Odds Generation — game · set + set · set (tennis)

Enter the match's Game Handicap and Game Total in Malay and its Moneyline in decimal — margins still embedded. Tennis scores are not counts: they are a nested race — point → game (win by 2 from deuce) → set (first to 6 by 2, tiebreak at 6-6) → match — so totals come out bimodal (straight sets vs a decider) with a parity comb (a set never totals 11 games; a 7-6 counts 13). The engine (ADR-030):

strips each bookmaker margin (Power method — every book here is two-way; tennis has no draw);
pins the serve sum p_A + p_B to the tour/surface prior (ATP ≈ 1.29, WTA ≈ 1.14) and bisects the serve difference against the Game Handicap — the match quotes leave the sum on a near-flat ridge, so alone they cannot fit it;
breaks the ridge with the Set-1 Total when the book quotes set 1 (Saba does): more serve ⇒ more holds ⇒ longer sets, so P(set-1 over) rises monotonically with the sum — stage Set 1 Σ bisects the sum against that quote and the prior becomes a fallback (a shape prior upgraded to a per-match reading, ADR-030 O2);
consumes the Game Total with whichever knob can reach it: τ, a match-level form mixture (matches shorten), or extra serve sum (matches lengthen; disabled once set-1 pinned the sum — τ only) — the difference re-solves at every trial;
never fits the Moneyline or the Set-1 Handicap: they are the cross-anchor consistency readouts (Δ ML, Δ S1hcp) — a large value means the book's quotes disagree with each other, a risk signal;
prices both scopes off the one solved distribution — set · set (Moneyline, Set Handicap, Set Total) and game · set (match game lines plus every per-set sheet, conditional on the set being played).

Stage 1 — the nested race: point → game (win by 2 from deuce) → set (first to 6 by 2, tiebreak at 6-6) → match (best of 3); every level is a race, no level is a count — Stage 1 — a race nested three levels deep: point → game (win by 2 from deuce) → set (first to 6 by 2, tiebreak at 6-6) → match — every level a race, no level a count

Each stage re-solves the same fair targets, so the stage switcher is a true before/after — flip iid @ prior / Set 1 Σ / Length and watch the totals hump move while the handicap holds (clear the Set-1 quotes to see the prior-only pipeline). Best-of-3 and best-of-5 are format configs (all four slams play a 10-point decider tiebreak since 2022); switching loads that format's seed quotes.

Game Handicap (Malay)LineHomeAway

Game Total (Malay)LineOverUnder

Moneyline (Decimal) — consistency readoutHomeAway

Set-1 Total (Malay) — pins Σ per matchLineOverUnder

Set-1 Handicap (Malay) — consistency readoutLineHomeAway

Serve-sum prior · marginsΣ priorATP 1.29Multi-wayShinTwo-wayladder

↑/↓ steps a field, Shift steps bigger; Malay pairs step their partner the opposite way. The Moneyline and the Set-1 handicap are never fitted — they are cross-anchor consistency readouts (Δ ML / Δ S1hcp); clear either to skip it.

The Set-1 total is the ridge breaker: when present it pins the serve sum per match (stage Set 1 Σ) and the prior becomes a fallback — clear it to see the prior-driven pipeline. Valid inputs: Malay in [−1, +1] non-zero with margin, decimals above 1, serve-sum prior in (1.04, 1.54).

p_A 0.6620p_B 0.6225τ 0.0292P(set 3) 41.8%TB 42.2%round-trip Δp 1.0e-7 / 4.7e-6Δ ML 0.03%Δ S1hcp 0.17%

Match distribution

p_A 0.6620p_B 0.6225Σ 1.285 ←S1τ 0.0292TB 42.2%E[games] 24.4

Final sets score — P(home s, away s) ×100

	A=0	A=1	A=2
H=0			17.5
H=1			16.9
H=2	40.7	24.8

Games margin pmf — P(home games − away games) ×100

-10	-9	-8	-7	-6	-5	-4	-3	-2	-1	0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11
0.1	0.3	0.7	1.9	3.1	5.2	6.7	6.2	4.7	3.5	3.8	4.1	6.5	9.9	12.7	11.4	8.4	5.9	2.8	1.4	0.4	0.1

Games total pmf ×100

13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39
0.1	0.5	1.6	3.0	5.9	7.0	8.2	7.8	5.3	8.2	6.1	1.2	3.4	4.6	3.2	3.9	4.2	4.4	5.0	4.9	3.1	2.3	2.9	1.7	0.3	0.6	0.5

two humps — straight sets vs a decider — with a parity comb: no set totals 11 games, a 7-6 counts 13

home awayone distribution, both scopes: the sets table is set·set, the pmfs are game·set

Market8 in scope

a match always has a winner — no tie leg

Outcome	Fair	Dec	Priced	Malay
Home	65.6%	1.525	1.492	+0.49
Away	34.4%	2.904	2.748	-0.57

How these numbers are computed

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

Game Handicap -2.50 → P(home covers)

	Home	Away
Malay quote	+0.82	-0.95
Decimal $d$	1.8200	2.0526
Implied $q = 1/d$	0.5495	0.4872
Fair $p = q^{1/x}$	0.5317	0.4683
Fair decimal $1/p$	1.8808	2.1353

overround Q = 1.0366 → solve

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

→ x = 0.9480 → fair P(home covers) = 53.17%

Game Total 22.50 → P(over)

