Quote:
Originally Posted by heehaww
I mean the geometric average. If I bet kelly and you bet 1.5x kelly, my plot of score vs time will be a steeper exponential curve than yours. (But if it's a close race as the players approach the finish line, then it can make sense to bet >kelly.)
As for the arithmetic avg time, that's infinity due to the nonzero possibility of ruin. If it's a given that we'll reach the goal, the time is minimized by betting all-in each round.
The green and red statements strike me as mutually contradictory. In particular, the green statement is much less intuitive to me than the red one. I tried producing both effects in C# code, but was only able to confirm my intuition that larger wagers produce steeper score VS time plots.
First to confirm the red statement, I ran a simulation of a game 1 million times for each integer percentage wager size as a linear function of the player's score. The game consists of rolling three dice per round, where the player loses their wager if the result is 10 or less, receives their wager as a net gain if the result is 11 through 14, nets double their wager for 15, nets triple their wager for 16, and nets quadruple their wager for 17. I ran a version of this game with a large jackpot in which the player nets 99 times their wager for 18, and a no jackpot version in which the player nets only 5 times their wager for 18. The x-axes represent percentage of score wagered each round, and the y-axes represent expected number of round required to reach the goal. The results were as expected (here is the code if interested:
https://pastebin.com/r1PBjMtk)...
Jackpot
No Jackpot
Now for the green statement, I ran a simulation of the same game 1 million times, but instead of iterating over different wager sizes, I created a separate score VS time (number of rounds passed) plot for each wager size. Here I computed the round-wise average scores for the first 100 rounds, so that the x-axes represent number of rounds passed, and the y-axes represent expected score. I removed the victory check from this version so that the steepness of the pure exponential growth can be observed. The results seem to disconfirm the green statement (here is the code if interested:
https://pastebin.com/etvvcv57)...
Jackpot
No Jackpot
I won't post these images but they show the same thing. I also omitted a lot of the images I made in order to prevent clutter, but what can be seen in all cases is that expected score VS number of rounds passed always gets steeper as wager size is increased. I tested this up to a wager size of 50% of the player's point total, which produces ruin far too often (about 95% of the time) to be viable, and never encountered a maximum steepness that could correspond to the wager size Kelly would pick out.
Is this experiment designed wrong? How can I understand the apparent contradiction between the green and red statements, and the counter-intuitive (and apparently empirically disconfirmed) nature of the green statement in its own right?
Last edited by Metronominity; 09-09-2021 at 03:37 PM.