Odds Generation — point · game + game · game (badminton)

Enter the match's Point Handicap and Points Total in Malay and its Moneyline in decimal — margins still embedded. Rally scoring makes the score a pure race — first to 21 by 2, capped at 30 (golden point at 29-29), best of three games — so one strength parameter would pin the totals the moment the handicap is known: the same one-parameter straitjacket that made Poisson wrong for basketball. The engine (ADR-030):

strips each bookmaker margin (Power method — all three books two-way);
bisects the rally-win share r against the Point Handicap on the exact race DP (win-by-2 barriers, golden-point cap);
fits γ, a score coupling ("restoring force": the rally probability tilts by γ · deficit), to the Points Total — γ > 0 keeps games close and lengthens them, γ < 0 shortens; a two-sided knob a mixture cannot give;
fits κ_M, a match-level form shock, to the Moneyline (toggle in the ML box) with a probed direction — counterintuitively κ_M raises the re-solved favorite's ML, and an ML outside the reachable band is a cross-anchor inconsistency surfaced as Δ ML, not a solver failure;
prices both scopes off the one solved distribution — game · game (Moneyline, ±1.5 game handicap) and point · game (match point lines plus per-game sheets, game 3 conditional on being played).

Stage 1 — rally scoring is a pure race: first to 21 by 2, capped at 30 with the golden point at 29-29, best of three games, solved on the exact race DP — Stage 1 — rally scoring is a pure race: first to 21 by 2, capped at 30 (golden point at 29-29), best of three — solved on the exact race DP

Each stage re-solves the same fair targets — flip Race / γ / κ_M and watch the totals humps breathe while the handicap holds. Race lengths are per-tournament config: the BWF approved 3×15 (cap 21) at the 2026 AGM with rollout pending, so both formats ship here; switching loads that format's seed quotes.

Point Handicap (Malay)LineHomeAway

Points Total (Malay)LineOverUnder

Moneyline (Decimal) · κ_M from MLHomeAway

MarginsMulti-wayShinTwo-wayladder

↑/↓ steps a field, Shift steps bigger; Malay pairs step their partner the opposite way. Quotes are whole-match; per-game sheets derive from the same solve, game 3 conditional on being played.

Valid inputs: Malay in [−1, +1] non-zero with margin, decimals above 1. The κ_M fit re-solves (r, γ) at every trial — allow it a second.

r 0.5296γ 0.0076κ_M 0.0391G3 40.4%round-trip Δp 1.0e-5 / 9.8e-6Δ ML 0.00%

Match distribution

r 0.5296γ 0.0076κ_M 0.0391G3 40.4%E[pts] 90.6

Final games score — P(home g, away g) ×100

	A=0	A=1	A=2
H=0			16.3
H=1			15.8
H=2	43.4	24.6

Points margin pmf — P(home pts − away pts) ×100

-21	-20	-19	-18	-17	-16	-15	-14	-13	-12	-11	-10	-9	-8	-7	-6	-5	-4	-3	-2	-1	0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11	+12	+13	+14	+15	+16	+17	+18	+19	+20	+21	+22	+23	+24	+25	+26
0.1	0.1	0.1	0.2	0.3	0.4	0.6	0.8	1.0	1.3	1.6	1.9	2.2	2.5	2.7	2.9	2.9	3.3	1.8	2.0	1.8	1.8	2.0	2.4	2.3	4.7	4.7	5.1	5.4	5.4	5.3	5.0	4.6	4.1	3.6	3.0	2.5	2.0	1.6	1.2	0.9	0.6	0.5	0.3	0.2	0.1	0.1	0.0

Points total pmf ×100

58	59	60	61	62	63	64	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80	81	82	83	84	85	86	87	88	89	90	91	92	93	94	95	96	97	98	99	100	101	102	103	104	105	106	107	108	109	110	111	112	113	114	115	116	117	118	119	120	121	122	123	124	125	126	127	128	129	130	131	132	133	134	135	136
0.0	0.1	0.1	0.2	0.3	0.5	0.7	0.9	1.2	1.6	2.0	2.5	2.9	3.4	3.9	4.2	4.5	4.6	4.5	4.2	3.8	3.2	2.4	1.6	1.7	0.8	1.1	0.4	0.7	0.2	0.4	0.1	0.2	0.1	0.1	0.0	0.1	0.0	0.1	0.1	0.1	0.1	0.2	0.2	0.4	0.5	0.6	0.8	1.1	1.3	1.6	1.9	2.2	2.4	2.6	2.7	2.8	2.7	2.5	2.3	2.0	1.7	1.4	1.2	1.0	0.8	0.7	0.5	0.5	0.3	0.3	0.2	0.2	0.1	0.1	0.1	0.1	0.0	0.0

two humps — a straight-games match vs a decider — the bimodality no unimodal count model can price

home awayone distribution, both scopes: the games table is game·game, the pmfs are point·game

Market6 in scope

a match always has a winner — no tie leg

Outcome	Fair	Dec	Priced	Malay
Home	68.0%	1.471	1.439	+0.44
Away	32.0%	3.122	2.926	-0.52

