Editor’s Note: Dr. Young is a respected analyst of defense issues with a long history of interest in submarine warfare matters. He had two previous articles published in THE SUBMARINE REVIEW. In the October 1985 issue he authored __Setting Goals for a Submarine Campaign__ and in the January 1987 issue his article was titled __Fighting in Defended Waters.__ This current piece treats the basic issue of the one-on-one warfare which is submarine against submarine operations. He characterizes the **Campaign Exchange Ratio** required by the relative strengths of the opposing submarine forces. As an example, he has set up a situation of a friendly force significantly outnumbered by the opposition and has derived several conclusions as to the potential for the friendlies to prevail.

There are lessons to be learned from careful attention to Dr Young’s conclusions: some editorial assistance has been provided to the reader by putting several of his observations in bold type. I hope this helps, but it cannot cover all that is to be gained as insight. A piece of editorial advice for a first reading is to accept the math he has provided and search for the implications of importance to the submarine community. For the mathematicians among us, as opposed to the engineers, Dr. Young has provided at the end of the article both the logic for his **Campaign Exchange Ratio** and his email address for those interested in a more detailed development of the formula.

There are implications of numerical strength of the force structure required and there are things to think about in ship, and weapons, characteristics, as the title implies. The operators have to look at doctrine and training to see if Dr. Young’s conclusion about the criticality of the.first attack is being served.

When one is satisfied that the important observations here have been duly noted, it might well be useful to reread his 1980’s articles in one ‘s complete library of SUBMARINE REVIEW issues. It should be remembered that at the time of those publications the Cold War was still going on and the focus of the US Submarine Force was on fighting in the Soviet Bastions to deny the USSR a viable escalation strategy. The October ’85 piece raised the point about the effect of **time requirements** on the campaign and what that meant for both force strength committed and tactics involved. Dr. Young’s January ’87 article described the probable effect on the **Campaign Exchange Ratio** of an in-place defense structure of enemy aircraft, surface units and other surveillance systems.

The question then seems to be whether our analysts can use these imbedded logics to make open source cogent arguments for both quality and quantity in our submarine force structure.

Suppose a force of 100 friendly units aims to eliminate a force of 300 enemy units in a series of discrete, one-on-one engagements. Small-unit actions against terrorist cells, certain forms of air combat, and prowling submarines locked in underwater combat, typify such search-and-destroy operations. In a fight to the finish in the example given, each friendly unit must destroy at least three enemy units, on average, for a minimally successful campaign. Setting aside the dynamic and probabilistic aspects of the campaign, the three-to-one exchange ratio translates into a demanding 75- percent average chance of victory in 400 individual battles.

This article relates campaign success to basic combat capabilities of friendly and enemy units and identifies stiff levels of performance for friendly units that guarantee victory, but also highlight its challenge.

**Top-Level-Description**

The 400 decisive battles can be sorted into clean wins, clean losses, and mutual kills, with fractional probabilities w, 1, and a, respectively, constrained by the equation, w+ 1+ a= 1 . In a large number of engagements, friendly units will destroy the fraction, w +a , of enemy units and, vice versa, lose the fraction, 1 +a , of friendly units, for a campaign exchange ratio it: which in our case, must equal three.

At this level of description, we can freely vary the probabilities, w, 1, and a, that characterize campaigns and find a relationship between a given value of one of these probabilities and allowed values of the other two. In particular, battles that end in mutual attrition- a one-to-one exchange ratio-tend to lower an otherwise higher ratio of clean wins to clean losses.

For instance, let E denote the ratio, w/1 , of clean wins to clean losses. Since w+1+a = 1, it follows that

w = E(1-a)/E+1 and 1= 1-a/E+1.

Then, the campaign exchange ratio w+a/1+a can be re-written as E+a/1+a set equal to three to fix the value of E necessary to achieve the minimum three-to-one exchange ratio for any given probability of mutual attrition (a).

