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<blockquote data-quote="Cerebral Paladin" data-source="post: 5984778" data-attributes="member: 3448">Yeah, the farther it is from even, the shorter the average length.&nbsp; Take the end case of 100% chance of success:&nbsp; the average (and uniform) time to the end of the walk will be 3.&nbsp; Next think about a high chance of success:&nbsp; if you have a 90% chance of success, better than 70% of the cases will resolve in three rounds, way more than the 21% at even.&nbsp; Then glance at the calculated values from Nagol, and you can see that even at just more than half (a +1 bonus), you have meaningfully more resolved games by round 9 than at even odds, and the remaining games are less likely to be at the midpoint of the walk.&nbsp; So the more you move away from random, the faster the walk resolves, with the average bounded at 9 rounds (for random, plus a little bit for ties) and 3 rounds for guaranteed success.&nbsp; But it will also be nonlinear in the time in between.The flip side is that (except when success is guaranteed) it's just an average.&nbsp; If the odds are 50-50, a meaningful chunk of the time it will go much longer than average.&nbsp; So if you want to bound it at a reasonable time length, you need to either impose a cap or make sure it's way off from 50-50.</blockquote>

[QUOTE="Cerebral Paladin, post: 5984778, member: 3448"] Yeah, the farther it is from even, the shorter the average length. Take the end case of 100% chance of success: the average (and uniform) time to the end of the walk will be 3. Next think about a high chance of success: if you have a 90% chance of success, better than 70% of the cases will resolve in three rounds, way more than the 21% at even. Then glance at the calculated values from Nagol, and you can see that even at just more than half (a +1 bonus), you have meaningfully more resolved games by round 9 than at even odds, and the remaining games are less likely to be at the midpoint of the walk. So the more you move away from random, the faster the walk resolves, with the average bounded at 9 rounds (for random, plus a little bit for ties) and 3 rounds for guaranteed success. But it will also be nonlinear in the time in between. The flip side is that (except when success is guaranteed) it's just an average. If the odds are 50-50, a meaningful chunk of the time it will go much longer than average. So if you want to bound it at a reasonable time length, you need to either impose a cap or make sure it's way off from 50-50. [/QUOTE]

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