Combat Maneuvers

Every man-to-man combat system begins with the assumption that both sides will take turns bipping one another with swords or lasers or whatever. The moment you allow for more complicated forms of interaction, things can get amazingly complicated in a hurry.

Our current campaign under the ACKS system uses a double-test for any special combat maneuver:

• make a normal attack with a -4 penalty, then
• give the target a saving throw vs paralysis

Effectively this makes success depend on three things: (1) attacker level, (2) defender armor class, and (3) defender level. (There's also a loose admonition to apply suitable penalties for size differential, but those are left to judge's discretion.)

The most controversial feature here is armor class. The system designer (Alex Macris) is pretty adamant about the innovation of a "touch AC" in D&D (or a CMD in Pathfinder) going to far in terms of negating the value of gear. The concept here is that being armored doesn't just block attacks, it also frees you up to defend in a totally different and far more confident way. A knight doesn't need to focus on anything except holding a sword and using it to beat someone else on the head. A light fencer needs to focus on survival-footwork first, and so is much more vulnerable to trickery. For example, he's going to constantly be using his sword to parry, which is going to make it easier to sunder that sword.

From a gamist perspective, there's another argument at stake here: "tanks" should be hard to control. If a heavily armored fighter steps forward, you don't want there to be a single right answer for how to neutralize him every encounter. "Trip him and make him fall down" is funny the first time, but if it happens every battle and becomes such an obvious tactic as to be mandatory, then it's negating the whole concept of having armor class as a numerical proxy for defense ability. The best tactic is then always to grapple the tank, then run past to murder everyone else. So special maneuvers shouldn't, in general, be able to disregard armor class, since that would negate the value (and fun) of stomping around the battlefield as an impregnably armored behemoth.

The bottom line is that I've decided it's probably best to keep using armor class as the defending stat for special maneuvers, rather than try to introduce an independent combat maneuver defense. For things like "sunder weapon" or "disarm", I think the best approach might be to define an alternate minimum armor class equal to level + strength-bonus, so that hulking shirtless barbarians can still hold onto their giant axes. High AC defenders could still use armor class, but low AC strikers could replace that with their raw force and tenacity.

Rate Of Attack

There's a split in RPG design between systems that represent a combat round/turn as being short (6 to 10 seconds) or long (a half-minute to a minute). This divide traces all the way back to 1st edition, where AD&D switched to the "long round". The latter interpretation requires that a "round" be interpreted as multiple extended exchanges with an aggregated outcome.

When playing ACKS last night, I was impressed by just how quickly the battle was over. Essentially from the moment cave entry until the fall of the dragon (split over two separate encounters), the fight took roughly 10 rounds, amounting to about a minute and a half. On that time scale you'd expect that an entire dungeon floor could be cleared out in 20 minutes. Granted, it would require amazing stamina to fight continuously for that long!

That carries over into the tactical battle system as well. A battle involving huge armies clashing still ends in a couple minutes. And that can involve a single hero wiping out multiple platoons, probably amounting to a kill every three seconds or better, under calculations based on the surprisingly generous cleave rules. Any class but a mage can cleave multiple targets, and even bows can get off a number of shots in that ten-second round under the same rule (which I guess means it should be called cleave/multishot!) Shortbows have a natural edge in this system, since they trade in their shorter range for a better cleave multiple, providing a definite incentive for an experienced fighter to downgrade to a lighter ranged weapon.

To provide some visual reference for the standard arguments for this compressed time scale, I did a search for rapid use of realistic weapons by modern-day specialists.

Here's a video of some Viking reinactors:

There's an absolute imperative in this style of fighting for taking shorter, weaker thrusting strokes, rather than the stereotypical wide and reckless swing. It's not particularly theatrical in appearance, relative to your average Hollywood movie, but the sharpness of sword-points makes it mandatory to create constant distance with any opponent. That requires both the sword and shield to be pushed forward at all time, moving in small, unpredictable arcs.

This is, in some sense, a justification of two tropes of fantasy wargaming at once: the short strokes happen fast (allowing substantial improvement in attack rate with experience). But they also aren't very effective at finding more distant kill zones, justifying the value of armor (as blocking instead of absorption) and the idea of whittling down health of a skilled opponent in small increments ("HP loss") to represent declining stamina or coordination-impairing bruise wounds to the sword-arm or shoulder. On the other extreme, an inexperienced or slow defender would take a single stroke to the neck and go down in a matter of seconds.

For ranged weapons, there are plenty of videos communicating the speed of a polished archer with a lightweight recurve bow. In a competition requiring careful aim, the limiting factor is always aiming, not the mechanics of notch and release. Here's a speed shooter firing on the move:

If you wait until the end, you can see that she's just burying the shots in a canvas screen with no attention to precision. Still, the idea of a company of composite shortbow archers unleashing this kind of barrage gives you a sense of how you really might get a chance to fight in the shade as you attempted to close the range on them.

All of the above videos involve normal humans, using mundane weaponry, in non-life-or-death situations. I'd wager that being an immortal elf defending your homeland with a magic bow is worth at least another factor of two in terms of attack capability. Five lethal shots in 10 seconds doesn't seem so crazy in that light.

