Civilization V: Military Power Curve

Mathematically, human intelligence trumps AI — in the end

Gandhi is a symbol of the series, known for his out-of-character aggression.

I am currently playing my largest game of Civilization V ever. I decided to do the One-City Challenge as the Shoshone on a Huge Continents map, Deity. I have played over 2000 turns and am currently in the middle of an endless war against a Franco-Aztec alliance. You can read more about it here.

The course of the war has caused me to assess the mechanics of the game. Despite having only one city, I am steadily winning a war against the team that owns most of the two supercontinents. Their military is so large that nearly every land tile on the map is occupied by one of their units. What is enabling me to win so far?

Some of you already have an answer: the AI. Rather, the artificial intelligence is not very intelligent. It often does not attack in the most efficient way, and it lacks the ability to form coherent, long-term goals. For example, I raided and destroyed most of my enemies’ coastal cities because it was the low-hanging fruit and would give me command of the seas. The game’s AI simply cannot plan things out in that way.

That said, we should not act like we have never lost a game to an AI, especially when playing on Deity. Occasionally, they prove too powerful to overcome. What might be helpful is to understand military power in this game and the curves that illustrate it.

I imagine military power in a game of Civilization V to follow a general S-curve, where f(a) represents the power of an AI at a high difficulty level and where f(p) represents the power of a human player. If we assume that both are products of a logistic function, then we might get the following formulae:

Let D represent the difficulty level of the AI (we will assume Deity), while I represents the intelligence of the player. Let a and p represent the respective sizes of the militaries at any given level of technological advancement. Meanwhile, k, which usually indicates the steepness of the curve, could represent the rate at which science is gained and perhaps how quickly units are produced. If we assume also that I > D in all cases, then we should get a graph that looks something like this:

You will notice there are two curves, one for the AI and one for the player. While military power is at least partially a function of size, both have different caps. This is because military power is partly a function of intelligence. Since the numerator in a logistic function represents the maximum value of a logistic function, and if the cognitive ability of the AI and the human player are the numerators in these functions, then the human player ultimately has the most potential for military power.

The reasons are simple. Anyone who has played the game long enough should know that the AI are not capable of complex tactics or strategy. This means that the increase in military power that an AI achieves by producing extra units hits diminishing returns sooner. A more intelligent actor can perform more complex actions with more units. Thus, a human army with 100 modern-era units will be more powerful than an AI army with 100 units of the same era.

Where a high-difficulty AI has an advantage is in the early game. You will notice that while the player’s curve has the most potential, the AI grows faster. This is because it is given advantages in science, production, and GPT that allow it to outpace the player, despite a relatively equal starting position. The challenge in the game then is for the human player to survive in spite of their deficient military power, until they are able to catch up.

When they do catch up, human intelligence allows the player to attain higher power than the AI. Of course, there is also an upper limit on human power, dependent on how intelligent a given player is, and even the most intelligent person in the world has finite intelligence. Finite intelligence, in theory, should therefore limit how complex and efficient uses of a military at a given size are, even with human players.

There is certainly more thought that could be put into this, but I think this is a good starting point for how to look at the strategy behind games like Civilization V mathematically. Are there other curves in the game besides that of military power?

Furthermore, what does this say about how we can assess real militaries (and civilizations) mathematically? After all, in 1940, the German Wehrmacht faced a more numerous and better-equipped Anglo-French force. Even the tanks, for which Germans became famous in World War II, were of higher quality on the Allied side. Why then, did the Third Reich defeat its adversaries in the outset and conquer Western Europe? Superior planning is the best explanation. What, then, is the power curve for this scenario?

Civilization V might be easy to write off as fundamentally different from the real word, as essentially a glorified spreadsheet, but so is real life. Everything we think and do is the result of specific and consistent values that pertain to matter, energy, space, and time. Thus, a sufficient simulation of Civilization V can actually teach us more about our own societies than we might expect.