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Subject: Fuzzy-Wuzzy valuation theory ? rss

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Mircea Pauca
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Anyone here has and can describe and comment the Fuzzy-Wuzzy theory article in The General ?
The only reference I've got is from this Russian academic site (?!):
http://dic.academic.ru/dic.nsf/enwiki/1194920

"Fuzzy Wuzzy Fallacy

The Fuzzy Wuzzy Fallacy is a name for a wargaming theory coined by Richard Hamblen in the September 1976 edition of Avalon Hill's "The General Magazine", loosely based on historical records of battles between the British and the Sudanese Mahdi. The Fuzzy Wuzzy Fallacy states that a single soldier with 2× firepower or attack strength is not equal to two soldiers with 1× firepower or attack strength. Instead, the soldier with 2× firepower is actually worth sqrt{2} of the 1× soldier, if either soldier can be killed in a single hit. This is another form of Lanchester's law.

As a result, tactics and strategy designed around this theory emphasize greater numbers and time, which the speed and mobility of the units in action can effect."


Then, it's very similar with my article based on Lanchester theory. It showed an 117 cruiser is really worth 1/2, not 1/4 of a 44x battleship.
The WAS article I sent to Nick Markevich's WAS Ladder site was deleted some time ago (I'll repost a revision here on BGG) but my similar article for Victory in the Pacific is still on John Pack's VITP Ladder site:
http://www.gameaholics.com/vitp_articles/vitp_unit_values.ht...
These were well tested in many games and situations.

Then, my lower-level (ship-to-ship) analysis based on Markov chains:
http://www.gameaholics.com/vitp_articles/markov/battles_of_n...
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Carl Olson
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BASIC PROBABILITY FOR WAR AT SEA (at sea with the Fuzzy-Wuzzy formula)

This is a brief synopsis of an article by that name written by Richard Hamblen for Vol 13 Number 3 of The General in Sept-Oct 1976.

I have not yet found the original "Fuzzy-Wuzzy" article he mentions from several years earlier, but it discussed the fact that doubling a unit's firepower does not double it's effectiveness. A target that can be killed by one bullet/die roll/etc. may be hit with several simultaneous shots.

Through a series of mathematical formulae, Richard arrived at the formula that the value of a unit was equal to the square root of it's combat value. Computing the combat value can be complicated, but the relationship holds. The defense value also holds to the square root formula.

Two sets of numbers were calculated for each ship: the chance of a ship winning the battle (which includes "Disabled" results) and the chance to sink ships, which is purely damage done. Richard allows there were necessarily some small fudge factors, but the results are fairly close to the expected results. [ed. They seem to have worked for 33 years]

The result was the following two tables. Note that Battle Points (BP) and Killing Points (KP) CANNOT be meaningfully compared to each other. You'll probably have to widen your window somewhat to see the correct formatting.

CHART A.

Battle Killing Battle Killing
Ship Factors Points Points Ship Factors Points Points

GERMAN FLEET ITALIAN FLEET
4-9-6 21 (17) 29 (21) 4-6-6 16 17
3-5-7 16 (13) 16 (11.5) 4-3-5 14 12
2-2-5 13 (10) 5 (3.5) 1-1-7 7 2
1-2-7 10 (8) 3.5 (2.5)
1-2-8 10(8) 3.5 (2.5) AND -8 to enemy
NOTES: The values in parentheses are for Axis ships WITHOUT the +1 fire bonus.


Battle Killing Battle Killing
Ship Factors Points Points Ship Factors Points Points

BRITISH FLEET AMERICAN FLEET
5-5-3 16 19 5-5-4 16 19
4-5-6 15 16 4-4-4 14 14
4-4-x 14 14 1-1-7 7 2
3-3-6 13 9
1-1-7 7 2 RUSSIAN FLEET
1-2-4 8 2.5 3-3-3 13 9
1-3-3 8 3

NOTES
All Air and U-Boat Factors are -4 per BP factor to enemy
Each Allied ASW point is +1 toward canceling U-Boat points (only)
Air and U-Boat factors only affect BP.



