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Modifying difficulty works in various ways within and between various roleplaying systems. But generally speaking, it seems added difficulty either modifies the target number (altering the check) or reduces the effectiveness of a roll (appending positive or negative modifiers, depending on how checks work).

Say you need to roll above 15. Let's say that in one situation, that difficulty bumps up to 17; in another situation, the target number is the same, but your roll has a −2 modifier. Superficially, the math looks like it would be the same: in either case, the roll becomes 2 more difficult.

I'm wondering whether there is actually a deeper, non-obvious difference between the two. Similar to how a 2d6 looks like a more complicated 1d12, but in reality, is made dramatically different by the rules of probability: the 1d12 gives a flat line of unweighted results while the 2d6 gives a curve naturally favoring middle values. Perhaps mathematics does similar witchcraft on these two styles of difficulty.

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The basic math does not change, no. 1d20+5-2 >= 15 is the same as 1d20+5 >= 15+2, and 3d6+5-2 >= 15 and 3d6+5 >= 15+2 are equal as well, in case you're wondering about those pesky curves.

However, there are five basic ways the outcome can still be different in certain cases.

  • The first is if there are any rules or abilities that kick in on a modified roll result that's below (or above) a particular value. Angelo Fuchs gives the example of crits in CyberPunk when spending luck: the luck-modified roll, if greater than a certain threshold, explodes.
  • The second way is if there's a rule that is activated when the DC or target number is above or below a particular threshold, like D&D 3.x's rule forbidding certain skill checks on DCs greater than 10 without ranks in that skill.
  • The third, which I've never spotted in the wild, would be if there's a way to reduce or remove unfavorable DC modifiers as such; this seems unlikely, given the way DCs are usually based on inherent aspects of the task.
  • The fourth is if there are rules that allow penalties to rolls to be subsumed by an existing penalty of the same type or removed by an ability; e.g., D&D 3.x applies only the largest penalty by default and allows feats to remove penalties regardless of amount.
  • And the fifth, the odd one out, is if it only affects rolls and there's an ability or rule to forgo rolling, like D&D 3.x's take 10 rule.
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  • \$\begingroup\$ One example of your third possibility might be curses or afflictions that give a penalty. These can be removed, which also removes the penalty. \$\endgroup\$ – indigochild Feb 17 '16 at 21:59
  • \$\begingroup\$ @indigochild: Those would usually be examples of the fourth, unless they're modifiers to the DC as opposed to the roll. \$\endgroup\$ – user17995 Feb 17 '16 at 22:05
  • \$\begingroup\$ Crits (at least in CyberPunk) are of the first kind: If you pay "luck" you increase your roll. If you roll 10 (or above) on a d10 (+luck) you roll again and add that up (repeatedly if applicable). \$\endgroup\$ – Angelo Fuchs Feb 18 '16 at 13:32
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    \$\begingroup\$ This answer is great, just what I was looking for. I particularly appreciate looking at it from a perspective of the rules ecosystem, which in hindsight looks more important than just what the numbers are doing. Also, though, everybody gave great answers to this, all of which are worth reading. \$\endgroup\$ – derektb Feb 18 '16 at 21:56
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Existing answers are already very good and this one does not mean to replace them. Wax Eagle and TuggyNE did a great job. I would like to offer an alternative, narrative point of view.

In games like D&D mathematics are supposed to express certain things and provide a difference in the mechanics where there is a narrative difference in actions modelled. For example, having a skill rank of 5 and beating a DC 10 expresses a different situation than having a skill of 20 and beating a DC of 25 - even though the odds and math are exactly the same! One situation is a simple task attempted by a non-skilled person, the other a very difficult task, but the character is an expert.

Altering the DC communicates that the task itself is more difficult. The lock is more complicated or maybe the cipher is harder to break.

Giving someone a modifier communicates that in this particular case the character has help or distraction from non-task circumstances. Maybe your arm is injured while picking your lock or you have discovered a partially deciphered message?

If you decide to change the DC, your players can expect that all locks of this type are of that new DC. If you add a modifier, your players will expect the action to be influenced by some (maybe hidden) circumstances. In another example, bumping a chance to jump a ravine by 2 says "This ravine is dangerous." Giving a modifier of -2 says "This ravine is not as dangerous, but the winds are high right now."

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  • \$\begingroup\$ This is fantastic. As a new DM, I've been trying wherever possible to link math to the story, and thinking about it this way really helps. \$\endgroup\$ – MikeyWard Feb 18 '16 at 18:43
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    \$\begingroup\$ @MikeyWard As a new GM I spent countless hours trying to do the same. As an old one I came to the conclusion it was unnecessary for the most part - unless you play a highly simulationist game. Now I play mostly high-abstraction games like Fate or Dungeon World. Props for analytical way of GMing though! \$\endgroup\$ – eimyr Feb 18 '16 at 19:46
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    \$\begingroup\$ I'm glad someone gave this kind of answer. I had originally tacked a "philosophical differences?" coda to the end of my question; I edited it out for clarity's sake (I's just wondering about the math), and because I already my own notions of what the two meant (DC reflects 'external conditions', modifiers 'character conditions'.) But this is a beautiful and wise answer, and I'm glad it's here to revitalize an otherwise pretty dry question of math. \$\endgroup\$ – derektb Feb 18 '16 at 21:39
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This depends heavily on the kind of adjustment you're making to the DC and to the roll. If the adjustments to the DC and roll are equal and static numbers, then there is no effective difference between a +2 to the DC and a -2 to the roll. Both of them affect the overall equation in the same exact way:

X + d20 - Y >= Z
X + d20 >= Z + Y

So a static modifier has no change.

