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I am looking for a defense system where the probability of hitting is proportional to the inverse of a quadratic polynomial with respect to defense. I don't care about attack scaling yet.

If you plotted the percent chance of success across a linear distribution of defense values the output of "% chance of hit" should be able to be mapped by "A/(Bx^2+Cx+D)" where A, B, C and D are constants, but we don't care what their specific values are--they can be anything.

This is not a bell curve. It is clear that each additional point of defense must add less percent increase to defense than the previous point. It would also be a curve where no matter how high your defense and how low your opponents attack the odds of evading will asymptotically approach 100% but never reach 100% (also can be seen as the odds of hitting asymptotically approaching 0% as defense increase but never reaching 0%).

I am looking for dice mechanics that mirror this behavior. If you are good with ANYDICE as well, I need help mapping this behavior for all defenses values for the following relationship:
defense equals attack to defense equals attack +10.

Why I want this is explained below.

The math is intensive and not needed to answer the question, only to understand the thinking behind it.


Recently while doing some math I realized that in many dice systems increasing your defense value actually makes you survive longer and longer exponentially, making them more and more efficient if your defense skill is under your control.

To show what I mean. Assume that attack and damage of the opponent, and your health, remains the same.
P(hitting) = probability of hitting
Number of rounds you will survive = [amount of hits it will take to run out of health given damage and health]/P(h).
This is true for any system where margin of success does not inflict extra damage (in a fashion which can take you out faster) and health has a static value.

Now for what I mean about increased efficiency of defense as defense improves. Take D&D for example.
If your opponent does damage such that it takes 2 hits to down you and you have an AC such that your attacker has to roll a 16 or better to hit you, you will be hit on in 4 times so it will on average take them 8 attacks to down you. If you improve your AC by 1 to 17 needed to hit you, now 1 in 5 hit or 10 attacks to down you (improvement of 2 survived attacks, or 25% improvement for 1 point of AC). Now increase it by 1 again to 18 or better: now we are looking at a bit over 13 attacks to hit you (3 improvement, or 30% improvement for 1 point of AC). Raise that to 19, and you are up to 20 attacks to hit you (improvement of 50%). Raise that so now they only succeed on a 20 and it jumps even more up to 40 attacks or a doubling of (+100%) improvement. As you can see this is cubic growth of efficiency.

This is true for other systems as well. Take FATE (which actually is what made me realize this). If an attack doesn't take you out, by forcing a player concession they go down if they have all their boxes checked, and they have to check another box. This means the formula for someone with a +1 or +2 Physique (3 physical hit boxes) is
4/P(hitting) = "number of attacks survived"
Assuming the norm of weapon rating > than armor reduction, if you and your opponent are equally matched (say you both have a good +2 attack and defense respectively) he will hit 58.44% of the time: survival = 6.8 attacks.
If your defense is 1 better (+3 skill) P(hit) = 41.56%% -> survival = 9.6 attacks.
Add another 1 (+4 skill) and P(hit) = 26.08% -> survival = 15.3 attacks.
add another 1 (+5 skill) and P(hit) = 14.13% -> survival = a whopping 28.3 attacks!
This raises even more exponentially than D&D.

In fact, in any system where skill adds a linear flat value on top of a static dice mechanic, not only will the efficiency of each point of defense increase per point, the rate at which it increases will increase faster the more bell curve shaped and tighter said bell curve is. Specifically defense efficiency increases by n^(number of dice +2).

To that end I am trying to find a system where for every point you increase your defense above your opponents attack value, you add a linear amount of extra attacks you can survive (proportional to the base number of hits that take you out based on damage and health). The idea is that each point of defense will be equally useful because it adds the same amount of extra time you can survive per point.

Doing some math I discovered that for this to be the case the probability of hitting plotted with respect to defense, with attack value remaining unchanged, must = A/(Bx^2+Cx+D) as stated above.

The problem to solve

I don't know what kind of dice mechanic will generate this curve. I am looking for help in creating that dice mechanic.

