Quick Dice Average

Quick way to find the average (expected value) of dice xDy: Take half of x, add one to y, then multiply the two. This is equivalent to what some refer to as "Gauss's formula" for summing a series. For example:

8d6 -> 4*7 = 28
12d4 -> 6*5 = 30

OED Reincarnate

I've been working on a compact collection of spells for Original D&D-style games. Here's one example -- actually the very last one I completed, the longest, and possibly the one that caused the most difficulty to find the proper balance.

Keep in mind that in my games, there are no clerics. That might seem to preclude raising dead characters, but recall that our wizards fortunately still have the 6th-level reincarnate spell:

Reincarnate
Range: Touch
Duration: Instantaneous

With this spell, the character brings back a dead creature in another body, provided death occurred no more than 1 week before the casting of the spell and the subject’s soul is free to return. The magic of the spell creates an entirely new young adult body for the soul to inhabit from the natural elements at hand. This process requires 1 hour to complete. When the body is ready, the subject is reincarnated.

The recipient of the spell must make a saving throw to return in the same body type as before (same race and abilities, appearance may change). If failed, the DM should instead choose a random humanoid race of the same alignment for the new body type (re-roll abilities, up to ogre size). It's quite possible for the change in the character's ability scores to make it difficult for the character to pursue his or her previous character class. The character’s level is reduced by 1.

A few comments. You'll see that the standard "stat block" elements are just range & duration; nothing else is really necessary for the majority of old-school spells. The first paragraph is just copied directly from the 3E SRD as provided by the OGL.

The second paragraph is text written by myself. Since this is our only raise-dead type spell, I wanted to have some chance that the character returned fundamentally unchanged and still playable. Is the saving throw the proper mechanic? I think so. On the other hand, I wanted to retain the chance that the character is lost in the process, and returning as a random humanoid makes it likely to be the effective case (perhaps socially speaking) -- although restricting it to humanoids makes it easier to run in play if the DM so wishes. The probability is generally similar to 1E "system shock" rolls for Constitution, but folds into the standard save mechanic. The character-level loss is retained from the SRD to make sure this doesn't become a routine (or cyclical) procedure. Other than that, there's quite a bit of flexibility for the DM to adjudicate this spell in the best fashion for his or her campaign.

The spell text above is designated open game content under the terms of the Open Game License v1.0.

OED Update (v0.5)

Minor update to the Original Edition Delta house rules (version 0.5). Mostly just editorial cleanups, I felt it was important to fit on 4 pages (i.e., one folded sheet of paper). Took out some details on spells and monsters that seemed unnecessary here. Revised exploration movement back to the core rules.

http://www.superdan.net/oed/

The Gray Zone: Convention Games

Let's consider three different contexts for playing D&D:

1. Home campaigns. Here you'll be playing with the same players & characters over an extended period of time. Characters will almost certainly be generated individually to player taste; they will advance and explore the world over time. Old-school “sandbox” style play basically requires this context.


2. Convention games. This is a one-shot adventure, possibly limited to a 4-hour time slot or something similar. Characters may be pre-generated or custom-made (consider RPGA point-buy rules or the old DMG Appendix P, which I still use). The characters won't advance in any mechanical way.


3. Tournament play. This is also a one-shot adventure, but in a competitive context. There will be multiple (possibly very many) playgroups run through the same scenario, with an eye towards scoring the best and picking a “champion”. Characters are almost certainly pre-generated (so as to give a level playing field to the competition).

Notice that I distinguish here between “convention games” in general and “tournament play” in particular (even though they have many coarse similarities, and tournaments are generally run within a convention gathering). Convention & tournament games are similar in that they both feature short one-off adventures, and they avoid any usage of the character-advancement rules. But they differ in that one is competitive and the other is not. Simple convention games, perhaps, have more of an incentive to let the players “win” (sometimes they are run as product-release promotions or trials, and have good reason to want the players to leave the table feeling like they “had fun” with the experience and the product).

Tournament games, meanwhile, have an excellent reason to be tough meat-grinders where the majority of the players “lose” (by acting as a strict filter, they make it easier to identify the one “champion” in the event that made the most progress; whereas if many people uniformly “win” it will be difficult to make that distinction). Compare to an interesting quote from recent cyberware games at West Point: the attacks designed by the NSA were made "a little too hard for the strongest undergraduate team to deal with, so that we could distinguish the strongest teams from the weaker ones." And this also explains why the earliest D&D published adventures all had a "killer DM" feel to them: they were all originally developed for competitive tournament situations.

