) -> Vec<(Vec<usize>, Vec<bool>)> {
fn permute(current: &mut Vec<usize>, k: usize, out: &mut Vec<Vec<usize>>) {
if k == current.len() {
out.push(current.clone());
return;
}
for i in k..current.len() {
current.swap(k, i);
permute(current, k + 1, out);
current.swap(k, i);
}
}
let mut orders = Vec::new();
permute(&mut (0..n).collect(), 0, &mut orders);
let patterns: Vec<Vec<bool>> = if strict_only {
vec![vec![false; n - 1]]
} else {
(0..(1u32 << (n - 1)))
.map(|mask| (0..n - 1).map(|i| mask >> i & 1 == 1).collect())
.collect()
};
let mut out = Vec::with_capacity(orders.len() * patterns.len());
for order in orders {
for pattern in &patterns {
out.push((order.clone(), pattern.clone()));
}
}
out
}
/// The full distribution over finishing orders, aggregated by rank vector.
///
/// Orders that differ only *within* a tied group describe the same finishing
/// order, so their probabilities are summed into one entry.
pub(crate) fn outcome_distribution(perf: &[Gaussian], margins: &Margins) -> Vec<(Vec<u32>, f64)> {
let n = perf.len();
let sampled = Sampled::new(perf, margins);
let mut aggregated: Vec<(Vec<u32>, f64)> = Vec::new();
for (order, tied) in events(n, margins.all_zero()) {
let p = order_probability(margins, &sampled, &order, &tied);
let ranks = ranks_of(&order, &tied, n);
match aggregated.iter_mut().find(|(r, _)| *r == ranks) {
Some((_, acc)) => *acc += p,
None => aggregated.push((ranks, p)),
}
}
aggregated.sort_by(|a, b| b.1.partial_cmp(&a.1).unwrap_or(std::cmp::Ordering::Equal));
aggregated
}
/// All permutations of `items`.
fn permutations(items: &[usize]) -> Vec<Vec<usize>> {
fn go(current: &mut Vec<usize>, k: usize, out: &mut Vec<Vec<usize>>) {
if k == current.len() {
out.push(current.clone());
return;
}
for i in k..current.len() {
current.swap(k, i);
go(current, k + 1, out);
current.swap(k, i);
}
}
let mut out = Vec::new();
go(&mut items.to_vec(), 0, &mut out);
out
}
/// Every (order, tie-pattern) event consistent with a grouping by rank.
///
/// Teams sharing a rank may finish in any internal order, so this is the
/// product of each group's permutations. Adjacencies inside a group are ties;
/// the adjacency joining one group to the next is not.
fn orders_for_groups(groups: &[Vec<usize>]) -> Vec<(Vec<usize>, Vec<bool>)> {
let per_group: Vec<Vec<Vec<usize>>> = groups.iter().map(|g| permutations(g)).collect();
let mut out = Vec::new();
let mut choice = vec![0usize; groups.len()];
loop {
let mut order = Vec::new();
let mut tied = Vec::new();
for (gi, group) in per_group.iter().enumerate() {
for (offset, &member) in group[choice[gi]].iter().enumerate() {
if !order.is_empty() {
tied.push(offset != 0);
}
order.push(member);
}
}
out.push((order, tied));
let mut k = 0;
loop {
if k == choice.len() {
return out;
}
choice[k] += 1;
if choice[k] < per_group[k].len() {
break;
}
choice[k] = 0;
k += 1;
}
}
}
/// Probability of one specific rank vector.
///
/// Ties in `ranks` mean the tied teams may finish in any internal order, so
/// this sums the orders consistent with the requested ranking rather than
/// picking one.
pub(crate) fn ranking_probability(perf: &[Gaussian], margins: &Margins, ranks: &[u32]) -> f64 {
let n = perf.len();
let sampled = Sampled::new(perf, margins);
let mut distinct: Vec<u32> = ranks.to_vec();
distinct.sort_unstable();
distinct.dedup();
let groups: Vec<Vec<usize>> = distinct
.iter()
.map(|&r| (0..n).filter(|&i| ranks[i] == r).collect())
.collect();
orders_for_groups(&groups)
.iter()
.map(|(order, tied)| order_probability(margins, &sampled, order, tied))
.sum()
}
/// A distribution over the ways a contest could finish.
