Archive for March, 2008

BATTLE OF TRAFALGAR

21 March 2008

This is what made Admiral Nelson great. It all came down to basic game theory.  It is a wonderful and beautiful thing.  I love this, and I hope you all enjoy it too.

Here’s the scene: The French Navy and the English Navy are sailing toward each other and there’s going to be a battle. Nelson is leading 40 English ships, but he counts 46 French ships in the distance.

His first calculation was as follows:

1. The Big Battle.

FRENCH: (46 . 1)2 = (46)2=2116

NELSON: (40 . 1)2 = (40)2=1600

2116-1600 =516

square root of 516 = 23.

Nelson’s first calculation shows that at the end of the battle, if they went head-on, all the English ships would be lost and there would be 23 French ship surviving.

Note that Nelson allocated a kill rate of one, and thought that the French ships were no better or worse than his (the kill rates cancel out).

Well, this was bad news for Nelson, he needed to do something.  He couldn’t get six or more ships, and he had no secret weapon to improve the kill rate by enough to match six ships.  He needed to divide and conquer, the question is in what proportion.  He did another calculation:

2. Three Battles to lose

Nelson splits the French in two exactly. He allocates 31 ships to fight one half, and the remaining 9 English to fight the other half.

BATTLE A
(9)2=81
(23)2=529
529-81 = 448
square root of 448 is 21.16 French
So all 8 of the English ships would be sunk and about 22 French ships survive Battle A.
meanwhile…

BATTLE B
(31)2=961
(23)2=529
1024-529= 432
square root of 432 is 20.78
So all of the 23 French ships would be sunk, and about 21 English ships would survive Battle B.

BATTLE C
After the first two battles, we have this one to see who wins. But this is no good as it is between 22 French and 21 English ships; the French are likely to win!

At this point Nelson realised that by dividing the French he has improved his chances, so he decided to try another calculation, but instead of the proportion being 9-31, he tried 8-32…

3. Three Battles to Even the Odds

BATTLE A
(8)2=64
(23)2=529
529-64 = 465
square root of 465 is 21.56
So all 8 of the English ships would be sunk and about 22 French ships would survive Battle A.
meanwhile…

BATTLE B
(32)2=1024
(23)2=529
1024-529= 495
square root of 495 is 22.25
So all of the 23 French ships would be sunk, and about 22 English ships would survive Battle B.

BATTLE C
This was an evens-Stevens match between 22 French and 22 English ships.

This was what Nelson had been looking for — he found a method of evening out the odds and making the battle more fair — it would be down to fate and the kill rate.

And that is how the Battle of Trafalgar was won.

Divide and Conquer (in the right proportion) will even things up when the odds seem to be stacked against you.

The only other variable is the kill rate.

Example of Kill Rate Calculation:

Three trained men fight 12 untrained men. What should their kill rate be to win? 2 or 4?

KILL RATE 2:

(3.2)2=62=36
(12.1)2=122=144
144-36 = 108
square root of 108 is 10.39 untrained survivors

KILL RATE 4:

(3.4)2=122=144
(12.1)2=122=144
144-144= equal chance of either group winning
so 3 trained men will win against 12 untrained men if their kill rate is more than 4.

There are valuable lessons to be learned from this. First, sailors or soldiers should not be told why and how, just given orders that they must blindly carry out.  Imagine how it would feel to be on one of the sacrificial ships, one of the eight — deliberately sent to your death for the bigger picture. Hence: “England expects every man to do his duty”.  You need blind obedience to win battles.

Second, that kill rate is very important in evening up odds.  You can see why a Gattling gun gave such an advantage over traditional guns. Martial Arts experts and other training gives an edge.  If you have more weapons, or more speed, the advantage is obvious.

A mixture of both is sometimes used to great advantage — divide and conquer can reduce your casualties if the kill rate is available. It’s obvious really, if you are going to fight a man, kill rate is the factor.  If you are fighting two men, it’s evens if you have double their kill rate.  If you don’t you have to divide and conquer.

I covered this on a course I did back in the early 1980s, and it has informed me ever since; I appraise outcomes of movies, review action novels, games and films in the light of the ideas and wee calculations.  You know the difference if you get the division wrong, in Nelson’s case, victory depended on a margin of error of just one ship. Oh, it could all have ended so differently.

It certainly has assisted me in understanding the world around me, so I hope you get as much out of it as I have.

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