I tried to determine the optimal constant Janowski cube life index by playing players with different cube life indexes against each other.
Interesting that the probability of win was well above 50% but on average lost points for x=1. It must be because when it wins it wins small, but when it loses it loses big. Presumably that's because the player gets doubled and takes when the game goes against it, but only doubles past the cash point when the game goes in the player's favor.
Next I tried to zero in on the winning range by running the same thing in a tighter range:
So it seems like 0.7 is indeed the optimal cube life index in terms of expected points per game, at least within +/ 0.05.
I started broadly by playing different cube life indexes against a fixed 0.7, since 0.7 is near where other people have quoted the cube life index.
Statistic  x=0  x=0.25  x=0.50  x=0.75  x=1 

Average points per game  0.30  0.14  0.04  0.00  0.09 
Probability of win  37.5  41.8  45.9  51.2  58.3 
Average cube  2.32  2.40  2.27  1.96  1.45 
Interesting that the probability of win was well above 50% but on average lost points for x=1. It must be because when it wins it wins small, but when it loses it loses big. Presumably that's because the player gets doubled and takes when the game goes against it, but only doubles past the cash point when the game goes in the player's favor.
Next I tried to zero in on the winning range by running the same thing in a tighter range:
Statistic

x=0.55

x=0.60

x=0.65

x=0.70

x=0.75


Average points per game

0.03

0.01

0.00

0.00

0.00

Probability of win

46.9

47.8

48.9

50.0

51.2

Average cube

2.23

2.17

2.11

2.04

1.96

So it seems like 0.7 is indeed the optimal cube life index in terms of expected points per game, at least within +/ 0.05.
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