I'd like to see the 'X' factor changed from a straight 0.0 - 1.0 ranged value to a weighted value. With this idea, it would still vary from 0.0 to 1.0, but by utilizing a weighted value -- a mean value which would vary along that 0.0 to 1.0 range -- would bias the 'X' factor towards itself. This varied mean value could be calculated by utilizing some combination of the SBDE, the "us vs. them" ratio, and/or a secondary 'X' factor to fit into a calculation using the standard deviation formula.

From wikipedia, "In statistics, the standard deviation (SD) is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values."

Refer to the standard deviation reference in Wikipedia for details on how to do this. I include two links at the bottom of this page that show how easy it is to do this. Depending on the various factors involved, one might have either a "normal distribution" or a weighted (skewed) distribution. I would like to see a skewed distribution based on the previously-mentioned combination of affecting values for the formulation of the standard deviation.

Example of a normal distribution of values determined by their standard deviation:


From Wikipedia, this is "a plot of normal distribution (or bell-shaped curve) where each band has a width of 1 standard deviation". Note that a standard deviation is denoted by the sigma Greek letter "σ". The deviation indicates the likelihood of a random value within a range is likely to occur. In the above normal distribution, the area under the curve represents the total likelihood of a random value (within a selected range of values) occurring within that band. Thus, a value of 0σ to -1σ is likely to occur about 34.1% of the time when a statistically random value is given (again, within that selected range of values).

This is another example of a normal distribution of a standard deviation.


In order to have the various factors (i.e., SBDE, the "us vs. them" ratio, a secondary 'X' factor, etc.) properly affect the mean of the standard deviation, you would need to skew the standard deviation to better-represent the true randomness of the 'X' factor value without removing the obvious benefits of having the standard deviation bias.

Example of a skewed standard deviation:

From Statistics How To, "Skewness is a measure of symmetry in a distribution. Actually, it’s more correct to describe it as a measure of lack of symmetry. A standard normal distribution is perfectly symmetrical and has zero skew."

Example showing both left and right skewness of a standard deviation:


Example of the details of a probability distribution of a skewed standard deviation:


From Asset Insights:

• Mean: In statistics, the mean is the average of a data set on a probability distribution. The mean is derived by sum of the values divided by the number of values.
  
• Mode: In statistics, the the mode is the number that appears most frequently in a distribution of numbers.
  
• Median: In statistics, the median is the middle point between the higher half and lower half of a probability distribution.
  

This short video shows how a new modified 'X' factor might determine the kind of standard deviation such that its weightedness is represented properly by the various inputs:


This is the basic formula for determining the single standard deviation value (the sigma "σ" value). It looks complicated, but it's really not so bad. See below on how to easily come up with the various values to put into the formula...as well as how to modify the formula to fit one's needs.


Simply put, the standard deviation is the square root of a sample set of the variation of values within a range. This easy to follow Math is Fun web page shows how to determine and calculate the "variance" of the standard deviation. This other easy to follow Math is Fun web page better explains how to easily create and use the standard deviation formula, in general.

Actually, I like your idea of having the veterancy affecting the 'X' factor. But instead of absolutes, let it be a skewing factor (positive or negative) upon the mean. And, the mean could be centered around the actual hit value, but then the possibility of the extreme possible attack values would necessarily be limited to a realistic value of the standard deviation bell curve....maybe up to 3σ either positive or negative. And, if the veterancy (and maybe other factors) skew the curve either positively or negatively, then those extreme deviations (3σ) would not necessarily be equidistant from the mean.

However, if this were to be the model, then there'd have to be some fancy artwork put together to put it into layman's terms for the typical player trying to figure out if his army's got a chance against another stack.

Boy, do I wish I could get in on the combat system design. I like your idea and I think we could really implement something like this to make the 'X' factor into something that really begins to simulate real life combat results.

Dice be danged, I want realism! And utilizing the statistical deviation model to control the 'X' factor would certainly go a long way towards that.

I'm not sure about the rest, but it occurs to me that in a 1v1 combat between a Commando and a Militia, even in a mountain range, this battle calculation is skewed too far into the weaker player's camp.

With just arbitrary numbers (not real to the situation), if the attacker is rated (with terrain, SBDE, etc.) as "8" and the defender is rated (with terrain, SBDE, etc.) as "2", then the lower-valued defending unit will deal a greater percentage of damage than it is entitled to.

For example, using the above arbitrary numbers, placed into the formula, you get:

//rewritten with better efficiency
maxChance = 0.3;
if (attackStrengh > defenseStrength) {
maxChance += 0.7*( min( ( attackStrength/(defenseStrength*20) )^2, 1 ) )
}

//the above gives:
//maxChance = 0.3 + 0.7*( min( ( 8/(2*20) )^2, 1 ) )
//maxChance = 0.3 + 0.7*( min( ( 8/40 )^2, 1 ) )

//maxChance = 0.3 + 0.7*( min( ( 1/5 )^2, 1 ) )
//maxChance = 0.3 + 0.7*( min( 1/25, 1 ) )
//maxChance = 0.3 + 0.7*( 1/25 )
//maxChance = 0.3 + 0.7*0.04
//maxChance = 0.3 + 0.028
//maxChance = 0.328

Suppose that the larger unit has a hit points of 20 while the smaller unit has a hit points of only 10. Then, using the arbitrary values from above, the "max" possible hit points that the larger unit can deal is 0.328 x 8 = 2.624. Then, in order to eliminate the smaller unit, the larger one must hit at least 4 times (4 x 2.624 > 10 ====> 10.496 > 10). And, in all likelihood, with this "X" factor representing the maximum "possible" hit, it could take more -- even many more -- than 4 rounds of fire to eliminate the smaller and weaker opponent.

