Forced consumption against declared reserve

01 · Counting hits is the wrong frame

Public conversation about saturation campaigns is almost always organised around a single figure: what the attacking projectile costs against what the intercepting one costs. The comparison is striking and makes for a headline, but it describes the problem badly, because it assumes that the attacker is buying destruction. He is not. What he is buying is defensive consumption.

A cheap and numerous means that forces a response with an expensive and scarce one does not need to reach its target to have paid off. Forcing the shot is enough. The material damage it produces may be irrelevant and the wave will still have been profitable, because it has moved inventory that is not replaced at the rate at which it is spent.

The defender, for his part, does not lose when he runs out of money. He loses when forced expenditure invades the part of the inventory he had committed against a threat he cannot afford to let through. We call that committed part the declared reserve. While margin over it exists, the defender sustains his posture. When there is none, he has to change it, and that change is the outcome.

Two properties that are often confused are worth separating from the outset. The predictable is a property of the world and is not promised. The tractable is a property of the model: it means that the model admits examination, that it can be run, sampled and inspected, and that every result can be traced back to explicit rules with a declared range of validity. What follows is an exercise in tractability.

02 · The structure proposed

The defender's territory is organised into three effect segments, with two sectors each. The response segment groups what sustains military and operational capability. The sustainment segment groups logistics and financing; damage there takes time to be felt and is partly recovered. The exposure segment groups visible infrastructure, which degrades no military capability at all and whose only function is to force coverage where militarily none is needed.

Each side has two arsenals. One cheap and numerous, replenished quickly. The other expensive and scarce, replenished on timescales that do not keep pace with the campaign. The attacker distributes effort between segments; the defender distributes interceptors. Neither has enough for everything, and that is where the decision lies.

The important symmetry is production with delay. Both sides decide today an allocation of industrial capacity whose effect does not arrive today but several waves later. In the present turn only what is already in store is available. That turns every production decision into a bet on a situation that does not yet exist.

None of the above contains real quantities, real timescales, budgets or inventories belonging to any actor. It is a structure, not a description.

03 · What is decided and what is imposed

The attacker decides two things. How he splits his production between the two munitions, with deferred effect, and how he allocates by sector what he already has available. The allocation is zero-sum over what is available: giving to one sector takes from the others, and his own strategic reserve cannot be allocated.

The defender decides three. How he splits his industrial capacity between recovering the most damaged sector and producing arsenal, and within production between the two classes of interceptor. How he distributes interceptors across the three segments, blind, before seeing the salvo. And what percentage of what he has allocated he actually employs once the salvo is in sight, with the option of employing none at all.

There is something that is not decided and that turns out to be central: the scarcity cut. When the defender has little margin left over his reserve, he cannot afford to fire two or three interceptors per threat, which is what guarantees the kill, and he is forced to fire one. The probability that something gets through rises. That is to say, the effectiveness of the defence worsens before the inventory runs out, simply because rationing becomes necessary. That cut is not chosen by the player; it is imposed on him by his own situation.

04 · How it is measured who can no longer sustain

The state of each side is summarised in a bounded exhaustion level, worth zero at the start and one at the point where the posture stops holding. The first to reach one loses.

The defender's level is the worse of two paths. The first measures how much of the margin that exists above the reserve has been burned, and reaches one exactly when the inventory falls to the reserve. The second measures yielding through accumulated damage, as a weighted combination in which the response segment weighs more than sustainment and exposure does not count as military damage. To that combination is added what was let through in a single blow against a critical sector, which is counted separately and is not recovered.

The worse of the two is taken because not defending is no way out either. It saves inventory on one side and pays on the other. Yielding is not linear: to yield is cheap at first and ruinous later, so that beyond a certain accumulated damage rationing stops being free and one has to spend precisely when one is worst off for inventory.

The attacker's level measures the deficit of his usable arsenal, the part above his own reserve, with the exposure inflicted counting as a cost to him. An attacker who punishes what is visible spends means he is not employing against what actually sustains the defender.

Properties, which is what can be declared without publishing the algorithm. The measure is bounded. It is monotonically increasing in each of its quantities. It grows through consumption only if more is spent than is replaced. Damage in different segments weighs differently. And, above all, the threshold does not depend on any price: there is not a single monetary quantity in the whole calculation.

05 · The anchor is a range, not a point

An anchor is an initial state, a question and a declared range of validity. If the initial state is fixed at exact values, repeating the campaign only rolls the dice again, and the dispersion observed is sampling and nothing else. That produces an illusion of rigour: most match-ups always return the same thing and the result is decided before the start.

That is why the anchor here declares an interval, of plus or minus forty per cent around the central value. Inventories on both sides, industrial capacity, maximum launch rate and starting damage to each sector are drawn within that range at the start of each campaign. The resulting dispersion is no longer merely numerical: it represents the fact that the situation one starts from is not known exactly, which is the normal condition of the problem.

This has an uncomfortable and deliberate consequence. With wide anchors, the variation attributable to the world exceeds the variation attributable to the decision. Which situation comes up matters more than which posture is adopted. That is a result about the anchor, not a defect of the measure, and it is published rather than hidden.

06 · The three variances and the claim of validity

With the question anchored and the space explored, what is obtained is not a result but a measure, and its uncertainty decomposes into three parts of different natures.

That of the world is irreducible. It comes from the anchor being a range, with each campaign drawing its own. It is represented, not eliminated.

That of the decision is the one that matters. It is how much the result changes according to the posture adopted, measured over the same drawn world for every posture.

That of the sampling is what remains: the noise of the dice within one and the same world. It decays with the number of runs.