Suppose the chances (in percents) for mutual attrition in one-on-one battles increase from zero-, to 10-, to 20-, to 30-percent. The associated sets of probabilities ( w, 1, a) of campaign outcomes that achieve a three-to-one exchange ratio are the following,

(75, 25, 0), (73, 17, 10), (70, 10, 20), and (68, 2, 30)

These result tell us that the three-to-one exchange ratio can be maintained only so long as clean losses can be turned into partial victories. At and beyond a 33-percent chance of mutual attrition, this becomes impossible.

**UNIT-LEVEL DESCRIPTION**

In any case, the top-level parameters w, 1, and a, that simply tally the outcomes of a campaign are not independent and freely variable, but collectively depend upon lower-level functional capabilities of friendly and enemy units in one-on-one engagements. In particular, at the next lower level of description, we might consider the following probabilities that characterize important unit combat capabilities:

- friendly units attack first upon contact (p) or not ( 1- p) ,
- first attacks by friendly units are lethal (f1) or not(l- f1)
- counterattacks by friendly units are lethal ( 1′!) or not(l-1’J)

Correspondingly,

- enemy units attack first upon contact ( 1-p) or not (p)
- first attacks by enemy units are lethal (/2) or not (1-f2)
- counterattacks by friendly units are lethal ( 1′!) or not(1-r2)

For the purpose of this discussion, a failure to counterattack and a counterattack that fails are both absorbed in the not-lethal outcome.

The parameters that describe the strengths of each side in one-on-one unit engagements still are collectively dependent on lower-level technical, human, and operational factors under each sides’ control and not freely variable. On the other hand, each side’s engagement parameters can be regarded as independent of each other for the following reason.

At any one time, the values of the engagement parameters are the product of complex interactions between specific systems and practices deployed on each side. However, these systems and practices, in tum, are under continuous development by independent actors in an action-reaction struggle not under either side’s sole control. **Since the smaller friendly force must ensure a minimum three-to-one exchange ratio, it is important to identify those sets of unit capabilities on each side that guarantee this ratio throughout the action-reaction struggle.**

In the interest of moving directly to data showing the connection between the campaign exchange ratio and the unit functional capabilities listed for each side, I’ll simply quote the formula for this relationship and make its easy derivation available separately to interested readers. The formula relating the campaign exchange ratio to unit engagement parameters is:

E = mf+(1-p)n1\mf+(1-p)n2

This expression for the campaign exchange ratio has a straightforward interpretation. In the numerator, friendly units destroy enemy units by successful first attacks (pf1) and counterattacks to enemy first attacks ( (1-p )r1 ). In the denominator, enemy units friendly units in successful counterattacks (Pl’l) and first attacks ({1-P}/2 ).

Before turning to illustrative data, the following qualitative points are evident from the formula for the campaign exchange ratio:

- Since the friendly side has the stiffest challenge, it needs high probabilities of both initiating (p) and winning (f1) firstattacks (pf1) and low probabilities (r2) of a lethal counterattack to its first attacks.
- If friendly units have a high probability {P) of making the first attack. then the probability of an enemy first attack (1- p) will be low, which simultaneously reduces the importance of friendly counterattacks (“!) and, to a lesser extend because of its sensitive location in the denominator, the legality of enemy first attacks (‘2) .

Data that bear on these qualitative observations are shown next.

**QUANTITATIVE IMPLICATIONS**

For easier reading, the probabilities in the following Tables are shown in percent form. Calculated exchange ratios are multiplied by I00 to show total potential kills by l 00 friendly units for comparison against the minimum campaign goal of 300 enemy units destroyed.

For the cases shown in Table 1, friendly and enemy units each have high 90-percent chances of a kill on first attack. Friendly units have 90-, 85-, 80-, and 75-percent chances of attacking first on contact and, in order to minimize the pressure on friendly units, enemy units have a low 5-percent chance of making a lethal counterattack. To assess the importance of a counterattack capability for friendly units, zero-, 10-, and 20-percent chances that friendly units will make a lethal counterattack are included.

**Campaign Exchange Ratios (x100)**

90-percent chance that friendly first attacks are lethal

90-percent chance that enemy first attacks are lethal

5-percent chance that enemy counterattacks are lethal

The numbers shown in Table 1 for potential kills by friendly units confirm both the expected strong advantage to friendly units of a high probability of initiating a lethal attack and- given this capability- the lesser importance of friendly units having a counterattack capability.

