Heroscape Magic Number / Sweet Spot Compilation

[mccombju]

There seems to be an agreement (dare I say consensus) among members of this forum about the importance of certain numerical thresholds as they pertain to Heroscape. For example:

The difference between three and four defense is one of the key defensive thresholds.

You need to draft at least two common squads to realize their true potential.

Often such assertions are based not only on a wealth of experience but can also be validated statistically. Practically, then, these numerical thresholds can be considered "magic numbers" or "sweet spots" when evaluating decisions or army cards in Heroscape.

My hope with this thread is to compile a list of verifiable "magic numbers" or "sweet spots" as they pertain to Heroscape.

Initial forays into creating such a list may not be entirely accurate, so it will be very important for members of this forum to challenge erroneous assumptions with statistical rigor (and/or other evidence when relevant).

Without further ado, the "magic numbers" presented below are based upon what I have read in other threads. Kroc originally contributed a number of calculations and summaries in the thread here, and I have tried to validate these figures with reference to other threads and the Heroscape Matchup Calculator. Finally, I calculated and inferred the figure count / life point number independently, based upon a list of tournament winning 500 point armies.

Heroscape "Magic Number"/“Sweet spot” Approximations:
Sisyphus' Contribution

Much of the subsequent analysis relies on the work done by Sisyphus in calculating the probabilities for wounds inflicted for given sets of attack and defense dice. If you are unfamiliar with his work, then I suggest you check it out.

The following figures are graphical representations of Sisyphus' work.

Probability of Inflicting a Wound:
*shows the probability of inflicting at least one wound OR of killing a defending squad figure



Probabilistic Damage Inflicted:
*shows the amount of damage inflicted upon defending multi-lifepoint heroes as determined by probability

The following figures depict the same information, albeit on different scales (vertical axes).







Heroscape "Magic Number"/“Sweet spot” Approximations:
mccombju's contribution

*Source: mccombju calculation
Army figure count / number of life points: ~25
- Seems to coincide with the number of starting zone hexes in the 500 point format

*Source: Forum discussion
Common squad army cards: 2 and up
- The rationale is that part of the point cost of common squads is attributed to the versatility of activating any of the common squad members (up to the number of units on a single card)

*Source: Grungebob stat tracking and forum discussion
Number of melee squad attacks per turn: 3 (or less)
-The rationale must have something to do with the melee figure's move speed, disengagement restrictions, and/or the hexagonal terrain configuration.

*Source: Heroscape Matchup Calculator
Anti-Stealth Dodge attack activations with Q9 Queglix Gun: 9x 1A
- Expected attack value
= (0.5 skulls rolled per attack) x (9 attacks) = 4.5 skulls rolled
- Expected Stealth Dodge defense value = 1 shields rolled
(i.e. one shield rolled will block all damage)

*Source: Heroscape Matchup Calculator
Anti-One Shield Defense attack activations with Q9 Queglix Gun: 3x 3A
- Expected attack value
= (1.5 skulls rolled per attack) x (3 attacks) = 4.5 skulls rolled
- Expected One Shield defense value = 1 shields rolled
(i.e. one shield rolled will limit defender to a maximum of one wound per attack)

*Source: Jexik
Median (Average) Melee Attack: 3
- Expected attack value = 1.5 skulls rolled
- Expected attack value from height (i.e. 4A) = 2 skulls rolled

*Source: Jexik
Median (Average) Melee Defense: 4
- Expected defense value = 1.33 shields rolled
- Expected defense value from height (i.e. 5D) = 1.66 shields rolled

*Source: Jexik
Median (Average) Ranged Attack: 2
- Expected attack value = 1 skulls rolled
- Expected attack value from height (i.e. 3A) = 1.5 skulls rolled

*Source: Jexik
Median (Average) Ranged Defense: 3
- Expected defense value = 1 shields rolled
- Expected defense value from height (i.e. 4D) = 1.33 shields rolled

*Source: mccombju threshold and above in "Probabilistic Damage Inflicted" figures shown below
Magic Number for Attack dice : 3
- Expected attack value
= 1.5 skulls rolled (melee base stats)
= 1.5 skulls rolled (ranged with height i.e. 2A plus height advantage)
- Expected value of median/average units’ defense
= 1 shield rolled (ranged)
= 1.33 shield rolled (melee)
(i.e. Expected value of attack > Expected value of median/average units’ defense)
(i.e. Skulls > Median/average units’ Shields)