D@W Campaigns: Possible House Rules

To summarize the previous post about why the Campaigns rules in Chapter 3 were difficult for me to use:

First, too many low-success recon checks slow down game play, and so the game over-invests in maintenance activities that don't involve players. Second, the modifiers that are present skew the dice roll results strongly toward one end of the spectrum, making them too predictable.

Looking at the table of recon modifiers, they fall into three general categories:

• Symmetric modifiers that affect both armies equally (distance, size, terrain)
• Asymmetric modifiers that affect just one side (magic, air, spies)
• Anti-symmetric modifiers that affect both sides equally but inversely (leadership, screening, familiarity)

For my own reference tables, I'll probably rearrange the modifiers in this order.

In principle, size and terrain could be asymmetric, but often they are not because both sides will seek to use them in similar ways. If there's forest around, both sides will hide in it. In a balanced game sides have similar sizes, and if there's an advantage to breaking down into sub-armies, then both sides will break down. In particular, they tend to both result in negative modifiers that make the majority of rolls almost useless (nothing but a 10+, or 11+, or even 12+ will find).

My first proposal is to turn the size modifiers into range modifiers, instead of dice roll modifiers. I would also make them explicitly symmetric, so that both the searching army and the opposing army would get the modifier of the smaller of the two. Then negative modifiers in these categories would simply suppress any long-range dice rolls involving small forces! The base range would be 3 24-mile super-hexes (that is, 12 6-mile hexes), and would range between 1 super-hex (for a -2 modifier with the smaller side under 600 troops) up to 6 super-hexes (for a +3 modifier with both sides over 72,000 troops).

To me, it makes sense for this modifier to be symmetrical and to affect range. Not only is a small force harder to find until it gets close, but it also can't spare as many soldiers to function as deep penetration scouts, and so the ones it can spare will cover a proportionally smaller area.

Also, one of the stranger effects of the current approach is that it is much easier to get a count of the number/type/strength of units in an army when that army is large, instead of when it is small. That seems backward, in that a small army should be easier to scout once it is discovered. That is, if you only see a couple hundred troops, it's easy to conclude "They're all orcs!", but if you see a hundred-thousand troops, all you can say is "The battalion closest to me looked like orcs, and beyond that, they were just an endless field of countless helmets and banners". Locating and evaluating an opposing army are distinct functions, and making size into a range modifier emphasizes their independence.

My second proposal is to remove the distance-between-armies modifiers from the roll itself, and allow it to only affect the recon result indirectly through the Degree of Success results. In effect, this modifier is currently being applied in two ways, since it both reduces success chance (on the roll) and then also changes what it means to have a "success" (on the results table). This is creating unnecessary overhead. If you want range to have strong penalties, it would make more sense to just confine it to one of these two places but intensify the consequences. That is, you could either double the roll penalty, but collapse "Proximity of Armies" to a single row, or else remove the roll penalty but make the variation with proximity more severe.

I think I like the latter option better, although I think distance effects might be too strong already, so the best option might be to just drop distance modifiers to the roll entirely. The advantage of keeping it on the Results table (instead of the roll) is that it doesn't create the too-frequent skew toward "Catastrophe" results.

The benefit of both proposals is that they eliminate effects that tend to push the 2d6 roll out of the balanced range 2-12, and skew it strongly toward one end. This increases the uncertainty of results, since players will have fewer expectations about whether they should be routinely falling on one end or the other. Instead of one or two reasonable outcomes, the full spectrum of five outcomes (from Catastrophe to Major Success) will seem plausible in most cases. If they are skewed, it will mostly be on account of hidden variables, like which side has more cavalry or better leadership.

This makes the Catastrophe result meaningful again! Instead of showing up over 50% of the time due to stacked negative size/distance modifiers, it will again become something rare and exciting, as players will really be unsure whether they have a Major Success or a Catastrophe.

Fantastic Frontiers: Cry Havoc, And Let Slip The Quantum Ogres!

A while ago, I read this interesting piece on the nature of freedom and player agency in a role-playing game. The basic question is whether it's reasonable to create a game that bends multiple choices to the same outcome, despite promoting the illusion that an alternate decision would have avoided that outcome. In this case, the outcome is a fight with an ogre, where the ogre just magically teleports to whetever location the players decide to go next. It prompted a number of helpful exchanges:

• Here's a proposal to solve the perceived problem from Courtney Cambell ("C") which emphasizes the role of information in making choices substantive. It's not wrong to have a quantum ogre as the outcome of a meaningless decision. But once players obtain information that should make that decision meaningful (which a good game should always offer!), then we have problem if the information isn't actionable.
• Here's some pushback, with a good set of comments by Courtney and Justin Alexander following it.
• And here's a nice overview from Alexander Macris of some fundamental questions related to maximizing player agency, and making a game feel (and actually be!) "fair".

All of these pieces are well worth reading, although the immediate problem being addressed is one that arises in rather different forms in other types of games. For an RPG, the problem is that the GM is engaging in a sort of trickery that confuses the players and might violate an implicit social contract to be a "fair judge" in the adjudication of choices. But for a computer game, the problem is often that the mechanics of the game are hidden behind an opaque GUI that needs to be probed by trial and error, and might be too complicated or feedback-deficient to ever allow for pattern recognition.