Okay, so getting closer to my point -- Having considered the different kinds of play contexts I've seen for D&D, two of them have seemed the most compelling, and one is rather more frail for me. We might ask the question, "Why are we playing; what do we gain at the end?" Two of these situations have a meta-reward, outside the game itself, that makes the experience deeper and more compelling. In case (1) Home campaigns, the meta-reward is largely character advancement; levelling up, accessing new powers and magic items. There's also exploration of a larger campaign world over time, but let's face it -- The #1 revolutionary, addictive development that D&D brought us was the idea of persistent, advancing characters over many game sessions, and this is almost solely accessible in terms of a home campaign. In case (3) Tournament play, the meta-reward is the competition with other teams playing in parallel to yours, and seeing one team at the end awarded with honor and a trophy (or somesuch). Personally, I love playing in a tournament, and love the heads-down, high-proficiency play that I see in that context.

So that leaves case (2) Convention games, and frankly, I can't figure out what the meta-game "point" is to them anymore. When I run one, I'm left a little bit bewildered at the end about what the payoff is. It seems very awkward if there's a TPK at the end, and it seems almost equally awkward if time simply runs out after a certain number of rooms are successfully looted.

One suggestion is that there needs to be a specific "quest" in a convention game -- The players are given an explicit (or obvious) assignment at the start, and if they can succeed in the time alloted, they are declared to have "won". A few problems here: (1) It's difficult to estimate in advance a perfect set of encounters that lead to a "win" at exactly the 4-hour mark. (2) The setup manages to frustrate the classic D&D architecture of open-ended exploration, multiple paths, resource management, wandering monsters, treasure and XP rewards, etc. (3) There's still no meta-game reward from this in-game "victory".

Now, I have a good friend Paul who recently ran an exceptional convention game a few weeks back. Philosophically, we tend to disagree about many of the high-level "whys and wherefores" of D&D, but I think we almost always agree about whether a given game we just experienced was good or not (sort of an "I know it when I see it" experience). In the past we simultaneously co-DM'd a campaign, and at least once our differing styles stomped ugly all over each other (Ettin-style?). He may run a better convention game than I do; the one he ran the other weekend was one of the most fun D&D sessions I've had in a long time -- hilarious characters, great encounters, well-paced, filthy humor (which I like), great ending. I was mulling over my troubles with convention games on the ride over, and lo, my friend snaps off one of the best such games in my memory.

Anyway, Paul wrote up his notes on that adventure on his blog over here. The thing I was surprised and a bit unsettled by was that the quest, locations, and NPCs were all being invented and moved around backstage on the fly, which is how our investigations managed to lead us to saving the girl at almost exactly the 4-hour mark. Made for a great, nigh-perfect gaming session -- and it's not something I think I'd ever be able to bring myself to do, as it goes against every grain I've been trained in as a game designer, thinking more in terms of published tournament-style adventures that we'd prefer to keep fixed, replicable, and fair if multiple groups are run through the same adventure over time.

So, what to do? Should I just give up on running one-off convention games (granted that they frustrate all the meta-game rewards that are the hallmark of D&D), and leave them to better narrative DMs? Is there any way to interface the classic rewards of D&D in an isolated, one-shot experience? Troubling questions, since at this point in my life the only opportunities I have for play are the infrequent one-off convention games: the "gray zone" in the middle, if you will.

What is the Best Combat Algorithm?

Throughout the history of D&D and RPG's (and more generally, any action/wargame), there have been a host of different algorithms to determining success in combat and other feats of skill and luck. For example: to-hit-tables, THACO, compare to increasing AC score, etc.

Within some very small tolerance for error (say +/-1 difference), all of these systems have been mathematically equivalent (i.e., result in "hits" for the same rolls of the d20 die). But which is the best algorithm? That is, treating the tabletop gamer's brain as a kind of natural "computer", which is easiest/ fastest/ most efficient/ least error prone? Is it one of the aforementioned algorithms, or something different?

First, let's establish the different components of the basic D&D "to hit" (or anything else) roll. They include: (1) a d20 die roll, (2) a basic attack proficiency, by class or hit dice, (3) the armor of the defender, (4) miscellaneous modifiers (positive bonuses being good for the attacker), and (5) the "baseline" chance to succeed at hitting, irrespective of other modifiers #2-4.

