Statistics questions

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Statistics questions

Postby Monty » 28 Mar 2008, 03:04

i'm going crazy with those statistics ;)
pascal, can you please explain (again) 2 statistics to me:

- (army) costs estimated
- (army) ratio (100,000$)

what are they trying to tell me and how are they calculated?
please give the units of measurement, too, because i was irritated when i tried to understand your formula:

Kroah wrote:Something to analyse, is the ‘army ratio ($100.000)’ statistics. It shows for the whole game the ratio between the kills and losses according to the army cost: (kills/losses*100)/(cost/100,000) (without taking account vanished armies). Needless to say, the highest, the better. The result is strange: among several games, Russia always had the best ratio (~45) and England the worst (~15). It seems the bad factors doesn’t handicap Russia too much according to the price.

what i don't understand about it is that the result (e.g. ~45 for russia) seems to be a percentage.
but how did you get a percentage out of that formula?

the numerator is clear: (kills/losses*100) results in % (120% in my latest game for JAPAN)
but the denominator...: (cost/100,000) would mean in detail [for JAPAN]: (300,000$/10,000men)/100,000
or do you simply meant cost per 100,000 men? then it would be 300,000$ * 10 / 10,000 men * 10
(because 300,000$ is the price for 10,000 men...)

but in any case, the result wouldn't be a percentage. in the first case it would be:
120%/(30$/men)*100,000 = 400,000%*men/$ = 4,000 men/$

in the second case it would be:
120%/3,000,000$/100,000men = 120%*100,000men/3,000,000$ = 120%*men/30$ = 0,04 men/$ (= 4,000 men/100,000$ if that's what it should tell)

but it's all very confusing to me and i don't know what it wants to tell me :shock:....help needed!
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Postby Kroah » 29 Mar 2008, 21:55

In this example, i'look at the army ratio of a random game between Germany and Japan.

Code: Select all
Germany
Kills     1684 (overall = attack + defense)
  Attack  1380 (when attacking)
  Defense  304 (when defending)
Losses    3179 (overall = attack + defense)
  Attack  2890 (when attacking)
  Defense  289 (when defending)

Ratio = (overall kills) / (overrall losses)
      = 1684 / 3179
      = 0.53


Code: Select all
Japan
Kills     1657 (overall = attack + defense)
  Attack  1190 (when attacking)
  Defense  467 (when defending)
Losses    1991 (overall = attack + defense)
  Attack  1690 (when attacking)
  Defense  301 (when defending)

Ratio = (overall kills) / (overrall losses)
      = 1657 / 1991
      = 0.83


Here, i want to compare Germany ratio with Japan ratio, taking into account the cost of the army.

If 2 ratios are equals, both army are equals if they have the same cost (the ratio is directly dependent of the power factors). If a major has a ratio of 1.00 (Germany) and another of 0.50 (Russia), both armies are equals because russian army cost half german army. So, to compare 2 ratios, they must be aligned to the same base cost. That's why "ratio/$100,000" is equal to the "ratio / cost".

To keep the ratio into a good scale, only the hundred of the cost is kept (Russia = 2) and the ratio is premultiplied by 100. We can multiple or divide ratios by any number if the same thing are done to all ratios: they keep their relatioship (x = a*y -> b*x = b*a*y where x and y are ratios, b the constant to keep good scale and a the relationship between both ratios). Example: 2 ratios 1.00 and 0.50 (1.00 = 2*0.50) have the same relationship to multiplied ones (by 100): 100 and 50 (100*1.00 = 100*2*0.50).

Returning to our main example, steps are the following:
1) multiplying all ratios by 100 (only in the statistics graphs):
Germany: 53
Japan: 83
2) dividing all ratios by their respecting cost:
Germany: 53 / 4 = 13
Japan: 83 / 3 = 28

Here, it seems the german army is less efficience than the japanese one.
Whether the german army should cost half its cost, or the japan one should cost twice its cost.

Now comes the estimated cost: I take the highest "ratio/$100,000" and set it to the reference:
Reference "ratio/$100,000": Japan (=28 )
Code: Select all
$300,000 -> 28
       x -> 13

x = 13*300,000/28 ~= 139,286
Estimated cost of Germany is $140,000.

Of course, this is an average, one game differs from another. Here, the sample is to low to conclude Germany is inefficient compared to Japan. But on lots of battles and games, it gives good average of army efficiency.