In a board game, there's no risk of deception or obfuscation. Everyone knows the rules, and has access to identical information. Instead, the problem is hiding enough of the information to make the choice feel unpredictable at all, without becoming totally random.

To make this more concrete, consider the following advice from a friendly wizard:

There is [a way around Mirkwood], if you care to go two hundred miles or so out of your way north, and twice that south. But you wouldn't get a safe path even then. There are no safe paths in this part of the world. Remember you are over the Edge of the Wild now, and in for all sorts of fun wherever you go. Before you could get round Mirkwood in the North you would be right among the slopes of the Grey Mountains, and they are simply stiff with goblins, hobgoblins, and orcs of the worst description. Before you could get round it in the South, you would get into the land of the Necromancer, and even you, Bilbo, won't need me to tell you tales of that black sorcerer. I don't advise you to go anywhere near the places overlooked by his dark tower! Stick to the forest-track, keep your spirits up, hope for the best, and with a tremendous slice of luck you may come out one day and see the Long Marshes lying below you and beyond them, high in the East, the Lonely Mountain where dear old Smaug lives, though I hope he is not expecting you. (The Hobbit, 7.145)

A perfect wall of impassible mountains. How convenient!

Here we have three choices: goblin-infested mountains, Necromancer-infested shadowlands, and You-Know-What-infested Mirkwood. Gandalf is presenting this as a sort of Hobson's choice - "do as I say or die" - but presumably he doesn't have the actual ability to compel the dwarves to do anything. He's just giving information that makes the best decision blindingly obvious. Thorin and company are apparently getting railroaded by world-design into some spider- and elf-related encounters! At times, one starts to suspect that the suspiciously long chains of encircling mountains and forests in Middle Earth that turn Erebor and Mordor into fortresses with a single door are Tolkien's way of keeping his hobbits from wandering off to escape his carefully-prepped arachnid miniboss events.

This is a real choice, though, if a bit artificially discrete. Discrete choices often run into the problem of making the answer too obvious, which feels like a loss of player agency. In a novel, this might be a good way to comment on the nature of fate, but in a game, it takes an interesting decision and makes it boring.

So I've decided to take the whole "quantum ogre" idea and make it as literal as possible! A quantum wavefunction is typically distributed in a neat Gaussian distribution, rather than occupying a single point like a classical photon, or being spread out uniformly in space like a traveling-wave field. This is reminiscent of the three distinct encounter types in RPG-style games: random, pre-existing, and roaming.

In a game like last night's Runebound, we have pre-existing encounters printed out on little cards. They're essentially discrete quantum ogres, in the sense that no matter where you go, you'll get "the next green encounter on the stack". This creates a zero-information Hobson's choice: take the next encounter, or don't. There's no way to scope out each of two possible encounters with scouts, or ask around at a local tavern, or anything like that. This is the sort of encounter model in an MMO, too, where you walk into a dungeon and it's full of fierce enemies standing stock-still -- in neat little battle-sized clumps, waiting for you to defeat them one by one, as their buddies just down the hall politely wait their turns indifferent to the nearby tumult. This creates a world that feels static and not very dangerous. Lots of strategic planning, but poor verisimilitude.

Swoosh! Outta nowhere!

At the other extreme, we have the Final Fantasy overland map. You wander around for a while, and suddenly there's a pixelated screen wipe and you're facing some Cactuars that weren't there a second ago. You can escape, maybe, but there was never any other choice at all. The world is a threatening, terrible place, and you're playing strategic defense 95% of the time. The random number generator is playing the fiddle, and you need to dance to it. This supports the theme of hostile wilderness, but gives you no strategic choices, not even illusory ones.

Probability Ogre distribution function.

A real quantum ogre needs to be the third kind of encounter, fuzzy around the edges, but with a discernible peak in the center. You can roughly spot the zone where the ogre might be, but you can only weigh some relative probabilities. The one certain fact is that the closer you get to his lair, the more likely you'll bump into him as he takes his daily constitutional (this one, or that one). Of course, you could just go around him, but maybe there's a quantum dragon over there, and the ogre is the lesser of evils. And here we have some interesting continuous decision-matrix material: Do we walk midway between them, to minimize the chance of any encounter at all, or do we walk closer to the ogre to avoid the dragon, at the cost of higher ogre-risk. And is it worth walking through the Dismal Swamp to avoid both? Now we have the same interesting narrative pressure that Gandalf gave Thorin, but with a range of possible player decisions to optimize that go beyond "stick to the path".

Mathematically, the information we have about each encounter should be equivalent to the width and height of the Gaussian distribution. We need to use narratively-meaningful encounter attributes to generate these values. (Well, technically, using discrete dice will give a binomial distribution, but we know they're equal in large-N asymptotic limit!)

In the Fantastic Frontiers rules, the logical variable to use for the distribution width is the encounter's movement speed, and the logical variable for the distribution height is the population size. Since the area rises quadratically on a 2D map, it makes sense to use linear-quad scaling (one of my favorite design elements!), and let the height rise as the square root of the population. Here's an example.

