Against Range Increments
Or: How 2-D maps spring from range increments
Because TTRPGs are wargames1, there are two essential elements overlooking all scopes of play. The first is space, and the second is time.
Though we’ve already extensively covered timekeeping2, the concern here with time is mostly in its relation to space. The interplay between time and distance is the foundation for both low-action logistics questions as well as high-intensity tactical considerations in combat.
Because neither time nor distance ever truly “act” independently of one another, the mathematical relationships that govern their nature force certain conclusions on us about practical range-tracking.
Relating Time and Distance
Consider a scenario where we are charged with delivering a package to the next village. Whether that village is 1 mile away or 240 miles away makes the delivery a vastly different proposition.
For the first case, we can easily walk there (and back!) with relatively little time or energy expended. For the second, we must lean heavily on technology (horses, trains, cars etc.) to even accomplish it at all!
Consider a different scenario where a spear-armed combatant is 50 yards from our character. He gives a battlecry and starts rushing towards us. How much time do we have to react? If we have a crossbow, how many shots can we get off before he’s upon us? What about a shortbow? What if we turned and ran—could we escape? In intense situations like this, time and distance are visceral concepts!
Since this second scenario is of more universal interest, let’s focus on range-tracking techniques with similar examples in mind.
Time Blocks and Speeds
An enemy is rushing at our character with some close-range weapon. He begins some distance D yards away.
In most TTRPGs, there is some smallest meaningful chunk of time depending on the circumstances and the questions being asked. This T is our smallest time block. In AD&D, T is usually called a “round” and has a duration of 1 minute (unless the question of segments comes up). In many games, T is a 6 second round. T is a 10 second round in ACKS and 15 seconds in Classic Traveller.
To link T (our time chunk) to D (our distance), many games have a Speed parameter we’ll call V (like “velocity”). If the enemy charging our character has a Speed of 30 feet, it means he can cover a distance of 30 feet per one time-chunk T—60 feet after 2×T and so on. The equation D = V × T covers these cases.
Using this simple system, if our assailant is 70 feet away and has a Speed of V = 30 feet, then in two rounds T of movement he can be a mere 10 feet distant from us.
We can see from this consideration that the distance, speed, and time rules are not really there to tell us about those numbers! Our real interest is in the room left for our character to respond.
The numbers are just a way to reconcile gameplay with gameworld limitations—people can only run at a certain speed or make a certain number of attacks in a certain time period etc. But the relevant consideration for us is whether we can respond to this assailant and how big our margin of safety is.
With that in mind, let’s see a different technique for range-tracking.
Range Increments
Because different games have slight discrepancies in using range increments, we will look specifically at Classic Traveller’s system. In Classic Traveller (CT), we have a list of five possible ranges of consideration:
Close: in physical contact
Short: at sword-length or spear-length (1–5 meters)
Medium: at pistol range (6–50 meters)
Long: at rifle range (51–250 meters)
Very Long: extreme range (251–500 meters)
The descriptive distances given are intended to be for visualization or illustration—not for calculations. Instead, the question of distances and speeds are converted into questions of time blocks of 1 round.
Close: 1
Short: 1
Medium: 3
Long: 4
Very Long: 5
For example, if we are at Long range (4) from our intended target but would rather move to Medium range (3), it will only take us one round of closing range—the difference between the two values. What if we wanted to move to Short range (1) from Long range (4)? It would take three full rounds of closing range! Running allows us to move at twice the rate per round (but comes with the risk of exhaustion).
This system can also take into account the enemy’s movement as well. If the enemy wants to move closer, and we want to move in closer, then our efforts are added together. Imagine we are at Long range (4) from an enemy. If we declare we are running closer, and the enemy declares they are moving closer to us, then the contributions (2 + 1) are added together. We’ve reached the full gap from Long range (4) to Short range (1) in a single round!
The Laws Increments Follow
Range increments are not intended to have the same resolution or fidelity as basic distance measurements, but there are still some principles they nevertheless must obey.
First, range increments have the symmetric property between characters. If character B is at Medium range from character C, then C must necessarily be at Medium range from character B. If B was within Close range of C, but C was at Medium range of B, we already have a contradiction—B can attack C with a melee weapon, but the reverse is not true! We could form many such contradictory conclusions without the symmetric property.
Second, Close range has the transitive property between characters. If C is at Close range to D, and D is at Close range to E, then C must be at Close range to E. Since ‘Close’ really just means ‘occupying roughly the same space,’ then that makes sense!
The other ranges do not have the transitive property between characters. If C is at Medium range to D, and D is at Medium range to E, then C is not necessarily at Medium range to E. But why is that?
We can make a counter-example, but why is this transitive property not true in principle? Despite our intuitions, a purely mathematical argument could be made that these ranges are, in fact, transitive! However, the Classic Traveller rules already head this off—by smuggling in a spatial dimension, the same way we did in our illustration!
Range increments are supposed to be abstractions, but CT explicitly talks about moving “toward” and “away from” enemies. This means there is an axis along which characters are traveling. Since the enemy’s “toward” and “away from” are simply reversed in direction, let’s choose a more universal “North” and “South.”