PROCEDURE
Total up the BP and KP for the area.

Add the total of these two factors to the totals for the area.

SPECIAL: If the U-Boats will fire on aircraft carriers, then for every -4
U-Boat points remaining, one carrier in the zone will be worth
only -3 per Air Factor, instead of -4



FOR EXAMPLE:
The Allies have two 3-3-6's and the 4-4-7 in the area, plus a carrier with 2 Air Factors. The Germans have the two 3-5-7's and 3 U-Boats available.

Allied German
Surface 40 BP, 32 KP 32 BP, 26 KP
Air -8 0
UB +6 -12 NET -6

A) If the German applies the UB to the surface ships, the Allies have
only 34 BP (40 - 6).
The Germans have only 32 - 8 = 24 BP.

B) If the 4 of the 6 net UB are used against the carrier, it subtracts only 6 from
the German total (3 per factor) leaving 26 BP.
The Allied BP are 38 (40 minus the 2 remaining net UB points)

The German is slightly better off by attacking the surface ships (-10 versus -12) but will probably lose the battle either way.
Adding a 2-2-5 (13 BP) to the German side makes it a pretty even battle.



CONCLUSIONS
I have left out charts B and C, which show the various totals by turn, and have summarized Richard's conclusions:

1. The Allies should arrange the forces such that they have a clear BP or KP advantage in every area. You probably cannot do both.

2. The Allies cannot gain BP superiority in 3 areas, except in Turn 1. However, the British have 3.7 times the KP of the Germans on turn 1. The ratio of KP goes down, but never drops below 2.4, except on turn 3, when it is 2.2. Then it remains at 2.6-2.7 for turns 5-8.

3. The Allies need a war of Attrition. No startling news here.

4. For the Germans, if you are facing a BP group, you must have a secure port. If you lose a battle in the Soiuth Atlantic and do not control the areas, the ships in the neutral port are toast. If you are facing a KP group, make sure you have a healthy BP advantage.

5. The loser of a battle will generally lose more ships than predicted, because the stragglers will be ganged up on, and they will probably have lost their bonus, if any, and may even be reduced to a single attack die.


Finally, Richard closes with this quote:
"Oh, yeah, and all of this discussion is based on an average performance; as I'm sure we all know, War At Sea is a game where a few die rolls can make a BIG difference."
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Mircea Pauca
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Many thanks !
These 'Battle points' are very close to my 'Control' points (article here on BGG:) - after scaling.
http://boardgamegeek.com/thread/403343
Mine are slightly easier to use because the common 44x is 10p.

For the Killing points, I assume these include the ratio between sinking a standard enemy and not being sunk itself; else it couldn't be a ratio of 29:5 between Bismarck 4+96 and the 2+25 pocket battleships.

I still wonder how exactly were these calculated. Was a standard enemy assumed, or a mix of all possible enemies ?
 
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Carl Olson
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ROMagister wrote:
Many thanks !
These 'Battle points' are very close to my 'Control' points (article here on BGG - after scaling.
http://boardgamegeek.com/thread/403343
Mine are slightly easier to use because the common 44x is 10p.

For the Killing points, I assume these include the ratio between sinking a standard enemy and not being sunk itself; else it couldn't be a ratio of 29:5 between Bismarck 4+96 and the 2+25 pocket battleships.

I still wonder how exactly were these calculated. Was a standard enemy assumed, or a mix of all possible enemies ?


The enemy is irrelevant, since the values only depend on the amount of damage the ship can deal out before, on the average, suffering the damage it can take. The German ships have two values, depending upon whether they have their +1 bonus or not, so the Allied ships will do worse against the undamaged German ships.

