The underrated power of the Immortal/Catapult/Spearman stack: a statistical analysis

[Jaguar]

The Mounted Warrior has long been favored over the Immortal by the warmongers of the civ community as the best ancient age attack unit because its retreat ability and its fast movement. Its retreat ability supposedly makes more of them survive. Immortals can survive just as easily.


Here is a calculations for a stack of 2 Immortals, a Spearman, and 2 Catapults vs 2 Spearmen:

Now suppose that all units in this game are regulars for now. The probability that the Immortal stack will take the town without casualties can be calculated as follows:

Strength of a spearman fortified in a town on plains: 2.7
Bombard strength of a catapult: 4


Chance that the catapult will hit (if the game chooses to go for units instead of improvements or citizens: 4/6.7 or 59.7%. That chance is halved to 29.8% because of the chance that the catapult rolls against improvements or citizens.

So, the odds that both catapults hit a spearman are .298 x .298, or 8.9%. The odds that both miss are .702 x .702, or 49.2%. The odds that one misses, the other hits, therefore are 100 - 58.1%, or 41.9%.

So, lets start with the one misses, the other hits scenario. Now we have two reg immortals vs. two spearman, one a regular and one with only 2hp. The odds that the first immortal will win are 67.7%. The odds that the second will win are higher, because the spearman wounded by the catapult is now defending. The odds are 81.2% that the second battle will be won. .812 x .677 equals the probability that the town will be taken with no casualties: 55.0%

If both catapults hit, there will be two 2hp spearmen: therefore the odds of both immortals winning are .812 * .812, or 65.9%.

If neither catapult hits, there will be two 3hp spearmen: odds of both immortals winning are .677 x .677, or 45.8%.

Now we get the total odds of taking the town with out any casualties...

(.550 x .419)+(.659x.089)+(.458x.492)

This comes out to .230+.059+.223, or 51.2%.

51.2% chance of having no casualties. And a spearman and two catapults to defend the town. And those are Immortal regulars that I was calculating the odds for.


Just for the record, if these immortals were vets and the spearmen were regulars, (common when attacking the AI,) these would be the odds:

(.910x.816x.419)+(.910x.910x.089)+(.816x.816x.492)
That's .311+.074+.328, or 71.3%.

A 71.3% chance of coming off with no casualties. I'd say that's a high survival rate.

[Jaguar]

Oh I forgot to show the significance of a spearman. Well you know how the AI likes archers in their cities for counterattacks? If there's a reg archer attacking the stack's spearman, the odds that it will win are extremely low. A catapult has bombard of 4, archer has attack of 2. The odds are therefore 66.7% that the catapult will hit it before it attacks the spearman. A nonfortified spearman on forest, for example (those catapults are wheeled, so you can't use mountain ranges.) has a defense of 2.5, so an archer has a 60.3% chance of being killed if the catapult misses. If it hits, he has a 76.6% chance of losing. Therefore, the spearman will have a (.766x.667)+(.603x.333) chance of winning. That's .511+.201, or a 71.2% chance that the spearman will win. That dispells the other complaint about immortals, the complaint that they are easy to kill. They aren't with a spearman defender.

It also shows that catapults are useful; the spearman's chance with catapult help is 10% higher than without. Not to mention the affect it has on the attack. (5% more chance of 0 casualties)

[Dominae]

You've demonstrated that Immortals are effective against Regular Spearmen defenders. You forgot to calculate the odds and survival rate of MWs doing the same thing at similar cost, and comparing the two UUs based on this information.


Dominae

[Jaguar]

I know but I realized I wasn't quite sure how the retreat bonus works. If someone could enlighten me I would do a similar calculation for MWs.

One thing that bothers me a bit: the MW's speed is always hailed as a great advantage... but they're absolutely abysmal at defending cities. If you want to hold a city, you often need to send a spearman with them, which slows them down.

[Dominae]

I thought retreat is a 50% chance when down to the last HP.

The key to defending with fast-movers is to bring the battles out of cities and into the open, where the extra mobility rules. The only problem is rugged terrain (Mountains, Hills, etc.).


Dominae

[Jaguar]

I'm pretty sure that there are some rules about elites retreating more often than vets, too.

[Arrian]

Dominae,

Retreat chance is modified by experience level (reg/vet/elite). I think it's about 50% for vets, a bit more for eiltes, a bit less for regulars.

But the real key problem with Immortals is their speed; or rather their lack of it. MWs can take a city w/o spending any time sitting in enemy territory waiting to be attacked. That is their advantage (at least on open ground). It gets even better with the Rider... which I think is the most powerful unit in the game (MW running a CLOSE #2).

-Arrian

[Jaguar]

Arrian:

Originally posted by Jaguar Warrior One thing that bothers me a bit: the MW's speed is always hailed as a great advantage... but they're absolutely abysmal at defending cities. If you want to hold a city, you often need to send a spearman with them, which slows them down.

How are you going to hold the city with a few mounted warriors who have been injured in the attack?


