True Strength of a Unit or Hit to Miss Ratio

[tabanli]

Let's say a Red Dragon is attacking bunch of light infantry in a level 1 city. So it is Strength 50 vs 10. Question is how many light infantry does a Red Dragon kill on average. And the answer is not 5 but it is 9. Be cause one factor is the chance of rolling a favorable dice, but the other factor is your opponent should miss.

There are several ways to calculate relative strength of units and chance to hit. Two of them are pointed out by KGB at the thread about Dread Knight, both of them valid and common sense, but if we continue the calculations to the full extend we will see that a third one is also valid. Let's call the Attackers strength A and Defender D.

a) Your relative chance to make a hit is proportional to your strength ratio. So you can divide Attacker strength by defender strength A/B. But in order to make a hit, not only you should roll a favorable dice but also your opponent should miss it. So chance to hit is inversely proportional to the miss ration. Which is (100-A)/(100-B)

Formula -> Relative Chance to produce a hit R=(A/B) / ((100-A)/(100-B))

Which is the formula using the first approach. But you can rewrite the final formula in two different ways.

b) R= (A*(100-B)) / (B*(100-A)
The logic of the second way is as follows. The attackers chance to hit is when he rolls a favorable dice ( A ) and the defender misses it (100-B). Defenders chance to hit is that he is rolling a favorable dice ( B ) and attacker misses it (100-A). The ratio of two values gives the relative chance to make a hit.

C) R= (A/(100-A)) / (B)/(100-B)

The term A/(100-A) is the attackers Hit to Miss ratio and I call this the true strength of a unit. By rewriting the equation, I can compare increase in my units strength with no regard to the opponents strength.

Strength 75 -> Hit to miss ratio = 75/25 = 3
Strength 60 -> Hit to miss ratio = 60/40 = 1.5
Strength 50 -> Hit to miss ratio = 50/50 = 1
Strength 33 -> Hit to miss ratio= 33/67 = 0,5
Strength 20 -> Hit to miss ratio = 20/80 = 0.25
Strength 15 -> Hit to miss ratio = 15/85 = 0.17
Strength 10 -> Hit to miss ratio = 10/90 = 0.11
Strength 5 -> Hit to miss ratio= 5/95= 0.05

Same thing is true for undead lore and ruin searching. Adding a scout increase your chance of survival a lot. So with level 1 ruin and 12 faced dice here is the true strength

Undead lore 12 -> Hit to miss ratio = 12 / 0 = infinity. You never lose.
Undead lore 11 -> Hit to miss ratio = 11/1 = 11. With 11 undead lore your true strength is also 11.
Undead lore 10 -> Hit to miss ratio = 10/2 = 5. Here comes the dramatic drop
Undead lore 9 -> Hit to miss ratio = 9/3 = 3
Undead lore 8 -> Hit to miss ratio = 8/4 = 2

Regardless of who you are facing at the ruin, going from undead lore 10 to 11 makes you more than twice strong. If you are searching a ruin with 8 undead lore, your chance to win is 20% of the hero with 11 undead lore.

[KGB]

Tabanli,

Regardless of who you are facing at the ruin, going from undead lore 10 to 11 makes you more than twice strong. If you are searching a ruin with 8 undead lore, your chance to win is 20% of the hero with 11 undead lore.

You've still got the math wrong here. You continue to improperly handle the enemy strength when you are doing your ratio's.

To see this consider fighting 1 skeleton (str 3) in a ruin with a hero with 11 UL and 8 UL.

