Maths (probability) for unit types fighting each other
PostPosted: 22 Nov 2016, 16:50
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.
[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!
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.
Its 27.5% chance for Sword to hit axe.
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!