Force-on-Force Dynamics Is Different from Olympic Competition
We show here that the consequences of gaining a small performance advantage, even if it is highly statistically significant, are likely quite different as regards force-on-force engagements than as regards Olympic competition.
In brief, a small performance advantage in force-on-force should generally result in a small change in the outcome, while in Olympic competition it can result in a large change in the outcome. We will illustrate the general principle with highly simplified, but quantitative, models.
We show here that the consequences of gaining a small performance advantage, even if it is highly statistically significant, are likely quite different as regards force-on-force engagements than as regards Olympic competition.
In brief, a small performance advantage in force-on-force should generally result in a small change in the outcome, while in Olympic competition it can result in a large change in the outcome. We will illustrate the general principle with highly simplified, but quantitative, models.
Lanchester’s Law for Force-on-Force Engagements
Lanchester, in 1916 [17] wrote down a simple model for the dynamics of a force-on-force engagement between a blue force A and a red force B. Let A be A’s numerical force strength (number of troops, e.g.) and B be B’s numerical force strength. Let kA be the effectiveness of A per unit force strength.
Lanchester, in 1916 [17] wrote down a simple model for the dynamics of a force-on-force engagement between a blue force A and a red force B. Let A be A’s numerical force strength (number of troops, e.g.) and B be B’s numerical force strength. Let kA be the effectiveness of A per unit force strength.
That is kA parameterizes A’s (hopefully) better equipment, training, situational awareness, and so forth. Correspondingly we have kB for B’s effectiveness.
Lanchester’s key concept, which defines the set of circumstances in which the model is applicable, is that B’s casualities are proportional to both the size and the effectiveness of A, while A’s casualties are proportional to both the size and the effectiveness of B. This gives immediately coupled differential equations that describes the drawdown of each force in the engagement:
Lanchester’s key concept, which defines the set of circumstances in which the model is applicable, is that B’s casualities are proportional to both the size and the effectiveness of A, while A’s casualties are proportional to both the size and the effectiveness of B. This gives immediately coupled differential equations that describes the drawdown of each force in the engagement:
Lanchester observed that these equations have a conservation law, namely that the difference of the squares of the force size (each times its effectiveness) is constant during the engagement, that is,
is the conserved quantity. Proof:
We can use Equation for example to calculate A’s casuality rate in the event that A prevails, that is, attrits B’s strength down to zero. If subscript i and f refer to initial and final values, respectively (Bf = 0), then
which can be rewritten as
Here CA is A’s fractional casualties. The approximation shown is valid when this fraction is small.
Some centuries-old rules of thumb for force-on-force combat can be found in Equation . For example, it is widely taught that at least a 3:1 numerical advantage is required for A to prevail over B in the case that B is in a fortified fixed position. One sees in Equation that this can be viewed as a statement about the relative effectiveness of forces in offensive versus defensive positions, that they differ by about an order of magnitude (∼ 32), in favor of the defense.
Also widely taught is that, for equal force effectivenesses, a numerical advantage of 3:1 will allow A to prevail over B definitively — wipe B out — while taking only acceptable casualities himself. Equation shows that this rule of thumb corresponds to the acceptable casualty rate being ∼ 5%, reasonably the case in all but very recent wars in which the U.S. has been involved.
Relevant to our application here, Equation shows that small fractional increases in A’s force effectiveness kA change A’s casualty rate (or, for that matter, ability to prevail) only by a small amount. In fact, the change is only half as much as would be achieved by the same fractional change in A’s force size:
Also widely taught is that, for equal force effectivenesses, a numerical advantage of 3:1 will allow A to prevail over B definitively — wipe B out — while taking only acceptable casualities himself. Equation shows that this rule of thumb corresponds to the acceptable casualty rate being ∼ 5%, reasonably the case in all but very recent wars in which the U.S. has been involved.
Relevant to our application here, Equation shows that small fractional increases in A’s force effectiveness kA change A’s casualty rate (or, for that matter, ability to prevail) only by a small amount. In fact, the change is only half as much as would be achieved by the same fractional change in A’s force size:
This result is illustrated for a 5% change in A s effectiveness in Figure.
As an example that we will use below, while increasing A’s effectiveness by ∼ 16% does allow A to prevail with ∼ 16% fewer casualities, the same decrease in casualties could be achieved by increasing A’s force size by 8%.
Lanchester’s law does not deny the utility of increased force effectiveness — in fact, it quantifies it. However, it shows why, in a situation where A intends to prevail at acceptable casualty rates, small changes in force effectiveness can never make decisive changes in the outcome or large changes in A’s casualty rate.
As an example that we will use below, while increasing A’s effectiveness by ∼ 16% does allow A to prevail with ∼ 16% fewer casualities, the same decrease in casualties could be achieved by increasing A’s force size by 8%.
Lanchester’s law does not deny the utility of increased force effectiveness — in fact, it quantifies it. However, it shows why, in a situation where A intends to prevail at acceptable casualty rates, small changes in force effectiveness can never make decisive changes in the outcome or large changes in A’s casualty rate.
Figure 2.2: The evolution of the fractional casualties of the two forces is shown for the case where force A has a 4:1 advantage in effectiveness (red curve) and the force levels are initially matched (Ai/Bi = 1), yielding a limiting casualty rate (e.g. where force B takes 100% casualties) of 13.4%. A small change in the effectiveness rate (blue curve, effectiveness ratio decrease by 5% to 3.8), causes a proportionally small change in the evolution of the conflict in terms of relative casualties.