A previously-spotted encampment of 12 ogres has movement speed 4. The square root of 12 is 3.46, so let's round that off to 4, and call that the base probability for entering the ogres' territory. (The wings of the bell curve will be chopped off, for simplicity.) Every step closer to the last location where the ogres were seen is a step deeper into a hex-ring of radius 4, and boosts the target chance by +1. To implement this as a bell curve, roll a couple of d20 against this target. (Just one die would give a linear distribution, not a bell curve.) If both results are at/under the target (for a roll-low check), then the ogres are encountered!

This makes the world feel a little more dynamic. The ogres are wandering around, instead of just waiting for us to come back and clobber them. But we still have the basic mechanic of being able to strategically plan for ogres, using our hard-won ogre-related information. It's the best of both worlds.

Long Marches

I was reading a bit about the difficulty of extended military marches, and found this comment by "Lars":

Regulations required a 10-minute break every hour. Marching and fighting at night was rarely done, so the march was generally limited to the time between dawn and dusk. Moving troops would make their breakfast and coffee well before moving out at first light, and they'd make camp while they could still see, and dinner after it was too dark to march.

Ten miles per day was fairly leisurely. Twenty was common, but pushing it. I've heard of thirty, and even forty, but a good fight had left the men after such a march. There are numerous incidents wherein a commander's excuse for not being aggressive enough was "my men had been marching hard all day and were wore out."

This is one of those areas where a game designer is confronted by the challenge of making a game feel "fun" despite the annoyance of having units that represent human beings that get tired and need to rest.

Almost every wargame I've every played just glosses this kind of detail. If a brigade has a movement of 4, it can move 4 hexes every day forever, and never needs to worry about camping over a weekend to wash and darn its socks. When controlling an army of dozens of troops, it would be laborious to commit to the bookkeeping of remembering when each individual brigade was too fatigued to fight well. Presumably the movement of "4" represents an averaging over longer days and R&R days.

Interestingly, you can find early RPGs trying to maintain this level of detail. The "party demands a 10 minute break every hour" was a feature of 1st edition dungeoneering (it's in every edition of Basic D&D up to the Rules Cyclopedia!) that must now seem curious to a modern player. I've noticed some similar drift from the early days of real-time computer RPGs (say, Baldur's Gate) where your party moves at some reasonable semblance of a walking pace, and more modern RPGs, where even the auction district of a capital city is populated by a more diligent set of joggers than the finish line of the Chicago half-marathon!

Fantastic Frontiers: Morale and Seaboxes

For a game that focuses on high-level decisions (resource management), instead of tactical decisions (who attacks what), it's important to have a fast combat system. Wargames have fast combat, usually based off a single dice roll, and so they can include lots of resource management. Computer games can have pretty complicated combinations of dice rolling, but as long as they hide it in the CPU, it's all instantaneous. Early RPGs also used pretty fast and abstract systems (before the days of miniatures and grids), at least compared to today, and could easily fit a dozen battles into a single session. Obviously this is the kind of effect we're going for, in a hex crawl board game.

But it's also important to emulate the feel of what would happen with a detailed miniature or man-to-man system. A group of knights in plate mail should feel different (and actually play different) than a group of footmen, or archers, or elven pegasus-riders. The trick is to get the feel of a more complicated system without the time expense. That requires some kind of "fast combat" system that's more akin to the single dice roll of a wargame, as opposed to the "moving dozens of little figures" feel of a miniatures system.

Fast, of course, is a relative concept. I don't mind if battles take 10 minutes or so. But it's important that they scale sensibly. That is, if it takes 5 minutes to fight a battle with twenty combatants on each side, I don't want it to take 50 minutes for a battle with 200 combatants on each side. I'd say that an upper bound of around 10 minutes for a complicated battle is my limit. Scaling needs to be much less than linear. This is the opposite of most tactical battle systems, which scale worse than linearly (doubling your armies more than doubles the time it takes).

The simplest option would be to just given each unit a combat factor, and then calculate odds and roll the result off an odds table. That would probably take only a couple minutes. But it's not very interesting, and it doesn't capture the different feel of having different types of armies. The strategy of the game reduces to "get the most factors on the board for the least money".

I'm shooting for a game that defaults to the age of discovery (say, 1450 to 1650), a time when heavy breastplate armor overlapped with primitive cannons. That's not quite medieval any more, but not quite Napoleonic either. There are a few concepts I think need to be included in any good simulation of early modern warfare:

• Matchlocks and crossbows
• Field cannons
• Melee weapons and armor
• A discipline/morale system

The last of the four elements is the most important one to me, so it's where I start. Discipline and morale are the reason that 500 conquistadors can hold their ground against 10,000 tribal warriors. Any simple combat-factor system can't capture that dynamic. The really important question isn't so much how large the sides are, but who actually shows up at the point of contact between the opposing lines. That's a question of concentration of force.

To imagine how this might work, I'll talk a little about sea boxes in World in Flames. Sea boxes are an abstraction in naval combat that represents how likely it is that a given ship will arrive in a particular battle (that is, "find" the battle). Ships with more mobility get to occupy a higher box, and a search roll determines whether they're included. This is a roll-low system, and everything under the sea box value arrives to fight.