Meanwhile, as the small unit is SLOWLY getting crushed, using the same arbitrary numbers, it deals a small amount of damage to the stronger unit with it's own max possible hit points which it deals at (using the numbers with swapped attack/defend values) only 0.307. But this value of "max" is not much less than the "max" of the much bigger unit (plug in the number "2" in place of "8" to find the truth in this). So, the smaller one will hit at most 2 x 0.307 = 0.614.

Now, that might not sound like a lot, but if the attacker takes 5 rounds to defeat the small unit, then it is possible for this pathetic little unit to deal as much as 5 x 0.614 = 3.07. While that might not sound like a lot when the bigger unit has 20 points, if the luck "X" factor" goes in favor of the smaller guy enough times, it could -- conceivably -- win the battle. Of course, though that is highly unlikely, it still is dealing far more accumulated damage than it ought.

In real life, a giant unit is going to squash a puny unit relatively quickly if not on the first shot, then within two or three at most (C'mon, how unlucky is the gunner behind that Heavy Tank aiming at the smallish RPG-mounted pickup-truck at 100 yards distance?) Meanwhile, even though the larger unit is going to come out with only partial damage, it is both held up and vulnerable to other strikes for the duration of several hours of play. This concept makes a blitz strategy virtually impossible without bumbling through completely-empty provinces.

As you can see, at least in single melee combat, unless the larger and more powerful unit is immensely greater (like a Heavy Tank vs. a child with a toy gun), it loses virtually all of it's advantage. Now think about this: if the larger unit and the smaller unit both only hit at a maximum of ~0.3 times it's potential damage, then the bigger unit is going to take a lot of smaller hits that add up FAR too much while it takes too long taking out the puny unit just because the puny unit will take at least three or more hits to eliminate (0.328 x 3 = 0.984) which is less than enough after three rounds to eliminate the other guy. In other words, the big unit is taking hit after hit after hit....and could eventually die the death of a thousand cuts (a little too fast, mind you), when it should be sailing through the battle like a hot knife through butter.

I guess I've got a problem with the max-chance aspect, out of principle. The idea that you can't get a lucky shot in (in either direction) that would wipe out an enemy unit in one hit seems a bit excessive. You shouldn't have to be 20 times greater than the enemy to be able to permit destroying them in the first one or two salvos. Sure, the units represent large(ish) groups of individual soldiers and units. But an effective battle plan can often eliminate an entire regiment or battalion in short order when the intelligence is good and current, the aim is true, your strength is good (though not necessarily 20 times greater than the other guy), and you have a little bit of luck.

Yet, the "1/3" factor removes luck, removes perfect aim, and removes battle superiority (except by the ridiculous ratios of 20+ : 1). I'm sorry, but history has shown many times that a single shot can wipe out a target, and sometimes even a bigger and more powerful target.

David beat Goliath because he hit him in his weak spot. Just the same, the chance that an Infantry unit (in real life) might take out a Heavy tank is slim -- especially in one or two salvos -- but technically possible. Though, frankly, the Heavy tank should be more likely able to occasionally take out the Infantry in one or two salvos. But the "1/3" factor prevents this in all instances. This is an absolute that doesn't belong in any simulation.

Perhaps, instead of the "1/3" factor being prevalent in all but very extreme lopsided ratios, this "1/3" factor could be amended to be a "usually 1/3" factor in which another random variable can determine whether the "1/3" factor should apply in the current salvo. Of course, to be fair, and to maintain the original intent of the "1/3" factor (to preserve game play beyond repeated skunking battles), the random factor could be a hard to hit number (i.e., like hitting snake eyes or double sixes with two dice).

That way, luck could simulate that occasional "hole in one" perfect shot. Also, the strength or likelihood of that random number happening would needfully be affected by the actual ratings ratio of the two sides. So, in a David vs. Goliath scenario, Goliath might be given 5 out of 9 chances of killing David in a single blow whereas David would only get a 1 in 9 chance of killing Goliath in a single blow (or maybe 1 in 100 or 1 in 1000).

I think this would be a fair compromise that would also increase the realism of the combat. Because chaos rules the battlefield, you can't just put everything into tidy simulations. Sometimes one side scores a great shot, sometimes the fight goes on and on, sometimes luck turns the tide in unexpected directions, and sometimes things wear down to a standstill or a draw...and sometimes even, both sides are destroyed.

Still, you make your own luck, so coming in with a much stronger force than your opponent should win the day in most cases while random chance proves that even the best prepared strategies can fail even when executed perfectly...and one well-aimed shot can also bring down even the mightiest of giants.

And in that case, even Goliath can fall when it might be argued that he was at least 20 times greater than David. Yet, David wasn't automatically defeated, so the "overwhelming force" principle in itself shouldn't be an absolute, either. Otherwise, it could be said that gigantic stacks could slice through thin enemies without a scratch, and that would lead to everyone wanting to build one giant stack which would alter the dynamics of this game and ruin the realism that it tries to convey.