The publishable claim is the ratio between the variance due to decision and the variance due to sampling. If that ratio exceeds one, the difference introduced by the posture chosen is greater than the one introduced by the dice within a single world, and the instrument then discriminates between decisions. In the measurements made on this model the ratio comes out at around two point zero in the attacker's role and zero point six in the defender's. In the attacker's role, therefore, the instrument discriminates between decisions. In the defender's, with this anchor, it does not: the posture chosen weighs less than the dice, and a measure like that, however much it looks like analysis, is a noise generator. Declaring it is part of the measure. Separating the three matters: mixing the variance of the world into that of the sampling sinks the ratio and makes the measure fail to measure what it says it measures.

For the comparison between postures to mean anything, all of them face the same drawn world. Without that precaution, part of the difference between postures would in fact be difference between starting situations.

07 · The window is part of the question

How long the campaign lasts is not a matter of presentation. It is part of the anchored question, and keeping it as a hidden constant was a defect.

With a twelve-wave window, rather more than half of the campaigns end unresolved; with fifty, a little over one in ten; with two hundred, practically none. Those are not survivors: they are censored. We do not know when they would have given way. Computing a mean duration only over those that did resolve is survivorship bias, and it produces an optimistic figure for the side that holds out.

When the window is widened, the censoring disappears and something else appears: the outcome changes hands. The attacker's exhaustion needs time to materialise, so a short window systematically favours the side that only has to hold out. The duration declared decides, in part, who wins.

The useful output over a long window is not a probability but a shape: the fraction still sustaining its posture at each wave, and the rate at which those still holding give way. A risk concentrated at the start, a constant one and one that grows with time are three different anchors that a single probability conflates.

The model's declared window is a bounded campaign. Beyond a few dozen waves, the result describes the structural behaviour of the anchor and not a plausible campaign. It is offered because it is informative, not because the model claims that a campaign lasts that long.

08 · What the model lets one see

Visible damage and military damage diverge The posture that maximises hits is not the one that bends the opponent most. Concentrating effort on visible infrastructure produces the highest number of unintercepted means of all the behaviours tested and, at the same time, the smallest effect on the defender's capacity to sustain his posture. A campaign can look devastating and not move the result.

Rationing early can bring the outcome forward Protecting the reserve from the start saves inventory but accumulates yielding. Since yielding is not linear, what was cheap at the beginning turns ruinous, and the posture that rations may cross before the one that covers generously. The saving has an expiry date.

Intelligence fails in a biased way, not a random one The only thing observable about the opponent is his posture, and posture does not coincide with his inventory. Whoever rations looks short long before he is; whoever covers generously looks deep right up to the moment he breaks. That is why the estimated band is not merely imprecise: it errs in predictable directions.

Certainty is not distributed like a bell The available evidence on strategic intelligence forecasts shows two features that the model reproduces. Estimates rarely fall in the central band, the least informative one, because the analyst leans to one side. And they tend to declare more uncertainty than necessary. The distribution of error has a valley at the centre, not a peak.

There is a threshold below which attacking is free and also useless There is a level of effort below which the campaign produces no effect on either side. The reason is that, as long as what is launched per wave does not exceed what industry itself replaces over that same period, the store fills up again and the attacker is not exhausted; and as long as the penetration that gets through is slight, the defender stops it with the cheap interceptor and never touches the expensive one. The result is a campaign in which nobody spends anything irrecoverable and nobody achieves anything.

In the model that threshold lies between twenty-five and thirty per cent of launch capacity, depending on the strategy. Below it, ninety per cent of campaigns end with both levels close to zero. It is not a defect of the formalisation but a consequence of having production with delay on both sides, and it appears nowhere in the debate about the comparative cost of the projectile, which takes for granted that every shot costs somebody something.

09 · What the model does not say

Neither quantities, nor timescales, nor budgets, nor real inventories belonging to any actor. No specific campaign, no theatre, no identifiable system. Per-shot effectiveness enters as wide bands because the published values are inconsistent with one another by large factors.

It does not model layered defence, nor electronic warfare as a separate layer, nor degradation through platform wear. It does not translate exposure into political decision: exposure is declared outside and without a threshold, because the political threshold is not a property of the defensive system but of the society that sustains it.

And it does not predict. The variance of the world is not reduced by working harder, because it belongs to the phenomenon and not to the model. What is within reach is that the variance of the model and that of the sampling be known and bounded: the first by submitting the assumptions to refutation and declaring the range of validity, the second by raising the number of runs until their contribution is negligible.

Nor does it declare a condition of total destruction of a sector. It was tried and withdrawn: in more than a thousand long-window campaigns the hit points of a sector never reached zero, so it was a rule promising an outcome that does not occur. A model should not announce conditions that its own dynamics never reach.

10 · The demonstrator

This note is accompanied by an executable demonstrator. A role is assumed, a strategy is determined and a campaign is conducted by deciding at each wave. Each side sees its own position exactly and the other's only through intelligence. When the campaign closes, the fog lifts and one can walk wave by wave through what each side decided and what happened.

A second mode repeats many campaigns over a declared anchor and returns the distribution, not a number. It also allows the behavioural tendency followed in a played game to be distilled and measured against the reference postures. The plays are not repeated, since they were responses to specific situations; the tendency is repeated, which is the only thing that can be repeated without being absurd.

The structure of the procedure is published: the steps of the wave, the decisions of each side, the intelligence die and the end conditions. What each measure measures and what properties it has are declared. The resolution table is withheld, which is the formulation of §04.

Non-commercial research repository. This describes no real campaign and does not constitute advice of any kind.