I was thinking some more over the last couple of days on how else to explain my point. It occurred to me that since I've been talking about resistance to illustrate that point, people assume it is dependent on how resistance is designed in the game. That isn't the case because even if resistance is designed in such a way that it works with the particular design described, that doesn't mean the rest of the game can or should do the same. I thus want to move away from that and consider a couple of other systems; as that further serves to illustrate what I've been saying I felt it was worth posting.

Percentile resistance like CoX's is an example of a system that provides growing returns on the property. Indeed, it is about as good an example as you will ever find, since there is an upper bound on it that has, in effect, infinite value, which forced the kind of growing returns on unbounded enhancement I was talking about. It is worth noting that the issue can potentially still exist on a mechanic with growing returns without such an upper bound; it just depends. However, I want to turn my attention towards mechanics that already have diminishing returns inherently and consider how the system would work on them.

An example of such a system is endurance reduction on powers. Generally speaking, if a power costs a certain amount to use, the amount of endurance that can be saved per use from slotting endurance reduction is bounded by that value. For example, if a power costs 10 endurance, presumably no amount of endurance reduction will ever allow you to save more than 10 endurance per activation. Also assuming the endurance savings are increasing with the number of endurance reduction enhancements (which is the only thing that makes sense), you must see diminishing returns on those enhancements, at least at some point. In particular, the system presented here uses the reducing power effect option. Note, however, that even with a linear increase in the Enhancement Multiplier you get these diminishing returns, so the square root function simply results in diminishing them further. What you find then is that with a power multiplier of 1, the first enhancement saves you 33.3% of the cost of the power, and the next three combined provide you with 16.7% reduction. In particular, the second enhancement is down to 8.1%, less than a quarter of the first enhancement, which is much more of a drop-off than looking at the square root function in isolation would suggest.

You might respond by saying that if you determine 8.1% reduction is too low on the second enhancement to make it meaningful, that you can increase the power multiplier to make it stronger. However, mathematically this is not the result. This is because increasing the power multiplier also increases the effect of the first enhancement, pushing you towards the upper bound of the effect (100% endurance savings) and leaving less room for further increases. Using some calculus to do the optimization, you actually find that the maximum possible value of the second enhancement under this system is about 8.4%, occurring when the power multiplier is the fourth root of 2, or about 1.1892. (Using the example of a power that costs 10 endurance again, this means that the most endurance the system can possibly allow you to save from the second enhancement is 0.84 per activation.)

Accuracy is another system that tends to see these diminishing returns, because typically you can look at the expected damage on an attack as being the probability of a hit multiplied by the power's damage. Thus, accuracy has an upper bound of 1, the probability at which the power always hits and the power's expected damage is equal to the power's damage value. A simple example of a possible way to implement accuracy is to define the probability of a miss as 1/a, where a is an accuracy value assigned to the power in question. If you suppose that a baseline hit rate is 75%, then the corresponding value of a would be 4 (so that probability of a hit is 1 - 1/4, which is 75%). Under such a system, if enhancements acted as an increasing power effect multiplier on this a value, then with a power multiplier of 1 and a base value of 4 we find that the first enhancement brings us from 75% hit rate to 83.3%, and the second to 85.3%. Those two enhancement are thus providing a value of 8.3% and 2.0% of the power's base damage.

For similar reasons as with endurance, the amount of benefit you get from the second enhancement is severely limited. Again doing some calculus, the maximum possible value occurs with a power multiplier of about 1.6818, at which point it provides less than 2.2% increase in accuracy.

In both of the cases above the issue results from the underlying system having diminishing returns on absolute differences, and further applying a diminishing returns formula to enhanced values. You can actually get smoother results by using something else, such as a linear formula on enhancement. (For example, if you simply multiplied the endurance cost by 1/1+0.25np, where n is the number of enhancements and p is the power multiplier, then for p=1 the first 4 enhancements save 20.0%, 13.3%, 9.5% and 7.1% of the power's endurance.)

Saying that the potential systems for endurance and accuracy don't work with these enhancement formulas doesn't imply that they are in any way broken. It just means the formulas don't suit them. In talking about resistance earlier I asserted that systems with inherent growing returns may (or may not) still see growing returns on enhancements; here I am asserting that systems with inherent diminishing returns may see those returns dwindle so quickly as to effectively eliminate the option of slotting more than one or perhaps two enhancements, since the "doubly" diminishing returns resembles the cliff that we are trying to avoid. The point here is that the various systems will interact with any particular formula in different ways. Without reworking what we have this could be only avoided by having all systems affected by enhancements behave in a very similar fashion, which I don't think is desirable. Instead, I would allow for different functions to be applied to different categories of underlying systems. For example, if one category needs to be bounded, use arctan or an exponential; if one category needs diminishing returns enforced, use something like square root; if a third category already has diminishing returns on it, use linear.