*Source: mccombju threshold and below in "Probabilistic Damage Inflicted" figures shown below
Defense dice: Defense of 4
- Expected value of median/average units’ attack = 1.5 skulls rolled
- Expected defense value = 1.33 shields rolled
- Expected defense value from height (i.e. 5D) = 1.66 shields rolled
(i.e. Expected value of the median/average attack 3 < Expected defense of 5 with height)
(i.e. Median/average units’ Skulls < Shields when on height)

*Source: Forum discussions
Anti-Q9 attack dice: 5 and up
- Expected attack value = 2.5 skulls rolled
- Expected Q9 defense value = 2.33 shields rolled
(i.e. Expected value of attack > Expected value of Q9 base defense of 7)

Probabilistic Damage Inflicted:
*shows the amount of damage inflicted upon defending multi-lifepoint heroes as determined by probability



The following figures depict the same data, albeit on different scales (vertical axes).





These three figures were created using Sisyphus' probability tables. This is a work-in-process.

The rationale behind these figures was to demonstrate the optimal use of attack power given knowledge of the number of defense dice that the defending player's multi-lifepoint hero has.

That is,

IF
the defending multi-lifepoint hero has K defense dice (e.g. 7)

THEN
you should optimally attack the multi-lifepoint hero with army cards and figures that enable you to use M or more attack dice (e.g. 5) in order to inflict more than 0.5 wounds per attack

SUCH THAT
you maximize the probabilistic damage inflicted through a combination of attack dice rolled and attack activations granted by attacking army cards.

As can be seen from the attached figure, the optimal (according to mccombju) use of attack power is shown by a trendline (all sets of (K,M) dice) hovering above 0.5 wounds inflicted per attack against a multi-lifepoint hero.

Optimal use of attack power occurs when the number of attack dice is greater than or equal to the cutoff values given by the threshold trendline. This trendline represents a greater than 50% chance to inflict one wound per attack against a defending multi-lifepoint hero.

Another way to conceptualize this is that below the threshold, more than 2 attacks are expected/required to inflict one wound.

Heroscape "Magic Number"/“Sweet spot” Rationalization:

I have included the hypothetical Attack Dice Magic Number and Defense Dice Magic Number as proposed in the compilation section and graphs shown above. One way to interpret these "magic numbers" is to suppose that they represent winning combinations against the median/expected attack or median/expected defense dice of opposing figures. For instance, if the armies you are facing always roll 3 defense dice, then 3 attack dice rolled (be it a base of 3A or 2A plus height) will represent the Attack Dice Magic Number.

More Detailed Aside:
One reason that 3 attack dice is a magic number, in above example, is that this is the lowest cost option (in terms of army card points) to achieve an optimal rate of dealing damage (power output). The rate of dealing damage is optimal (0.75 wounds inflicted per attack), because fewer than 2 attacks are probabilistically expected/required to inflict one wound. 3A is the lowst cost option, because higher numbers of attack dice would cost more points; the numbers of attack dice are a function of the points on the army cards themselves.

Moving on, we look at the case for "magic numbers" or "sweet spots" in Heroscape on the basis of the statistic distribution of attack and defense dice. When discussing a statistical distribution, it is important to understand that the median approximates center of the data, and the distribution considers how closely the rest of the data is clustered around said center of data.

Heroscape has been designed so that figures have relatively low attack and defense dice numbers, on average (e.g. 4A is the highest attack value of a non-boosted ranged figure and 3A is the highest attack value of a non-boosted ranged squad figure). As Jexik pointed out, the median attack/defense statistics for tournament-viable melee and ranged figures are 3A/4D and 2A/3D, respectively. Even the attack and defense dice values of units considered non-tournament-viable are clustered in a tight distribution around the median values.

As such, we can define the attack dice magic number and defense dice magic number in Heroscape as follows.

Definition of Attack Dice Magic Number (by mccombju):

An Attack Dice Magic Number is one that is probabilistically above the threshold proposed (by mccombju) for an optimal rate of inflicting wounds given the expected median defense dice of opposing figures.