///
/// Each entry pairs a rank vector — the same shape [`crate::Outcome::ranking`]
/// takes, with equal ranks meaning a tie — against its probability. Entries
/// are ordered most likely first, and cover the whole outcome space, so the
/// probabilities sum to one.
///
/// The rank vectors compose directly with inference: feeding one to
/// `Game::ranked` asks "what would we believe if *this* happened", which is
/// what an expected-information-gain calculation needs alongside the weight.
#[derive(Clone, Debug, PartialEq)]
pub struct Prediction {
outcomes: Vec<(Vec<u32>, f64)>,
}
impl Prediction {
pub(crate) fn new(outcomes: Vec<(Vec<u32>, f64)>) -> Self {
Self { outcomes }
}
/// Every possible finishing order and its probability, most likely first.
pub fn outcomes(&self) -> impl ExactSizeIterator<Item = (&[u32], f64)> {
self.outcomes.iter().map(|(r, p)| (r.as_slice(), *p))
}
/// The single most likely finishing order.
#[must_use]
pub fn most_likely(&self) -> Option<(&[u32], f64)> {
self.outcomes.first().map(|(r, p)| (r.as_slice(), *p))
}
/// Probability of one specific finishing order, or zero if it cannot occur.
#[must_use]
pub fn probability_of(&self, ranks: &[u32]) -> f64 {
self.outcomes
.iter()
.find(|(r, _)| r.as_slice() == ranks)
.map_or(0.0, |(_, p)| *p)
}
/// `P(team i finishes strictly first)`, for each team.
///
/// Sums to less than one exactly when the top place can be shared; the
/// shortfall is [`Prediction::shared_first_place`].
#[must_use]
pub fn win_probabilities(&self) -> Vec<f64> {
let n = self.outcomes.first().map_or(0, |(r, _)| r.len());
let mut wins = vec![0.0; n];
for (ranks, p) in &self.outcomes {
let leaders = ranks.iter().filter(|&&r| r == 0).count();
if leaders == 1 {
let winner = ranks.iter().position(|&r| r == 0).expect("a rank-0 team");
wins[winner] += p;
}
}
wins
}
/// Probability that two or more teams share first place.
#[must_use]
pub fn shared_first_place(&self) -> f64 {
self.outcomes
.iter()
.filter(|(r, _)| r.iter().filter(|&&x| x == 0).count() > 1)
.map(|(_, p)| p)
.sum()
}
/// Total probability mass, which should be one.
///
/// Exposed because it is a genuine check on the numerics rather than a
/// formality: the outcome space is exhaustive and disjoint by construction,
/// so any drift from one is integration error and nothing else.
#[must_use]
pub fn total(&self) -> f64 {
self.outcomes.iter().map(|(_, p)| p).sum()
}
}
#[cfg(test)]
mod tests {
use super::*;
fn g(mu: f64, sigma: f64) -> Gaussian {
Gaussian::from_ms(mu, sigma)
}
fn flat(n: usize, eps: f64) -> Margins {
Margins::new(n, |_, _| eps)
}
/// Exact two-team result: `P(a first) = Phi((mu_a - mu_b - eps) / sd)`.
fn closed_form_two(a: Gaussian, b: Gaussian, eps: f64) -> (f64, f64) {
let sd = (a.sigma().powi(2) + b.sigma().powi(2)).sqrt();
(
phi((a.mu() - b.mu() - eps) / sd),
phi((b.mu() - a.mu() - eps) / sd),
)
}
#[test]
fn two_team_win_probabilities_match_the_closed_form() {
for (ma, sa, mb, sb, eps) in [
(0.0, 6.0, 0.0, 6.0, 0.0),
(3.0, 6.0, -2.0, 1.0, 0.0),
(0.0, 6.0, 0.0, 6.0, 2.0),
(3.0, 6.0, -2.0, 1.0, 1.5),
(40.0, 1.0, 0.0, 1.0, 0.0),
] {
let perf = [g(ma, sa), g(mb, sb)];
let got = win_probabilities(&perf, &flat(2, eps));
let (wa, wb) = closed_form_two(perf[0], perf[1], eps);
assert!(
(got[0] - wa).abs() < 1e-7 && (got[1] - wb).abs() < 1e-7,
"mu=({ma},{mb}) sigma=({sa},{sb}) eps={eps}: got {got:?}, want [{wa}, {wb}]"
);
}
}
/// The identity that a wrong-but-plausible implementation cannot fake:
/// with no draw margin, exactly one team finishes first.