The "4-box" means a 4-in-10 chance of being included in the battle.

Once the battle begins, the casualties are mostly deterministic, with some adjustments for which side is more surprised. But mostly, it's a matter of figuring out what ships arrive at the point of contact and participate. That's exactly the idea I want!

Imagine that instead of having box numbers representing search capability, for ships wandering around on an empty ocean, we have boxes that represent morale and military discipline. A high "box" represents lots of morale, and the unit will bravely assault in the front lines. A low "box" indicates that the unit is cowering in the rear. With a 2d6 roll-low morale check, that would amount to giving each unit a morale "box" value between 1 (never participates) and 12 (always participates), with a bell curve of possibilities. This morale value could reflect both intrinsic properties (like training) and also situational ones (like weather or supply level).

This post is running a bit long, so next time I'll give an example of how this concept might work in practice. This approach to combat will be a little more complicated than WiF's system, since it will need to include not just melee engagements, but also the opening volleys of guns and artillery that can themselves adjust morale modifiers (nothing saps morale like seeing a row of musket barrels aimed at you!)

Challenge Systems: Quantitative Analysis

A core element of many games (both board games and RPGs) is the binary challenge resolved by a dice roll. This might represent an attempt to hit a target with an artillery piece, an attempt to jump over a chasm, or an attempt to persuade a guard to let you past a security door. Previously I discussed the four possible ways to implement the roll itself: rolling over a target versus under it, and modifying roll results versus modifying targets. For example, a roll-high and modify-result system would involve something like "If your 1d20+3 roll is greater than or equal to 15, then you succeed".

This is mostly an aesthetic issue. From a gameplay perspective, a more important question is how the probability of success changes as modifiers are added. This depends on whether how many dice are rolled (a multi-dice bell curve generates diminishing returns in both wings), what modifiers are applied, and how targets are set. There are an infinite number of different possible systems, but I'd like to focus on four distinct approaches in existing RPG games. The systems are:

1. Moldvay's B/X edition of D&D, which used a simple "roll a 1d20, with result at or under the attribute to succeed"
2. The system proposed by Wesley Ives in Dragon #1 (June, 1976), for use with original 70s-era D&D. This system is rather baroque in presentation, even by the standards of that era, involving rolling lots of dice both to set the target and then to test it.
3. The modern convention of a "skill system", with small modifiers based on attributes. Here I've chosen the SEIGE engine (from Castles & Crusades) as a representative example, using a non-prime target of 12, although most other post-3.0 systems in the OGL d20 line work in a similar way.
4. The use of multiple dice against a single target. In this case, I've elected to use the D6 engine from 80s-era West End Games products (like Star Wars), also using a target of 12. (d20 attribute points have been converted one-for-one into "pips" for the calculations below.)

One comment on the "skill system" approach. In practice, this system has substantial modifiers based on custom design, which shift the chance of success up linearly as "skill points" are invested in a category. In practice, however, this kind of system involves illusory improvement, in that the targets (DC's) are simply adjusted higher at the same rate! (In fact, I think the 3.5 DMG explicitly instructs the DM to do precisely this.) So the skill system, rather than producing real improvement, just places characters on a leveling treadmill where they start to fall behind in any category where they fail to pay the appropriate "skill tax".

Here are the results of a computer simulation of 100,000 dice rolls using each method, to show the qualitatively different effects that this choice has on the game system.

The Moldvay approach creates a linear success rate, and the skill-based approach flattens the chance out substantially without altering linearity. (Of course the DM can rather easily shift the entire line upward or downward, by the use of DC-setting fiat powers!) Attributes matter much less in a modern game, apparently.

The Ives system, despite apparent complexity, has a simple effect. It takes the Moldvay system and forces it to saturate at high skill levels, bending it down into sublinearity. This means that even when using his optional bonuses (including level), the chance of success improves very slowly beyond a certain point. This is an excellent example of the design principle of diminishing returns! I rather like this feature, despite the obscurantist feel of the convoluted mechanics.

The multidice system involves a true bell curve of results, which shows up here as an S-curve in the binary check. Note that below a certain point, success is basically impossible, but even at the top end there's a small chance of failure. An alternative system is to replace the roll-high system with a roll-low system (as used in, say, Ninja Burger or Kobolds Ate My Baby), which reverse the curve ends to always provides a small chance of success at the low end, with guaranteed success at the high end. This friendly option works well for beer-and-pretzels games, where you never want a check to be impossible.

Overall, it's hard to beat the Moldvay roll-under-attribute system for simplicity, but I think I prefer the Ives system on philosophical grounds. I wish I could find a way to implement it with less complexity, though!

Curious Feature Of Naval Thunder

There are three basic dice mechanics in this game system:

1. A "to-hit" check, using "roll high, modify target".
2. A "penetration" check, using "roll high, modify result". Just to make things extra confusing, for this one it's necessary to beat the target and not just match it.
3. A damage control (or command) check, using "roll low, modify result".

