Lanchester's Game

I've been kicking ideas around about using Lanchester's Laws as a way to create a point-value system (1) for balancing sides in a game, and one of these involves designing a game around this mathematical principle. Usually point system seem to evolve after the game is created; I want to create the point system first, and build a game to fit the math. I'm not sure this is possible, but it should be interesting to try. So I set out to write out the rules for a simple game to demonstrate the difference between Lanchester's Linear, Square, and Logarithmic (see footnote 2) Laws, which I will abbreviate with L1, L2, L0. This turned out to be a really good idea, because it gave me an insight about how these laws arise, and a simpler way to explain them.

[A note aside: I wrote most of this post two months ago, and it gave me many good ideas for other post in the process, but I never quite figured out how to finish this one. Now I need this basic discussion to go along with some new posts I'm working on, so I'm making a second effort to finish this one.]

The Game
This game is really very simple, actually more of a thought experiment that explains Lanchester's laws.

Map, Movement, and Range: There is no map, and so no movement, and no range. This is an abstract game, and attacking another unit depend on which Attack Rule is being played (see below). It would be a more interesting game with these elements, but they only complicate the discussion. Maybe I will try to add these back in for another post. (Also good discussion points.)

Forces: Two sides, Blue and Red, each side have a number of tokens (B or R tokens, respectively) or markers representing the strength of each force (These might represent soldiers, tanks, etc.). Each player should start with 20 to 30 tokens, but not necessarily the same number.

Lethality: an attack is resolved by rolling a die: success kills one enemy, remove that marker/token. Assume equal lethality for simplicity, or allow to be different for completeness. Each attack has a lethality, or probability of a kill equal to b for Blue and r for Red. Lethality does not have to be the same for each side, but it simplifies this discussion if it is. For a good demonstration this should be a fairly small probability, so that the game will last 10-20 turns. The following discussion will assume a lethality of b = r = 1/6, so a roll of 1 on 1d6 can be used to resolve this easily.

Attacks: Every turn each player makes one or more attacks. The number of attacks a player makes depends on the Attack Rule in play, and could depend on the current size of each force (B or R).

Sequence of play:
1) Set up the game, decide force sizes, lethality, and Attack Rule.
2) Begin turn: players make one or more attacks, as determined by the Attack Rule in play.
3) Resolve attacks for each player based on the size of their force at the beginning of the turn.
4) Remove destroyed forces.
5) If both played still have forces remaining, go back to step 2 and play another turn. Play continues until one side is eliminated.

Attack Rules:

L1: Each player makes one attack every turn. No matter what casualties occur over the course of the game, each player will have same same total number of attacks, and this number will be proportional to the small of the two forces. This is exactly what is expected under the Linear Law.

L2: Each player makes one attack for every 5 tokens they have remaining (round up or carry fractions o the next turn). Over the course of the game the total number of attacks will be larger for the player with the larger initial force. The ratio of total Blue attacks to Red attacks will be proportional to (B/R)^2 [the ratio B/R, quantity squared]. (It does not have to one attack for every 5 tokens, it only need be some small proportion of the current size of the force. 5 was just convenient).

L0: Each player makes one attack for every 5 (convenience again) of the other sides tokens. Here the number of attacks made against you is proportional to the size of your own force (see footnote 2 again). This seems like a strange rule, but war in unhealthy! Putting your army in the field makes if subject to direct and indirect threats. Starvation, disease, accidents, mules kicks, artillery and bombing, are all hazards that put the entire force at risk. Sometimes the more you bring, the more you lose.

Discussion:
Each of these "attack rules" will lead to distinctly different outcomes for the game. More importantly, a form of one or more of these rules is inherently present in all war games and combat simulations. Even if it is not written explicitly, but it will still arise from how the game plays.
Game combine these rules in interesting ways. For instance, terrain, stacking rules, and range limits will tend to restrict some units in a game to the L1 attack rule. Other units will have a clear field of fire to attack (and be attacked) will use the L2 rule. Some units might stay in relative safety and threaten the other force from afar (like artillery) and subject the other side to the L0 rule. A unit firing from a bunker might only be attacked under the L1 rule, may be able to attack other using the L2 rule. so it's not necessarily the same rule in effect for both sides.

I have read many papers trying to model data from historical battles as if there is a new rule that somehow combines two or more of these rules. From a certain standpoint that is the wrong approach. These rules might mix, affecting different parts of armies in different ways, but there is no rule that says you will always get the same sort of mixture every time. In fact, you will almost certainly get a little different mixture in every battle. Lanchester's laws are not something over which either side has total control - they are something that happen to you during the battle. In a close fight, the army that is better able to exploit the rules is more likely to win.


Footnotes:
(1) Such as BattleValue in Battletech, which is the one I know the best. Ogre/GEV has a very simple point system. Warhammer 20K+/-20K has a point system but I know nothing about it other than it exists. If you can suggest other games that use point systems please post or email me about it so I can look into this topic further.
(2) The "Logarithmic" law arose from attempts to fit actual data of battle casualties to either the Square or Linear law, and finding that sometimes neither one is a very good fit. I have avoided mentioning it thus far in order to simplify discussion. The interpretation of the Logarithmic Law implies that the casualties suffered by one force are proportional to the size of one own force (not the opposing force). This seems unusual, but is sometimes observed in historical data describing large scale battles. See Fricken (1997) for the an excellent discussion and justification for the Logarithmic law.