(That concludes the main thrust of the argument. What follows are essentially footnotes to deal with complaints that may pop up.)

Firstly, I didn't include calculations on the calculus because it just isn't worth posting them. Anyone can check the math if they want, but they are backed up by a quick check with Excel.

Secondly, I should repeat that the systems under consideration are [I]examples[/I] to illustrate the point that their behaviour influences the behaviour of enhancing things with a particular function. The example of accuracy is intentionally a simplification; it only considers base chance to hit without defensive potential, and you could potentially have an accuracy system for which expected damage is [I]not[/I] bounded by the power's base damage (if, say, values of accuracy over 100% resulted in bonus damage).

Thirdly, I had to make a decision on how to measure returns on investment in order to calculate anything, but what I used aren't the only ways. You might look at endurance reduction in terms of how much it increases the time you can go all out before running out of endurance, though using CoX as a point of comparison I think it was generally assumed you would slot enough endurance reduction at high levels to continue indefinitely, so you really measured the returns by how much it saved you until you had enough to do that. For accuracy, expected damage is a good metric for players that are primarily interested in that value. In some cases you might be more concerned with reducing the chance of a miss. A character using CC powers might be a good example of this, since you are generally interested in the probability of maintaining your control and reducing incoming damage. In that respect accuracy is more of a defensive property not unlike resistance.

Finally, in my conclusion of saying different functions are needed, I did consider that a single function could be used, but then a mapping is applied on all the underlying systems to faithfully produce the diminishing returns behaviour that is desired. However, I think that complicates the systems without being effectively any different than just using a variety of functions in the first place. Further, the particular function in question (square root in this case) essentially becomes irrelevant anyway, since any time it isn't quite right you just "wash it out." At that point you may as well do away with it altogether and have each underlying system directly determine how a particular number of enhancements are applied, though perhaps still with a variable serving the same purpose as the power multiplier.

Redlynne wrote:

Um ... we've been trying to tell you for a while now that 1/(1+X), the Endurance Reduction formula, performs very differently from ANY kind of "just add the percentages together and make sure they don't get to 100%" system like City of Heroes used for Defense and Resistances. Furthermore, we've also been at pains to explain to you that the 1/(1+X) formula used for Endurance Reduction is itself one that has built in Diminishing Returns ... *unlike* the "just add the percentages together" system that City of Heroes used for Defense and Resistances, which resulted in Increasing Returns. This is why we kept telling you that what you plug the Enhanced effects of Powers into made all the difference in the world, and that since such mechanics hadn't been settled yet, arguing over how they worked when they hadn't been nailed down yet didn't make a whole lot of sense (and was really the subject of a completely different topic thread).

I find that to be backwards. I've quite clearly recognized that the system you plug enhanced values into makes all the difference in the world; that's been my point from the beginning. I've been trying to tell you that [I]because[/I] the system into which you plug enhanced values makes such a difference, there is absolutely no point in looking at the square root function and saying "yep, looks about right," because by the time you plug it into a power it may not in any way resemble the returns you thought you were looking at.

Further, because the mechanic it goes into makes such a difference, the returns you get can be completely different from one case to another. If you were looking at a replacement for ED in CoX for example, then consider that as the returns on damage, resistance and endurance reduction all behave in a categorically different manner, there simply [I]isn't[/I] a single function that will give you the behaviour you want for all three. Tailor the function to one and it won't work for the others. The way ED "solved" this problem (in a way that I actually agree is unsatisfactory) is by saying that [I]all[/I] mechanics will see effectively no returns after 3 enhancements, regardless of what they got before that.

Let me say yet again that I'm [I]not[/I] arguing about what the systems should look like. I don't care how they work right now; I only care whether or not they all work the same way, because unless they do there is no single function that can fit where you've put square root that will do what you're trying to do. Admittedly, I'm working under the assumption we don't [I]want[/I] all mechanics to be behaviourally similar, but even if we did there's no way to tell what the function in question should be without knowing what they look like.