Industry: Defense Product: ModelRisk Application: Combat Simulation
A deterministic Lanchester calculation for a company-sized assault — 30 Blue effective units against a prepared Red defence of 24, with single-shot kill probabilities of 0.16 and 0.11 — returns one tidy number: about 13 Blue units survive. The force commander reads that as a decisive win with a healthy remnant and signs the plan. The Monte Carlo version of the same engagement, run 100,000 times in ModelRisk, says the mean remnant is 19 units but the P10 is 8, and in 9.3% of battles the surviving force drops below the 25% combat-effective floor. Roughly one assault in eleven that the deterministic model calls a win comes back unable to hold the objective. The single number hid the one outcome the commander most needed to see.
A defense analysis cell rebuilt its force-on-force engagement model in ModelRisk to replace point-estimate attrition arithmetic with a probability distribution over outcomes. The engagement is modelled as a stochastic Lanchester-style duel: each round, every surviving shooter on each side fires, and round attrition is a Binomial draw on the number of shooters at the side's effective kill probability. Across 100,000 simulated battles the surviving-force distribution the deterministic plan never produced looks like this:
A deterministic Lanchester model multiplies average kill rates by force sizes and reads off a survivor count. It cannot represent the two things that actually decide a firefight: the spread of outcomes around the average, and the fact that on any given day every unit on the battlefield shares the same conditions. The model here draws one "fog-of-war" factor per battle — a LogNormal common factor representing terrain cover, weather, communications and surprise — that scales the kill probability of every Blue shooter together, with a paired defensive-cover factor for Red. That shared factor is what keeps the outcome genuinely uncertain even with 30 units committed; without it, the law of large numbers would collapse 30 independent duels into a near-certain result and reproduce exactly the false confidence of the point estimate.
Return to the distribution at the top. The mean survivor count is 19.2 units, but the deterministic plan's own arithmetic put it near 13.5 — and neither number is the decision-relevant one. The decision-relevant readouts are the P10 of 8 units and the P90 of 26: an eighteen-unit gap between a comfortable win and a force that has been gutted. The dotted line marks the 25%-of-committed-force combat-effective floor; 9.3% of battles end below it. That tail — the assault that nominally succeeds but leaves a remnant too weak to consolidate — is invisible to a mean and is precisely the outcome that gets people killed on the objective.
The headline question for the planner is how the probability of a mission win moves with the initial Blue:Red force ratio. Sweeping the ratio and re-simulating at each point gives a clean dose-response curve, shown for baseline Blue lethality and for a degraded-lethality case (Pk cut 15%, e.g. a night assault or a maintenance-degraded weapons fleet):
At the baseline 1.25:1 ratio the win probability is already high, but the curve shows that an 80% win probability needs a force ratio of about 1.11:1 at baseline lethality and 1.19:1 if Blue lethality is degraded. At parity (1.0:1) the win probability is only 70% baseline and 60% degraded — a 10-point penalty for a lethality shortfall the deterministic model treats as a rounding error. The curve also flattens hard above 1.5:1: committing extra force past that point buys almost no additional win probability, which is exactly the kind of diminishing-return boundary a commander wants quantified before requesting reinforcements.
Varying each input across its plausible band and measuring the shift in win probability ranks where the risk really sits:
The single biggest mover is Red force size / intelligence error — a ±3-unit error in the estimate of enemy strength swings the win probability by ±7.4 percentage points, more than any friendly-force lever. Own Blue force size is next at ±6.3 points, then Blue and Red weapon lethality at roughly ±4.7 and ±4.6 points, and the fog-of-war dispersion at ±3.5. The ranking is an intelligence-investment argument in chart form: reducing uncertainty in the enemy order of battle does more for the win probability than a comparable improvement in friendly weapons, which is the opposite of where procurement attention usually goes.
Plotting the force-exchange ratio (Red losses per Blue loss) against the surviving Blue force exposes a tail the aggregate win rate conceals:
The median exchange ratio across battles with any Blue losses is a favourable 2.67 Red units lost per Blue unit lost, and the headline decisive-win probability — Red destroyed and a combat-effective remnant retained — is 90.7%. But 2.5% of battles are Pyrrhic: Red is destroyed yet Blue is left below the combat-effective floor. Those runs sit in the bottom band of the scatter and never appear in either the mean survivor count or the win/loss tally. Quantifying them is what lets a commander pre-position a reserve sized to the actual tail rather than to the average.
A combat plan built on an average attrition figure is a forecast of one battle that will never happen exactly. Built on a distribution, it is a forecast of the range of battles that might — and of the specific bad ones worth planning a reserve against. ModelRisk is what turns the firefight from a single number into a shape the commander can actually command against.