A lot of good points above. I have incorporated as much of that as possible in my next post. Riders: this applies to the basic combat system, LL. The following may be old hat, or not.

How many knives should you bring to a gunfight? Obviously, if your army has exactly the same characteristics as the enemy army, then they have the same combat power. But in general, having more weaker units can compensate for a deficit of stronger units, and vice-versa.

The old hands know that there is an Army Power Function 2*U+C that works quite well, where U is the number of hit points and C is the total combat strength. Of course, this must be an approximation, because the combat strength is applied in each combat round, while the hit points count only once, so there must be at least one more factor in the function. However, if the combat lasts 3 rounds or less, then you should be able to calculate the result exactly, while hardly any combat lasts longer than, say, 6 rounds. The function may be analytical (based on analysis) or empirical (based on observations), but it turns out that it is exactly equal to a good, simple analytical function, up to a scaling factor, provided that there are 6 rounds of combat.

Initial considerations indicate that we should look for an Army Power Function that has the form

P = P(U,C,N) ,

where U is the number of hit points, C is the total combat strength, and N is the number of rounds of combat. We try

P = U + N*(f*C/6) ,

noting that it takes a combat strength of 6 to inflict each hit point.

N = U'/(f*C/6), where U' is the number of units in the enemy army, and f is a factor to be determined.

It quickly becomes apparent that there are unexpected and undesirable features to this trial function. According to this function, the power of your army increases when attacking stronger armies, because the number of rounds of combat increases! Indeed, the power is equal to the total number of units in both armies! In general, the Army Power Function depends on the exact nature of the enemy army. Therefore, we abandon the attempt to find a general function that calculates the power of an army, and instead seek only a function that calculates the power of an army *at balance*, that is, when your army is *as good as* the enemy army. Having that, then we can add a couple of units or make a couple of units stronger, because it is known from experience that that increases the probability of winning a battle from 50% to about 95%. (That applies to Low Luck conditions.)

It should be clear that the factor f cannot be unity, because the total combat strength of an army drops each round, unless it is padded with 0-strength units. The rate at which the combat strength drops is nonlinear, because the rate at which the enemy army inflicts casualties decreases each round, unless it is padded.

In order to keep the solution simple but general, we linearize the problem by assuming that the combat strength of each army at balance drops linearly from its maximum value to zero. The expression f*C is equal to the average combat strength of the army, and for this linear decrease, it is equal to half the maximum strength. Thus f = 1/2.

The practical problem at hand is what we need to have in order to equal a specific army, i.e. how many knives do we need to bring to this gunfight. Thus in general, U' and C' will be known, and the average combat strength of your units C/U will be roughly known, so the problem reduces to determining U.

N = U'/(C/12) = U/(C'/12) , so

U/U' = C'/C = (U'*C'/U')/(U*C/U) , so

(U/U')^2 = (C'/U')/(C/U) , so

U/U' = SQRT(c'/c) ,

where c and c' are the average unit strengths of the respective armies.

Let's look at one example to see how this works. The enemy has 10x4. How many 1s do you need to equal this? U/U' = SQRT(4/1) = 2 . Bring 20 units for balance, and 2 more for the win. The power of these armies is P = U + U' = 30.

The number of combat rounds is N = 12x10/20 = 12x20/40 = 6. This is an approximation. Direct calculation shows that each army kills exactly 1/3 of the opposing army in the first combat round and 2/3 of that in the second combat round, for a total of 5/9 in the first 2 rounds. Thus 10 units are reduced to 4.4 in 2 rounds and 2 in 4 rounds and .9 in 6 rounds, so the true number of rounds is about 8.

The army power can also be expressed as P = U + 6xC/12 = (2*U + C)/2 , which is the same as the legacy function, except for the constant scaling factor. Although the legacy function is less accurate, it is easier to use, and does not involve fractions.

Let's see how well the legacy function predicts the number of units you need for balance. The legacy function has 2*U+C = 2*U'+C' , so U*(2+c) = U'*(2+c') , so U/U' = (2+c')/(2+c) . The reader can verify that, for the smallest quality mismatch c=c', U=U' for both functions. Also, for the largest likely mismatch c=1 and c'=4, U/U' = 2 for both functions. For some other cases such as c=1.5 and c'=3, the results are almost identical. Otherwise, the legacy function gives a slightly inaccurate result.