Let's look at one example, say, the THACO mechanic from 1E-2E. In the form of an inequality, the basic algorithm is:

THACO ALGORITHM: d20 + mods ≥ THACO* - AC

* The THACO was itself determined (pre-game time) from tables in the core books. But in essence, for fighter and monster-types, this incorporated the "baseline" success chance (Normal Men need to roll ~20 vs. AC 0) and a +1 bonus per fighter/monster level. In other words, THACO = (~20 - level). Let's substitute and see all 5 terms plainly:

THACO ALGORITHM: d20 + mods ≥ 20 - level - AC

Now, if we proceed to search for other, variant algorithms, we can apply the basic algebraic "rebalancing" operations to make any of these terms appear on either side of the inequality that we wish. For example, we could add a "level" term to both sides (canceling it on the right and appearing as an addition to the left). Or, we could subtract the "mods" from both sides (thereby appearing as a subtraction on the right).

In fact, since there are 5 terms, and each can appear on either of the 2 sides of the inequality that we wish, there are in fact 2^5 = 32 different formats for this inequality (by the fundamental principle of counting) that we could consider. Here are just a few of those 32 possible variations:

TABLE ALGORITHM: d20 + mods ≥ (20 - level - AC)
[Encapsulated in table]

THACO ALGORITHM: d20 + mods ≥ (20 - level) - AC
[Encapsulated in THACO]

d20 SYSTEM ALGORITHM: d20 + mods + level ≥ (20 - AC)
[Defined as New AC]

"SUBTRACT ALL" ALGORITHM: d20 ≥ 20 - level - AC - mods

"GO NEGATIVE" ALGORITHM: 0 ≥ 20 - level - AC - mods - d20

Etc...

Now, obviously, those last few were for humorous illustrations only, and I assume not many people would want to use those systems. But what criteria can we use to choose the "best" possible system? Let's consider the following as guiding principles (and we'll back each of them up with results from experiments in cognitive psychology as we proceed):

(1) Additions are easier than subtractions. Although mathematically equivalent (and using fundamentally the same operation in digital computing systems), most people find subtraction significantly harder than addition. For example, addition is commutative (the order is irrelevant) whereas subtraction is not. See the paper by MacIntyre, University of Edinburgh, 2003, in a test of 2-digits sums and differences (six of each) given to 8th-graders (N = 135): "Addition tasks are clearly completed in a much more confident manner than the subtraction items, with over 50% of the study group gaining maximum credit. Subtraction items appear to have presented a much bigger challenge to the pupils, with around 50% of them having 3 or more questions wrong." See also Kamii et. al. 2001 (a study of N = 33 kindergartners, and N = 59 first and fourth-graders).

(2) Round numbers are easier to compare than odd numbers. In other words, when comparing which of two numbers is larger (the final, required step in any "to hit" algorithm) it will be easier if the second number is "20" than, say "27". This follows from the psychological finding that it's faster to compare single digits that are farther apart; see Sousa, How the Brain Learns Mathematics, p. 21: "When two digits were far apart in values, such as 2 and 9, the adults responded quickly, and almost without error. But when the digits were closer in value, such as 5 and 6, the response time increased significantly, and the error rate rose dramatically..." In our case, setting the second digit to zero would maximize the opportunity for a large (and thus easy-to-discern) difference between the numbers.

(3) Small numbers are easier to compare than large numbers. This has also been borne out by a host of psychological experiments over the last several decades. Again from Sousa, p. 22: "The speed with which we compare two numbers depends not just on the distance between them but on their size as well. It takes far longer to decide that 9 is larger than 8 than to decide that 2 is larger than 1. For numbers of equal distance apart, larger numbers are more difficult to compare than smaller ones." Again, this is true for human computers only, not digital ones (ironically, the digital processor "compare" operation is really just an application of the same "subtract" circuitry).

Okay, so let's think about applying these principles to find the cognitively-justified best tabletop resolution algorithm. Applying principle #3 means that we'd generally prefer dealing with smaller numbers rather than larger. Before considering anything else, it's clear that it will be hardest for people to mentally operate in a d% percentile system, easier in a d20-scaled system, and easier still on a d6-scaled system. We should pick the easiest of these that gives the fidelity necessary to our simulation, and the d20-scale does seem like a nice medium.