#[test]
fn win_probabilities_sum_to_one_without_a_draw_margin() {
for perf in [
vec![g(0.0, 6.0), g(0.0, 6.0)],
vec![g(5.0, 6.0), g(0.0, 3.0), g(-5.0, 1.0)],
vec![
g(8.0, 2.0),
g(3.0, 6.0),
g(0.0, 1.0),
g(-3.0, 4.0),
g(-8.0, 6.0),
],
] {
let sum: f64 = win_probabilities(&perf, &flat(perf.len(), 0.0))
.iter()
.sum();
assert!(
(sum - 1.0).abs() < 1e-7,
"{} teams: sum = {sum}",
perf.len()
);
}
}
/// A rival with a tiny sigma is a step function in disguise. Fixed-node
/// quadrature steps over it and lands ~1e-2 out while still looking like a
/// probability; this is the case that rules that approach out.
#[test]
fn win_probabilities_survive_a_rival_with_a_tiny_sigma() {
let perf = [g(0.0, 0.001), g(0.5, 6.0), g(-0.5, 6.0)];
let got = win_probabilities(&perf, &flat(3, 0.0));
let sum: f64 = got.iter().sum();
assert!((sum - 1.0).abs() < 1e-6, "sum = {sum}, probs = {got:?}");
}
#[test]
fn a_stronger_team_is_more_likely_to_win() {
let perf = [g(10.0, 3.0), g(0.0, 3.0), g(-10.0, 3.0)];
let p = win_probabilities(&perf, &flat(3, 0.0));
assert!(p[0] > p[1] && p[1] > p[2], "not monotone: {p:?}");
}
#[test]
fn identical_teams_are_equally_likely_to_win() {
let perf = [g(1.0, 4.0), g(1.0, 4.0), g(1.0, 4.0)];
let p = win_probabilities(&perf, &flat(3, 0.0));
for probs in p.windows(2) {
assert!((probs[0] - probs[1]).abs() < 1e-9, "asymmetric: {p:?}");
}
}
/// Every realisation sorts into exactly one finishing order, so the whole
/// distribution must sum to one — with or without a draw margin.
#[test]
fn outcome_distribution_sums_to_one() {
for (perf, eps) in [
(vec![g(0.0, 6.0), g(0.0, 6.0)], 0.0),
(vec![g(0.0, 6.0), g(0.0, 6.0)], 2.0),
(vec![g(0.0, 6.0), g(0.0, 6.0), g(0.0, 6.0)], 0.0),
(vec![g(5.0, 6.0), g(0.0, 3.0), g(-5.0, 1.0)], 1.5),
(vec![g(0.0, 0.05), g(0.5, 6.0), g(-0.5, 6.0)], 1.0),
(
vec![g(6.0, 2.0), g(2.0, 6.0), g(-2.0, 1.0), g(-6.0, 4.0)],
1.0,
),
] {
let n = perf.len();
let dist = outcome_distribution(&perf, &flat(n, eps));
let sum: f64 = dist.iter().map(|(_, p)| p).sum();
assert!(
(sum - 1.0).abs() < 1e-6,
"{n} teams, eps={eps}: sum = {sum} over {} outcomes",
dist.len()
);
assert!(dist.iter().all(|(_, p)| *p >= 0.0), "negative probability");
}
}
/// With two teams the distribution is the exact win/draw/loss triple.