Oh, and critical hits use a hash-table with a bell curve, placing less damaging criticals (guns, engines) in the center of the distribution, and more dramatic results (bridge, magazine) out in the tails.

Three different systems, three totally different implementations. That's pretty unusual in games today. The only mechanic missing from the list is the rarely-encountered "roll low, modify target" method, which is the one I actually like best!

I'm kind of curious why the designer used such a piecemeal approach, but I think I have my own idea. For the to-hit roll, it's often necessary to roll identical guns from the same ship against two different targets. For the penetration roll, it's necessary to roll different guns against the same target. This is something I encountered during the design of a morale system for the fast-combat system I wrote over the summer. All of a sudden, I wanted to apply a single roll against a variety of different targets, and suddenly a "modify target" system made perfect sense. And so I can respect the decision.

I think that so many games today use result modification instead of target modification that even when a target-mod system makes logical sense, designers still try to shoehorn everything into the industry standard. I chalk it all up to lingering trauma over the complicated "THAC0" experiment, which tried to mix negative bonuses to the target with positive bonuses to the result, all at once. That was probably a bridge too far, even by my standards.

Classifying Methods of Randomization

Previously I compared some methods of randomly generating binary outcomes with dice. But of course there are non-binary contexts where dice can be useful as well. Roughly speaking, there are three ways to transform a dice roll into a result:

1. Binary: Roll a result greater than (or less than) a given target to succeed. Usually also includes "equal to", so that the target value is the success-threshold rather than the failure-threshold.
2. Spectrum: Roll a result, then look up the outcome of that result from a tabulated list of all possible outcomes, arranged in some ordinal schema (best to worst, etc).
3. Hash: Also by reference to a table, but for outcomes that aren't (perhaps can't be) arranged in any meaningful order.

The boundary between a "spectrum" system and a "hash" system isn't always a sharp one, but roughly speaking, if the system is amenable to having a bonus or penalty applied to it (as a modifier) it probably counts as some kind of a spectrum. A true hash table result would be senseless to modify, since being adjacent to another item in a disordered hash table has no significance.

A major trend in games over the last couple decades has been the absolute decline of spectrum systems in favor of hash systems. Almost all complicated simulationist wargames still use spectrum tables (with a battle resulting in complicated combinations of losses, retreats, and other adverse effects), but most other board games and most RPGs have switched entirely to binary rolls for everything except "combat damage".

This bothers me enough that it sometimes intrudes on my ability to enjoy a game. I need a few spectrum results to make the game's theme feel "real" to me. Plus, there's something intrinsically exciting about a system that allows not just a success, but a critical success.

For example, early editions of D&D from the 70s had a ubiquitous "social roll" system that utilized a 2d6 spectrum as its common mechanic. Here are a few examples (courtesy of Brendan at Untimately):

Social Rolls

2d6	Negotiation	Reaction	Morale
2	Refused, -1 future	Attacks	Surrenders
3-5	Refused	Hostile	Flees
6-8	Will consider another offer	Uncertain	Fights, check again
9-11	Accepted	Friendly	Fights
12	Accepted, +1 future	Proactively helpful	Fights, no further check

Note that the system is intentionally open-ended. What does "hostile" mean? In an arbitrated game, that's for the judge (the DM, for D&D) to decide. A system like this is intended to encourage creativity and flexibility, adapting to the particular circumstance while retaining the thematic pattern of "bad to good outcomes, with a bell curve distribution". In effect, every time you see a table like this, the game is saying "you're making a social roll" -- as opposed to a combat roll, or an encounter table, or a skill check. Social interactions, by their nature, have lots of possible outcomes that make them much more interesting than a "can I jump across this pit?" binary system!

In the near future, I'm intended to use some spectrum mechanics for the intelligence and logistics systems in my Guadalcanal campaign hack of Naval Thunder. In the long term, I'd like to use the specific "social roll" system to build the social subsystems of my (work in progress) hex crawl game like negotiation and recruiting, helping to give it some "old school" flavor.

Design Criteria

One of the things I'd like to do here is review games (in particular, wargames and other strongly thematic games). That requires me to have some kind of criteria for evaluating the quality of design choices, which in turn implies that I need to outline some kind of design philosophy to explain the rubric for that evaluation. So here's a set of categories for evaluating whether a particular game has a good design:

1. Player Agency: A common early childhood game model is the roll-and-move, something like Chutes and Ladders (or Snakes and Ladders, if you're a Brit). This kind of game creates a structured experience ideal for young children, but it isn't really "played" in any meaningful sense. Instead, it plays itself without the need for any outside decisions. There are no choices to be made, and so the game lacks player agency.
   
   Providing meaningful player agency is actually tougher than it seems. Just providing choices doesn't suffice. For example, suppose you're on a game show where you get to pick two unmarked doors. One of them has $50 randomly behind it, and the other has $500 behind it. You get to choose the door, but since the selection is based on zero information there's still nothing to "play". You might as well flip a coin. Now suppose, instead, you have two marked doors. Does that provide agency? Not really, since now the correct choice is obvious. That's a trivial choice that might as well be automated and made by default.
   
