Maths (probability) for unit types fighting each other

[Lord Dan]

The following covers calculating how strong units are against each other in direct 1v1 battles, including how fast the winner should win, and how many of the weaker units in a row would be needed to beat 1 of the stronger.

I will be releasing a proposal prototype for units and balancing for Knight's Province soon.

First I want to establish a good example of 2 units of different levels fighting each other.

In summary, the process has been:
A. See chance for each to hit each other, using Kistrel’s KAM_proportions.exe
B. Calculate swings required for each unit to have about 50% chance to kill each other.
C. See how quick the winner usually wins.
D. See how many weaker units in a row are needed to win.
E. Calculate how many the stronger could kill if full health each time.

The classic Axe Fighter vs Sword Fighter.
That is: 2 attack, 1 armor vs 3 attack, 2 armor.

Part 1.

By the calculator, its 11.67% chance for Axe to hit Sword.

Axe fighter on Sword fighter.png

Its 27.5% chance for Sword to hit axe.

Sword fighter on Axe fighter.png

If Axe fighter swings 5 times, his chance to hit 1 time or more should be:

[Chance to not hit to the fifth power, and then minus this from one]:
Ie: 1 – ( (1 - 0.1167) ^5)
= 1 – ( 0.8833 ^5 )
= 1 – ( 0.5377 )
= 0.462 = 46% chance to hit at least once.
If Sword fighter swings 5 times, his chance to hit 1 time or more should be:
1 – ( (1 - 0.27.5) ^5)
= 1 – ( 0.725 ^5 )
= 1 – ( 0.2 )
= 0.8 = 80% chance to hit at least once.
Conclusion here – about twice as much damage?

There is a chance they will hit 2 times or more.
This requires more math than is worth showing – using a binomial calculator (http://stattrek.com/online-calculator/binomial.aspx) I used for axe fighter:

0.1167 chance to hit (success).
5 swings (trials)
2 hit (success required)

Chance for at least 2 hits from 5 swings: 10.7%
For Sword fighter, it is: 42%
More than twice as much damage?

Now, chance to kill – 3 hits from 5, is:
Axe Fighter: 1.3%
Sword Fighter: 13%
Ten times as much damage?
Both a low chance, but big difference! (in chance of sudden death) We begin to see why strong units can hold off weaker units 1v1 for a very long time!

More realistically, let’s use 10 swings.
Axe Fighter: 10% chance to kill
Sword Fighter: 55% chance to kill
Could you argue that a Sword Fighter could kill 3-6 Axe fighters in a row, in 1v1?

Lets see how many swings from Axe Fighters it typically takes to kill a Sword Fighter:

20 swings from Axe Fighter: 42% chance to kill
25 swings from Axe Fighter: 57% chance to kill
30 swings from Axe Fighter: 70% chance to kill

Therefore with about the same chance to kill, 55%,
a sword fighter needs to swing 10 times.
an axe fighter needs to swing 25 times.
From this you could estimate 2-3 axe fighters would be needed to win (to get up to 25 swings).
This is conclusion #1.

Part 2:

However if the sword fighter is given time to heal, he could hold off about 5-6 axe fighters before he is more likely to die. Right?
Or is it 10x?

Let’s look at it this way – calculate the chance that a single axe fighter can win in 1v1 directly.

Aka, this = 1.00 minus chance of Sword Fighter to win.

Chance of Sword Fighter to win calculation:
0.275 chance to hit (success)
25 swings (trials), because this is the amount to give Axe fighter 55% chance to kill.
3 hits (success required)

Calculator gives 98.27% chance of Sword Fighter winning in 25 swings or less.
A single Axe fighter thus has
1.00 – 98.27% chance
= less than 2% chance to win in 25 swings or less.
Overall, this should be <2% chance of surviving to have a 57% chance of winning.
so there is more likely only a ~1% chance of axe fighter winning 1v1.

Putting that 1% chance of winning into the calculator, then how many axe fighters fighting a full health sword fighter would be needed for over 50% chance of winning?

Calculator’s closest estimate is 69 Axe Fighters needed to clear 1 Sword Fighter.
Therefore on average, a full heath Sword Fighter can hold off an average of 69 Axe Fighters!!
Or you will need probably 70+ Axe Fighters to kill off 1 Sword Fighter if he is allowed to regen health!

Wow, this is a special game!