How these numbers are computed

1 · Strip the bookmaker margins (Power method — every book two-way)

Point Handicap -5.50 → P(home covers)

	Home	Away
Malay quote	+0.88	+0.99
Decimal $d$	1.8800	1.9900
Implied $q = 1/d$	0.5319	0.5025
Fair $p = q^{1/x}$	0.5149	0.4851
Fair decimal $1/p$	1.9420	2.0616

overround Q = 1.0344 → solve

q_1^{1/x} + q_2^{1/x} = 1

→ x = 0.9511 → fair P(home covers) = 51.49%

Points Total 78.50 → P(over)

	Over	Under
Malay quote	+0.80	-0.93
Decimal $d$	1.8000	2.0753
Implied $q = 1/d$	0.5556	0.4819
Fair $p = q^{1/x}$	0.5375	0.4625
Fair decimal $1/p$	1.8604	2.1622

overround Q = 1.0374 → solve

q_1^{1/x} + q_2^{1/x} = 1

→ x = 0.9468 → fair P(over) = 53.75%

Moneyline → P(home wins) — the κ_M anchor (stage 3)

	Home	Away
Decimal quote	1.45	2.99
Decimal $d$	1.4500	2.9900
Implied $q = 1/d$	0.6897	0.3344
Fair $p = q^{1/x}$	0.6797	0.3203
Fair decimal $1/p$	1.4713	3.1216

overround Q = 1.0241 → solve

q_1^{1/x} + q_2^{1/x} = 1

→ x = 0.9622 → fair P(home ML) = 67.97%

Standalone strips: Margin — two-way (Power).

2 · The race — rally scoring is a pure best-of race

Every rally scores a point, so a game is a race to 21 by 2, capped at 30 (golden point at 29-29), best of three. A single strength r would pin the totals once the handicap is known — the same one-parameter straitjacket that made Poisson wrong for basketball — so the rally probability carries a score coupling:

P(\text{A wins the rally at } x\text{-}y) = \operatorname{clip}\!\left(r + \gamma\,(y - x)\right)

γ > 0 is a restoring force (the trailing player lifts — games stay close and run long, deuces multiply); γ < 0 lets games run away and shorten. The DP walks the exact (x, y) lattice, so the coupling costs nothing; the match folds games carrying the points margin and total pmfs, the games joint and each game's grid. Validated against a 200k-run Monte Carlo.

3 · Stages — γ from the total, κ_M from the Moneyline (direction probed)

Stage	r	γ	κ_M	Δ cover	Δ total	Δ ML
Race	0.5224	0.0000	0.0000	4.1e-6	3.7e-2	0.83%
γ	0.5167	-0.0096	0.0000	9.4e-6	7.9e-6	5.42%
κ_M	0.5296	0.0076	0.0391	1.0e-5	9.8e-6	0.00%

κ_M is a 3-node form mixture on r. Counterintuitively it RAISES the re-solved favorite’s ML — the mixture widens cumulative point margins faster than the game race, so the handicap re-solves to a stronger r — hence the direction is probed at the bracket ends, never assumed. An ML outside the reachable band keeps the nearer endpoint and leaves Δ ML large: a cross-anchor inconsistency (alertable), not a solver failure.

4 · Periods — game 3 is a random period · formats

Game 3 exists with probability 40.4% here; its markets are void if it is not played, so its sheet prices conditional on being played — and under the κ_M mixture the conditioning re-weights the form nodes (deciders come from the close-form nodes). The same fold yields the game·game scope for free (Moneyline, ±1.5 game handicap, correct score). Race lengths are per-tournament config: the BWF approved 3×15 (cap 21) at the AGM on 2026-04-25 with rollout pending, so 3×21 and 3×15 will coexist — the format toggle above re-prices the same latents under either race.

5 · Read “Moneyline” · margins · notes

Full match: P(home wins the game race) — no tie leg. Per game: the game winner, conditional on the game being played (void otherwise). Fair groups are re-margined exactly as in goal-regular (Power ladder two-way, Shin multi-way).

targets  tH = powerStrip(HDP pair)   tO = powerStrip(OU pair)   tW = powerStrip(ML pair)

stage iid   : γ = 0, κ_M = 0        bisect r until P(A covers | dist) = tH
stage γ     : bisect γ until P(over | dist) = tO      (r re-solved inside — restoring
              force lengthens games, runaway shortens; two-sided, unlike a mixture)
stage κ_M   : probe ml(0) vs ml(κ_max) at re-solved (r, γ) → bisect toward tW
              (κ_M RAISES the re-solved favorite's ML; unreachable target ⇒ Δ ML alert)

dist: game DP on the (x, y) lattice — race to 21 by 2, cap 30 (golden point) —
      folded best-of-3 with margin/total pmfs, games joint, per-game grids
scopes: ONE dist prices game·game (ML, ±1.5) and point·game (match + per-game, conditional)

Engine + invariants: docs/src/lib/odds/badminton.ts, __tests__/badminton.test.ts; decision record ADR-030 (extends ADR-029).