Rule for Smaller factors become more important when....

[blueshifter]

Hi all,

Is there a rule in math or logic that states something like this:

When Skill is the Primary factor in determining an outcome,
When two opposing people are closer in skill, then smaller factors become more important?
Lets say two warriors are battling it out and are close to being equally skilled, secondary factors or advantages become more important.

Lets say one warrior's armor protects him 40 percent more... the closer the two warriors are in skill the more important that armor might become. Now the statistical discrepancy of the armor and winning might not be that noticeable if the skill level between the two is quite different, but the armor statistically might become more and more important if the primary determination: aka "skill" is very close between the two.

Or in Physics perhaps those gravity race cars friction is the primary difference between each car.. but when friction becomes closer and closer to optimal between cars... then center of gravity becomes more and more important.

Hopefully this gives you the idea of what I am looking for... These are probably not the best examples, but I think you guys will catch on to what I am looking for.

It could be a logic, math, or statistic rule I'm thinking of... but if you know anything similar from any field of study please let me know. Thanks for your time any help would be greatly appreciated.

 

[Stephen Tashi]

I think the proper terminology for what you want is not a "rule". What you want is a mathematical model for competition that represents various aspects of the combatants, some of which are "skills" and you want the formula in the model to have the property that when the combatants are nearly equal in some set of skills then small differences in those (or other) properties become important in determining the outcome. To get suggestions for a specific model, you have to be specific about what an "outcome" is. Is the "outcome" a single win-or-lose result? Is it determined by a single random draw? Or are there degrees of winning (or losing), like scores in a football match?

 

[FactChecker]

Yes. This is a concept that is very important in statistical analysis. In your first example, suppose you did a statistical analysis that predicted success based on skill. You can then remove all the influence of skill and see how strongly the remaining success was related to the quality of armor. Remember that more skilled warrior probably tends to also have better armor. This says that skill and armor quality are correlated. So a valid statistical model would only allow a warrior's armor quality to influence success to the extent that armor quality was above or below the expected armor quality for his skill level. There would be another factor which is the combination of skill and quality. The overall success is influenced by the combination of both skill and armor quality and the two can be traded off. A lesser skilled warrior might still win if his armor is much better. The subjects of step-wise linear regression and variance analysis put these ideas into formal mathematical methods.

 

[WWGD]

I don't know if this is what FactChecker was referring to, but PCA (Principal Component Analysis) and Factor Analysis may be methods that interest you, which derive, as FactChecker observed from (co)variance and (auto)correlation methods.

 

[FactChecker]

but PCA (Principal Component Analysis) and Factor Analysis may be methods that interest you,

Yes. Those are two more good examples. In fact, I think that the original question is fundamental to most statistical methods.