PS: Any changes done to combat rules can involve big changes to army efficiencies.
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Postby Monty » 30 Mar 2008, 11:00

hi pascal!

you are right when you say that the values stay comparable to each other if every operation is done to each of them.
but: the meaning of the result is of course affected.

so, if you have the value of 53 for GERMANY i know that it means "kills per losses" (for every 2 lost men GERMANY kill 1 man).
but when you divide those ratios of every major by, let's say 20 apples, they are still comparable, but what is the new meaning of them?

and like the 20 apples, 400,000 is not a plain value, in detail it is 400,000$ per 10,000 men".

my problem of understanding was that i didn't know what your result of 13 meant.
and that was because imho you give it with the wrong unit. you give 13 as a percentage, but its unit is men/$.
my confusion was enhanced by the fact that your value seems to be wrong by the factor of 1000.

explanation:
you have the ratio of 0,53 and divide it by 400,000$/10,000men (= 40$/men)
-> 0,53/(40$/men) = 0,53men/40$ = 0,01325 men/$
now you can multiply numerator and denominator with 100,000 and get
1325 men per 100,000$

this would be the ratio with respect to the cost. but what is the meaning?
it means that for every 100,000$ that GERMANY are investing in their army they can kill 1325 enemies.
that is the sentence that stupid me needs to fully understand what that value means.:roll:

JAPAN would then have a ratio/cost of 2767 men per 100,000$

apart from the wrong dimension your values are right because your way of calculation is of course correct.

to show the wrong dimension in your calculation i will "factor out" the units. in the end we have to get to 53/4 (your formula):

Code: Select all
source is 0,53/(400,000/10,000)    [(1/($/men)) = (men/$)]
= 0,53/40 [men/$]    /multiply with 10
= 0,53/4  [10men/$]  /multiply with 100
= 53/4    [1000men/$]
= 13      [1000men/$]


estimated costs:
Kroah wrote:Now comes the estimated cost: I take the highest "ratio/$100,000" and set it to the reference:
Reference "ratio/$100,000": Japan (=28 )
Code: Select all
$300,000 -> 28
       x -> 13

x = 13*300,000/28 ~= 139,286
Estimated cost of Germany is $140,000.

well, i still don't understand the meaning of that value. you know, again i need an understandable sentence ;)

you say that an input of 300,000$ brings JAPAN 28 whatevers (according to my theory it should be 2767 killed enemies)
but it also should be 100,000$ which bring JAPAN 2767 killed enemies, shouldn't it?
and it's again 100,000$ which bring GERMANY 1325 killed enemies

now the only reasonable question would be (imho): how many $ has GERMANY to spend to kill as many enemies as the ratio-leader (JAPAN) kills with 100,000$?

that is not a very spectacular question, because it just sets the previously calculated ratio/cost-values in relation.
it then would be:
Code: Select all
100,000$ -> 1325 killed men (GERMANY)
x        -> 2767 killed men (best ratio [JAPAN])
x = 208,830$ (or 835,320$ compared to the real costs of 400,000$ per 10,000 men [GERMANY])


but if i got it all completely wrong and you wanted to state something completely different, then what is the meaning of 140,000$? it seems to be cheap if you keep the bad ratio in mind...?

sorry for bugging the hell out of you...:oops:
Monty
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Postby Kroah » 30 Mar 2008, 22:15

Thank you very much for taking the time to give the good unit to this statistic. I admit i haven't took this time before.

Monty wrote:explanation:
you have the ratio of 0,53 and divide it by 400,000$/10,000men (= 40$/men)
-> 0,53/(40$/men) = 0,53men/40$ = 0,01325 men/$
now you can multiply numerator and denominator with 100,000 and get
1325 men per 100,000$

this would be the ratio with respect to the cost. but what is the meaning?
it means that for every 100,000$ that GERMANY are investing in their army they can kill 1325 enemies.

Now that you gave me the good unit, i've update it's name and scale in the upcoming release.

Monty wrote:you say that an input of 300,000$ brings JAPAN 28 whatevers (according to my theory it should be 2767 killed enemies)
but it also should be 100,000$ which bring JAPAN 2767 killed enemies, shouldn't it?
and it's again 100,000$ which bring GERMANY 1325 killed enemies

Exactly, i've done a mistake (because i haven't the unit). It's 100,000$ for 2767 kills (and not 300,000$).

