A clearer example might be that of a bonus which halves your chance of failure changing 80% success likelihood to 90% success (20% failure to 10% failure)
The problem with this is it only makes sense when you have a high chance of success.
Suppose I attempted to blow up the Earth. Normally, I’d have an approximately 0% chance of success. Would that bonus increase it to 50%?
That’s a 49.995% chance of failure, and a 50.005% chance of success. Also 0.49995 is much closer to 50% than to 49%.
In any case, it should be nowhere near 50%. Increasing the log probability by log(2) would approximately halve the probability of failure if you’re very likely to succeed, but it would double the chance of success if you’re very likely to fail.
To answer the earlier question, an alteration which halved the probability of failure would indeed change an exactly-0% probability of success into a 50% probability of success.
If one is choosing between lower increases for higher values, unchanged increases for higher values, and greater increases for higher values, then the first has the advantage of not quickly giving numbers over 100%. I note though that the opposite effect (such as hexing a foe?) would require halving the probability of success instead of doubling the probability of failure.
The effect you describe, whereby a single calculation can give large changes for medium values and small values for extreme values, is of interest to me: starting with (for instance) 5%, 50% and 95%, what exact procedure is taken to increase the log probability by log(2) and return modified percentages?
Edit: (A minor note that, from a gameplay standpoint, for things intended to have small probabilities one could just have very large failure-chance multipliers and so still have decreasing returns. Things decreed as effectively impossible would not be subject to dice rolling or similar in any case, and so need not be considered at length. In-game explanation for the function observed could be important; if it is desirable that progress begin slow, then speed up, then slow down again, rather than start fast and get progressively slower, then that is also reasonable.)
what exact procedure is taken to increase the log probability by log(2) and return modified percentages?
The simplest way is to use odds ratios instead of log probability. 5% is 1:19. Multiply that by 2:1 and you get 2:19 which corresponds to 9.52%. If it’s close to 100%, you get close to half the probability of failure. If it’s close to 0%, you get close to double the probability of success.
This can be done with dice by using a virtual d21. You can do that by rolling a higher-numbered die and re-rolling if you pass 21. Since the next die up is d100, you can combine two dice to get d24 or d30 the same way you combine two d10s to get a d100. Alternately, use a computer or a graphing calculator instead of a die, and you can have it give whatever probabilities you want.
The problem with this is it only makes sense when you have a high chance of success.
Suppose I attempted to blow up the Earth. Normally, I’d have an approximately 0% chance of success. Would that bonus increase it to 50%?
I think it’d increase it to ~49% (if you have a 0.0001 chance of success, you have a 0.9999 chance of failure, and 0.9999 / 2 = 0.49995).
That’s a 49.995% chance of failure, and a 50.005% chance of success. Also 0.49995 is much closer to 50% than to 49%.
In any case, it should be nowhere near 50%. Increasing the log probability by log(2) would approximately halve the probability of failure if you’re very likely to succeed, but it would double the chance of success if you’re very likely to fail.
To answer the earlier question, an alteration which halved the probability of failure would indeed change an exactly-0% probability of success into a 50% probability of success.
If one is choosing between lower increases for higher values, unchanged increases for higher values, and greater increases for higher values, then the first has the advantage of not quickly giving numbers over 100%. I note though that the opposite effect (such as hexing a foe?) would require halving the probability of success instead of doubling the probability of failure.
The effect you describe, whereby a single calculation can give large changes for medium values and small values for extreme values, is of interest to me: starting with (for instance) 5%, 50% and 95%, what exact procedure is taken to increase the log probability by log(2) and return modified percentages?
Edit: (A minor note that, from a gameplay standpoint, for things intended to have small probabilities one could just have very large failure-chance multipliers and so still have decreasing returns. Things decreed as effectively impossible would not be subject to dice rolling or similar in any case, and so need not be considered at length. In-game explanation for the function observed could be important; if it is desirable that progress begin slow, then speed up, then slow down again, rather than start fast and get progressively slower, then that is also reasonable.)
The simplest way is to use odds ratios instead of log probability. 5% is 1:19. Multiply that by 2:1 and you get 2:19 which corresponds to 9.52%. If it’s close to 100%, you get close to half the probability of failure. If it’s close to 0%, you get close to double the probability of success.
This can be done with dice by using a virtual d21. You can do that by rolling a higher-numbered die and re-rolling if you pass 21. Since the next die up is d100, you can combine two dice to get d24 or d30 the same way you combine two d10s to get a d100. Alternately, use a computer or a graphing calculator instead of a die, and you can have it give whatever probabilities you want.
Thank you!