The 11 UL hero chance to hit is (11/12*9/12) = .6875
The skeleton vs the 11 UL hero is (3/12*1/12) = .0208
The 8 UL hero chance to hit is (8/12*9/12) = .5
The skeleton vs the 8 UL hero is (3/12*4/12) = .0833

In battle the 11 UL hero hit chance is .6875/(.6875+.0208) = .6875/.7083 = .9706
In battle the 8 UL hero chance to hit is .5/(.5+.0833) - .5/.5833 = .8572

The skeleton beats the 11 UL hero if he gets 2 hits (2 straight or 2 hits and one on himself which happens 2 ways. That works out to .0294*.0294 + 2*(.0294*.0294*.9706) = .0008 + .00167 = .00247

The skeleton beats the 8 UL hero if he gets 2 hits (2 straight or 2 hits and one on himself which happens 2 ways. That works out to .1428*.1428 + 2*(.1428*.1428*.8572) = .0203 + .0349 = .0552

So the 11 UL hero wins 99.8% of the time and the 8 UL hero wins 94.5% of the time. So the 8 UL hero is not 20% as likely to win against 1 skeleton. He is .945/.998 = 94.6% as likely IN THIS PARTICULAR BATTLE ONLY.

Now as the defenders change (more skeletons, Ghouls etc) the odds are going to change due to needing to defeat more/stronger defenders. The number will *never* be something as simple as 20% as likely unless you are considering a very specific battle with set defenders.

KGB

[tabanli]

Your calculations are correct. Based on that

Against 11 Undead lore hero, Skeleton has approximately 3% chance to hit. 100-97,06
Against 8 Undead lore hero, Skeleton has approximately 15% chance to hit. 100-85,72

As we may see, his chance to hit is improved by 5 fold. Which is what I was giving at the table. So I am not talking about % chance to kill or a simple formula about percent chance to hit. I am comparing only the "RELATIVE" percent chance to hit. I understand that using the terms strength is misleading and I should probably use another term, like POWER.

Since the killing percent is proportional to the square of the chance to hit (must produce two hits), we can check whether if Skeleton improved his chance by 25 fold?

11 UL wins 99.8 times out of 100. Skeleton has 2 per thousand chance to win.
8 UL wins 94.5 times out of 100. Skeleton has 55 per thousand chance to win.

It is still low but as we can see, his chance is improved "RELATIVELY SPEAKING" 25 times.

[tabanli]

Let's compare two hypothetical creature, A has 75 strength and 1 hit and we have B with 50 strength and 1 hit. A is attacking a city with 3 B's in it. In order for A to capture the city, he must hit 3 times consecutively.

A chance to hit (75/100*50/100)=.375
B chance to hit (50/100*25/100)=.125

A hit chance is (.375/(.375 +.125)) =.75
B hit chance is (.125/(.375+.125))=.25

A's hit chance is 3 times more than B's hit chance. But that doesn't translate directly to the win %. Crowd has slight edge against a single attacker.

The only time A kills, is when he hits 3 times consecutively so

% Chance to win =(.75)*(.75)*(.75)= .42

However as the attacker and defender size increases, % chance to win rapidly converges to 50%. So already with 2 A's Attacking 6 B's % chance to win is (.75^6)+(7*(.25*.75^6))=.49

(.75^6) is the chance that battle lasts in 6 rounds with consecutive hits by A and both A's are surviving.
(7*(.25*.75^6)) is the chance that the battle last 7 rounds with B is providing only one hit in either of the seven rounds.

So 1 unit of 75/2 is approximately equivalent to 3 units of 50/2. So 75 Strength is 3 times more POWERFUL than 50 Strength.

[KGB]

Tabanli,

As we may see, his chance to hit is improved by 5 fold. Which is what I was giving at the table. So I am not talking about % chance to kill or a simple formula about percent chance to hit. I am comparing only the "RELATIVE" percent chance to hit. I understand that using the terms strength is misleading and I should probably use another term, like POWER.

Calling it power makes much more sense.

You have to be careful when talking about the relative chance to hit. In the skeleton example the relative chance to hit is 2 different units (Str 11 and Str 8) vs a Str 3 where in your later example (str 75 and str 50) you are comparing 2 units directly against each other. This is where you have to be careful to distinguish exactly what you are comparing.

11 UL wins 99.8 times out of 100. Skeleton has 2 per thousand chance to win.
8 UL wins 94.5 times out of 100. Skeleton has 55 per thousand chance to win.

