Sagacious' Unit Ranking System

The purpose of this post is to set up a standardized system for ranking military units based upon their comparative performance in combat. Because this is a work in progress, much of the information presented in this post is subject to revision.

You will require a reference for unit statistics in order to make use of the various equations presented in this post. Although most online sources are now outdated (their statistics derive from versions prior to 1.03), some are still somewhat reliable, such as the one found here.

Period of One Killing Cycle

The greatest and worst aspect of Age of Mythology is the strategic depth of its game play. Although it requires players to think strategically, its depth and complexity can often prevent them from making informed decisions. This is especially true in the case of units, which are extremely diverse, and consequently, difficult to compare and contrast. However, although the properties of each unit can vary widely from one another, you will find that the ultimate measure of a military unit's value is its performance in combat – its killing speed. As you will soon see, a unit's killing speed is a reflection of almost every attribute it possesses.

How long would it take for unit A to kill unit B?

The equation below can be used to calculate the time taken for unit A to kill unit B.

It should be noted that this equation does not account for shortcomings in range, nor does it account for units whose accuracy in combat is less than 100%. As a result, this equation should only be used if unit A's range is greater than or equal to that of unit B and unit A has an accuracy of 100%. Also, it is essential that unit B's armor correspond to the type of damage inflicted by unit A. Lastly, this equation always assumes that unit A initiates the first attack in the sequence of attacks exchanged between the two units.

I will illustrate the proper use of this equation in the following example, where I will use it to calculate how long it would take for one Hoplite (unit A) to kill another Hoplite (unit B). This scenario is illustrated in the image below.


Hoplites possess 115 HP, which means that they can sustain 115 damage before dying. In order to discern how long it would take for one Hoplite (Hoplite A) to kill another Hoplite (Hoplite B), you need to divide the amount of damage that Hoplite B can sustain (115 damage) by the amount that Hoplite A inflicts every second (8 damage/second), as demonstrated below.

However, before dividing, you must account for any modifiers to the damage that Hoplite A inflicts.

Ask yourself the following questions:

• Do Hoplites receive a bonus to the damage they inflict when battling other Hoplites?
• How resistant are Hoplites to the type of damage that other Hoplites inflict?

Hoplites receive no bonus to the damage they inflict when battling other Hoplites. As a result, the modifier to Hoplite A's damage per second will be equal to 0%, as demonstrated below.

Hoplites are 35% resistant to hack attacks, which means that the percentage by which Hoplite A's damage is reduced is equal to 35%, as demonstrated below.

Once you have accounted for any modifiers to the damage that Hoplite A inflicts, you can proceed with the arithmetic.


Note that these are the base statistics for each unit without the inclusion of any upgrades.

One might speculate that a unit A's resilience in combat, that is, its ability to sustain damage, rather than deal it, is also of vital importance. However, doing so would overlook the fact that unit A's survival is actually dependent upon the killing speed of the unit it faces (unit B), and thus, equal to the killing speed of the unit it faces (unit B). As a result, it is by calculating the period of one killing cycle for unit B that we discern the resilience of unit A, which leads us to the next equation used in this system of ranking: the equation for the period of one killing cycle for unit B.

How long would it take for unit B to kill unit A?

The equation below can be used to calculate the time taken for unit B to kill unit A.

It should be noted that this equation does not account for shortcomings in range, nor does it account for units whose accuracy in combat is less than 100%. As a result, this equation should only be used if unit B's range is greater than or equal to that of unit A and unit B has an accuracy of 100%. Also, it is essential that unit A's armor correspond to the type of damage inflicted by unit B. Lastly, this equation always assumes that unit B initiates the first attack in the sequence of attacks exchanged between the two units.

I will illustrate the proper use of this equation in the following example, where I will use it to calculate how long it would take for one Hoplite (Hoplite B) to kill another Hoplite (Hoplite A). This scenario is illustrated in the image below.


Hoplites possess 115 HP, which means that they can sustain 115 damage before dying. In order to discern how long it would take for one Hoplite (Hoplite B) to kill another Hoplite (Hoplite A), you need to divide the amount of damage that Hoplite A can sustain (115 damage) by the amount that Hoplite B inflicts every second (8 damage/second), as demonstrated below.

However, before dividing, you must account for any modifiers to the damage that Hoplite B inflicts.

Ask yourself the following questions:

• Do Hoplites receive a bonus to the damage they inflict when battling other Hoplites?
• How resistant are Hoplites to the type of damage that other Hoplites inflict?