We can also apply principles #2 and #3 to discard a key change brought about to D&D in the 3rd Edition: Ascending AC numbers. While it has its proponents (and is of course mathematically equivalent to all the other 32 permutations of the core mechanic inequality), it forces us at the end of our algorithm to run a comparison against a relatively large, and frequently odd, number, such as AC 15, or AC 27. By using instead descending ACs, they will always be a single digit (and therefore easier to manipulate according to finding #3), and we'll also see below that we can arrange a rule such that the final comparison is always run against a fixed, round number (and therefore preferred according to principle #2 as well).

Applying principle #1 indicates that we'd prefer to have all of our operations be additions, and do away with any subtractions (as in the THACO system). Returning to our 32 different options for presenting the basic resolution inequality, this is easily accomplished: simply add back all the terms on the right-hand side of the inequality, and all those terms become simple additions on the left. Having done this, we'll see that we're left with a nice round number to compare that addition to (fortuitously complying with principle #2, as mentioned above). We'll call this the "Target 20" algorithm:

TARGET 20 ALGORITHM: d20 + level + AC + mods ≥ 20

Now, you may have guessed where I was going with this if you'd read previous blog entries of mine supporting the idea. While never presented this way in TSR/WOTC core rulebooks, I'm quite confident that this is the most mentally efficient representation of the core d20-based resolution mechanic: Add d20, your fighter level, your opponent's single-digit descending AC, and miscellaneous bonuses; a number equal to or greater than 20 then indicates a "hit". It satisfies all of our 3 psychologically-verified guiding principles: (1) additions are easier than subtractions, (2) round numbers easier to compare than odd ones, and (3) small numbers easier to manipulate than large ones (particularly in the form of single-digit, descending ACs).

Like a lot of things in our hobby, the Original D&D rule was pretty close to optimal, but not quite perfect in this sense. If I won the lottery it might be interesting to definitively prove which method is best by running a series of psychological experiments; but since the result just follows from already-proven principles, I'd also want to set up a betting pool and recoup some of my money from the WOTC chief designers of the last several years.

See more recent development of the Target 20 system at the OED Games website.

The Golden Rule

Comments on the last post made me once again recall what I consider to be the most important passage in all of Gygax's writings on D&D. I use the following as my "golden rule" when thinking about game design for D&D:

ADVANCED DUNGEONS & DRAGONS is first and foremost a game for the fun and enjoyment of those who seek to use imagination and creativity. This is not to say that where it does not interfere with the flow of the game that the highest degree of realism hasn't been attempted, but neither is a serious approach to play discouraged.

- DMG p. 9: "The Game: Approaches to Playing Dungeons & Dragons"

Now, in the interest of being as clear as possible, allow me to unpack the latter two clauses and clean up the double negatives. If we do so, we read this:

(1) The highest degree of realism has been attempted (so long as it does not interfere with the flow of the game).

Again, the double-negatives make the passage slightly hard to parse on first viewing. In fact, we do seek the highest degree of realism -- claims that D&D has "never been realistic in any way" are totally false. Purely abstract systems are not of interest to us. However, if a conflict arises, then what must take precedence? Definitely, the flow of the game. Both elegant gamesmanship and realistic modelling, working in synergy, are the zenith of game design; but if those goals come into conflict, then gamesmanship must clearly, (narrowly) win out.

(2) A serious approach to play is encouraged.

We can allow ourselves to be serious about our gaming. Critiques that "you're thinking too hard about fantasy" can generally be ignored as meaningless. And at the same time, if some of our friends are most interested in the fantastical or even comical elements of our gaming, then that should be seriously respected, as well.

Games-Within-Games

Here's an important aspect of early D&D I've been meditating on lately: the propensity for it to be an ongoing construction of games-within-games. Let's consider a few exemplary examples that spring to mind, starting with D&D and some of my most-favorite computer games:

(1) Dungeons & Dragons. In some sense, OD&D can itself be thought of as the “discovery” that the CHAINMAIL rules contained an even more interesting sub-game with its fantasy combat at the man-to-man scale (not to mention its even more refined system for jousting competitions). In the initial “White Books” you had both the standard dungeon exploration, as well as separate and distinct rules for large-scale wilderness exploration, castle-building, aerial combat, and ship-to-ship naval engagements.