   But suppose that behind the marked door with $500 there's also an angry rabid badger, and someone just handed you a knife. Now you have meaningful agency, since the choices are nontrivial to rank - based here on your own self-assessment of badger-fighting prowess!
   
   Agency requires intermediate levels of information. If you can't see the chessboard (you have zero information), deciding whether to move the knight or the pawn is just a coin flip. But if you're playing against a deterministic computer program that you wrote yourself (you have perfect information), then you can anticipate how every set of moves will end the game and select only winning moves. There's some optimal intermediate level of information that maximizes agency.
   
   Agency also requires intermediate levels of granularity. If an entire battle is resolved by a single dice roll (low granularity) with odds representing the power of each side, there are no decisions left. But if you need to allocate the firing pattern of each gun on a battleship, then nothing short of a supercomputer will allow you to intelligently predict the outcome of shifting your aim a dozen inches to the right. Again, there's some optimal balance between these extremes.
   
   Some games try to hide their lack of agency by creating the appearance of agency within a narrative that distracts the player from the substantive content of the game. (In RPG circles, this is called "railroading".) This is not always a bad thing, since it can support other aspects of the game that are primarily concerned with theme and narrative (see verisimilitude, below). But this kind of agency deprivation needs to be carefully monitored to make sure that it doesn't violate the social contract of the game, a special problem in the context of RPG and computer games where the social contract often involves a third party (either as referee or as content provider).
   
   
2. Emergent Complexity: You're playing a computer RPG. At level 1, you fight grey rats with 10 health, and do 3-5 points of damage with your dagger. At level 100, you fight neon pink mega-rats with 10,000 health, and do 3000-5000 points of damage with your ion sword. Curses! You've been duped into running on a treadmill!
   
   A very simple game can involve repeating a single task until the point of mastery, but most complex games need a sense of forward progress that rewards you for good task completion. You start in Civilization with a settler caravan, and found a city with a militia defending it. After building for a while, you have a dozen cities, and an army of phalanxes and chariots. A while after that, you have wonders, and ships, and irrigation, and dozens of mechanized armies waging war in every direction. You're playing a game that started simple but became complex while operating under a fixed rule system. This is usually a highly addictive experience.
   
   Complexity can develop in terms of the raw scale of the game. Once a difficult task is mastered, the game asks you to do it three times, all at once. But it's important that the numerical scaling be nontrivial. If you make the same attack with a hundred star dreadnoughts as you do with a single star dreadnought, then the game is just making you roll extra dice after you make the same decision. Adding ships to your star navy needs to open up new tactical possibilities to make the mechanical cost of running the larger naval more than a narrative novelty. A better (but more difficult) model is to allow larger-scale play to include new rules that are inclusive of the old ones, like a strategic scale map that forces you to move around squadrons of 4 to 12 dreadnoughts each using a custom logistic system to engage some of them in coincident battles as part of a larger war in a persistent world.
   
   Another form of complexity is depth of the decision matrix. So, say, initially your warrior has three abilities available each turn: basic attack, dodge, or parry. After you gain a couple levels, you now have two more abilities to choose: power attack, or sweep. Maybe some of these cost you more energy, or can only be used occasionally. The deeper menu of choices forces you to balance more objectives. But if one ability dominates over the others, then it quickly smothers the complexity and reboots the player to a simple game. This is a terrible problem in computer MMOs, where players constantly search for a way to collapse their personal decision matrix back into an "instant win button".
   
   The most difficult form of complexity to manage is dynamic inconsistency, the way in which the relative value of game objectives changes situationally. This is a fairly abstract concept that takes a number of concrete forms, like the changing weight of risk versus reward based on victory criteria. For example, if your opponent is about to win the game on the next turn, it suddenly becomes very appealing to make a high-risk desperation attack! There are many more subtle examples of the same phenomenon, serving to disrupt the usual equilibrium of the game.
   
   
3. Verisimilitude: When used in the narrative sense, this word denotes the faithfulness of a story to the reality it depicts. In game terms, however, it's pretty important to realize that a second layer of simulation is often present. In development of a game's theme, it is often the case that game events are intended to simulate not a reality, but a fictional world (from literature, movies, etc) that is intentionally fantastical. This creates a strange paradox of the game needing to be faithful to something that is intentionally unfaithful to reality. In many cases, actual "realism" can instead become the enemy of that desired fidelity to source narrative, and should be avoided.
   
   The standards for evaluating the quality of a theme's development is another aspect of social contract. If you call your game "Swords and Sorcery", then you suddenly have a player assumption that your game will allow them to replicate the feel of a Howard novella. Every element of the game, not just the title, makes some intangible contribution to shaping the social contract: art, writing style, game mechanics, even the manual's font.
   
   The overall quality of these presentation elements is often less important than their ability to define the social contract in ways that makes the outcome of the game feel satisfying. It's acceptable to have the majority of your local gaming group's Paranoia or Call of Cthulhu adventures end in a spectacular catastrophe. It's probably less acceptable these days to create a Pathfinder campaign that slays multiple characters in the first battle (as my friend Jeff was complaining to me over Christmas break). And if you're a nostalgic DM who wants to haul out an old tournament classic like Tomb of Horrors, it's essential to find ways to communicate to your players that they're about to play the first kind of game, not the second.
   