[Lord Dan]

Ok, few more tasty examples of classic matchups:

1 Lance Carrier vs 1 Axe Fighter
1 Axe Fighter vs 1 Pikeman
2 Axe Fighters vs 1 Sword Fighter

These will become rather complicated calculations, but I aim for best accuracy, and a clear explanation of each.
We are using KAM_Proportions.exe, and
a binomial calculator:
http://stattrek.com/online-calculator/binomial.aspx


First the easy 2 sets:
1 Lance Carrier vs 1 Axe Fighter

12.5% hit vs 17.5% hit
(a fairly even fight?)
Both have 3 Health, so its 3 hits to win.

At 12.5% chance, needing 3 successes (X), we put this into the calculator
and keep raising “n”, which is the ‘number of trials’,
until the “Cumulative Probability” (X>=3) goes above 50%. (to establish a benchmark chance for both units)
Result:
The calculator shows at 22 swings (n=22),
the Lance Carrier has 53% chance to kill (3 or more hits)

The same test for the axe fighter at 17.5% hit
p=0.175
n=?
X=3
raise ‘n’ until P(X>=3) goes above 50%.

at n=15, the probability of kill reaches 50.3%

Conclusion here:
Lance Carrier needs average 22 swings to kill Axe Fighter
Axe Fighter needs average 15 swings to kill Lance Carrier.

- Not that close a battle, but not an extreme difference.

Q1. So how many Lance Carriers to kill average axe fighter?
22 divided by 15, is about 1.5 Lance Carriers. If they fight one after the other with no healing time.

Q2. What chance does each have to win?
Well, if the Lance Carrier needs 22 swings for 53% chance to win,
his chance to get there is 100% minus chance of Axe Fighter to win at 22 swings.

Chance of Axe Fighter to have won:
p=0.175
n=22
X=3
P(X>=3) = 76.7% chance.
1.00 minus 0.767 = 23% chance for Lance Carrier to still be alive after 22 swings.
At that point he has 53% chance to have won.
Therefore chance to win at this point is about
0.53 x 0.23 = about 12%

I think you can conclude the Lance Carrier has about 12% or less chance to win any 1v1.

From here I think the math gets too complicated to be more accurate – you would need to compare binomial distribution graphs of both units.
At least we can see he has a much better chance of beating 1 Axe Fighter, than the Axe Fighter had against the Sword Fighter!


Next Matchup!
1 Axe Fighter vs 1 Pikeman
Will this be a much closer battle?
11.67% hit vs 17.5% hit
From the outset, you can see its not even as close as previous comparison! The Pikeman’s extra armor will win it in 1v1.
Axe Fighter needs average 23 swings to kill.
Pikeman needs average 15 swings to kill.
Almost the same conclusion as the last matchup.
The Axe Fighter is equally stronger than the Lance Carrier, than the Pikeman is to the Axe Fighter. Roughly

 
2 Axe Fighters vs 1 Sword Fighter
Now things get a lot more complicated!

First I need to establish – this matchup has 4 outcomes, not 2 outcomes like 1v1.

In 2v1, the outcomes are:
A. Axe Fighters win – no deaths.
B. Axe Fighter #1 dies, but they win
C. Axe Fighter #2 dies, but they win
D. Sword Fighter kills both of them

Interesting, right?

So we need to calculate the chance of all 4 outcomes.
Or all 3 scenarios (because B+C are the same, mathematically)

First A. (Axe Fighters win – both alive)

Note: We are now including the bonus damage that occurs when one Axe Fighter is attacking the front right/left of the Sword Fighter (until one of them dies)

So for A. the chance to hit of the Axe Fighters is 1x11.67% PLUS 2x11.67%

Therefore, with 3 chances per swing, we need to use 1/3 as many swings.

How many swings for the Axe Fighters to kill, usually?

p=0.1167
n=?
X=3
P(X>=3) >= 50% chance.

We use the calculator to test for ‘n’.

n=23, chance of kill reaches 51.3%
so the average Axe Fighter needs 23 swings to win.

If there are equivalent of 3 Axe Fighters, then to win without either dying,
its 23 divided by 3. Which is about 8 swings (or >7).

That means the Sword Fighter has average 7-8 swings before killed by them.

What is the chance the Sword Fighter can kill at least one of them in 7 or 8 swings?

p=0.275
n=7 (swings)
X=3
P(X>=3) >= 29.7% chance.

p=0.275
n=8
X=3
P(X>=3) >= 38.5% chance.