For those who can follow mathematical arguments arguably too closely, it is actually clear only after adding this spatial dimension why this example plays out this way. If combatant D is Medium due North from combatant C and Medium due South from combatant E, then we’ve reproduced the illustration above and explained the non-transitivity.
CT makes this spatial dimension addition even more explicit by introducing the line grid.
(pg. 29)
Because the effects of range are so important, and because the ranges between specific characters can vary greatly, it is suggested that the complex combats be mapped out on a line grid …
The line grid is exactly what we would imagine—a countable axis along which the directionality and magnitude of motion is well specified. Thus, even if we try to think of range increments as an abstract design, it is still a 1-dimensional “distance” metric under the hood.
Multiple Groups and Constraints
Range increments work well if we focus on singular characters or groups that stay together, but what happens if a combat situation has a large number of independent actors?
Let’s cut our increments down to three: Close (transitive), Medium, and Far. Now imagine a situation with four players [A, B, C, D]. We will list their ranges one by one:
Player A
Far from B
Medium from C
Far from D
Player B
Far from A (constrained)
Medium from C
Close to D
Player C
Medium from A (constrained)
Medium from B (constrained)
Medium from D
Player D
Far from A (constrained)
Close to B (consrained)
Medium from C (constrained)
Notice how with each successive character, the symmetric property places constraints on our possible options. If we are going down the list [A, B, C, D…], then the Nth character will have N - 1 constraints on it due to the symmetric property.
Let’s see this example in a different form:
In each column of this matrix, we are specifying the relation of the column head to each of the other characters. Naturally, each column is ‘Close’ to itself. For example, the B column tells us that character B is Far to A, Close to B, Medium to C, and Close to D. The symmetric property does something interesting here; it creates what’s called a symmetric matrix.
Note how the purple and green shaded regions are correspondent—if we reflected their values across the diagonal, we would have exactly the same matrix. Put a different way, if we specify a value in the purple region, a value in the green region is automatically set—and vice versa.
Range Increments as Radial Distances
Since we have demonstrated that range increments are magnitudes and directions (vectors under the hood), our mathematical intuition is nearly forced to guess that they may conform to radial distance. In simple terms, each range increment corresponds to a circle of a certain radius.
Let’s take the example above and start with character ‘A,’ drawing circles around A’s location to represent the three range increments [Close, Medium, Far]. We have drawn a circle for ‘Close’ purely for fun since we intuit that—as a transitive-property range—it must be a circle of radius 0. We know, reading down the ‘A’ column, that A is Far from both B and D and is Medium from C.
Here is a naive attempt to place the other characters; this will get us into trouble:
If we move onto the next character, we know that B must be Close with D—this illustration doesn’t represent that correctly, and we reject it for that reason. But, is there a principled way to place the other characters with respect to A so that we get a picture that matches up with the range increments?
Let’s try a different method. We will place ‘A’ and draw its circles representing our ranges. Then we will place only B. Since there are no apparent restrictions, we place B at an arbitrary point along the ‘Far’ circle. Now we think very carefully and consider placing the next character ‘C.’ We know that C must be at Medium with respect to both B and A. But look what happens when we draw this picture:
There are only two possible choices for where C could be—the intersections of the ‘Medium’ circles of ‘A’ and ‘B.’ If we continue this exercise, taking care to accurately draw circles and to properly represent the ‘Close’ circle as radius 0, we will come to the conclusion that the illustration is heavily restricted. We could have placed ‘B’ anywhere along A’s Far circle, but we still have C₁ vs. C₂ as our only remaining choice.
This pattern of adding characters successively rapidly results in zero-choice placements—fixed illustrations! It’s worthwhile to note that even if we “thicken” our circles into circular bands, the same phenomenon applies; we will end up with increasingly constrained illustrations that, from a macroscopic point of view, are forced upon us.
Thus, when we define the relative range increments of a group of characters, we not only are guaranteed the ability to represent this as a 2-D illustration, we are actually forced into acknowledging a very specific and highly constrained 2-D illustration! The range increments define, and essentially stand in for, a 2-D map.
Lone Ranger
Logically consistent range increments necessarily represent an equivalent 2-D map. Because of the nature of maps, it will almost certainly be the case that the annotated map is more useful and more informative than the list of range increments.
It is sufficient for most purposes to describe the terrain features and relative position of important units; theater-of-the-mind is both efficient and engaging. But with complex circumstances that are difficult to convey with brief descriptions, a 2-D map with dots or other symbols denoting important units and terrain features can communicate the most information with the least time while having minimal impact on the engagement of imagination.
As designers, it is important to understand tools before using them. Range increments can seriously cut down on time, and they excel in simplistic situations. With multiple actors and increased relevance of directionality (for flanks or other maneuvering), proper 2-D maps are a superior tool. Keep building, and never settle for lesser solutions.
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Discussed thoroughly in The Nature of TTRPGs.