Excursions
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Excursions

Defending an Expanding Empire

To defend itself, should a pre-modern state grow larger or smaller? Well, it depends on what threats it faces….

In a previous article I examined the trade-offs a defender faces when dividing up his defensive forces. We assumed the size of the empire was fixed. But what if it isn’t, and its leaders can pursue strategies to grow the territory or shrink it? Which is better, under what circumstances?

At first blush, it may seem like bigger is always better. After all, very successful historical empires were constantly trying to expand. But this may have been for reasons other than defense; and there are costs as well as benefits.

Basic Trade-offs

The main trade-off is the number of forces an empire can raise versus the time it takes those forces (when well positioned) to meet invading forces — what I call interception time. We know these two properties scale at different rates: troop availability should scale roughly with the area of the empire, while interception has more to do with its length or radius.

Since r² beats r, this also suggests that if we expand our empire, we’ll gain troops faster than we lose interception ability — making expansion the best choice. But things are more complicated than this because of how we must divide our empire and respond to the particular threats around us.

Measuring Interception

Let’s imagine that our enemies reliably hit us with attacks of a set size. To be effective, our defenders must match this size. The only question is how many of them we can have. More units means closer positioning to the border, and therefore better interception. But we have to balance this against the difficulty of patrolling a larger empire.

Previously we determined the interception distance formula for an invading force and an ideally placed defender (where a is the sector angle):

interception = radius * sin(a/2) / (sin(a/2) + 1)

Using this, we can look at the total effect of a changing radius, with four different sizes of attacker (the units are arbitrary):

A few things to note:

  • At very small radii, the lines disappear because the empire cannot defend itself (and may be destroyed).
  • A larger radius ultimately helps all empires, with better interception times.
  • But there are intermediate values where things appear to be worse.
  • Finally, a saw-toothed pattern is seen, with sudden changes between sizes.

What’s Going On?

We can explain the saw-tooth shape by rounding: although area varies continuously, an empire has to sort its troops into a discrete number of units (1, 4, 30, etcetera). If it increases its size just a little bit, it will gain troops — but not enough to create a new unit of the minimum size (based on what attacks it suffers). The same number of units now has to patrol a larger territory, and interception suffers. If the empire keeps increasing, though, it will eventually be able to add a new unit, and will quickly improve its interception by re-positioning them. See the following example:

An empire expands once with no unit gain, then expands again and gains a unit.

This effect is most extreme with a smaller number of sectors, as positioning changes a lot by adding one more. This explains a lot of the big upward climb at the beginning of each line on the graph: transitioning to 2 sectors takes twice the area, but doesn’t change interception at all (because the ideal defensive position is still at the center, just like with 1 sector; as we calculated earlier).

Once over the big hump of having 2 sectors, an empire sees rapid gains by adding more units. But this slows down, because they’re already being positioned very close to the edge. If you already have 30 border forts, making it 31 doesn’t really change things much.

The Decision to Increase

Let’s take a step back. We’ve been plotting interception time against radius. But empire’s don’t expand by increasing a big “radius” dial in the throne room. In fact, an increase of+1 radius is very different for small and for large empires: because it’s the area of territory that actually has to get conquered to colonized. And how easy is it to conquer some more land? That will probably depend on how big the empire already is.

So really we should be asking how an empire’s situation would change if it increased its area by some percentage. I’ve plotted this below for 10% incremental increases:

This re-scales our original graph, and more accurately portrays the decision-pathway an empire would take when growing.

We can see that the period where interception grows (i.e. is worse) is not a momentary hassle for a growing empire, but a long difficult period. Even the smaller saw-tooth hurdles look substantial. And still, the attacker dictates a lot of this by the size of his forces.

Geopolitical Implications

It could be very hard for a smaller empire to grow through its “adolescent” period, where it has only a few defenders that it must keep close to home. During this period, its outer expanse is open to raiding and invasion; newly conquered provinces might quickly be taken away again.

Meanwhile, larger empires don’t suffer the same problems, and see continual advantage in growing. This could bifurcate empires in a region: the small ones are stuck at their current size, even though they would like to expand; and the large ones that will gobble anyone who tries.

Of course, this dynamic is created by neighboring empires using large attack forces. If this isn’t common, empires are more free to grow. So what would an attacker like to do best?

Strategy and Damage

Attackers are not merely interested in interception times, because they launch multiple attacks. Unless they’re attempting conquest, they might like to maximize their damage. Here’s what that would look like:

This is using a first-approximation damage model we discussed previously:

damage = interception * size^.5 * number of attacks

Each damage curve is similar. Relative to an empire’s size, attacks that are large or small are not inherently better or worse. And thus, the dynamics for large and small empires are similar.

But from an attacker’s point of view, this still creates challenges. When attacking a particular victim, he will have to decide what force size to use — that is, which line to be on. Let’s take an example, with a defending empire that has expanded about 55 times from its humble beginnings:

Just sticking with the options I’ve already graphed, let’s say the attacker has four distinct choices (red circles). The two bottom ones obviously do less damage and have no advantage. But the other two are both viable.

  • The upper-most point does the most damage. This is done with moderate attacking forces, not the largest ones possible.
  • The second point comes from those largest attacks, and falls just on the edge of what the defender can possibly resist — he would certainly need to organize his troops into a single force. This option does less damage, but places the defender on the fourth line, where expansion worsens his defensive position.

Thus, an attacker may choose between damage and containment, if he wants to stop a rival empire. Another interesting result coming from simple math and abstract thinking!

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Casual mathematical explorations of games, history, and science fiction

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Jasper McChesney

Jasper McChesney

Data, graphics, games. So You Need to Learn R.

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