Conclusion – Sword Fighter has about 36% chance of killing at least one of the Axe Fighters?
Therefore, chance of Axe Fighters winning, with neither dying is
100% – 36% = 64%?

So this is an approximate answer for Outcome A: 64%


Now lets examine the other two parts: B+C, and D. Together they should make up about 36% as above.

Let’s try to answer Outcome D.
(if this gets too confusing, maybe skip to the end and work your way backwards

How many swings does the Sword Fighter need on average, to kill 2 Axe Fighters?

Average to kill 1 Axe Fighter is 10 swings (54% chance).
So he needs about 19-20 to kill both?

No – using the calculator to answer correctly:
p=0.275
n=21 (swings)
X=6
P(X>=3) >= 53.85% chance

about 21 swings to kill 2 Axe Fighters (6 hits) are required.

Now we need to breakdown the conditional outcomes for IF the Sword Fighter kills 1 Axe Fighter with 10 swings (which was 36% chance which we will need later).
Remember the two axe fighter have had 30 swings so far.

The conditional outcomes are:
1. The Axe Fighters have hit 0 times so far
2. The Axe Fighters have hit 1 time so far
3. The Axe Fighters have hit 2 times so far.
4. The Axe Fighters have hit 3 times or more so far and won. (which is the remainder %)

What is the likelihood of these four conditional outcomes?

1.
p=0.1167
n=30
X=0
P(X=0) 2.4% chance

2.
p=0.1167
n=30
X=1
P(X=1) 9.6% chance

3.
p=0.1167
n=30
X=2
P(X=2) 18.4% chance

4.
p=0.1167
n=30
X=3 or more
P(X>=3) 69.6% chance

(these cover all four conditional outcomes – they total 100%. We want to use the first 3 – that is – the Sword Fighter hasn’t been killed)

Now we take 1-3 further.
We will need to total these conditional outcomes for later – see table below.
They total 30.4%. (2.4%+9.6%+18.4%)

IF the Sword Fighter isn’t killed, he must have been hit 0, 1 or 2 times already.

Lets see how many swings he can survive in each of these scenarios.

1. (Hit 0 times so far)
p=0.1167
n=?
X=3 (needs 3 hits to win)
P(X>=3) >= 50% chance.

We use the calculator to test for ‘n’.

n=23, chance of kill reaches 51.3% (this is an identical test to earlier – Axe Fighter fighting fresh Sword Fighter)
So Sword Fighter can survive average 22 swings in conditional outcome 1.

2. (Hit 1 time so far)
p=0.1167
n=?
X=2
P(X>=2) >= 50% chance.

We use the calculator to test for ‘n’.

N=14, chance of kill reaches 49.8%.
So Sword Fighter can survive average 14 swings in conditional outcome 2.

3. (Hit 2 times so far)
p=0.1167
n=?
X=1
P(X>=2) >= 50% chance.

We use the calculator to test for ‘n’.

N=6, chance of kill reaches 52.5%.
So Sword Fighter can survive average 5 swings in conditional outcome 2.


Lets table this so far!

If Sword Fighter has killed 1 Axe Fighter.png

Summary: IF the sword fighter kills 1 Axe Fighter, he has a 40.4% chance to win from there.

Therefore, the second Axe Fighter has a 59.6% chance to win.

Next, If we use the original outcome of A.
(which was 64% chance for Axe Fighters to win without either dying,)
then the above 2 scenarios have a total 36% chance of happening.

So of that 36%, its 59.6% for Axe Fighters to still win. Or 21.5% overall.
of that 36%, its 40.4% for Sword Fighter to still win. Or 14.5% overall.

So we have our answer for average outcomes!

A. Axe Fighters win – no deaths. 64% Chance!
B. Axe Fighter #1 dies, but they win.
C. Axe Fighter #2 dies, but they win.
(these are the same thing, so B or C, losing 1 Axe Fighter to win, is 21.5% Chance!)
D. Sword Fighter kills both of them. 14.5% Chance!

So to answer another way – in 2v1:

85% Chance of 2x Axe Fighters to win!
15% Chance of Sword Fighter to win!

Further tests to come later!

--------------------------------


Hope you enjoy, and comment on this everyone its fairly heavy maths - if you see any inaccuracies let us know

This system above leaves great room for calculating much more!
eg. different unit combinations,
teaming up on single units,
attacking with bonuses from the side and behind,
and even group battles!

