Quote:
Originally Posted by raiders72002
Ganch- You're getting lazy. Where's the math? Just lower juice. You 're starting to talk like me.
Show them hold percentages and just don't say less juice.
|
I had thought it was considerably too trivial a matter with which to bother. Nevertheless, since I'm bored, drunk, and not quite ready for bed ...
<hr>
Let F = US fave odds < -100
Let D = US-style dog odds ≥ +100
Let W = US-style market width > 0
Let M = decimal-style fave odds
Let N = decimal-style dogs odds
Let H = theoretical hold (juice) > 0 (as long as W > 0)
Goal is to show:
∂H/
∂D < 0
- By the definition of US-style market width:
F+D = -W
F = - (D+W)
D = - (F+W)
- By the definition of definition of decimal odds:
M = 1 - 100/F
N = 1 + D/100
- By the definition of definition of theoretical hold:
H = 1 - NM / (N+M)
- Combining (2) & (3):
H = 100*(D+F)/(DF+200F-10000)
- Combining (1) & (4)
H = 100W/(D2+WD+200D+200W+10000)
- Taking the partial of (5) wrt D gives us:
∂H/∂D = (-100*W*(200 + 2*D + W))/(10000 + 200*D + D2 + 200*W + D*W)2
- As the denominator of (6) is strictly positive, the partial derivative will be negative iff:
-100*W*(200 + 2*D + W) < 0
W*(200 + 2D +W) > 0
which will always be true as by definition W, D > 0.
Hence, holding US market width constant, theoretical hold (juice) always decreases as line magnitude increases.
QED
Expanding the proof to include F ≤ D < -100 is left as a similarly trivial exercise for the drunk and bored reader.