A dynamic modifier to either the roll or the DC would have a much more difficult to predict effect. Typically you get a dynamic modifier on the roll, such as in D&D 5e where you can gain advantage or disadvantage. This creates a curved distribution of roll values and finding a DC equivalent to the original is actually quite difficult (because it depends on how difficult you want the DC to be).

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The subtle difference in math is the range of possible results. In most cases this is not important, because you are only checking success, which is a binary code: true or false. As skill checks do not have a minimum result, allow negative totals and there are no exceptions like "1 fails", "20 automatic success", the probability of true or false is the same for adding DC or subtracting from the roll.

One significant exception in DnD 3.5 is the craft skill. A successful check result translates to a monetary value so deducting from the roll decreases the monetary result of a successful check.

The other difference is much more obvious: bonuses and penalties can stack. A change of DC can come as a modifier (e.g. in the case of the feat Spell Focus, or several consecutive Tumble attempts) or as an alternative DC for a different task. If you got several alternative DC's you have to choose one, but this kind of increased difficulty does not stack.

Example: Crafting a martial weapon is DC 15 (still DnD 3.5), crafting an exotic weapon is DC 18. So this difficulty is increased by three steps. Crafting a masterwork weapon is DC 20. But it does not make a difference if you are crafting a martial masterwork weapon or an exotic masterwork weapon. In case the difference of martial or exotic weapons would be handled by a modifier, this would be different.

In some cases, this leads to strange results. The Concentration skill asks for several checks for each kind of distraction:

If more than one type of distraction is present, make a check for each one; any failed Concentration check indicates that the task is not completed.

Vigorous Motion DC 5

Grappling DC 20

Weather is wind-driven hail DC 10

So casting a 1st level spell while grappling is DC 21. Casting a 1st level spell while grappling aboard a ship in a hail storm takes three checks, but incurs no negative modifier. So for a caster with a concentration skill modifier of +10 it makes no difference if he casts while grappling or while grappling on a ship in a hail storm. This would certainly be different if stacking modifiers were used to handle the situation.

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  • \$\begingroup\$ Good catch on the craft skill keying on the modified result; that's a good example of more or less my first possibility, but I couldn't think of such an example off the top of my head. \$\endgroup\$ – user17995 Feb 17 '16 at 21:38
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Yes in Some Dice Pool Systems

Modifying difficulty works in various ways within and between various roleplaying systems.

Indeed, not all systems are the same, and what is a distinction without a difference for a single-die engine that uses a d100 or d20 is a huge difference for a dice-pool-and-TN system like the Storyteller system (World of Darkness) or Exalted. I will show how and why it makes a difference in those systems.

Please note that this answer is written in reply to the system-agnostic variant of the question, as seen circa 2019-01-25, not in reply to system-specific versions.

How Storyteller Dice Rolls Work (Used as an Example)

The system uses a number of d10 for all rolls. The number of dice is determined by the character's sum of attribute and skill (usually the total ranges from 1 to 10), and some roll modifiers (e.g. using supernatural effects to increase Dexterity and/or aiming for extra turns before taking a shot can give you a few extra dice; conversely, being wounded penalises your roll, meaning you get fewer dice). The difficulty (target number) is dictated by the rules or the GM. For example, hitting someone with a punch is difficulty 6, while hitting someone with a punch in the dark is difficulty 8.

When you make a roll, you see how many dice come up equal target number or higher; this is the number of successes you scored. Each 1 rolled also subtracts one from your number of successes. Enemy actions, such as dodging, can further subtract from your number of successes. (There are potential other complications that I won't go into for the purposes of this example.) But so long as at least one success remains, you've . . . succeeded to some degree at your action. If you score more than one success, you succeeded better at your action - e.g. that may grant you a damage bonus on a to hit roll, or mean you not just finished making your film but also achieved more fame and earned more money from it than the expected minimum.

Why in Such Systems, the Difference Matters

Mathematically, on average, one expects a number of successes approximately equal to dice×(10-target number)/10. (Don't use this formula for precise calculations, especially for difficulty 9 or 10.)

This means that when the target number is low, adding a roll modifier (such as an attribute increase, or scoped aiming) will significantly increase the expected number of successes (almost by one success per die). Conversely, when the target number is high, like 9, adding extra dice has very little effect on the final result.

A similar relation exists on the other side: when the dice pool is low (such as due to having a -5 roll penalty from wounds), changes to difficulty have relatively little effect on the outcome, and you won't be scoring many successes even if lucky; when your dice pool is huge (e.g. 15 for a highly optimised sniper character taking a well-prepared shot), a single level of difficulty (target number) difference can provide more than one success worth of difference in expectation.

There is a fair amount of criticism against such a dice engine, which resulted in some subsequent systems by the same group of developers to have fixed target numbers (e.g. Chronicles of Darkness' Storytelling system has the target number fixed at 8 for all rolls, only using roll modifiers for everything), and a split in the community over which is more enjoyable in play.

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    \$\begingroup\$ There is a graphic depiction of this in another answer, I'll try and find it and link it in a comment. \$\endgroup\$ – KorvinStarmast Jan 25 at 20:36

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