Odds Generation — run · inning (baseball)

Enter the full game's Moneyline in decimal and its Total and Run Line quotes in Malay — margins still embedded. Baseball runs are counts but not Poisson: an inning is scoreless ~72% of the time yet carries a fat "big-inning" tail, so a team's game variance (~10) is more than double its mean. The engine (ADR-029):

strips each bookmaker margin (Power method — all three books are two-way);
models each inning as a zero-inflated geometric (mean μ/9, tail q) and back-solves (μ_H, μ_A) by nested bisection — the Total fixes the run environment, the Moneyline fixes the split;
applies the sport's endgame rules — the corrections here are not a statistical tilt but exact stopping rules: the bottom of the 9th is skipped when home already leads, a walk-off stops play the moment the lead is taken (home wins compress toward +1), and extra innings replay until the tie breaks — then re-solves;
optionally fits the big-inning tail q to the Run Line (toggle in the RL box) — the RL is pure dispersion information at a fixed Moneyline, exactly as the ML pins basketball's σ_M;
derives First 5/3/7 innings and Inning 1 markets by convolving the same per-inning distributions — the inning is the period unit, no scaling knobs.

Stage 1 — nine-inning convolution: a zero-inflated geometric inning (~72% scoreless, big-inning tail) convolves nine times into the game distribution — Stage 1 — an inning is scoreless ≈72% of the time with a big-inning tail: zero-inflated geometric innings convolve ⊛9 into the game — σ² > 2μ, counts but not Poisson

Each stage re-solves the same fair targets, so the score grid's stage switcher is a true before/after — flip Convolution / Endgame / RL q and watch the home-win-by-1 stripe fatten.

Moneyline (Decimal) — incl. extrasHomeAway

Total quote (Malay)LineOverUnder

Run Line (Malay) · q from RLLineHomeAway

Shape priors · marginsq (tail)pre-RLextras ×ghost runnerMulti-wayShinTwo-wayladder

↑/↓ steps a field, Shift steps bigger; Malay pairs step their partner the opposite way. Quotes are the full game (extra innings included); First-N periods derive by convolving the same per-inning distributions.

Valid inputs: decimals above 1, Malay in [−1, +1] non-zero, each book carrying margin, q in (0.15, 0.8). The solve walks all 18 half-innings — allow it a second.

μ_H 5.6090μ_A 4.3818q 0.6669extras 8.6%round-trip Δp 5.0e-9 / 5.5e-9 / 1.2e-6Convolution → Endgame → RL q

Score grid — P(home = h, away = a) ×100

μ_H 5.609μ_A 4.382q 0.667skip-9 49.6%walk-off 9.2%extras 8.6%

	A=0	A=1	A=2	A=3	A=4	A=5	A=6	A=7	A=8	A=9	A=10	A=11	A=12	A=13	A=14
H=0	0.0	1.8	1.6	1.4	1.2	1.0	0.8	0.7	0.5	0.4	0.4	0.3	0.2	0.2	0.1
H=1	3.9	0.0	1.3	1.1	0.9	0.8	0.6	0.5	0.4	0.3	0.3	0.2	0.2	0.1	0.1
H=2	2.4	2.3	0.0	1.1	1.0	0.8	0.7	0.5	0.4	0.4	0.3	0.2	0.2	0.1	0.1
H=3	2.1	1.3	2.2	0.0	1.0	0.8	0.7	0.5	0.4	0.3	0.3	0.2	0.2	0.1	0.1
H=4	1.9	1.1	1.1	1.8	0.0	0.8	0.6	0.5	0.4	0.3	0.3	0.2	0.2	0.1	0.1
H=5	1.6	1.0	0.9	0.9	1.5	0.0	0.6	0.5	0.4	0.3	0.2	0.2	0.1	0.1	0.1
H=6	1.4	0.8	0.8	0.7	0.7	1.2	0.0	0.4	0.4	0.3	0.2	0.2	0.1	0.1	0.1
H=7	1.2	0.7	0.6	0.6	0.5	0.5	0.9	0.0	0.3	0.3	0.2	0.2	0.1	0.1	0.1
H=8	1.0	0.6	0.5	0.5	0.4	0.4	0.4	0.6	0.0	0.2	0.2	0.1	0.1	0.1	0.1
H=9	0.8	0.5	0.4	0.4	0.3	0.3	0.3	0.2	0.4	0.0	0.2	0.1	0.1	0.1	0.0
H=10	0.7	0.4	0.4	0.3	0.3	0.2	0.2	0.2	0.2	0.3	0.0	0.1	0.1	0.1	0.0
H=11	0.5	0.3	0.3	0.3	0.2	0.2	0.2	0.1	0.1	0.1	0.2	0.0	0.1	0.0	0.0
H=12	0.4	0.3	0.2	0.2	0.2	0.2	0.1	0.1	0.1	0.1	0.1	0.1	0.0	0.0	0.0
H=13	0.3	0.2	0.2	0.2	0.1	0.1	0.1	0.1	0.1	0.1	0.0	0.0	0.1	0.0	0.0
H=14	0.3	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.0	0.0	0.0	0.0	0.1	0.0

home win tie away winsettlement grid — extras resolved; note the fat home-win-by-1 stripe (walk-offs)

