Lanchester’s Laws

Note: This obviously does not apply to very narrative games, where combat is highly abstract and these kind of considerations are taken care of on a totally different level.

This is about how you can twiddle with the deadliness of combat as the GM. Of course, this isn’t all-encompassing, but understanding these principles will help with both setting up the situation and adjusting the number of opponents.

Frederick Lanchester was a polymath, who founded a car company, did much work with aerodynamics and was one of the founders of operations research. Since operations research might be new to many of you, its the study of using math to help decision-making. Normally, we are using higher mathematics, but today we won’t be needing anything more complicated than powers.

One of the things Lanchester was interested in was how the rudimentary WWI planes faired in battle. He noticed that the battles were fought very differently than other combats. His major discovery was about the amount of killed.

Back in the day, when wars were fought in lines of people, who tried to kill the guy just in front of them, the number of killed would simply be the difference in the sizes of the groups. Assuming similar training and equipment, and no surrendering a group of ten soldiers would win against a group of seven soldiers and three of the group of ten would survive. Obviously, battles would never drag so long that everyone died from one group, but if they would, this is how they’d end up.

Since then, how combat is conducted has changed very much. Its not man against man in straight lines, but people with access to automatic fire or, in the case of planes, no clear battlelines. Therefore both sides can better utilize their firepower instead of limiting it to one person.

Now, in a combat, the actual difference between the forces isn’t linear anymore. So, if we give planes to the soldiers mentioned before (and trained them), the situation wouldn’t be 10 against 7, but those numbers would be squared instead, so its 100 agains 49. The larger group would still have sqrt(100 – 49) = 7,14 or seven people plus someones arm or something.

Proof (not real mathematical proof, but hopefully enough to convince you):

Assume both sides are firing with everything they’ve got. Each unit within the group has a 10% chance of killing an opponent during each “round” of combat.

round group 1 group 2
1 10 7
2 9,3 6
3 8,7 5,07
4 8,193 4,2
5 7,773 3,3807
6 7,43493 2,6034
7 7,17459 1,859907
8 6,988599 1,142448
9 6,874355 0,443588

There is of course some variance. As you can see this differs from the figure above, but its close enough. Its because of the nature world with rounds in it when compared to the world where time flows seemlessly (well, close enough).

In short: If you double the opposition, its power will actually quadruple.

The actual power used in today’s military science isn’t actually 2, but 1.5 because most battles don’t fall cleanly into either category here. Also, this is about symmetric warfare, which is becoming increasingly rare as well.

Size of the Group, Mathematical Approach

With two people, there is one interaction. If we bring in a third person, there are two more interactions, one with each earlier member. With the fourth member, three new interactions emerge. In other words…

x = 0 + 1 + 2 + 3 + … + n-1 = (n – 1) * (n / 2) (I’m going to be lazy and not bother with the proof.)

How can we put this into use?

Depends on what we want to do. If we want a game where the players themselves produce much of the narrative, we need more of these interactions, whereas if we want to keep the amount of time used on bickering to a minimum, we want less of these interactions.

With three players we have (3 – 1) * (3 / 2) = 3 interactions. With five players, we have (5 – 1) * (5 / 2) = 10 interactions. Therefore by adding two people into the group, they will interact with each other over three times as much as the three player group, thus taking much more time and potentially bogging down the game.

Obviously, its not always this straightforward. Certain players will use more time than others, certain players will only agree with other players and so forth. Also, different genres will work differently. Combat-oriented games with little planning will not work the same way and adding players will not break it in the same manner.

Still, understanding this principle and how it will change the picture, should be taken into account when planning games or designing adventures, although I do doubt anyone in the world does so.