Nuclear Warfare I – Strategy Of Assured Destruction

On 11 February 2010, the newspapers worldwide reported a newsflash. An airplane based LASER had brought down a missile fired far away. It raised some eyebrows and provided an item for cheesy geek discussions around water coolers. But for the most part the world missed the vision behind this weapon and the potential of this weapon to begin a new era in strategic warfare worldwide.

To understand the importance of that event, we have to travel back in time for about 65 years.  On July 16, 1945, the words of Bhagwad Gita reverberated through Robert Openheimer’s mind  “Shining light of thousand suns, I span the space between the sky and the earth. Then I become death, destroyer of this universe.” He was watching the first nuclear test, Trinity test, in the deserts of New Mexico and he and his fellow scientists were in disbelief as they witnessed the lethal power of the weapon they had just created. A new age started on earth. Atomic age.

Atomic weapons are strategic weapons. They have far reaching impact on war. The warfare using atomic weapons is very much a game of chess, where brain is as important as might. The missiles, bombers and submarines are new knights, rooks and queens of this game.

To understand the nuclear strategy, let us analyze that as a very simple two player, two choice game between two countries, America and Russia. Each country has a strategic objective, to defend themselves from the nuclear attack from the other country. Each country does that by guaranteeing the other country a complete reciprocal nuclear wipeout, termed Mutually Assured Destruction, in case the nuclear war is started by another country.

Each country has two choices, whether to attack another country, or to remain quiet.

Let us say NA means no attack, A means America, R means Russia. Then the game is described in a table as follows.

A-NA, R-NA	A-A, R-NA
A-NA, R-A	A-A, R-A

America will try to deny the third choice (America-No Attack, Russia-Attack) to Russia by ensuring that if Russia attacks using nuclear weapons, America will retain enough nuclear weapons to launch a second strike and destroy Russia. Russia will do the same to America. So as long as each country guarantees the other forcing to fourth choice, the world always remains in first choice. So a very simplistic Nash equilibrium of this game exists at A-NA, R-NA.

What is meant by destroying a country? Let us say to make sure that country suffers maximum damage, we drop nuclear bombs all over. Let us try to put it in equation.

Total area of  USA is 10 million sq km. Say one nuclear bomb destroys 50 km x 50 km, that is 2500 sq km. That means to destroy 10 mil sq km, Russia will need 10,000,000 / 2500 = 4000 bombs.

But if these bombs were being launched from missiles, with the probability of the missile hitting the target being 0.5, you would need twice as much bombs to achieve the strategic objective of nuclear destruction. So Russia needs 8000 bombs.

But if America strikes first, not all bombs will survive the destruction. So to be able to launch attack after first attack by your enemy, you need more weapons. Say the probability of survival of your nuclear weapon was 0.5, then now you need 16000 nuclear weapons to assure guaranteed reciprocal nuclear destruction of America.

In reality one nuclear power will not plan to carpet bomb the other power across the entire nation. Things will more complex and will be planned differently. But we are just creating a simple mathematical model here.

Lets try to put it in equation. Say each country prepares for two attacks. First attack is your plan to destroy them before they hit you, either because you think you are invincible or because you know for sure they are launching nuclear attack soon. Second attack is when you have been caught off guard and suffered heavy destruction and now you are all out to teach a lesson to your enemy.

Let’s assume the following,

TAa = total area of USA

N1a, N2a  = no. of first strike and second strike weapons with USA.

PS1a, PS2a = probability of survival of  first strike and second strike American nuclear arsenal in case of enemy attack.

BAa = area destroyed by one American bomb.

PH1a, PH2a = probability of successful target strike of first and second strike weapons of USA.

So now we have an equation for the strategic objective.

Strategic objective for USA = SOa = have enough capability to destroy Russia in first strike as well as second strike.

Super simplified SO:  area destroyed by one bomb x number of bombs > total area of enemy. Ensure this much and your enemy will be scared of you.

Considering probabilities, SO:  (probability of survival of a bomb) x (probability of a bomb hitting the targt)  x (area destroyed by one bomb) x (number of bombs) > (total area of enemy).

This is strategic objective of any one strike.

SO for USA , SOa:  SOa1 (first strike strategic objective) & SOa2 (same for second). Note the use of “AND” boolean operation and not “OR”

SOa = SOa1 & SOa2

SOa1 = (N1a x PS1a x PH1a x BAa > TAr )

SOa2= ( N2a x PS2a x PH2a x BAa > TAr)

We could obtain strategic objective of Russia, SOr by interchanging prefix “a” with prefix “r”.

One country will not attack another if it knows for sure that the second country is meeting its strategic objective. So putting it back in game table again.

SOa = true & SOr = true, so R-NA & A-NA	SOa = true & SOr = false, so A-A, R-NA
SOa = false& SOr = true, so R-A & A-NA	SOa = false& SOr = false, so R-A & A-A

What the above table shows us is that the probability of small scale nuclear war between USA and Russia was higher if each country had only a limited nuclear weapons. But the world remained peaceful because both powers had enough nuclear weapons to wipe out the other one.

Let’s examine the strategic objectives equations again.

SOa1 : N1a x PS1a x PH1a x BAa > TAr

SOa2: N2a x PS2a x PH2a x BAa > TAr

SOa = SOa1 & SOa2

SOa = (N1a x PS1a x PH1a x BAa > TAr)  & (N2a x PS2a x PH2a x BAa > TAr)

We make two interesting observations here.

1. As we see, if you do not have enough control over PS, probability of survival of your weapons, the only strategy for you to ensure SO = true is to increase N, the total number of weapons you have. Thus number of nuclear weapons worldwide increased drastically throughout sixties and seventies.
2. The increased number N gives you strategic advantage, but only to certain extent. After you have comfortably passed the condition, with safety margin, there is not much strategic advantage by building more nuclear weapons. So the making of nuclear weapons peaked in eighties and nineties. Not because suddenly the powers wanted peace, just because there was not much strategic gain.
3. If you are not yet attacked, you have all your weapons, so probability of survival is 1. Thus you need less weapons for first attack objective. But probability of survival of your weapons is lot less if you are attacked, which is the case for second attack. What it means then is, every country allocated far more weapons for second attack capability than first attack.
4. Also doubling the probability of survival means you need half the weapons. It is a factor with high sensitivity to strategic objective.

It is interesting that even a simple mathematical model can show us so many things. It highlights the factors of high sensitivity. It can spot and explain trends, like we saw this model explained the trend in number of nuclear weapons. It can drastically improve the quality of your guesses.

This post completes the analysis of first phase of nuclear weapons race. In second post we will some interesting technological developments that happened in parallel with nuclear race.