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 it is a prior, not a fit;
consumes the Game Total with whichever knob can reach it: τ, a match-level form mixture (matches shorten), or extra serve sum (matches lengthen) — the difference re-solves at every trial;
never fits the Moneyline: on the ridge it is the cross-anchor consistency readout (Δ ML) — a large value means the book's ML disagrees with its own game lines at this prior, 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 / Length and watch the totals hump move while the handicap holds. 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

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 is never fitted — it is the cross-anchor consistency readout (Δ ML in the diagnostics); clear it to skip the readout.

Valid inputs: Malay in [−1, +1] non-zero with margin, decimals above 1, serve-sum prior in (1.04, 1.54). Quotes are whole-match; per-set sheets derive from the same solve, conditional on the set being played.

p_A 0.6650p_B 0.6250τ 0.0300P(set 3) 41.5%TB 42.8%round-trip Δp 2.0e-7 / 2.6e-6Δ ML 0.03%

Match distribution

p_A 0.6650p_B 0.6250Σ 1.290τ 0.0300TB 42.8%E[games] 24.4

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

	A=0	A=1	A=2
H=0			17.6
H=1			16.8
H=2	40.9	24.7

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	4.8	3.5	3.8	4.1	6.6	10.0	12.8	11.4	8.4	5.9	2.7	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	5.8	6.9	8.3	7.9	5.3	8.3	6.2	1.1	3.4	4.6	3.1	3.9	4.2	4.3	5.0	4.9	3.1	2.3	3.0	1.8	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%	1.524	1.491	+0.49
Away	34.4%	2.909	2.752	-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%

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) = 0.853, hold(B) = 0.785). 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 — why the sum is a prior and the Moneyline is a readout

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. So the sum comes from the tour/surface prior (1.29 here; ATP ≈ 1.29, WTA ≈ 1.14 — per-set quotes identify it per match, ADR-030 O2), and the totals anchor is 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, fixing the iid model's known favorite overconfidence — or by extra sum when the market runs longer than the iid baseline.

Stage	Σ = p_A + p_B	p_A	p_B	τ	knob	Δ cover	Δ total	Δ ML (readout)
iid @ prior	1.2900	0.6642	0.6258	0.0000	—	1.4e-7	6.4e-2	2.75%
Length (τ)	1.2900	0.6650	0.6250	0.0300	tau	2.0e-7	2.6e-6	0.03%

The ML is deliberately not a fitting target: with the prior right it lands on its own (ΔML ≈ 0 on the seed quotes); a large ΔML means the book's ML disagrees with its own game lines at this prior — 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 41.5% here (set 3 with 41.5%). 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)   [ML stripped → readout only]

stage iid    : sum = PRIOR(tour/surface)       bisect diff  until P(A covers | dist) = tH
stage length : model over > tO ? bisect τ ∈ [0, .06]   (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).