It is still low but as we can see, his chance is improved "RELATIVELY SPEAKING" 25 times.

Correct. But what's important isn't the fact that the 11 UL is relatively speaking 25x as good as the 8 UL. What's important is that the winning % is 94.5 and 99.8. So in this example it's pointless to wait to add extra scouts for this battle because they aren't needed.

% Chance to win =(.75)*(.75)*(.75)= .42

However as the attacker and defender size increases, % chance to win rapidly converges to 50%. So already with 2 A's Attacking 6 B's % chance to win is (.75^6)+(7*(.25*.75^6))=.49

(.75^6) is the chance that battle lasts in 6 rounds with consecutive hits by A and both A's are surviving.
(7*(.25*.75^6)) is the chance that the battle last 7 rounds with B is providing only one hit in either of the seven rounds.

What's interesting is when I run this through my battle simulator the 1 A vs 3 B's comes out at 41.8 which is the theoretical value you show. However when I run 2 A's vs 6 B's the number comes out around 44.5%. This happens on repeated runs of 10000 tries. Then I realized why. Your second part of the calc is using 7 ways. But there are only 6 ways (you can't get a hit on A after giving 6 hits to B because the battle is already done). When you change that 7 to 6 you get the correct 44.4%.

When I then run 3 A's vs 9 B's the winning percentage for A increases to 46.5. Still not quite at 50 yet.

KGB

[tabanli]

I think this was the reason why Monk was so powerful before you made adjustments and tone him down in DLR. Classical Dragons or blessed power units with 9 regular strength often ends up with +1 to +3 bonus maximum in hero battles and gets to be 10-12 strength (equivalent to 50-60 here), but Monk with all the spells and items could get 15 strength (equivalent to 75 here). We watched with Bugra, battle after battle as Monk rising as the sole survivor. The only set who had equal chances was Sasquatch's Summoner-Shaman-Green-4 Black Dragon-Cave Wyrm set with Berserker. However that required a very large map and sufficient time to build. As you may remember he had 60+ consecutive wins streak with it.

So the moral of the story is, 75/4 Barbarian is notoriously hard to kill without ambush. On average he will take down 5-6 units of 50/2 before dying and you shouldn't be surprise if he kills all 8 of them. Keep that in mind before engaging him in a hero battle.

But a high level Assassin sitting in a crowded city is also notoriously hard to kill, even if the units are trash. So what do you know, it's a tough game.

[KGB]

Tabanli,

Yeah a 15/4 Monk is a real beast when he got those spells going that increased his hits/strength. After I adjusted the heroes the Monk can still reach 15/4. In fact he's probably a better solo fighter than he was because his spells were redesigned for that including some personal Trample power! He just doesn't have quite the same stack skills (still has the Group Acid though . The updated Barbarian can also reach very high strengths and hits just as he can in Warbarons.

Bugra won a ton of battle purely on that 15/4 because everyone else was using a Wizard or Summoner that maxed out at 6-8 strength and 2 hits so they got killed by the Monk. Also it's not well known but flying 4 hit heroes are immune to archery (they aren't considered flying) and they don't have +3 archery bonus against them which also really helped the Monk.

I remember Sasquatches set and 60 game win streak. I lost to him in there at least once. The only other hero that would have been better would have been a Vampire with +2 Lifedrain combined with a Wizard (Chain Lightning), Red and Blue Dragons and Siege unit of choice (even with the acid bug). But that would require even longer to build that Sasquatches hero stack. It's truly the ultimate hero combo due to the Lifedrain at +2 but it's also virtually impossible to ever get as no one would allow the time for the Vampire to level up that far. My adjusted Vampire is better than the original but still takes a long time to level up.

And yes, it's definitely a tough game. You don't want to face down a Barbarian with 75 strength and 5 hits (or 7. Yes, I have had 7 hits before when I acquired the Gabriels Heart item) unless you have ambush. Nor do you want to attack a city full of units guarded by an Assassin.

KGB