A couple more concepts that fall under the categories above, which I look for when reviewing a game:

Diminishing returns (DR) is the idea that a particular player choice, if repeated, eventually yields a lower return on investment of resources. Including a DR mechanism is an easy way to balance a game's economy (allocation of resources toward objectives), when it doesn't naturally emerge as a result of competition between multiple players. Many solo or cooperative games absolutely need a strong mechanism of this sort, and will become "broken" if it isn't present. A simple way to enforce DR is create sub-linear relationships between resource investment and outcomes.

Mechanical diversity, the creation of dissimilar rule subsystems for handling dissimilar situations rather than using a uniform design model, tend to boost the verisimilitude of a game at the cost of additional rule complexity. I am usually in favor of this model, so long as the subsystems are comparable in complexity to one another and the enthusiasm doesn't go overboard. Modern game designers often look askance on this approach, but there's something very satisfying about the idea that a land battle should involve a totally different sequence of actions from a naval battle. It also gives the game a rawer, more organic feel that creates flavor and excitement. It also tends to promote asymmetries in dynamic inconsistency, where the player of one strategy responds rather differently to an event than the player of another strategy would.

Binary-Outcome Dice Resolution Methods: Roll-High vs Roll-Low

In many modern games, particularly RPGs and miniature wargames, randomization is represented in the form of a dice roll that has two binary outcomes. This has largely replaced the older model of wargames in which dice results are used to reference a table with a wider range of variable results. For example, many RPGs involve a "skill check" or a "to-hit check" that rolls dice, performs some fixed situational adjustments ("modifiers"), and then returns either a success or a failure for the attempted action.

The general technique is this: (1) Set a target, (2) Roll dice, (3) Perform modifications to the target value and/or the dice roll, and (4) Compare them to one another. But the comparison can be done in one of two ways. In a "roll high" system, the dice result (after modifiers) must be higher than the target. In a "roll low" system, the result must be lower. Although early games often oscillate between each method for different subsystems, modern games usually insist on enforcing one approach or the other uniformly throughout the entire game.

To further complicate the picture, note that the situational modifiers may be applied to either the target or to the roll result - or both! This effectively crates four possible approaches to choose between or combine. Let's suppose we have a 20-sided die, and we want to attempt an action that typically has a 75% chance of success, but in a situation where the odds are improved by an additional 25%. Here are the four equivalent implementations:

1. Roll-low, and modify target: The base target is 10. Any result less than or equal to the target is a success. Give a +5 modifier to the target.
•  Equation: 1d20  ≤  10 + 5
2. Roll-low, and modify result:  The base target is 10. Any result less than or equal to the target is a success. Give a -5 modifier to the result.
• Equation: 1d20 - 5  ≤  10
3. Roll-high, and modify target: The base target is 11. Any result greater than or equal to the target is a success. Give a -5 modifier to the target.
• Equation: 1d20  ≥  11 - 5
4. Roll-high, and modify result: The base target is 11. Any result greater than or equal to the target is a success. Give a +5 modifier to the result.
• Equation: 1d20 + 5  ≥  11 

It's not difficult to confirm that all of these give an identical 75% chance of success. However, that doesn't mean all of them are equally intelligible. To me, the first approach is the only one which makes the success probability obvious to me based on the way it's written. I can easily see that the target (10+5=15) is 75% of the maximum die roll of 20. This strongly predisposes me to method #1. Even method #2 isn't much worse, since it just requires a bit of algebra to slide the negative modifier to the other side of the equation.

The two roll-high methods are more opaque to me. The problem lies with the fact that a target of 6 doesn't look much like a 75% chance of success, on the face of it. Obviously it is, since there are 15 out of 20 results that succeed. But the simple "divide target by max result to find probability" trick no longer applies.

Unfortunately, this system seem to have won for psychological reasons. Rolling high feels more like a "win", since big numbers correspond to higher scores in most sports and games. So clarity is sacrificed for psychological pay-off. This occurs most dramatically in 3rd Edition D&D, where all previously existing roll-low mechanics are rewritten to be exclusively roll-high.

One additional problem with this psychological appeal is that it tends to motivate the system to grow in an unbounded way. Roll-high systems effectively can keep stacking modifiers on both the target and the result, making the situation even harder to interpret. When rolling a 1d20 for a DC32 check (i.e., target of 32) with a +14 modifier to the dice roll, you need to perform two steps of mental algebra just to extract the probability. Quick, perform that operation in your head! What are the odds? Go ahead and grab a pencil and paper, I'll wait.

OK, the result isn't quite that hard. You can subtract that +14 from the target to get 18 (switching to the slightly less confusing method #3), and then recognize that results of 18, 19, or 20 all succeed, for a probability of 15%. But why be forced to do mental subtraction, when you could use method #1 all along?

It's probably too late to eliminate the general preference for roll-high systems. Interestingly, a few games that use percentile d100 rolls have managed to retain roll-low mechanics, suggesting that the real obstacle is that few players today can even due a single division (100%*15/20 = 75%); once the system uses any other dice type, it completely gives up all interest in retaining the intelligibility of its odds.