Lanchester's equations and the structure of the operational campaign: Between-campaign effects

The main thrust of this paper is to reconcile common combat modelling practice with the evidence stemming from the analysis of historical battle outcomes. In many aggregated models variants of Lanchester's ‘Square Law’, or a ‘Square-Linear Law’, are used to represent the direct fire attrition process. These place a heavy premium on the concentration of force with, other things being equal, the balance of attrition strongly favouring the side with the greater number of combatants. However, almost without exception, the relationships actually observed in collected samples of historical battles are in line with a ‘Log-Linear’ version of Lanchester's equations. This would suggest that, in attrition terms, concentration of force should positively be avoided. The greater the number of combatants assembled on the battlefield, the greater will be the likely number of casualties accruing to that side. In this paper it is pointed out that the battles which feature in historical samples are self-selecting. By definition, these samples do not feature those instances where a would-be assailant chose not to launch an attack because he calculated that his chances of success were negligible. Nor do they include those occasions where the defender chose to abandon his position because he saw that defeat was almost inevitable. For those cases that remain a key task for the attacking commander would have been to assemble the resources he deemed necessary to ensure a reasonable chance of mission success. In practical terms this means that he would be prepared to enter battle with a smaller force ratio if he perceived the opposing forces to be militarily ineffective; if he had confidence in the prowess of his own troops; if he felt that he had the edge in terms of weapon effectiveness; and/or if he judged that the chances of local concentration and other terrain features were in his favour. These sorts of considerations are in line with the between-campaign relationships actually observed in the main historical data base assembled by UK analysts. This paper describes a process of theoretical modelling and simulation, based on evidence from live trials and from battle. The results suggest that, even though the affrays within a campaign may obey a version of Lanchester's ‘Square-Linear Law’, the mechanisms outlined above will ensure that their outcomes will appear to obey a ‘Log-Linear’ relationship when they are aggregated over a collected sample of campaigns. This effect will be enhanced if casualties from the direct fire battle are simply combined with those from other quasi-independent sources, such as those due to air power or the taking of prisoners.