How Lanchester Models Can Predict War Trends: A Practical Overview

Modeling | Change Modeling 01: How to Use Models to Depict War Trends?

1.1 Lanchester Model

We desire peace, yet wars occur; the 2022 Russia‑Ukraine conflict is a recent example. Scholars use mathematical perspectives such as the famous Lanchester combat models to analyze possible future war trajectories.

The Lanchester models form a family of equations that differ based on 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 trajectory depends only on these factors, with combat technology and attrition rates treated as constants.

1.2.2 Symbols

Let the forces of side A and side B be represented by their soldier counts.

Let \(\alpha\) and \(\beta\) denote the combat attrition rates that side A inflicts on side B and vice versa; these rates measure how quickly one side reduces the other's combat personnel.

1.3 Model

Within the general war‑model framework, specific model forms are discussed according to different combat types.

1.3 Conventional War Model

1.3.1 Assumptions

If a soldier of side A is within the observation and fire range of side B, the loss of any soldier causes the remaining firepower to concentrate on the surviving opponents, and the same applies reciprocally.

The larger a force and the higher its combat efficiency (e.g., technology), the greater the attrition it inflicts on the opponent; attrition is proportional to both force size and efficiency.

1.3.2 Symbols

\(\gamma\) denotes the unit‑time killing capability of a single soldier from side B against side A (and similarly for the opposite direction).

1.3.3 Model

(Model equations are presented in the original source.)

1.4 Guerrilla War Model

Both sides employ guerrilla forces. Soldiers of side A operate in concealed areas unseen by side B; side B fires at the area without knowing exact casualties. Consequently, side A's attrition rate depends on both sides' force sizes and increases with side A's troop count because more soldiers in a limited area lead to higher losses.

1.4.1 Assumptions

The attrition rate of side A is proportional to the product of the numbers of side A and side B soldiers.

1.4.2 Symbols

\(\delta\) represents the effective combat coefficients for both sides.

1.4.3 Model

(Model equations are presented in the original source.)

1.5 Mixed War Model

One side (A) uses guerrilla tactics while the other (B) employs conventional forces. Using the same symbols and assumptions as the previous models, a combined model is constructed.

1.5.1 Model

(Model equations are presented in the original source.)

2 Numerical Example

Consider both sides using the conventional‑war model with initial forces of 15 (A) and 5 (B). The unit‑time killing rates are 0.001 for A against B and 0.002 for B against 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 plotted below.

3 Summary

Lanchester's models are simple, considering only force sizes and combat effectiveness; forces decrease due to combat and non‑combat attrition and increase via reinforcements. Combat power depends on shooting rate, hit probability, and war type (conventional or guerrilla). These models ignore political, economic, and social factors, so they cannot reliably predict overall war outcomes, though they can offer insight for specific battles. More importantly, the modeling approach demonstrates how mathematical models can be applied to social‑science problems.

The article provides an introductory overview of several Lanchester combat models, with more detailed discussions to follow.

References

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