Guest Post: The Power of +1

Several weeks ago, I was approached by one of my players who wished to write a guest post for the blog. He plays a Ranger in my Dungeons & Dragons 4th Edition campaign, and his character has the wonderful genre-breaking trait of hating the outdoors and anything associated with the wilderness. I found my player’s concept for the post interesting, and it built off a conversation regarding uneven leveling that has sprouted up in our games from time to time. Below is his column, which was shaped with a bit of feedback from yours truly and The Hydra DM who shares similar interests in dissecting the building blocks of a game – including Experience Points – and theorizing about what the results mean for those playing each session. During the life of The Id DM, I have hosted one previous guest post on the motivations of a Power Gamer. Enjoy the guest post below . . .

Power (Non-Outdoorsy) Ranger

Garrick, famed Urban Ranger, penned the following column.

As an introduction, I have played with The Id DM in a 4th edition game for almost a year.  I am almost ashamed to admit that after playing for that long I only recently examined this site. [Iddy’s note: he has not yet been punished for such insolence!]  I was impressed with how well put together the site was and how well written the articles were.  The reason I visited this site for the first time was because an old discussion was restarted about uneven party member leveling and the associated benefits and consequences of giving some party members varying experience for activities or actions completed, which in turn results in some players leveling before others.  The Id DM wrote an article that listed reasons to avoid uneven party leveling while another player in our campaigns, Dungeon Maestro, listed reasons to embrace uneven party leveling.

An interesting thing is mentioned which pops up all the time in d20 systems and can be summarized by the following statement, “It’s +1 so your hit chance is increased by 5%.”  The most misleading thing about this statement is that it is absolutely true.  I wanted to take a moment to examine some of the math inherent in D&D to point out the flaw in logic that comes associated with the absolutely true fact that +1 is a +5% chance in a d20 system.

It is an elementary exercise in probability to calculate that a +1 lowers the number you need to roll on a twenty sided die by one to reach your target value, which means that it increases your chance of success by 1/20 or 5%.  Despite the simple math behind this calculation a logical fallacy almost always becomes associated with the 5%.  I will list two common oversights made with the 5% rule in the format of truth and associated fallacy.

Truth: +1 to hit increases my chance to hit by 5%

Fallacy: +1 to hit increases the amount I hit by 5%

Truth: +1 to AC decreases my chance of being hit by 5%

Fallacy: +1 to AC decreases the amount I get hit by 5%

I like to use 4e as my example but the following tends to hold true in other d20 systems such as 3.5 and 3rd edition. The target number to hit a monster in 4e combat is typically about ’11′ on a d20.  In reality it is closer to ’9′ or ’10′ on a d20 but the math is simpler if we use ’11′.  Players, NPC’s, and monsters (in 4e especially) are designed so that they hit and get hit about 50% of the time.  This means that on a roll of 11 or greater the target is hit and 10 or lower is a miss.  Now if the player or monster has a +1 to their roll then their target number on a d20 is ’10′ so 10 or higher becomes a hit and 9 or lower a miss.  The player’s hit chance on a d20 is increased by 5%.

The often missed step in the math is calculating how much the amount the attacker hits increased by.  Previously the attacker hit on an 11 or greater which is 10 numbers on the d20 or 10/20 which is 50% of the time.  By increasing his to hit by +1 he now hits 11/20 or 55% of the time.  To calculate the increase in the amount the player hits, one takes the difference between the player’s former hit chance and the new hit chance divided by the former hit chance.

Fractions:

Percentages:

The amount the player hits increased by 10%.  At first this can seem confusing but logically it makes sense, the player used to hit on 10 numbers now he hits on 11 numbers which is 10% more.  The difference in your target number does have an effect on this equation. For example, if the target number is 8 and a +1 moves one’s target number to 7 the increase in the amount one hits is 8.33%.

Fractions:

Percentages:

It is interesting to note that the +1 has diminishing returns on the amount one hits.  The same follows for the +1 to AC since in general creatures have a 50% chance to hit you the +1 to AC (or defenses) results in getting hit 10% less.  I will let the reader work out the math on this one but it is essentially identical.

The numerical output of hitting and getting hit is damage.  It follows that by increasing the amount one hits by 10% increases damage output by ~10%.  It also follows that reducing the amount one gets hit by 10% decreases damage taken by ~10%.  There are definitely some liberties taken in this math.  There are other things going on here such as critical hits or any mechanics that increase the damage of a hit after it is confirmed.

Summary

The purpose of this article is not to get bogged down in these details but more to point out that the effects of a +1 are usually underestimated and the 5% ‘effect’ is patently not true, the actual effective amount is almost always considerably larger.

What does +1 really mean?

As someone who also runs games of D&D and other d20 systems I think it is important to make some conclusions from this information.  The power gamer at the table is unbalancing the game more than one might think.  His +2 higher hit chance causes him to do ~15-20% more damage than the other players at the table.  I typically try to help my less experienced players close this gap by making recommendations to them on their character builds.

Uneven leveling causes a larger gap than people initially assume - especially in 4e.  Party members who are at an even level ahead of other party members receive a +1 to hit and a +1 to defenses.  This means they do ~10% more damage (all other things being equal) but they also take ~10% less damage. This imbalance causes a ~15-20% gap between them and the lower level player, and I am not even including things like the increase to hit points and ability scores as well as new feats and abilities learned. When one considers the other benefits provided by level advancement in 4e, this gap becomes even larger.  Five even leveled players should be able to easily outmatch six identical characters who are a single level lower in 4e.  When someone says ‘it’s only +1′ it might be worth the time to see if they understand what that really means.

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About The Id DM

The Id DM is a psychologist during the weekdays. He DMs for a group of fairly loyal and responsible PCs every other Friday night. In the approximate 330 hours between sessions, he is likely anxious about how to ensure the next game he runs doesn't suck.
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5 Responses to Guest Post: The Power of +1

  1. thehydradm says:

    Since it’s got my name at the top, I feel it’s important to point out that the reason I recommended changing the “Truth: +1 to hit increases my chance to hit by 5%” line to “Truth: +1 to hit increases my chance to hit by 5 percentage points” (something that ultimately was discarded as a change). That’s because, literally speaking, that line is committing the very fallacy it claims to avoid because of the way it’s written. I know what it’s supposed to mean, but for Joe Reader it may be confusing.

    So, for anyone that’s confused: 50% chance to hit increased by a +1 on a d20 would become 55% chance to hit; that’s an increase of 5 percentage points (50+5=55), but not 5% (50*1.05=52.5).

  2. thehydradm says:

    Alas, poor Yorick, I hit reply before I was done!

    My thoughts on this are conflicted. In theory I’m in full agreement: the disconnect between 5 percentage points and 10% better is hard enough to grasp, but when you start doing “power gamer geometry” and finding the “power volume” of these geometric characters (by combining their damage, defenses, etc. stats together to get a superior whole not unlike combining the length, width, and depth of a regular prism, in this case by multiplication) it gets crazy what a +1 can really do, and that’s discounting feats or powers!

    At the same time, on the coat-tails of a campaign where I let PCs’ levels run wild (a level 8 even adventured alongside a level 1 at some points) I didn’t find much deficit in practice except in seriously wide level gaps (such as the aforementioned level 8 + level 1).

    Sort of reminds me of the Chris Perkins dice trick, in that way. Sure having a theoretical distribution of several dice is great and all, but in practice will it ever really matter that much compared to the time you save only rolling a single die for damage?

    When the theoretical model and practical results don’t really seem to line up like this is where the real science gets done; I say we all give it a thorough investigation!

  3. I will take that bet… 6 level 10 characters > 5 level 11. =)

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