Insufficient rules, more attacks. -Sun Tzu's Art of War

Not enough to defend, but enough to attack? Isn't that weird? This sentence has puzzled countless sages for thousands of years, and some even simply changed it to "more rules and less attacks." However, in my opinion, this sentence is not wrong. In fact, in the case of "insufficient defense", it is indeed possible to "attack everything." Don't believe it? Let's look at the victim's analysis.

Even if the attacker is short of troops, he will not necessarily lose.

There used to be such a topic in the "Game Theory" course of Princeton University in the United States:

If you are given two divisions, you can break through the city defended by three enemy divisions. It is stipulated that both sides can only mobilize troops for the whole division. There are always two roads to this city: A and B. When you attack, if you are stronger than the enemy, then you win. If your strength is less than or equal to the enemy, then you will be defeated. How will you make a siege plan?

Maybe you will think this is an "impossible task": even if the troops are equal, you will still be defeated, not to mention that the other side has a full 50% more troops than you. However, this situation is relatively real. In actual combat, the defensive side does have more supplies than the offensive side. Moreover, if all kinds of fortifications are used again, as long as the difference between the scientific and technological level and the quality of troops between the two sides is not too great, under the same strength, the defensive side's combat power is always higher than that of the offensive side.

But even so, the probability of both sides winning this battle is 50%. Don't believe it? Let's take a look at the possible strategic deployment of both sides.

There are four plans for the deployment of the enemy's three divisions:

A 3 divisions are all arranged on road A;

B 2 is divided and arranged in A, and 1 is divided and arranged in B;

C 1 is divided and arranged in A, and 2 is divided and arranged in B;

D 3 divisions are all arranged on road B.

Our army has the following three attack plans:

1, both divisions attack from road a.

The 2 nd and 2 nd divisions attacked from A and B respectively.

All divisions 3 and 2 attacked from road B.

Let's make a table according to these schemes:

If we assume that both sides are equally likely to adopt each scheme, then the probability of the attacker winning is 6/ 12 = 1/2, which is 50% as we said earlier.

Wait a minute! This is our conclusion under the assumption that the defensive side will equally adopt four strategies: A, B, C and D. However, if we look at the two schemes A and D, we will find that the attacker is more likely to win when adopting these two schemes. Further contrast, plan B is definitely better than plan A, and plan C is definitely better than plan D. So what is the reason for the defender to choose plans A and D? In other words, in fact, the defender has only two choices, B and C, and their possibilities are 50% each. Then in this case, the attacker's chances of winning are only 1/3.

But wait a minute! The defender is not a fool, but the attacker is an idiot? If the defender will only adopt two schemes, B and C, then the attacker will not adopt the failed scheme 2. So in fact, the attacker only has 1, three options to choose from, and their probability is also 50%. So we can get a new form:

The probability of both sides winning is still 50%. Isn't it a little incredible? The defenders have many troops and good operational conditions, but the chances of winning are equal.

In the case of equal strength, the attacker has a better chance of winning.

I'm sure readers will be curious when they read this. If the offensive and defensive forces are equal, what are their chances of winning? We might as well cut the garrison troops by one division, and then look at the garrison's strategy at this time:

A complete defensive armor of the 2nd Division.

B 2 division defends a and b separately

C 2 division fully defends B.

Attackers can still use the above three strategies: 1, 2, 3. Then the game between the two sides at this time is as follows:

It can be seen that no matter which strategy the defender adopts, the probability of the attacker winning is 2/3, so the total probability of winning is also 2/3. This result is very interesting. On the surface, it seems that the game that is beneficial to the defensive side is dominated by the offensive side. This just confirms the sentence: "offense is the best defense."

Why is offense the best defense?

But why? The answer may be summed up in six words in Jia Yi's book On Qin in the Han Dynasty: "The offensive and defensive trends are different." For the attackers, their purpose is just to "attack the city." In this game, no matter which way they get the advantage in A or B, they can achieve their goals. In contrast, the task of defenders is much more arduous. They must make arrangements for various offensive strategies. In this case, they must resist two attacks. In other words, the operational rules in a war may be beneficial to the defensive side, but they are partial to the offensive side in terms of victory conditions. "The Art of War" said: "If you can't win, keep it; But the winner can attack. " This sentence is just in front of the sentence "insufficient rules, more attacks", which just confirms our theory.

Let's take StarCraft II as an example: if you just want to stick to it, you must guard against all possible attacks. Opponents may use early blasting flow or late technical flow; It is possible to attack from the front, or it is possible to bypass the rear and drop ground troops. In this case, you should not only ensure that you have enough troops to deal with rush in the early stage, but also climb the technology to fight in the later stage; It is necessary to build ground combat units and fortifications as well as air defense. Without knowing the other side's strategy, with the same amount of resources, the attacker only needs to concentrate on building one unit, and you need to build almost all units and buildings! In the end, all this is in vain, just as Sun Tzu's Art of War said, "You can do anything if you are prepared."

It is worth mentioning that strong information gathering ability can pull back a game for the defender, just like playing StarCraft II. If we open the whole picture and know the enemy's production situation in detail, we can carry out targeted defense. However, information asymmetry game is not the content of this paper.

label

At this point, although it is not excluded that we misinterpreted Sun Tzu's original intention according to the thinking of science and engineering, after all, his old man's sentence "If there is not enough code, there will be enough attacks" has been well interpreted by die rationalists. Perhaps the best way to deal with "the world is attacked on three sides" is, as Shakespeare said in King John,

"Give them a head-on blow."

This paper explains why attack is the best defense from a specific angle with a simple but very novel calculation. Of course, the actual war is far from being comprehensively summarized by simple calculation in this paper. After all, the city of three miles and the country of seven miles are sometimes besieged and blocked.

Select the futures competition in the menu and sign up for the competition!