[Esthlos]

Interesting!
In case you care for some experimental validation, here's a test map (modified to only have duels; it will only test your final result; to change the number of axe fighters, edit the constant cAxeNumber at the start of the script):

aaa Test Fight 2.zip

Results:
500 battles
1 sword fighter vs 69 axe fighters, only duels (railroaded by using unwalkable tiles)
Axes won 500 battles (100%)
Average survivors: 63 Axe Fighters (min 27, max 69)
Average duration: 115 seconds (shortest 7 seconds, longest 727)

So, it seems that on average axe fighters only need 6 duels to kill one sword fighter

[Lord Dan]

Thanks! How do I run this? I put in the multiplayer maps folder, but it doens't appear anywhere that I can open it in the game.

Regarding holding off massive amounts like ~70 Axe Fighters, that was with allowing healing time. Which should be about 30 seconds with no one attacking the Sword Fighter.
If consecutively fighting, yes, he should only last 3-6 Axe Fighters on average.
Right now I am doing the math on fighting 2 of them at the same time.

[Esthlos]

I put in the multiplayer maps folder, but it doens't appear anywhere that I can open it in the game.

Put it in the singleplayer folder, then run it as a singleplayer map.
(Here is the unmodified/more versatile version: http://www.knightsandmerchants.net/foru ... 584#p47584 )
(Run them as the default player, the red player 1)
(F11 -> Speed x300 to run the test in a short time)

Regarding holding off massive amounts like ~70 Axe Fighters, that was with allowing healing time. Which should be about 30 seconds with no one attacking the Sword Fighter.

Oh. Right.
Modified (again) the map to add that healing: (it actually kills and respawns the sword fighter whenever he scores a kill; it's easier this way)

aaa Test Fight 2.zip

Results:
500 battles
1 sword fighter vs 69 axe fighters, only duels (railroaded by using unwalkable tiles), sword fighter respawns when he kills one axe fighter
Axes won 500 battles (100%)
Average survivors: 62 Axe Fighters (min 21, max 69)
Average duration: 146 seconds (shortest 6 seconds, longest 893)
The sword fighter was respawned 7 times per battle on average.

(I honestly expected the results to be more different than those of the first test... )
(I'm sure it kills and respawns the correct sword fighter; here is the code)

  Code:

iID is the Sword Fighter's UnitID, iX and iY are the coordinates right behind him, pA1 is his owner Actions.GroupOrderLink(Actions.GiveGroup(pA1, aArmy[pA1].aStats[0].aType, iX, iY, 4, 1, 1), States.UnitsGroup(iID)); Actions.UnitKill(iID, True);

[Lord Dan]

Respawning isn't healing, right?
He's only dying because they are wearing down his HP from 3 to 0, I think.
Can you run a 1v1 500+ times and see how many of those times the Axe Fighter wins? It should be around 1% by the math

[Esthlos]

Respawning isn't healing, right?

It is: units spawn with max HP

Can you run a 1v1 500+ times and see how many of those times the Axe Fighter wins? It should be around 1% by the math

Sure

Results:
5000 battles (yes, I can set this as high as I want; you GUI-using peasants are capped at 999 instead )
1 sword fighter vs 1 axe fighter
Axe won 619 battles (12%)
Average duration: 18 seconds (shortest 6 seconds, longest 106)

[Lord Dan]

Excellent - so 12% chance to win. Maybe not the 2% or less I said near the beginning.

If that's the case, then average chance of 1 or more win/6 battles should be a 53%. OR Sword Fighter kills average 6 Axe Fighters, regardless of health.
This matches your previous data.

That is alot higher than I had. I think I was calculating with the presumption of the Sword Fighter's survival to be long enough to kill in more extreme cases (25 swings).
Or vice versa - presuming it was impossible for the Axe Fighter to kill in under 25 swings.

Can you do a 2v1 to test my latest theory from this evening? 2 Axe Fighters vs 1 Sword fighter.
(but the Axe Fighters must be next to each other, so neither of them can be on the Sword Fighter's flank?)

Feel free to do these too if easy for you
Lance Carrier vs Axe Fighter 1v1
Axe Fighter vs Pikeman 1v1
Militia vs Knight
3 Militia vs Knight (attacking his front 3 squares only)
(and any other good simple matchups you fancy)

Thanks!

