In my previous article on the geometry of empires, I considered two essential strategies for organizing the defense of a territory. In one, troops were positioned along the border, ready to intercept enemies but spread thin. In the other, a single, centrally-located force was powerful but slow to respond. These are extreme approaches, though, lying at the ends of a spectrum filled with intermediate strategies.

We also saw that while the size of a territory affects what strategy should be chosen, the tactics of enemies is paramount. In particular, a state's defensive armies must match the size of a typical invasion (neglecting many important matters, like morale). This means its total defensive forces cannot be divided into too many sub-units. But as long as they don't cross that limit, the state wants as many sub-units as possible so they can react quickly and address multiple threats if necessary.

For example, the Saluzar king knows his defensive forces ought to consist of at least 500 men to counter the largest raiding parties he thinks any of his enemies can muster. If the kingdom supports 2,500 men, obviously he wants no more than five forces; but he can have five, and he will.

Placing Defenders

The real question is where defensive forces should ideally be placed. We may continue to assume a circular territory that can be attacked from anywhere; that is, all positions on the border are equally likely to be attacked so must be defended. Thus, forces should be placed so the average distance from border to force is minimized.

In the extreme cases, the answer of placement is not hard. A single force must be in the center of the realm so that one side is not favored over another. And an infinitude of extremely small forces should be strung out along the border itself. For some intermediate number of forces though, the question becomes harder. To make it approachable, let's take the reasonable step of dividing our circular territory into a number of wedge-shaped zones, each protected by a defensive force; all are the same, and the number of zones depends on how many forces are desired.

Circles divided into 1, 4 and 16 wedges

This makes things a little simpler, as we only have to think about placing one force at a time. Still, an attack could occur anywhere along the border, and our force would need to respond.

Quarter-circle with various lines connecting the edge to a random point on the interior

You can see from the above image that any random position will likely be poor, because some part of the sector's edge will be farther away than another. The exception is if the defending force is placed exactly on the midline of the sector.

Quarter-circle with point on its midline

Position on the Midline

Where on the midline should a defender be? If the sector is small, clearly near the edge; if it is large, clearly near the focus (the center the empire). But what of the intermediate strategies? As an empire is divided into smaller sectors, how quickly are its forces logically pulled to the edge? Very quickly, it turns out.

Steeply rising curves that then level out

The above graph shows ideal positions, based on the number of defenders, that is, sectors. Perhaps surprisingly, the ideal distance (from the center of the empire) very rapidly increases as defender number increases: with only three or four sectors, defenders should be placed around 90% of the distance away from it. (With 100% being the empire's radius; the result does not depend on empire size.)

The ability of an ideally situated defender to intercept any attacker also increases quickly, though not as quickly. (The graph shows the average distance to the sector's edge, subtracted from 1, the worst possible distance a defender would have to travel.)

Trade-offs

The discussion so far has assumed that the number of defenders is set; decided by various factors. But we are now in a position to see what the benefits of various intermediate defense strategies would be. Interception ability, as shown above, is the benefit of splitting one's armies into many defending forces. The disadvantage is a drastic reduction in each force's strength. The graph below shows both these effects in relative terms.

A steeply rising curve and steeply falling curve intersect near 3 and then both level out

If interception ability and unit strength are considered equally valuable, in general, we can sum the two for the overall benefit of any given strategy, as has been done on the graph in grey. This shows that either extreme strategy has clear benefit, as do many involving a high number of defenders. However, when an empire has two or three defenders, unit strength has already fallen steeply while the benefit of interception ability is still modest; these strategies are perhaps, then, the worst to adopt, all else being equal.

Conclusions

Rarely, of course, will the benefits be valued exactly equally. Strategy depends on one's enemies, or rather one's best guesses about them: How likely is an attack of very high strength? How likely are multiple attacks in the near future? The severities and probabilities of such scenarios must be evaluated in some complex calculus. However, the above analyses make several broadly important points.

  1. An increase in the number of defenders does not linearly change their ideal position: very quickly, they should be located close to the perimeter.

  2. Not all possible strategies are created equal: while none are good at everything, some are good at very little (though this does not mean they should never be employed, given the right circumstance). Namely, avoid having 2 to 4 defending units.

  3. If unit strength is paramount, an empire has few options: it must sacrifice interception ability almost entirely to concentrate its forces as much as possible. Conversely, if interception is important--because of multiple attacks--an empire has more viable strategies to consider. A canny opponent might take advantage of this fact.

Many assumptions have been made so far; later articles may address these, as well as explore other related matters.

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