#[test]
fn two_team_distribution_matches_the_closed_form() {
let perf = [g(3.0, 6.0), g(-2.0, 1.0)];
let eps = 1.5;
let dist = outcome_distribution(&perf, &flat(2, eps));
let (wa, wb) = closed_form_two(perf[0], perf[1], eps);
let find = |ranks: &[u32]| {
dist.iter()
.find(|(r, _)| r == ranks)
.map_or(0.0, |(_, p)| *p)
};
assert!(
(find(&[0, 1]) - wa).abs() < 1e-6,
"a wins: {}",
find(&[0, 1])
);
assert!(
(find(&[1, 0]) - wb).abs() < 1e-6,
"b wins: {}",
find(&[1, 0])
);
assert!(
(find(&[0, 0]) - (1.0 - wa - wb)).abs() < 1e-6,
"draw: {}",
find(&[0, 0])
);
}
/// Asking for one ranking must agree with that ranking's entry in the
/// full distribution — the two use different code paths to the same value.
#[test]
fn ranking_probability_agrees_with_the_distribution() {
let perf = [g(5.0, 6.0), g(0.0, 3.0), g(-5.0, 1.0)];
let eps = 1.5;
let margins = flat(3, eps);
let dist = outcome_distribution(&perf, &margins);
for (ranks, expected) in &dist {
let direct = ranking_probability(&perf, &margins, ranks);
assert!(
(direct - expected).abs() < 1e-9,
"ranks {ranks:?}: direct {direct} vs distribution {expected}"
);
}
}
/// Tie mass is controlled by the draw margin. Only the *all-tied* outcome
/// is monotone in it: every one of its constraints is a window that widens
/// with the margin. A partially-tied outcome like `[0, 0, 1]` is not, and
/// must not be asserted to be — widening the margin makes its tie easier
/// but its "and the last team is strictly behind by more than the margin"
/// clause harder, so it peaks and then falls.
#[test]
fn all_tied_probability_grows_with_the_draw_margin() {
let perf = [g(0.0, 4.0), g(0.0, 4.0), g(-8.0, 2.0)];
let mut previous = 0.0;
for eps in [0.0, 0.5, 1.0, 2.0, 4.0, 8.0, 24.0] {
let p = ranking_probability(&perf, &flat(3, eps), &[0, 0, 0]);
assert!(p >= previous, "eps={eps}: {p} < {previous}");
if eps == 0.0 {
assert!(p < 1e-12, "a tie needs a margin, got {p}");
}
previous = p;
}
assert!(
previous > 0.9,
"a very wide margin ties everyone: {previous}"
);
}
/// The converse, stated as the non-property it is: a partially-tied
/// outcome is non-monotone in the margin. Pinning this down stops a future
/// change from "fixing" it into monotonicity and quietly breaking the model.
#[test]
fn a_partially_tied_outcome_peaks_in_the_middle() {
let perf = [g(0.0, 4.0), g(0.0, 4.0), g(-8.0, 2.0)];
let sweep: Vec<f64> = [0.5, 2.0, 4.0, 8.0, 16.0]
.iter()
.map(|&eps| ranking_probability(&perf, &flat(3, eps), &[0, 0, 1]))
.collect();
let peak = sweep
.iter()
.enumerate()
.fold(
(0, 0.0),
|(bi, bv), (i, &v)| if v > bv { (i, v) } else { (bi, bv) },
)
.0;
assert!(
peak > 0 && peak < sweep.len() - 1,
"expected an interior peak: {sweep:?}"
);
}
/// With no draw margin a tie has probability exactly zero, and the
/// enumeration must not waste work pretending otherwise.
#[test]
fn ties_are_impossible_without_a_draw_margin() {
let perf = [g(0.0, 4.0), g(0.0, 4.0), g(0.0, 4.0)];
let dist = outcome_distribution(&perf, &flat(3, 0.0));
assert_eq!(dist.len(), 6, "expected only the 6 strict orders: {dist:?}");
assert!(dist.iter().all(|(r, _)| {
let mut seen = r.clone();
seen.sort_unstable();
seen.dedup();
seen.len() == r.len()
}));
}
}