Hoplites receive no bonus to the damage they inflict when battling other Hoplites. As a result, the modifier to Hoplite B's damage per second will be equal to 0%, as demonstrated below.

Hoplites are 35% resistant to hack attacks, which means that the percentage by which Hoplite B's damage is reduced is equal to 35%, as demonstrated below.

Once you have accounted for any modifiers to the damage that Hoplite B inflicts, you can proceed with the arithmetic.


Note that these are the base statistics for each unit without the inclusion of any upgrades.

In this case, because unit A and B are both Hoplites, the period of one killing cycle for unit A is equal to that of unit B. Thus, in the event that either of the two Hoplites initiates the first attack, the time taken for Hoplite A to kill Hoplite B will be identical to the time taken for Hoplite B to kill Hoplite A.

As previously stated, these equations do not account for shortcomings in range. Because this system of ranking evaluates units based upon their comparative performance in combat, the advantage of additional range should be taken into account, otherwise, units with ranged attacks will appear less valuable than they actually are. This leads us to the next equation used in this system of ranking: the equation for the period of one killing cycle for unit A in the event that its range is less than that of unit B.

How long would it take for unit A to kill unit B if unit B's range exceeded that of unit A?

The equation below can be used to calculate the time taken for unit A to kill unit B if the range of unit B exceeds that of unit A and unit A be must cover ground before engaging unit B directly in combat.


It should be noted that this equation does not account for units whose accuracy in combat is less than 100%. As a result, this equation should only be used if unit A has an accuracy of 100%. Also, it is essential that unit B's armor correspond to the type of damage inflicted by unit A. Lastly, this equation always assumes that unit A initiates the first attack in the sequence of attacks exchanged between the two units.

I will illustrate the proper use of this equation in the following example, where I will use it to calculate how long it would take for one Hoplite (unit A) to kill one Toxotes (unit B). This scenario is illustrated in the image given below.


In this example, the Hoplite (unit A) is out-ranged, and consequently, must first cover a certain amount of ground before it can engage the Toxotes (unit B) in direct combat. The question is: How much ground does the Hoplite have to cover?

In order to discern the distance that the Hoplite (unit A) must travel before it can engage in direct combat with the Toxotes (unit B), you must discern both the range of the Hoplite (unit A) and the Toxotes (unit B), and then find the difference between these two values. Toxotai have a range of 15 meters, while Hoplites possess a range of 0.3 meters. This means that the Hoplite must travel a minimum of 14.7 meters before it is within range of attack, as demonstrated in the equation given below.

Once you have discerned the distance that the Hoplite (unit A) must travel before it is within range of attack (14.7 meters), you must determine how long it will take the Hoplite to travel this distance by dividing it by the speed at which Hoplites travel (4.2 meters/second), as demonstrated below.

Toxotai possess 60 HP, which means that they can sustain 60 damage before dying. In order to discern how long it would take for the Hoplite (Unit A) to kill the Toxotes (Unit B) once it was within range of attack, you need to divide the amount of damage that Toxotai can sustain (60 damage) by the amount that the Hoplite inflicts every second (8 damage/second), as demonstrated below.
However, before dividing, you must account for any modifiers to the damage that the Hoplite (unit A) inflicts.

Ask yourself the following questions:

• Do Hoplites receive a bonus to the damage they inflict when battling Toxotai?
• How resistant are Toxotai to the type of damage that Hoplites inflict?

Hoplites receive no bonus to the damage they inflict when battling Toxotai. As a result, the modifier to the Hoplite's damage per second will be equal to 0%, as demonstrated below.

Toxotai are 15% resistant to hack attacks, which means that the percentage by which the Hoplite's damage is reduced is equal to 15%, as demonstrated below.

Once you have accounted for any modifiers to the damage that the Hoplite inflicts, you can proceed with the arithmetic.


Note that these are the base statistics for each unit without the inclusion of any upgrades.

As previously stated, unit A's survival is dependent upon the killing speed of the unit it faces (unit B), and thus, equal to the killing speed of the unit it faces (unit B). As a result, it is by calculating the period of one killing cycle for unit B that we discern the resilience of unit A. Thus, if unit B is out-ranged by unit A, such an advantage must be taken into account because it will extend the period of unit B's killing cycle, rendering unit A more resilient. This leads us to the next equation used in this system of ranking: the equation for the period of one killing cycle for unit B in the event that its range is less than that of unit A.

How long would it take for unit B to kill unit A if unit A's range exceeded that of unit B?

The equation below can be used to calculate the time taken for unit B to kill unit A if the range of unit A exceeds that of unit B and unit B be must cover ground before engaging unit A directly in combat.


