Calculating Political Power is hard!

My friend James is especially good at asking questions on WorldBuilding that require algorithmic answers. He is kind enough to offer bounties and I am especially good at getting those bounties.

This time, his question was about calculating the power scores for nation-states. No big deal right? One nation is bigger and stronger than the others, it’s going to have a higher number. Well, yeah, it’s easy at that level of abstraction. What happens when you get into the weeds of assigning numbers to amorphous characteristics like cultural power?

Getting an answer to this question was hard and took a lot of twists and turns in the writing. I’d like to share some of that process here. Have a look if you’d like to see the finished answer.

As I was working this problem out, I kept thinking of real life examples of military, cultural and economic power. The Mongols make an excellent example of all three. For their time on the steppe, the Mongols were completely unbeatable with their combined manifest destiny, martial discipline, mobility, firepower, logistics advantage, and superior officers. If the Khans hadn’t kept dying all the time, Europe would be speaking a very different language now. After conquering everything they saw, they set about building an enormous empire rich enough to generate large caravans packed with luxuries and valuables. While the Mongols were able to militarily and economically dominate the Chinese, it was Chinese culture that ultimately beat Mongol culture. Genghis Khan was raised in native Mongol culture while Kublai Khan was raised in Chinese culture. Oops.

I started off with a definition of power exactly the same as the physics definition.

Power = Work/Time

Power = Force * Displacement / Time

Power = ( Mass * Acceleration * Displacement) / Time

Initially, this made a lot of sense. An organization who can move a lot of mass, really quickly, over a great distance in a small amount of time would have to be really powerful. By the physics and intuitive definition, this is absolutely correct.

However, this runs into some problems when you try to assign a ‘mass’ to how a human thinks and what they think about. Thoughts don’t weigh anything but their influence is profound. Further, how do you measure the acceleration in the change of someone’s thinking? I couldn’t come up with a way.

At this early stage, I was also very interested in power projection at great distances. After all, the march of civilization is the march of how far away you can kill someone (which also goes back to our displacement term in the power equation). Someone who can kill you at 1000 meters with a single bullet is much more powerful than someone who kills you at 100 meters with a crossbow bolt.

At this point in the process, I was still trying to work with the strict physics definition of power and tried to incorporate the quadratic fall-off of power observed in lighting. I couldn’t find a satisfactory way to incorporate this metric into the power equation so abandoned this approach. About this same time I stopped using the physics power equation as a model and made it more of a guideline.

When I’m working on problems like this, I really like to form fundamental ontologies of nouns and verbs that can be used to build up the rest of the ideas in that system. It helps to organize things. The first verbs were “Think”, “Buy”, and “Kill”. We can see examples of this kind of competition all over the place and frequently at the same time. I don’t think any examples are needed.

One of James’ sub-questions is “Can each nation’s score exist independently or is it necessary that they be part of a larger ecosystem?” I went back and forth on whether you could measure power independent of opposition. At first, I thought “Sure, you can build a plane that can deliver troops to a point 5000km away. That’s power right?” And while yes, that is powerful, it is only a single data point in measuring a nation-state’s power nor does it include the enemy’s anti-aircraft capabilities or the difficulty in deploying troops in hostile airspace.

I went back to the original physics definition of power and realized that work is done in opposition to another force; the old “for every action there is an equal and opposite reaction”. Work as we typically think of it is always in opposition to gravity. Well shoot, power is always measured in opposition to something else. Now how to measure that.

By David Benbennick — Own work, Public Domain,

Around this point I considered building K-graphs to measure all the various tensions between competing political entities but discarded this idea since drawing out a picture then labeling or coloring all the lines seemed overly complicated though could turn out very pretty. K-graphs are okay when keeping track of smaller lists of nation-states but are wholly inappropriate for extensive lists. I suppose if you had the right software tools to do so, you could model global and regional power structures but I didn’t have the time or tools to explore that approach. Maybe someone else will.

Ultimately, what you’d want from your k-graph is a scored hierarchy of which nation-state was more powerful than another and by how much. We’ll come back to this idea later.

The word “sphere of influence” popped into my head and so I looked it up. Of particular interest was the phrase “ level of cultural, economic, military, or political exclusivity”. Aside from affirming that my power categories were correct (I grouped political and cultural power together), the key phrase was “exclusivity”. While there are almost always areas of contested control between nation-states, any functioning nation-state has an area of exclusive control. Measuring that area provided a good hard metric to work with.

Oddly enough, the Richter Scale for measuring the strength of earthquakes made an appearance here. While more accurate earthquake scales are now in use, the logarithmic measurement of earthquake energy inspired a similar measurement of exclusive control. Also, a logarithmic distance measurement means that power levels can be meaningfully calculated from the reach of a single unaugmented human (power level = 0) all the way up to the radius of the observable universe (power level = 26). Most modern nation-states will have a power level between 6.0 (a little smaller than Ireland) and 7.2 (half-way around Earth). The equation to calculate power level is P=log_10(size of sphere of influence). At the most simplistic, spheres of influence are imagined as rings on the world’s surface. This is never true in real life and rarely true in fictional worlds. Still, like spherical cows, it makes the math easier.

I’ll admit, I punted here. I didn’t come up with a method for calculating how to assign a number to cultural power or military power. (Trying to come up with a number for that would easily consume my life for the next couple of years. It’s just too big a problem to accurately handle.) Fermi to the rescue! Based on information gleaned from the work with K-graphs, I realized that I just need to work out comparative power levels between nations instead of absolute scores (as much as the latter would be to have).

Given the linear relationships between the various terms in the power equation, I figured I could use linear relationships between the terms involved in calculating nation-state power. While the displacement measures are on a log scale, as long as all the displacement measures were on the same log scale it wouldn’t matter. I chose to avoid the question of what to do with the time term from the power equation for the same reason that I avoided coming up with hard absolute numbers, it would take too long, there is too much variation and didn’t offer much value.

I also didn’t forget that this tool was for the construction of fictional worlds by an author who has absolute control over all aspects of the world. I don’t need to create a model that can account for the near infinite dimensions of detail that the real world offers, I just need something that an author can use and extend to meet their own needs. With that in mind, I came up with the following method:

  1. Make a spreadsheet and make a column for all the nation-states of interest.
  2. Make three columns for military, three for cultural and three for economic power.
  3. Calculate a power points pool size. I chose to use 5 * count of nations in list. The author has complete power to use different points here.
  4. In the first column under each power category, assign points. 5 points is average power. Greater powers get more than 5 points.
  5. The second column is for power level. Calculate the power level using the equation from earlier. It is in this column that you establish who are the global players and who are merely regional players.
  6. Multiply the power points column by the power level column and store the result in the third column.
  7. Repeat this process for cultural and economic power.
  8. Work a little spreadsheet magic to sort the each category from strongest to weakest.
  9. Write a short one paragraph description for each nation-state. This was the most fun for me. It was very clear what kind of influence each nation had and how far they could spread it.
  10. If there are major discrepancies between what was intended and the result then make adjustments to power points or power levels till the results are satisfactory.

There’s no reason that more power categories couldn’t be added. A magic or political power category would be easy to add. The same sorting and hierarchical sorting would equally apply to those categories.

The limitation of this approach compared to the graph approach is that the spreadsheet doesn’t cope well with mapping out regional influences. It can only say that a nation is a regional power not who are its regional rivals.

I had a lot of fun coming up with this method and it’s why I enjoy WorldBuilding so much. These kinds of questions really stretch my brain and help me see the world more completely.