Indeed, if friendly chances of making the first attack decrease from 90-to slightly less than 80-percent, then the drop-off in potential kills throughout the Table-600-plus to less than 300-is striking. **Unless the probability of initiating attacks is nearly 80- percent or better, the minimum campaign goal cannot be met, even when enemy units have only a 5-percent chance of a lethal counterattack. **

Table l also confirms that, given high chances of friendly units making the first attack, a friendly capability for counterattack has a modest impact on campaign effectiveness. Raising the chances of a lethal counterattack by friendly units from zero- to 20-percent increases campaign effectiveness between 3-percent in the least favorable case (15 kills more than 600 in the first column) and 8- percent in the most favorable case (20 kills more than 260 in the last column). To better focus on main effects, friendly units have no capability to counterattack in subsequent Tables.

Table 2 continues the assumptions in Table 1 that friendly and enemy units each have high 90-percent chances of success in a first attack and, to quantify the impact of a counterattack threat to friendly units, includes chances of a lethal enemy counterattack of 5-, 10-, 15-, and 20-percent.

Campaign Exchange Ratio (x100)

90-percent chance that friend first attacks are lethal

90-percent chance that enemy first attacks are lethal

Table 2 illustrates the seriousness of the threat of lethal counterattacks-especially if friendly units lose some initiative for first attack- and a trade-off between probabilities of a friendly first attack and an enemy counterattack.

If friendly units initiate 90-percent of the battles and win 90- percent of these, the campaign is compromised once the chances for a lethal enemy counterattack rise above 20-percent. Continuing on, if friendly units initiate 80- versus 90-percent of the battles, then the chance of a lethal enemy counterattack must be kept below 7- percent, illustrating the small room for fall-offs in friendly combat capabilities. **From the standpoint of campaign effectiveness, friendly units highly capable of initiating attacks have a small need for a counterattack capability, but a critical need to suppress, deflect, evade, or harmlessly absorb enemy counterattacks.**

As for the trade-off between probabilities in Table 2, the data show the following pattern. Starting from any position in the Table, the improvement in a campaign outcome from an increase of five percentage points in the probability of first attack can be nearly matched by reducing the chances of a lethal counterattack by the same five percentage points. A closer analysis shows that reducing the chances of a lethal counterattack by six or seven percentage points would match any gain from raising the probability of first attack by five percentage points.

Although this indicates a small advantage from increasing the probability of first attack versus reducing the threat of lethal counterattack, the practical problems of making a small percentage increase in a large probability of first attack or or a large percentage reduction in a small probability of lethal counterattack are vastly different.

Table 3 shows campaign results in a side-by-side comparison of 90- and 75-percent chances of a lethal first attack by enemy units, given a 90-percent chance of success in first attacks by friendly units. At high probabilities of first attack by friendly units, even though first attacks by enemy units are infrequent, their lethality appears in the denominator of the campaign exchange ratio and deserves review.

Campaign Exchange Ratios (x100)

90- and 75-percent chances (side-by-side) that enemy first attacks are lethal

90-percent chances that friendly first attacks are lethal

The value of a counterattack capability to friendly forces (nonexistent in Table 3) tends to increase when enemy forces are less lethal in first attack. For the cases shown, however, increasing the chances of a lethal counterattack by friendly units from zero- to 20- percent raises all results in Table 3 by less than 7-percent in the best case, and the percent increases in campaign effectiveness as shown hardly at all.

As the lethality of enemy units in first attack falls from 90-to 75- percent, the gains in potential kills by friendly units (between 6- and 17-percent) are significant, especially for cases in the top half of the Table. For example, a gain of 30 potential kills is equivalent to the production of 10 friendly units and most of the gains in potential kills shown exceed 30 enemy units.

ls shown exceed 30 enemy units. Furthermore, the tight lower bounds (80-percent or better) on the chances for a first attack by friendly units and upper bounds (20- percent or less) on the chances of lethal enemy counterattacks are both relaxed by about five percentage points by less capable enemy units. **Consequentlyt methods to blunt the lethality of infrequent enemy first attacks are helpful. **

Lastly, Table 4 examines the effect ofless-lethal first attacks by friendly units. Side-by-side comparisons of 90- and 75-percent chances of lethal first attacks by friendly units are shown, given a lower 75-percent chances oflethal first attacks by friendly units are shown, given a lower 75-percent chance of a lethal enemy first attack. The lethality of enemy units in first attack is fixed at the lesser of the two friendly capabilities in keeping with the fact that the smaller friendly force must be generally superior in capability to the enemy force to have any chance of meeting its campaign goal.