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closed as too broad by Purple Monkey, SevenSidedDie Jun 3 '18 at 19:21

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • \$\begingroup\$ It's been a while since I've mathed... but am I reading that "x" in the equation A/(Bx^2+Cx+D) would be the actual defense 'rating'? \$\endgroup\$ – Ifusaso Jun 3 '18 at 9:13
  • \$\begingroup\$ I don't understand your maths. In the D&D example you give, # of attacks goes 8-10-13.3-20-40 as you say, but this is not cubic growth. It's a reciprocal function: 40/(21-r), where r is the number required to hit. Later you describe FATE as "even more exponentially than D&D", which doesn't make sense - either a function is exponential, or it isn't (and D&D isn't). Also, I don't see how the probability you state would "add the same amount of extra time you can survive per point" - this would add an increasing amount of time with every point. \$\endgroup\$ – Geoffrey Brent Jun 3 '18 at 10:25
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    \$\begingroup\$ If your hit probability is A/(Bx^2+Cx+D) then the expected time (# of attacks) required to inflict n hits is n(Bx^2+Cx+D)/A. The improvement in survival time between defence=x and defence=x+1 is then given by n(B(x+1)^2+C(x+1)+D)/A-n(Bx^2+Cx+D)/A which simplifies to (n/A)(B(2x+1)+C) which obviously gets larger with each increase in x. This is not consistent with your goal that "each point of defense adds the same amount of extra time you can survive". \$\endgroup\$ – Geoffrey Brent Jun 3 '18 at 10:32
  • \$\begingroup\$ I made a first attempt edit due to the run on nature of the prose, in an attempt to clearly pull out the problem you want solved. Please review the edit and revise again as needed to better clarify the problem you want solved. What dice are you intending to use? d6, d12, d20? \$\endgroup\$ – KorvinStarmast Jun 3 '18 at 13:36
  • \$\begingroup\$ Can I sum up the main question as "If I plot my defensive value vs the number of rounds I can survive combat, it should be linear"? So that doubling your defense also makes you live twice as long? \$\endgroup\$ – Erik Jun 3 '18 at 14:50
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If I understood your question correctly, your actual goal is this:

I am trying to find a system where for every point you increase your defense by above your opponents attack value, you add a linear amount of extra attacks you can survive[.]

If so, instead of trying to complicate the "chance to hit" formula, it would probably be easier and more effective to just have your defense attribute contribute to something that already naturally scales linearly with the expected number of attacks you can survive.

Luckily, it turns out that many RPG combat systems already have something like that. In D&D and many other similar systems it's called hit points. In FATE it would be stress boxes. In fact, pretty much any system that doesn't directly model injury as a consequence of each hit (or that doesn't just straight up have characters go down from one hit) will have some abstraction like that.

So a very simple system that behaves like what you want would be one where each item or effect that improves your defense simply adds +n to your hit point total.

There are various ways to embellish such a system further. For example, you could treat the extra hit points provided by, say, a suit of armor as the hit points of the armor, and require the attacker to first take out the defender's armor before they can deal damage to actual character protected by said armor. One natural implication of such an interpretation would be that damage absorbed by armor could not be naturally healed, but would instead have to be fixed by repairing the armor. Also, when a character took off their armor, the damage dealt to that armor would stay with it.

On the other hand, in some cases it might be more reasonable to assume that the extra hit points provided by armor just represent extra survivability conferred to the wearer, but that any hit point loss still represents injury to the character, not to the armor. That way, you wouldn't need to repair armor or track damage taken by it, although it does raise the question of what should happen if a character really low on hit points take their armor off.

Also, a natural counterpart to this mechanic would be to keep attack hit chance more or less fixed (possibly at 100%, to keep things simple) and have attack bonuses simply increase the average damage dealt per attack. A dice pool system, where each bonus adds a die to the pool rolled for damage, would seem a natural fit here. (If you wanted to keep the damage amounts lower while still using normal six-sided dice, you could e.g. count only one point of damage for each 6 rolled, and zero points for anything else.)

In any case, I would not recommend trying to make any such mechanics too detailed or "realistic". Rather, just try out various simple mechanics and try to find one that gives the right "feel" in terms of how combat plays out and how various bonuses affect it, and then conceptualize it in relatively abstract terms.

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