Odds Generation — run · inning (cricket)

Pick the format (T20/ODI), say who bats first, and enter the batting-first innings total in Malay plus the Match Winner in decimal — margins still embedded. A limited-overs innings is overs of six balls scoring {0, 1, 2, 3, 4, 6}, so the model is compound from the ball up (ADR-029):

strips both margins (Power method) to fair targets;
builds ball → over → innings under a phase curve (powerplay / middle / death — cricket's quarter weights) and back-solves the two innings abilities: the innings line fixes μ₁, the Match Winner fixes μ₂;
independent overs come out far too thin (σ ≈ 15 vs ≈ 28 real), so a day-form mixture G = 1 + τZ scales every over and τ is solved to the dispersion prior — collapses and flat tracks correlate overs;
the chase truncates: the second innings stops at the target (won chases pile up just past it, failed ones fall short), scores level is the tie — the 1X2's third leg, settled by the super over in T20 — so 2nd-innings Over/Unders price off the settled distribution, not the ability;
group runs (the powerplay) and single-over markets read the same phase curve and mixture, consistent with the innings by construction.

Stage 1 — ball → over → innings: per-ball outcomes {0,1,2,3,4,6} compound six to an over and twenty to the innings under the powerplay/middle/death phase curve — Stage 1 — compound from the ball up: {0, 1, 2, 3, 4, 6} per ball, ×6 into an over, ×20 into the innings — weighted by the powerplay / middle / death phase curve

Wickets and resource decay are folded into the phase curve and mixture rather than modelled per ball — the honest limit of a quotes-anchored page; ball-level state belongs to the ADR-029 phase-2 engine.

Home bats first

Innings 1 total (Malay) — Home batsLineOverUnder

Match Winner (Decimal)Homebats 1stAwaychases

Shape priors · marginsσ inningsdispersionMulti-wayShinTwo-wayladder

Phase shares % (PP · middle · death)ov 1–6middledeath

↑/↓ steps a field, Shift steps bigger; the Malay pair steps its partner the opposite way. Quotes: the batting-first innings total + the match winner; the second innings and every segment derive from the same model.

Valid inputs: Malay in [−1, +1] non-zero, decimals above 1, each book carrying margin, σ above ~8. The solve mixes 7 form nodes over 20 overs — allow it a second or two.

μ₁ 166.52μ₂ 162.47τ 0.1371σ₁ 28.00tie 1.0%round-trip Δp 4.9e-7 / 1.9e-7Overs → Form τ → Chase

Innings runs — P(runs ∈ bin) ×100 · bins of 10

μ₁ 166.5μ₂ 162.5τ 0.137σ₁ 28.0tie 1.0%

	90	100	110	120	130	140	150	160	170	180	190	200	210	220	230	240	250
Innings 1 (Home)	0.2	1.0	2.7	5.4	8.4	11.1	12.9	13.5	12.8	10.9	8.5	5.9	3.6	1.9	0.9	0.3	0.1
Innings 2 (Away) — settled	0.5	2.1	5.6	10.5	14.9	17.2	16.6	13.6	9.4	5.5	2.6	1.0	0.3	0.1	0.0	0.0	0.0

the settled second innings piles up just past the target (won chases stop) and spreads low (failed chases fall short)

Market2 in scope

the third leg is scores-level — a T20 tie heads to the super over

Outcome	Fair	Dec	Priced
Home	53.5%	1.868	1.800
Draw (tie)	1.0%	100.693	49.393
Away	45.5%	2.199	2.109

How these numbers are computed

1 · Strip the bookmaker margins (Power method)

Innings 1 total 165.5 → P(over)

	Over	Under
Malay quote	+0.90	+0.92
Decimal $d$	1.9000	1.9200
Implied $q = 1/d$	0.5263	0.5208
Fair $p = q^{1/x}$	0.5028	0.4972
Fair decimal $1/p$	1.9888	2.0113

overround Q = 1.0471 → solve

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

→ x = 0.9335 → fair P(over) = 50.28%

Match Winner → P(chasing side wins)

	Home	Away
Decimal quote	1.80	2.10
Decimal $d$	1.8000	2.1000
Implied $q = 1/d$	0.5556	0.4762
Fair $p = q^{1/x}$	0.5403	0.4597
Fair decimal $1/p$	1.8509	2.1752

overround Q = 1.0317 → solve

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

→ x = 0.9547 → fair P(chase wins) = 45.97%

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

2 · Ball → over → innings, and why independent overs are not enough

A ball scores {0, 1, 2, 3, 4, 6}; six balls convolve into an over, and the innings convolves 20 overs under a phase curve — powerplay / middle / death get 29/41/30% of the runs across 6/9/5 overs (cricket's quarter weights). But independent overs give σ ≈ 18.8 — real innings run σ ≈ 28.0 because collapses and flat tracks correlate every over. The fix is a day-form mixture: every over's mean scales by $G = 1 + \tau Z$ , and τ is solved so the innings dispersion hits the prior:

Stage	μ₁	μ₂	τ	σ₁	tie	Δ over	Δ ML
Overs	166.0	163.4	0.000	18.8	1.5%	1.9e-7	4.3e-7
Form τ	166.5	162.5	0.137	28.0	1.0%	4.9e-7	1.9e-7
Chase	166.5	162.5	0.137	28.0	1.0%	4.9e-7	1.9e-7

Each stage re-solves both fair targets: the innings line pins μ₁ (size) and the Match Winner pins μ₂ (the chasing ability) — with ties heading to the super over, the ML read is $P(\text{chase wins}) + \tfrac{1}{2}P(\text{tie})$ .

3 · The chase — the second innings truncates at the target

Whoever bats second stops the moment the target falls: with a first-innings score $r_1$ and chasing ability $A_2$ ,

r_2 = \begin{cases} A_2 & A_2 \le r_1 \;\; \text{(failed chase — or the tie at } A_2 = r_1\text{)} \\ r_1 + 1 + U & A_2 \ge r_1 + 1 \;\; \text{(won — } U \text{ is the winning hit's overshoot)} \end{cases}

which is why the settled second-innings pmf (the Chase stage in the panel) piles up just past the target and never shows the ability tail — and why 2nd-innings Over/Unders must price off the settled distribution, not the ability. Who wins is truncation-free, so μ₂ matches the Form stage. The tie mass P = 1.0% is the 1X2's third leg.

4 · Read “1X2” · margins · notes

Three legs: batting-first side wins, scores level (the tie), chasing side wins — read off P(A₂ vs the target). Fair groups are re-margined exactly as in goal-regular (Power ladder two-way, Shin multi-way).

targets  tO = powerStrip(innings pair)     tW = powerStrip(match-winner pair)

stage ∈ {Overs (τ=0), Form τ, Chase}:          # each re-solves the same targets
  sweep: bisect μ₁ until P(runs₁ > line) = tO
         bisect τ  until sd(innings₁)   = σ prior          # Form/Chase stages
  bisect μ₂ until P(chase wins) + ½·P(tie) = tW

innings: ball pmf(mean) → over (6-fold conv) → phase curve over 20 overs
         → mix over day-form G = 1 + τZ (7 nodes)
chase:   r₂ = A₂ if short; = target + overshoot if won; level = tie (super over)

Wickets/resource decay are folded into the phase curve and mixture rather than modelled per ball — the honest limit of a quotes-anchored page (ADR-029 phase 2 owns ball-level state). Engine + invariants: docs/src/lib/odds/cricket.ts, __tests__/cricket.test.ts.