Margin — multi-way (Shin)

Multi-way Shin-model margin tools, paired on one page. Strip recovers fair probabilities from priced decimal odds; Apply injects margin into a fair distribution to produce priced quotes. The two are exact inverses (apply ↔ strip round-trips to ~1e-12). This is the same three-way strip the Odds Generation engine runs on the 1X2 quote before it fits the draw lift φ; the two-way analog uses the Power method — see Margin — two-way (Power).

Recover fair probabilities from a multi-way priced market (1X2, Correct Score, AOS — any market with three or more outcomes) via the Shin model. The solver finds the insider-fraction parameter z ∈ (0, 1) such that the implied fair probabilities satisfy pᵢ = (√(z² + 4(1−z)qᵢ²/Q) − z) / (2(1−z)) and Σ pᵢ = 1, where qᵢ = 1/dᵢ and Q = Σ qᵢ is the bookmaker overround. Shin is the canonical multi-way solver. Production serves both directions via precomputed lookups; this demo solves live by bisecting on z — the per-iteration solve is shown in the Bisection trace below.

Worked example — 1X2, decimals (2.10, 3.60, 3.40)

Implied qᵢ (0.4762, 0.2778, 0.2941) — overround Q = 1.0481 → m = 0.0481
→ bisected z ≈ 0.024083
→ fair pᵢ (0.4587, 0.2626, 0.2787) → fair decimals (2.18, 3.81, 3.59).

Priced decimal odds (input)

Decimal oddsdᵢ

One decimal odd per line. Spaces / commas / tabs / newlines all work. Each value must be > 1.

3 ≤ n ≤ 26 outcomes

n = 3 parsed

Derived

Countn

Count of parsed decimal odds. Must be in [3, 26].

OverroundQ

Q = Σ 1/dᵢ. The bookmaker overround. Q > 1 required (otherwise there is no margin to strip).

Shin zz

Insider-fraction parameter, bisected so Σ pᵢ = 1. z → 0 means no margin to strip.

Equivalent marginm

m = Q − 1. Equivalent margin scalar for cross-reference with the two-way (Power) tool.

Bisection trace — solving for z (Σ pᵢ(z) → 1)

#	z lo	z hi	z mid	value → target	move
1	0.000000	1.000000	0.500000	0.6008	hi←mid
2	0.000000	0.500000	0.250000	0.7885	hi←mid
3	0.000000	0.250000	0.125000	0.9021	hi←mid
4	0.000000	0.125000	0.062500	0.9623	hi←mid
5	0.000000	0.062500	0.031250	0.9929	hi←mid
6	0.000000	0.031250	0.015625	1.0083	lo←mid
7	0.015625	0.031250	0.023437	1.0006	lo←mid
8	0.023437	0.031250	0.027344	0.9968	hi←mid
9	0.023437	0.027344	0.025391	0.9987	hi←mid
10	0.023437	0.025391	0.024414	0.9997	hi←mid
11	0.023437	0.024414	0.023926	1.0002	lo←mid
12	0.023926	0.024414	0.024170	0.9999	hi←mid
… converges to z ≈ 0.024083 after 45 iterations (\|value − target\| = 1.78e-15)

Fair probabilities (output)

#	decimal in	q implied (1/d)	p fair	fair decimal (1/p)
1	2.10	0.4762	0.4587	2.18
2	3.60	0.2778	0.2626	3.81
3	3.40	0.2941	0.2787	3.59