It should be noted that this equation does not account for units whose accuracy in combat is less than 100%. As a result, this equation should only be used if unit A has an accuracy of 100%. Also, it is essential that unit B's armor correspond to the type of damage inflicted by unit A. Lastly, this equation always assumes that unit A initiates the first attack in the sequence of attacks exchanged between the two units.

I will illustrate the proper use of this equation in the following example, where I will use it to calculate how long it would take for one Hoplite (unit B) to kill one Toxotes (unit A). This scenario is illustrated in the image given below.


In this example, the Hoplite (unit B) is out-ranged, and consequently, must first cover a certain amount of ground before it can engage the Toxotes (unit A) in direct combat. The question is: How much ground does the Hoplite have to cover?

In order to discern the distance that the Hoplite (unit B) must travel before it can engage in direct combat with the Toxotes (unit A), you must discern both the range of the Hoplite (unit B) and the Toxotes (unit A), and then find the difference between these two values. Toxotai have a range of 15 meters, while Hoplites possess a range of 0.3 meters. This means that the Hoplite must travel a minimum of 14.7 meters before it is within range of attack, as demonstrated in the equation given below.

Once you have discerned the distance that the Hoplite (unit B) must travel before it is within range of attack (14.7 meters), you must determine how long it will take the Hoplite to travel this distance by dividing it by the speed at which Hoplites travel (4.2 meters/second), as demonstrated below.

Toxotai possess 60 HP, which means that they can sustain 60 damage before dying. In order to discern how long it would take for the Hoplite (Unit B) to kill the Toxotes (Unit A) once it was within range of attack, you need to divide the amount of damage that Toxotai can sustain (60 damage) by the amount that the Hoplite inflicts every second (8 damage/second), as demonstrated below.
However, before dividing, you must account for any modifiers to the damage that the Hoplite (unit B) inflicts.

Ask yourself the following questions:

• Do Hoplites receive a bonus to the damage they inflict when battling Toxotai?
• How resistant are Toxotai to the type of damage that Hoplites inflict?

Hoplites receive no bonus to the damage they inflict when battling Toxotai. As a result, the modifier to the Hoplite's damage per second will be equal to 0%, as demonstrated below.

Toxotai are 15% resistant to hack attacks, which means that the percentage by which the Hoplite's damage is reduced is equal to 15%, as demonstrated below.

Once you have accounted for any modifiers to the damage that the Hoplite inflicts, you can proceed with the arithmetic.


Note that these are the base statistics for each unit without the inclusion of any upgrades.

The Unit Ratio

By calculating the killing speed of both unit A and B, you can compare the killing speed of each unit, and thus, calculate how many of unit B are required in order to kill one of Unit A.

How many of unit B are required to kill one of unit A if the range of each unit is equal?

The equation below can be used to calculate the unit ratio for unit A if the range of Unit A is equal to that of Unit B.


It should be noted that this equation does not account for units whose accuracy in combat is less than 100%. As a result, this equation should only be used if both units have an accuracy of 100%. Also, it is essential that each unit 's armor correspond to the type of damage inflicted by the other. Lastly, this equation always assumes that each unit initiates the first attack in the sequence of attacks exchanged between the two units.

How many of unit A are required to kill one of unit B if the range of each unit is equal?

The equation below can be used to calculate the unit ratio for unit B if the range of Unit B is equal to that of Unit A.


It should be noted that this equation does not account for units whose accuracy in combat is less than 100%. As a result, this equation should only be used if both units have an accuracy of 100%. Also, it is essential that each unit 's armor correspond to the type of damage inflicted by the other. Lastly, this equation always assumes that each unit initiates the first attack in the sequence of attacks exchanged between the two units.

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If the range of Unit A exceeds that of Unit B...

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The equation below can be used to calculate the unit ratio for unit A if the range of Unit A exceeds that of Unit B.

h1		r1 – r2
	+
(1 – a1) (d2 (1 + m2))		s2
h2
(1 – a2) (d1 (1 + m1))

Where:

a1	=	The percentage by which unit B's damage is reduced (unit A's hack/pierce/crush armor)
a2	=	The percentage by which unit A's damage is reduced (unit B's hack/pierce/crush armor)
d1	=	The amount of damage dealt by unit A per second
d2	=	The amount of damage dealt by unit B per second
h1	=	The amount of damage that unit A can sustain before death (unit A's HP)
h2	=	The amount of damage that unit B can sustain before death (unit B's HP)
m1	=	The percentage by which unit A's damage is increased by modifiers
m2	=	The percentage by which unit B's damage is increased by modifiers
r1	=	Unit A's range
r2	=	Unit B's range
s1	=	Unit A's speed
s2	=	Unit B's speed

It should be noted that this equation does not account for units whose accuracy in combat is less than 100% or who have special attacks. As a result, this equation should only be used if both units have an accuracy of 100% and possess no special attack. Also, it is essential that each unit's armor correspond to the type of damage inflicted by the other.