Also, 1 was added to the ship defense before the square root was taken for calculating defense, since it requires 5 hits to sink a defense value of 4.

Richard admits that using the square root is not exactly the same as using the stepwise reduction method, but it is pretty close.
 
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Mircea Pauca
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The enemy *is* relevant because too much fire just overkills a small target.
That's why Bismarck's 4+ is NOT 4x as useful as four 1+ cruisers, even as raw firepower (not even having 4 different hulls, which helps again).

How exactly does work the 'stepwise reduction method' ?

I also did some time ago a spreadsheet with a more sophisticated approach, based on Computable General Equilibrium. This takes into account all possible states of damage a given unit may reduce to (e.g. a 44x can have 1, 2, 3 or 4 damage) and their further values.
These values are so defined that expected reductions in enemy value are proportional to own value - the same principle Lanchester's equations were based on.
This results in a *massive* system of equations linking all values together, and gives relative values (or prices) compared to one reference. Excel's Solver found a solution by successive approximations.

If I'd not be that shy/lazy on those matters between 'officialdom' and hobby, I could publish it at an Operations Research or related symposium ;-)
 
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Carl Olson
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ROMagister wrote:
The enemy *is* relevant because too much fire just overkills a small target.
That's why Bismarck's 4+ is NOT 4x as useful as four 1+ cruisers, even as raw firepower (not even having 4 different hulls, which helps again).

How exactly does work the 'stepwise reduction method' ?



Richard did reduce the damage done by each ship to take this factor into account, but he doesn't say exactly how he did it. In a one-on-one battle it would matter a lot more. Given that these are general-case values, he probably assumed/calculated the average defense factor to be about a 3. He does give the values he used for the attacks, which are: combat factor 1= firepower 1, cf2 = fp 1.8, cf3 = fp 2.5, cf4 = 3.1, cf5 = fp 3.8.

I think what he means by "stepwise" is that the expected damage is not technically the square root of the attack factor times the average damage die roll. To be absolutely (and painfully) correct, you would have to calculate the steps (i.e. each face of each die individually) because the values are discrete.

It is also unclear from the article whether the U-Boat and Air factors apply to the Killing Points totals. It seems that they do not, but yet they should. The ASW and UB values are complicated because they only fire once, and some of the air factors may not get used.

Unfortunately, Richard passed away several years ago, so we can't ask him.
 
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Carl Olson
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ROMagister wrote:

I also did some time ago a spreadsheet with a more sophisticated approach, based on Computable General Equilibrium. This takes into account all possible states of damage a given unit may reduce to (e.g. a 44x can have 1, 2, 3 or 4 damage) and their further values.
These values are so defined that expected reductions in enemy value are proportional to own value - the same principle Lanchester's equations were based on.



How did you account for the combat factor going to 1 when hits = defense value? The combat value doesn't change with every point of damage.
 
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Mircea Pauca
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carlj wrote:
How did you account for the combat factor going to 1 when hits = defense value? The combat value doesn't change with every point of damage.


Actually they are different types of objects that can transform into the lower ones. (e.g. 43x taking 2 damage becomes 41x)
44x, 43x, 42x, 41x use the exact firing probabilities of 4 dice.
10x uses the exact firing probabilities of 1 die.

Solutions for values are:
44x = 10 (control points) as reference
43x = 9.75
42x = 9.53
41x = 9.33
10x = 4.90

The reference enemy for both sides is a 44x.
 
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Steve Sabean
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The article in The General that Hamblen was referring to was in vol. 6, no. 4. It was entitled "Fire Power, the Fair Fight, the Fuzzy Wuzzy Fallacy," by William J. M. Gilbert.

To me, the most interesting part of Hamblen's article is the quote that you mention. It would be interesting, say, to do an analysis of the variance of an Axis fuzzy-wuzzy based attack against an Allied Bo1 strategy, for example. It would be easy enough to do. The variance of a binomial distribution is easy to calculate.
 
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