Monty wrote:now the only reasonable question would be (imho): how many $ has GERMANY to spend to kill as many enemies as the ratio-leader (JAPAN) kills with 100,000$?

The meaning of the statistics i wanted to have was: "How much should cost my army if i had the ratio of Japan (best ratio)".

Code: Select all
100,000$ -> 1325 killed men (GERMANY)
100,000$ -> 2767 killed men (best ratio [JAPAN])

Theory: 400,000$ -> 11068 killed men (2767*4) (best ratio)
Real:  (400,000$)->  5300 killed men (actual ratio)
Estim.:       x$ ->  5300 killed men (projected ratio)
x = 5300 * 400,000 / 11068 = 191,543$

German army should cost 191,543$ to kill the same amount of men if his army was as efficient.

Do you agree with me ?
My mistake was to use "300,000$ -> 11068 killed men" instead of 400,000$

Thank you again,
Pascal
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Postby Monty » 31 Mar 2008, 08:06

i think you're right.
i'm always a bit confused about it because for every major the kills per 100,000$ include different losses.
but after a long way i got the same result you got in a short way ;)

for 100,000$ JAPAN get 3333 men who can kill 2767 enemies (10000/3=3333 men).
for 100,000$ GERMANY get 2500 men who can kill 1325 enemies (10000/4=2500 men).

now, if GERMANY could kill 2767 enemies for 100,000$ (like JAPAN), they needed 5221 losses for that (because of their ratio) (2767/0,53=5221)
-> 100,000$/5221men = 2767 killed enemies (GERMANY)

now we need to get the costs per 10,000 men.
we already know JAPAN's costs are 300,000$ which is the max (best ratio)

to get the costs for GERMANY we have to multiply the 100,000$ with (10000/5221 = 1,91) = 191,000$/10000men (= your result)

so, the lower the value the weaker your army is...
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Postby Monty » 31 Mar 2008, 09:08

Monty wrote:we already know JAPAN's costs are 300,000$ which is the max (best ratio)

correct sentence has to be: we already know JAPAN's costs are 300,000$ which is the max for JAPAN (best ratio/costs)

of course GERMANY's estimated costs can be higher than 300,000$ because their max would be 400,000$

example:
if JAPAN had a ratio of 1,0 and GERMANY 1,1
then JAPAN still had the best ratio/costs = 3333 men (GERMANY = 2750 men)

JAPAN's estimated costs would be still 300,000$ while
GERMANY's estimated costs would be 330,033$

(just to make it clear for myself)
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Postby Kroah » 31 Mar 2008, 13:51

Glad to see you got the same result.

Concerning vanished troops not included in the calcul of the ratio and estimated cost:

Actually, only kills/losses from the "damage" done by the battle system are used. Others are put in the vanished category (retreat, no home, ...).
As i said you, i did this to not skew the result by only keeping kills/losses done by the efficiency of the troops (power factor and cost).

Can you give me your point of view? Should the vanished army (the whole or part of it) be included in the real kills/losses? This may give big differencies in the result for the ratio and estimated cost, but which one is closer to reality?

Thank you,
Pascal
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Postby Monty » 31 Mar 2008, 14:52

i already thought of including it into the calculation when i tried to find out why RUSSIA always had (one of) the best ratio/costs. i thought it must have something to do with not including the vanished troups into the calculation because normally RUSSIA can't have the best ratio/cost (personal guess, i have to admit).

but when i tested it back then, my assumption of the meaning of "vanished" was wrong (see the "vanished"-thread), and therefore i thought it would contain the troups that were forced to vanish by my army. so i just added that amount to the kills(attack), but it didn't change significantly the ratio/cost-result. i then left that track for the time being.

now, to check it again in the right way, i think we needed a categorisation of "vanished"-types to do that. at least the amount of men we have lost through vanishing (already existing) and the men we forced into vanishing are needed.

but i don't know if it's the riddle's solution, we will just have to prove it.

on the other hand i'm still thinking about an explanation for a significant rising of RUSSIA's ratio/cost-graph at one specific point that caught my attention after a complete game with JAPAN against all CPUs. it seemed that it dramatically rose when i finally started to attack them, but i can't interpret it so far, because i killed them very quickly i think...(and at the same time i started to attack FRANCE and ENGLAND, too, and it did nothing special to their graph, if i remember correctly)

best regards,
monty
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