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If the range of Unit B exceeds that of Unit A...

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The equation below can be used to calculate the unit ratio for unit A if the range of Unit A is less than that of Unit B.

h1
(1 – a1) (d2 (1 + m2))
h2		r2 – r1
	+
(1 – a2) (d1 (1 + m1))		s1

Where:

a1	=	The percentage by which unit B's damage is reduced (unit A's hack/pierce/crush armor)
a2	=	The percentage by which unit A's damage is reduced (unit B's hack/pierce/crush armor)
d1	=	The amount of damage dealt by unit A per second
d2	=	The amount of damage dealt by unit B per second
h1	=	The amount of damage that unit A can sustain before death (unit A's HP)
h2	=	The amount of damage that unit B can sustain before death (unit B's HP)
m1	=	The percentage by which unit A's damage is increased by modifiers
m2	=	The percentage by which unit B's damage is increased by modifiers
r1	=	Unit A's range
r2	=	Unit B's range
s1	=	Unit A's speed
s2	=	Unit B's speed

It should be noted that this equation does not account for units whose accuracy in combat is less than 100% or who have special attacks. As a result, this equation should only be used if both units have an accuracy of 100% and possess no special attack. Also, it is essential that each unit's armor correspond to the type of damage inflicted by the other.

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Averaging Unit Quotients

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If you calculate a particular unit's unit ratio for every other type of unit, you can assess its average efficiency in combat and compare it to that of other units. For example, let's say I calculated a Hoplite's unit ratio for each of the basic human units accessible to civilizations during the classical age:

	0.6470329670
	0.8492307692
	1.0000000000
	1.0451979120
	1.0454545455
	1.3479853480
	1.3760683761
	1.6000000000
	1.6890031210
	1.7331240188
	2.7132926059
	3.4502334744

Note that unit ratios displayed in red are likely inaccurate, and have not yet been confirmed in-game by experiments. Also, note that these are base statistics for each unit without the inclusion of any upgrades.

Notice how the table above lists the units in order of increasing unit ratio. This allows you to see exactly how much more or less effective Hoplites are against each type of unit in a way that was not possible before. For example, we can observe that Hoplites are least effective against Axemen and most effective against Turma. If you divide these two ratios, you can actually see how much more or less efficient (Hoplites are 81% less effective against Axeman than they are against Turma). You can also use the unit ratio to determine the number of Hoplites you would require in order to be successful, were you to deploy them against a certain unit type.

Another interesting thing you can do is average these ratios to see which civilization Hoplites excel at fighting against.

Civilization
Greek	1.1794112967
Norse	1.2564172120
Egyptian	1.6978165306
Atlantean	2.0318960066

Note that unit ratios displayed in red are likely inaccurate, and have not yet been confirmed in-game by experiments. Because they derived from the previous table of statistics, these too may be somewhat inaccurate.Also, note that these are base statistics for each unit without the inclusion of any upgrades.

We can see from these averages that Hoplites are the most effective against Atlanteans and least effective against Greeks.

You can also calculate a units average unit ratio by averaging its unit ratio for all other types of units. For example, the average of all the above statistics would be 1.5413852615. This overall average would obviously include many more units, but it could then be compared to other units (which would all have an average unit ratio as well) in a distribution.

You can also evaluate units based upon their cost and whether or not they pay for themselves (more information on this is pending).

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Unit Ratio Percentiles

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After compiling the various average unit ratios to be found in the game, you could plot them all on a distribution and then assign each unit a percentile, telling you even more about how it compares to all other units in the game.

If this explanation seems incomplete, that's because it is. I have a lot of things to contribute, but I am very limited on time.

Some of the things that are pending include:


Questions are always welcome.

On a side note, all numbers used are rough approximations. Depending on your computer, or your own personal estimations, you will have different variations in answers. For example, your computer may be running slow, and this may cause the unit speeds to decrease. It is also noted that the hills in a game of AoM can cause your units to go faster or slower, depending on the height of the incline.

[This message has been edited by Sagacious (edited 12-22-2011 @ 06:53 PM).]