Campaign Exchange Ratios (x100)

90- and 75-percent chances (side-by-side) that friendly first attacks are lethal

75-percent chance that enemy first attacks are lethal

Table 4 shows substantial reductions in campaign effectiveness should the chances for lethal first attacks by friendly units fall from 90- to 75- percent. For example, even if the probability of initiating first attacks is 90-percent, the less-lethal friendly units cannot meet their campaign goal when the enemy has a 17-percent chance of a lethal counterattack, but the more capable friendly force easily can. The same is true whatever the chances that friendly units make the first attack.

If their lethality in first attack falls from 90- to 75-percent, friendly units must increase their chances for initiating attacks or reduce the chance of a lethal enemy counterattack by less than five percentage points to maintain a three-to-one campaign exchange ratio. A failure to follow up high probabilities of first attack with a high probability of a kill risks defeat when enemy units have moderate chances for a lethal counterattack.

**BOTTOM LINE**

**If a friendly force is numerically over matched three-to-one or worse in a search-and-destroy type operation, then it must initiate and convert first attacks with probabilities both better than 80-percent and, at the same time, hold the risk of a lethal enemy counterattack below 20-percent in the most favorable case and perhaps 10-percent in modestly unfavorable cases.**

These bounds can be relaxed by some five percentage points if enemy lethality in first attack falls from 90-to 75-percent or tightened by the same five percentage points if the lethality of friendly units in first attack falls from 90- to 75-percent.

**Whether it is feasible in practice to achieve winning combinations of such tightly constrained and demanding unit functional capabilities for combat certainly depends upon the kind of search-and-destroy operation and the comparative technical and operational strengths of each side.** However, if the opposing sides are roughly comparable in capability, then the prospects for victory by the smaller force most likely are dim.

**In search-and-destroy operations against a responsive enemy, it is asking a lot for quality to overcome a serious quantitative inferiority. **

Interested readers can have a short development of the formula for the campaign exchange ratio by contacting me at hank.young@verizon.net.

**CAMPAIGN GOALS AND UNIT CAPABILITIES**

The text describes the combat capabilities of each friendly unit by the probabilities of: (I) attacking first upon contact, p; (2) destroying the enemy unit on first attack, f1; and, (3) reacting to an enemy first attack with a lethal counterattack, r1, With the same interpretations, the independent probabilities 1 – p, f2, and Ti characterize the combat capabilities of enemy units.

Depending upon which side attacks first, the independent probabilities for clean wins, clean losses, mutual losses, and a possible no-Oecision, are the following (overbars denote complementary probabilities, i.e. a= 1-a ):

From the standpoint of the friendly side, the total probability of a clean win on contact is the sum of a clean win on first attack and an enemy clean loss on its first attack or w1 + 12 . Similarly, the total probability of a friendly clean loss is the sum of a clean loss on a first attack and an enemy clean win on its first attack or, 1 1 + w2. The total probability of a mutual loss on contact is the sum of mutual losses, ai + a2 , and the total probability of a no-decision on contact is the sum of no decisions, m1 + m2 As seen by the friendly side, these probabilities are:

By definition, the campaign exchange ration, E, is the fraction of enemy units destroyed, w +a, per fraction of friendly units lost, 1 T a, in all encounters, or E = w+a/1+a.

Adding the expression for w and a- taking into account that Pf1 r2 + pf1r2 “”p/1 and pf 211 + Pf21’JPrl. because the uncommon.

factors in two products sum to one-w + a – pfi + Pl. Adding the expressions for I and a (making the same kinds of simplifications as before), 1 +a= fJT2 + pfi. Finally, by substitution,

E = fp1+fp2/fT1+fT2 as quoted in the article.