How Can Lanchester Models Predict War Outcomes? A Step-by-Step Guide

Modeling War Dynamics 01: How to Use Models to Depict War Trends?

1.1 Lanchester Model

We desire peace, yet wars occur. The 2022 Russia‑Ukraine war is a recent example. Scholars use mathematical models to analyze possible future war trajectories, notably the Lanchester combat models.

The Lanchester models are a family of equations that differ according to whether the conflict is conventional or guerrilla, and whether supplies are available.

1.2 General War Model

1.2.1 Assumptions

Non‑combat attrition rates are proportional to the total force size of each side.

The battle dynamics depend only on these factors, with combat technology and attrition rates treated as constants.

1.2.2 Notation

Let A and B denote the troop numbers of the two sides.

α and β represent the combat attrition rates inflicted by A on B and by B on A, respectively.

1.3 Model

Within the general war framework, specific models are discussed according to combat type.

1.3 Conventional War Model

Additional assumptions:

1.3.1 Assumptions

If a soldier of side A is within the observation and fire range of side B, the fire of side B immediately concentrates on other A soldiers once one is killed; the same applies symmetrically.

The larger the number of soldiers, the greater the attrition inflicted on the opponent; higher combat efficiency (e.g., technology) also increases attrition. Thus, troop numbers and efficiency are proportional to the opponent’s attrition rate.

1.3.2 Notation

γ denotes the per‑soldier killing rate of side B against A and vice versa.

1.3.3 Model

… (model equations would be placed here) …

1.4 Guerrilla War Model

Both sides use guerrilla forces. Side A operates in concealed areas unseen by side B; side B fires at the area without knowing actual casualties. Consequently, A’s attrition rate depends on both A’s and B’s troop numbers.

1.4.1 Assumptions

The attrition rate of side A is proportional to the product of A’s and B’s troop numbers.

1.4.2 Notation

δ represents the effective combat coefficients of both sides.

1.4.3 Model

… (model equations would be placed here) …

1.5 Mixed War Model

One side employs guerrilla forces (A) while the other uses regular troops (B). Using the same symbols and assumptions as above, a combined model is constructed.

1.5.1 Model

… (model equations would be placed here) …

2 Numerical Example

Consider both sides fighting conventional wars with initial forces A=15, B=5. The per‑unit‑time killing rates are 0.001 for A on B and 0.002 for B on A; non‑combat attrition rates are 0.0001 for both. Reinforcement rates are 0.1 for A and 0.05 for B. The resulting 40‑day force trajectories are shown below.

3 Summary

Lanchester’s models are simple: they consider only troop numbers and combat effectiveness. Forces decrease due to combat and non‑combat attrition and increase through reinforcements. Combat power depends on shooting rate, hit probability, and war type (conventional or guerrilla). The models ignore political, economic, and social factors, so they cannot predict overall war outcomes, though they can offer insight for specific battles. More importantly, the modeling approach provides a template for applying mathematical models to social‑science problems.

A preliminary introduction to several forms of the Lanchester combat model is presented; future work will expand the discussion.

References

Wu Mengda. *Mathematical Modeling Tutorial* [M]. Higher Education Press, 2011.