What do I consider some of my most memorable D&D adventures? How about module X10, with its unique strategic-level world-warfare game (in parallel with PC-based diplomacy/adventure scenarios – including possible sidetracks to other X-series modules). Or M5, with a points-based diplomacy roleplay between imperial powers at the adventure's climax. Or even module S3, with its special system for trying to manipulate high-tech artifacts (among other things).

(2) Sid Meier's Pirates! Man, did I play a lot of this game on my cousin's Commodore 64 one summer. In some sense I consider it to be the near-perfect game – and, a lot of my design efforts wind up looking like attempts at replicating this classic. One of the strengths is that it has a completely different sub-game for each skill you might perform in your career as a privateer in the Carribean. Strategic sail navigation, taking a sun-sighting, fighting by cannons, personal swordplay on the deck, invading towns, choosing crew and cargoes, puzzling over map fragments, and wooing the daughters of prominent mayors, are all simulated in distinct sub-games. And almost all of them are both flat-out wonderful, and interface perfectly with all the rest (to the extent that only at this late date can I recognize them as sub-games at all).

(3) Mechwarrior. The original Mechwarrior was another game I played and re-played a whole lot of times. It's the first game I played that had both (a) “sandbox” play, and (b) “plot” based threads. The “sandbox” allowed you to progress as a mercenary captain, taking randomly-generated combat missions, improving your team and equipment over time. The “plot” (for lack of a better word) allowed you to follow up on clues that you were the member of a deposed royal family, and potentially win back your family's home. Some great (and dare I say Gygaxian) aspects of this: (1) you could play the mercenary sandbox indefinitely, (2) it was actually fairly hard to discover that there was a “plot” based mystery to follow up on in the first place, and (3) you still had to do some random mercenary missions in order to build up the strike team you needed at the end of the plotted scenarios. The exact time and sequence of events is impossible to predict in a game of Mechwarrior.

Now, some of this should be well-known to players of current computer game “sandbox” designs (Grand Theft Auto, anyone?), but since I don't play modern consoles, I can't comment directly on those. The thing I want to emphasize is that we don't lose the willingness to allow games-within-games in our classic tabletop RPGs.

Consider a few other examples from TSR/WOTC. In the old Star Frontiers Knight Hawks space combat game (by Doug Niles, who deserves his own blog acclaim), there was a brilliant scaling rule: for 15+ ships, use the coarse, Basic rules for the game; for 5-14 ships, use the more detailed Advanced rules; for 2-4 ships, use the Advanced rules with the individual characters' piloting & gunnery skills detailed. In the more recent d20-based Star Wars game, the spaceship rules were entirely done by analogy to the stock character-to-character system – which I was rather appalled to see when I read it.

Post-2000, there's been a bit of an over-reaction by my left-brained brothers and sisters, often times feeling that all activities in a particular game need to be abstracted out into one single universal mechanic. While this might be nice in theory, in practice I consider it be an abject failure (see the Star Wars example above). Even AD&D is not immune to criticism – when it converted overland movement rates from hexes to miles-per-day (so as to be usable with any campaign map scale; compare DMG p. 58 to OD&D Vol. 3, p. 16), it should have been emphasized that each DM really needed to manipulate those numbers and turn them back into spaces-per-turn on their personal map scale. Unfortunately, it did not. Here we see how frequently the attempt at abstraction interrupts the gamesmanship that we need at any scale of action.

Hence we have a few criticisms of the current branding of D&D: Action at different scales should have different mechanics that support the distinct flavor appropriate to each. Likewise, character classes that represent very different approaches to adventuring (magic vs. martial arts) should have different mechanics supporting each. We lose a lot when the game is reduced to a single kind of action scale (6-second moves on 5-ft squares), and the willingness to include sub- and super-games is prohibited (such as castle-building, tactical mass warfare, etc.) And, we have even more reason to avoid fetishizing character development, because we have to be willing to lose those characters abruptly if we play out an encounter at a larger scale (see 3E's Tome of Battle for the mangled result of being unwilling to allow for this).

Much of the addictive beauty of the original D&D game comes specifically from its flexibility as a model of developing games-within-games, both above and below the “normal” scale of action. It's a more interesting and more challenging enterprise than writing either "story" or "sourcebook" supplements, which add nothing concrete to our gameplay. But likewise, we should avoid being dogmatic, and try to engage our expansion systems only when it makes sense to do so (perhaps taking Doug Niles' SFKH as a canonical, concise example).