Market5 in scope

settles after extra innings — the endgame construction leaves no tie mass

Outcome	Fair	Dec	Priced	Malay
Home	58.7%	1.703	1.667	+0.67
Away	41.3%	2.423	2.339	-0.75

How these numbers are computed

1 · Strip the bookmaker margins (Power method — all books two-way)

Moneyline (extra innings included) → P(home wins)

	Home	Away
Decimal quote	1.65	2.30
Decimal $d$	1.6500	2.3000
Implied $q = 1/d$	0.6061	0.4348
Fair $p = q^{1/x}$	0.5873	0.4127
Fair decimal $1/p$	1.7026	2.4233

overround Q = 1.0408 → solve

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

→ x = 0.9410 → fair P(home ML) = 58.73%

Total 8.50 → 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%

Run Line -1.50 → P(home covers)

	Home	Away
Malay quote	-0.80	+0.72
Decimal $d$	2.2500	1.7200
Implied $q = 1/d$	0.4444	0.5814
Fair $p = q^{1/x}$	0.4307	0.5693
Fair decimal $1/p$	2.3218	1.7565

overround Q = 1.0258 → solve

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

→ x = 0.9627 → fair P(home covers) = 43.07%

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

2 · The inning model — zero-inflated geometric runs

Runs are not Poisson: an inning is scoreless ≈ 72% of the time yet carries a fat “big inning” tail, so a team's game variance (≈ 10) is more than double its mean (≈ 4.6). Each inning draws

P(0) = \pi = 1 - m(1-q) \qquad P(k \ge 1) = (1-\pi)(1-q)\,q^{\,k-1} \qquad m = \mu/9

— the mean per inning $m$ carries the team rate and the tail $q$ is the big-inning knob (this run: home $\pi = 0.792$ , q = 0.667). A team's nine innings convolve into its runs distribution, and the two teams multiply into the score grid. The Total is the size and the Moneyline is the split — the same nested bisection as goal-regular recovers $(\mu_H, \mu_A) = (5.609,\ 4.382)$ so both fair targets hold on the settlement grid.

3 · Endgame rules — the corrections are the sport's own stopping rules

Basketball needed a tie-inflation ι; baseball's corrections are deterministic rules, applied on the inning walk: the bottom of the 9th is skipped when home already leads (49.6% here — the home total is truncated); a walk-off stops play the moment home takes the lead, compressing home wins toward +1 (margin mix {1: 75%, 2: 14%, 3: 6%, 4: 5%}); extra innings (8.6%) replay until the tie breaks, at 2.00× the inning rate (the ghost-runner era). Every stage re-solves the same fair targets:

Stage	μ_H	μ_A	q	skip-9	walk-off	extras	Δ ML	Δ total	Δ RL
Convolution	4.952	4.108	0.460	0.0%	0.0%	0.0%	8.9e-9	1.1e-8	1.2e-3
Endgame	4.948	4.091	0.460	48.0%	10.8%	9.7%	6.1e-9	1.9e-9	4.2e-2
RL q	5.609	4.382	0.667	49.6%	9.2%	8.6%	5.0e-9	5.5e-9	1.2e-6

The RL stage bisects q until the grid reproduces the stripped Run Line — the RL is pure dispersion information at a fixed ML, exactly as the ML pins basketball’s σ_M. Note μ's are hypothetical full-nine rates: the settled total mean sits below them because skipped bottom-9s and walk-offs remove runs.

4 · Periods — first-N-innings convolutions

Baseball's period model needs no scaling knobs: the inning is the unit. First-5 markets convolve the same per-inning distributions five times; ties are real outcomes there, so the F5 moneyline pushes on a tie (R = 18.2% here, fair home = $W/(1-R)$ = 57.68%). The endgame rules never touch innings 1–8, so first-N grids are rule-free by construction. Inning weights (starters vs bullpen, top of the order in inning 1) are a calibration refinement — ADR-029 O1.

5 · Read “Moneyline” off the grid · margins · notes

Home sums cells with h > a. Full game: the endgame construction leaves no tie mass. First-N periods: a tie pushes, fair = W/(1−R). Fair groups are re-margined exactly as in goal-regular (Power ladder two-way, Shin multi-way).

targets  tW = powerStrip(ML pair)      tO = powerStrip(Total pair)      tR = powerStrip(RL pair)

stage ∈ {Convolution, Endgame, RL q}:          # each re-solves the same targets
  bisect μ_total (size)  until P(over T | grid)  = tO     # outer — goal-regular's solveLambdas
    bisect μ_H   (split) until P(home ML | grid) = tW     # inner
  RL stage: bisect q until P(home covers | grid) = tR

grid: per-inning ZIG(m = μ/9, q) → innings 1–8 convolve → top 9 → bottom 9
      skipped if home leads · walk-off stops play (margin mix 1–4) · extras ×2.00
periods: first-N convolutions of the same inning pmfs (no endgame rules)

Grid runs 0..34 runs a side (tail mass clamps into the last cell). Engine + invariants: docs/src/lib/odds/baseball.ts, __tests__/baseball.test.ts; decision record ADR-029.