	Over	Under
Malay quote	+0.85	-0.98
Decimal $d$	1.8500	2.0204
Implied $q = 1/d$	0.5405	0.4949
Fair $p = q^{1/x}$	0.5232	0.4768
Fair decimal $1/p$	1.9114	2.0972

overround Q = 1.0355 → solve

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

→ x = 0.9496 → fair P(over) = 52.32%

Set-1 Total — stripped as the SERVE-SUM anchor (the ridge breaker)

	Over	Under
Malay quote	+0.71	-0.83
Decimal $d$	1.7100	2.2048
Implied $q = 1/d$	0.5848	0.4536
Fair $p = q^{1/x}$	0.5668	0.4332
Fair decimal $1/p$	1.7642	2.3086

overround Q = 1.0383 → solve

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

→ x = 0.9450 → fair P(set-1 over) = 56.68%

Moneyline — stripped for the CONSISTENCY READOUT, never fitted

	Home	Away
Decimal quote	1.50	2.79
Decimal $d$	1.5000	2.7900
Implied $q = 1/d$	0.6667	0.3584
Fair $p = q^{1/x}$	0.6560	0.3440
Fair decimal $1/p$	1.5245	2.9067

overround Q = 1.0251 → solve

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

→ x = 0.9616 → fair P(home ML) = 65.60%

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

2 · The nested race — point → game → set → match

Tennis games are not counts: every level is a race with absorbing barriers. A server holding with point probability $p$ wins the game with the closed form

P_{\text{hold}}(p) = p^4\!\left(1 + 4q + 10q^2\right) + 20p^3q^3\,\frac{p^2}{1-2pq} \qquad q = 1-p

(deuce is a geometric race; this run: hold(A) = 0.849, hold(B) = 0.781). A tiny DP walks the tiebreak over the real serve rotation (F, S, S, F, F, …; the 6-6 deuce closes as $\alpha/(\alpha+\beta)$ ), the set lattice alternates servers by game — a set totals {6..10, 12, 13} games, never 11, and a 7-6 counts 13 — and the match folds sets carrying the games margin and total pmfs, the sets joint and every per-set grid. The pre-match serve coin toss is averaged. Everything is exact DP; the engine was validated against a 200k-run Monte Carlo.

3 · The ridge — the sum is a prior until a Set-1 quote reads it per match

The handicap pins the serve difference, but (HDP, OU, ML) leave the serve sum and the form shock τ on a near-flat ridge: a weak-serve iid fit can match all three quotes yet misprice every tiebreak-sensitive tail. Without more quotes the sum comes from the tour/surface prior (1.29 here; ATP ≈ 1.29, WTA ≈ 1.14). With a Set-1 total quote (Saba quotes Set 1), stage Set 1 Σ bisects the sum against it — more serve ⇒ more holds ⇒ longer sets, so P(set-1 over) rises monotonically with Σ — and the prior becomes a fallback (ADR-030 O2: a shape prior upgraded to a per-match reading). The match totals anchor is then consumed by $\tau$ — a 3-node mixture $(p_A + z\tau,\; p_B - z\tau),\ z \in \{-\sqrt3, 0, +\sqrt3\}$ that preserves the sum but compounds over sets — or, only when no Set-1 quote pins Σ, by extra sum when the market runs longer than the iid baseline.

Stage	Σ = p_A + p_B	Σ src	p_A	p_B	τ	knob	Δ cover	Δ total	Δ S1tot	Δ ML (readout)
iid @ prior	1.2900	prior	0.6642	0.6258	0.0000	—	1.4e-7	6.4e-2	3.3e-2	2.75%
Set 1 Σ	1.2469	S1	0.6410	0.6059	0.0000	—	2.6e-7	4.7e-2	7.2e-7	1.58%
Length (τ)	1.2845	S1	0.6620	0.6225	0.0292	tau	1.0e-7	4.7e-6	2.1e-3	0.03%

The ML — and the Set-1 handicap when quoted — are deliberately not fitting targets: with the sum right they land on their own (Δ ≈ 0 on the seed quotes); a large readout means the book's quotes disagree with each other — a risk signal, not a solver failure.

4 · Periods — sets are random, so per-set prices are conditional

Unlike basketball quarters, set 3 exists only with probability 41.8% here (set 3 with 41.8%). Per-set markets are void if the set is not played, so their fair prices condition on it — and under the τ mixture the conditioning re-weights the form nodes (deciders come disproportionately from the close-form nodes). Set 1 is unconditional. The same fold also yields the set·set scope for free: the sets joint prices the ML, set handicap and set totals with zero extra parameters.

5 · Read “Moneyline” · margins · notes

Full match: P(home wins the set race) off the sets joint — no tie leg. Per set: the set winner, conditional on the set 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)   tS1 = powerStrip(Set-1 OU pair)?
         [ML and Set-1 HDP stripped → readouts only]

stage iid    : sum = PRIOR(tour/surface)       bisect diff  until P(A covers | dist) = tH
stage set1   : (with tS1)  bisect sum          until P(set-1 over | dist) = tS1   — ridge broken
stage length : with tS1  : bisect τ ∈ [0, .06] only (extra sum would contradict the S1 reading)
               without   : model over > tO ? bisect τ (form mixture — matches shorten)
                                           : bisect sum upward (more serve — matches lengthen)
               — the difference re-solves inside every trial (same-targets discipline)

dist: hold(p) closed form → tiebreak DP (serve rotation) → set lattice (alternating serve;
      totals ∈ {6..10,12,13}, 7-6 = 13 games) → match fold over (sets, first server)
scopes: ONE dist prices set·set (ML, set HDP/OU) and game·set (match + per-set, conditional)

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