(BTW I know it works when you have 15 of them, because they can just move on and kill anything that heads for the city. However, I would argue that if you had 15 Immortals you owuld be equally invincible.)

[Brizey]

Combat Calculator

JW:

I have attached my combat calculator (requires Excel). It includes all bonuses and retreats. Note that it is only for non-army, non-naval, non-bombardment combat. I have some pretty tricky Excel poop in there, so be careful making changes.

Open up the standard BIC with the editor to see the chances of retreat.

If you wish, look at all the worksheets in the workbook. Several of them show a tree of every possible result during the combat.

Arrian:

I agree about the Rider. The Chinese are one of my favorite civs.

[Dominae]

If an invincible stack of 15 MWs confronts an invincible stack of 15 Immortals, what happens?!



Arrian, thanks for the info about retreating; I never knew that it varied with experience.

Dominae

[Brizey]

Another thing about my calculator:

If you play a modded version, just change the numbers in all the table worksheets, the macro and control page reads the data from them. You can even delete lines or add lines to the tables (handling this is one of my tricky Excel poop tricks).

[joncnunn]

By having previously left a city purposely undefended (important: NOT near the path your MWs are going to take) to sucker the AI into sending all it's offensive units after it.

Don't keep the MW in the town you captured for long, just long enough to heal and send them to the next target.

The AI force that was attracted to one of your undefended cities now turns around and starts marching back towards the undefended city you just took.

Ignore that force for now, and take another city, because you can almost certaintely do so before that AI force gets back.

By now, the enemy forces may be close to reaching the cities you just took so sue for peace for the best possible deal and use the time to bring up defenders to the newly acquired acquistions and build culture improvemnts to them while the core area builds settlers to replace the auto-razed cities and fill in unoccuplied territory followed by building more MWs for the next war to start in about 20 turns.

Originally posted by Jaguar Warrior
Arrian:

How are you going to hold the city with a few mounted warriors who have been injured in the attack?

(BTW I know it works when you have 15 of them, because they can just move on and kill anything that heads for the city. However, I would argue that if you had 15 Immortals you owuld be equally invincible.)

[Catt]

Jaguar Warrior - nice analysis (lots of math!).

But I think this is one of those comparisons (immortals+catapults+spears vs. MWs) that is more relevant when taking into account non-mathematical variables: who is the target? what is the terrain like? what map features are prominent? etc.

The key strength of an MW attack is its speed. Only rarely need one worry about defending cities with MWs if conducting a well-planned campaign. The MWs will almost always be on offence, and the border expansions that come with taking cities reinforces this. Slow moving spears take up the rear and reinforce behind the blitzkrieg.

On the other hand, with very rough terrain (taking away the MW speed advantage), an opponent that likes to build fast-movers (e.g., Iroquois, Egypt), and/or map features that promise a more scattershot approach rather then a linear march of conquest, a slowmoving, mixed-unit force that is great on offense and defense makes a heckuva lot more sense.

I find that part of the fun of Civ is that it's so da*n hard to say that MWs or the approach you analyze is the better -- it all depends on circumstance!

Catt

BTW - some testing by etj4Eagle and sumthinelse at CFC seemed to indicate pretty clearly that the bombardment targeting ratios have changed with PTW -- it now seems that bombardments will target a unit (rather than a city improvement or pop point) only 30% or 33% of the time -- it seemed to be 50% of the time in vanilla Civ - so the catapult damage percentage in your post should be a small bit lower).

Edit for spelling.

[Arrian]

Originally posted by Jaguar Warrior
Arrian:


How are you going to hold the city with a few mounted warriors who have been injured in the attack?


(BTW I know it works when you have 15 of them, because they can just move on and kill anything that heads for the city. However, I would argue that if you had 15 Immortals you owuld be equally invincible.)

Simple. First, when I open my attack, I do bring some spearmen along. I stack up my units at the border (hopefully at the good angle, where you can do a "move 1 tile and hit" attack even on a city with a cultural influence of 2). Then, my mounted troops move into enemy territory, attack, and take the city. Then, my spearmen (2 or 3 at the start of the campaign, prolly*) use the road (which is now mine) to enter the city and fortify. Yes, I know this requires roads, but there usually are roads built by the time I unleash a major mounted attack. Otherwise, I send in the spearmen/swordsmen 1 turn earlier, such that they are adjacent to the enemy city when my mounted troops attack.

* - I often ditch the spearmen in favor of swordsmen upgraded from warriors. They can hit and defend. Newly captured cities poprush regular spearmen and are left behind.

-Arrian

[Jaguar]

Originally posted by Dominae
If an invincible stack of 15 MWs confronts an invincible stack of 15 Immortals, what happens?!

Stalemate. The MW's speed ensures that it can always be on the attack when it faces the Immortal. They win about 3/5 to 2/3 of the time attacking Immortals on flat ground. However, the key is "on flat ground." The immortals would head for the mountains, and the MW general would unsuccessfully try to lure them onto flat ground.