I've come up with a formula to help with determining what number of armies your bonus must give.

B = (Bt * (Nt / Tt)) * X

B = The number of armies you want the bonus to give.
Bt = The amount of territories inside the bonus.
Nt = The amount of ways the bonus territories connect to territories outside of the bonus.
Tt = The total amount of territories on the map.
X = The army density, a number you can choose yourself, for lots of armies choose around 10, for very little around 5, you may pick it yourself.

I've tried it on my Europe map, and I encourage everyone to try it on their maps too to see if it has potential.

I like the concept, but how do you define connections? If one territory in the bonus connects with two territories in other bonuses, is that one or two connections? And vice versa (two or more territories in the bonus border with one territory outside the bonus).

that means a 50 territory bonus, on a 100 territory map, should only have a 12.5 bonus, if it only had 5 connections

and on a 100 territory map, a 1 territ bonus must have 20 connections in order to have a bonus of 1.. -10 connections if you use his guestimated 10, for alot of armies-

To balance bonus values the following questions might help:
a) How many territories are within the bonus? (doh!)
b) How many turns does it take to complete the bonus? (If you are thorough you should calculate this for every entrypoint to the bonus individually and then calculate the average)
c) How many territories inside the bonus have a common border with territories outside the bonus? (= how many territories are threatened from the outside? The fewer, the easier to defend the bonus, since you can concentrate your forces.)
d) How many territories outside the bonus have a common border with territories inside the bonus? (= how many territories threaten from the outside? Note: not necessarily equal to c), since there may be multiple connections to individual territories. The more, the harder to defend the bonus (doh! again ...))
How many connections are there in total? (= sum of all connections from outside to inside. Note: theoretical maximum for this value would be c) x d); I'm not sure if this value is of any real significance, since for example a total of 16 connections might mean anything from 4 inside + 4 outside territories, all interconnected with eachother, to 16 inside + 16 outside, all with a single connection. The latter bonus would be much, much harder to defend)

Now how best to weigh all those numbers once you have them ..? I don't know :-)

Interesting proposition, Moros. I thought I'm probably the only person who does this by formula! I often modify maps (especially the world map) according to:

A = (B + 0.5*C)/2

where:
A = the number of armies that the bonus should give
B = the number of territories inside the bonus
C = the number of territories inside the bonus that are on the border of the territory

Seems to work well, and easy to calculate in your head. On the world map, this leaves most of the bonuses unchanged.

I think it makes more sense empirically more sense to add B and C (instead of multiplying). I also don't know why one would want to include a term for the total number of territories in the map.

@ Perrin: it would be an extremely weird 100 territory map with a 50 territory bonus that has only 5 connections, but I guess it is possible. Maybe Moros's formula just works for "normal" maps.

I'd go for Bytjie's formula as well. Makes sense.
However, the amount of turns needed to take the territory aren't taken into account.
Then again I suppose that influence is rather subjective.

We'll talk about subjectivity when I smash you with the bonus armies from my 3-turn-Greenland while you are still struggeling to complete your 4-turn-Europe :-)

It's subjective because it all depends on where you start/attack the bonus from. Europe CAN be taken in 3 turns (start in Poland, France, Germany or Italy) and are lucky with attacks.

Also, its that very reason that Europe has a higher bonus than Greenland.

Edit:

C = the number of territories inside the bonus that are on the border of the bonus

(Obviously.)

@Richard: You are right, the starting point of your attack is relevant. That is why I said previously if you wanted to judge the average number of 'turns to complete' a bonus and thus its 'grade of difficulty' correctly, you would have to look at each entry point to that bonus separately.

Example:
Medium Earth, Scandinavia:
Danmark Havn: 3 turns
Germany: 3 turns
Ukraine: 3 turns
Moscow: 3 turns
Murmansk: 2 turns
Average: 14/5 = 2.8 turns

As for Europe and Greenland: I was also referring to Medium Earth, which is the norm for ladder games. Europe and Greenland both give 5 bonus armies there, despite the fact that Europe is harder to conquer and to defend:

Greenland:
6 territories in the bonus, 4 of which are border territories
4 adjacent territories, belonging to 3 different bonuses
4 connections total
average turns to complete (TTC): 14/4 = 3.5 turns

Europe:
7 territories in the bonus, 5 of which are border territories
7 adjacent territories, belonging to 6 different bonuses
7 connections total
average TTC: 26/7 = 3.7 turns

As you can see Europe is worse in every aspect, unless you want to argue that the vicinity of more adjacent bonuses is rather good than bad because it offers more expansion options.

How about:

　x^2
------
  x+y

Where:

x = number of territories inside the bonus
y = a number you can choose yourself

if you all looking for a flawless thought little vlue use x=y

IronHeart: It is x^2/2x = (x*x)/(2*x) = x/2.

x^2 / (x+y) is asymptotic to x-y.

See also my thread http://warlight.net/Forum/Thread.aspx?ThreadID=3077.

i meant territories equal bonus was a flawless simple formula

Here is the completely made-up formula that I used on the Bloomridge map:

Bonus Armies = ((Outside Connections/3)+#Territories in Bonus) / 2

(Dividing by two is the last thing you do, for those who can't remember.)

For reference, if applied to Medium Earth it would reduce a few of the isolated bonuses, such as Greenland and Australia, but would keep the other bonuses pretty much the same.

Worth considering for anybody balancing a new map, with a few nudges here and there.

Just used this complex formula in comparison to Muppet's and Bytjie's on Excel to test bonuses on my new map:

=((A/3)*100+(B*100+C*90))/(3*100)

Where ;
A is the Surrounding Borders (Not Connections)
B is the Territories in the Bonus itself
C is the Borders that Connect to other territories outside the bonus, (Borders you'd have to defend in the bonus)

---

The *100 and *90 are just to give C less weight. This should give you a rough guideline of the bonuses, also take in to account the bonuses values nearby, and ultimately decide yourself. :D

Mine, Muppets and Bytjie's all work well, if want an even better number, take the average of all 3 formulas.

Haven't tried Moros's though.

The areas the averages of 3 highlighted on my map, were that:
Belarus needs to go up to 5,
West Ukraine up to 5,
Southern France up to 4,
England and Wales go down to 3,
Iceland down to 2 (It was 2.44)
Northwestern Africa down to 3.

Some of these will be changed now, the formulas were very useful. :)

I still think you should make a *X in your formula, with X as the army density. That is fairly high on, for example, the Eastern Asia map, but very low on World of Hetacaeus)

Ironheart, x=y is a highly flawed bonus system...

it makes small territories much more valuable than large territories... Imho a simple formula that works is closer to x=y-Z z being a number determined by the maker... it's not a 100% formula ofc', there will still be stronger and weaker relative bonuses, but it's good for a simple formula...

Another formula

5/17 agm(a, 5b/17) + 10a/17

Where

agm (x, y) is Arithmetic-geometric mean (http://en.wikipedia.org/wiki/Arithmetic-geometric_mean) of x and y
a is number of territories
b is number of connections