[Esthlos]

Can you do a 2v1 to test my latest theory from this evening? 2 Axe Fighters vs 1 Sword fighter.

Results:
5000 battles
1 sword fighter vs 2 axe fighters, frontal
Axes won 3997 battles (80%)
Average duration: 15 seconds (shortest 4 seconds, longest 112)

Feel free to do these too if easy for you

How about you use the "versatile" version of the map and run the tests yourself?
(Everything should be explained in in-game messages; I've also changed the way the number of battles is set up so that it accepts up to 99900 total battles)
(Alternatively, you can ask directly for the troop proportions to get roughly 50% win chance, as they are all in the code for Kistrels' tool - whom I do not absolutely know and by the way I think that his nickname is truly awful )

aaa Test Fight.zip

[Lord Dan]

Ok, thankyou I had 85% chance for 2 Axe to beat 1 Sword - so it's close. It's probably to do with picking averages, instead of covering all distributions (distribution curves). Either my assuptions aren't all right, or I'm not using fancy enough maths.

I'd love to use the system - sorry but I barely understand how it works. How do I access these instructions? or messages?
I really haven't used the map editor before, but I checked and see no way to edit unit battles, select unit types etc.
So far I've only been able to run the map, and run it at high speed, to see the results build up in the on screen text.

[Esthlos]

It's probably to do with picking averages, instead of covering all distributions (distribution curves).

Probable.
(The test map could be used to collect those too, but exporting that data might prove... problematic.)

I'd love to use the system - sorry but I barely understand how it works. How do I access these instructions? or messages?

Just start the map as a singleplayer game; order weapons (or trade wares) to change the numeric options (you'll see them change in real time in the overlay, so you'll know which is which).
(If they don't change, then the auto-computation of Army 2 is on and what you are trying to change are that army's values)

To change the toggle options, just toggle the repair/delivery of any building (every one does something).

To start or reset the simulation, just demolish any building (doesn't matter which one).

P.S. I've exported from the tool the ratios to get roughly 50% win chance; this was obtained with a variation of the test map (was modified so that it would simply keep adding units to the losing side and restarting the test until either side got 45%-55% win rate)

50chance.jpg

[Esthlos]

I encourage to disregard the image I posted... just tried to run a test with the axes vs swords ratio, and got axes winning 88% of the battles (79% the second time).
Knights vs pikemen seems nearly correct though (knights won 46% of the battles the first time, 58% the second).

No idea why; maybe I got the numbers wrong one of the times they have been copied... anyway, only use that table with extreme caution (if at all).

[The Dark Lord]

Before spending precious time on these walls of text I'd like to know what your purpose is? In my opinion Knight's Province should have a completely different fighting system; KaM's system is far from ideal. Do you know how Krom thinks about this?

[Lord Dan]

Ok, yes I've made things very complicated -

my goal in this thread it to try to figure out how to use math to determine outcomes of battles, to compare strengths between units.

Calculations ideally will show accurately:
Who beats who in 1v1, 2v1, 3v2, even 5v2 etc including breakdown of how many survivors.
How many swings to win on average (how long battles will last - longer allows flanking time, shorter allows tactical rushing etc - this means strategy!)
What % of the time chance to win, chance to lose - variance of outcomes is important (we want unpredictability, but not too much! Balance between being interesting but not frustrating!)
To see the resistance of units to being attacked on his side - I believe the bonus damage should be tweaked down just a little on the front corner.

If we can determine the math to answer these,
Then if/when we tweak the system, units, we will know exactly the change that occcurs in all the above!
eg. what if we raise HP by 1? what if we have a new unit with 3 defense? what if we change the spears vs cavalry bonus?

We will be able to calculate the effects immediately and accurately!

Or even better - we can do it in reverse.
We can say "What change do I want to have in this unit?" etc and then adjust an aspect of his design according to the answer the math gives.

So I hope this thread can aim to determine a math system to do all these things but maybe we'll need an expert. Or we can only use it for simple problems, and rely on the test running system (which is also great!)

[sado1]

I didn't read anything before the last post, however I think that this will not be that useful, given how you're not really creating real world scenarios that way. I'm afraid it's impossible to actually create something useful for automated testing, given how fighting in this game works. I'm about to post a new thread where I explain how does the fighting system in KaM work in practice. I'm looking forward to your opinion.