Definition of Defense Dice Magic Number (by mccombju):

A Defense Dice Magic Number is one that is probabilistically below the threshold proposed (by mccombju) for an optimal rate of inflicting wounds (i.e. a greater ability to prevent wounds from being inflicted) given the expected median attack dice of opposing figures.

The above analysis is work-in-process and represents how I have tried to rationalize "magic numbers" and "sweet spots" in Heroscape. Of course, the preliminary conclusions I have drawn must be subject to debate and constructive criticism. Other interpretations are most welcome.

Heroscape "Magic Number"/“Sweet spot” Approximations
Kroc's Contributions (vs. Squad Figures)

Kroc's rationale for his contributions is that the player must optimally deploy the attack dice of activated army cards when combating squad figures. The emphasis in his analysis is on pairing attack numbers with defense numbers so that attack power isn't wasted (i.e. avoid squad figure overkill).

*Source: Kroc
Attack dice vs squad figures: Attack dice # equals Defense dice #
e.g. 3A vs. 3D
- Expected attack value = 1.5 skulls rolled
- Expected value of defense = 1 shield rolled
*Corroborated by mccombju. See figure below.



Attacks below or equal to Kroc's Rule of Thumb (shown in blue) and above the mccombju threshold (shown in green) represent optimal attack power when combating squad figures.

That is,

IF
you have X attack dice (e.g. 5 attack dice)

THEN
you should attack a squad-based defending figure with Y or more (e.g. 5 or more) defense dice

SUCH THAT
the (X,Y) combination does not lie on the mccombju threshold where Y is greater than 3.
(e.g. 5A > 5D is optimal; 5A > 6D is optimal; but 5A vs. 7D lies on the threshold and is bordering on suboptimal)

*Source: Kroc
Anti-Shields of Valour attack activations with Q9 Queglix Gun: 9x 1A
- Expected attack value
= (0.5 skulls rolled per attack) x (3 attacks) = 4.5 skulls rolled
- Expected Shields of Valour defense value = 2.66 shields rolled
*Validated by mccombju.
*Contradicted by Heroscape Matchup Calculator (HMC must be incorrect)
Anti-Shields of Valour attack activations with Q9 Queglix Gun: 3x 3A
- Expected attack value
= (1.5 skulls rolled per attack) x (3 attacks) = 4.5 skulls rolled
- Expected Shields of Valour defense value = 2.66 shields rolled



Any combination of attack dice and defense dice between Kroc's Rule and the proposed 'variant/complement to Kroc's rule' is an optimal use of attack power.

*Source: Kroc
Anti-Gift of the Empress Aura attack: ~Defender's defense doubled=6
- Expected attack value = 3 skulls rolled per attack
- Expected defense value = 2??? shields rolled
(i.e. 6A vs. 2x3D=6D???)
*mccombju opinion: Kroc overestimated number of optimal attack dice
Threshold selected by Kroc:
Greater than 70% chance of wound inflicted, but this is greater than the thresholds for his other Rules of Thumb.
*variant proposed to complement analysis by Kroc.



Any combination of attack dice and defense dice between the 70% cutoff and the proposed 'variant/complement to Kroc's rule' is an optimal use of attack power.

*Source: Kroc
Anti-Gorillinators attack: 5
- Expected attack value = 2.5 skulls rolled
- Expected defense value = 1.66??? shields rolled
(i.e. 5A vs. 5D???)
*mccombju opinion: Kroc almost overestimated number of optimal attack dice.
Threshold selected by Kroc:
Roughly 70% chance of wound inflicted, but this is on the higher end of the thresholds represented by other Rules of Thumb.
*variant proposed to complement analysis by Kroc.



Any combination of attack dice and defense dice between Kroc's Rule and the proposed 'variant/complement to Kroc's rule' is an optimal use of attack power.

Stinger Drain Analysis:

Stinger drain ONLY ever makes sense statistically when all of the following criteria are met:

1. The Stingers can't get height advantage. (Exceptions: Stingers on height vs. Major Q9 with Raelin or Captain America with Raelin)

2. The Stingers are contending with a figure that has:
a. Sufficient defenses to withstand numerous attacks from the Stingers (i.e. a multi-lifepoint hero)
b. High offensive output per Order Marker
2. (contd') meaning that this opposing figure can kill its points worth in Stingers before it dies. (i.e. Stingers vs. a multi-attacking hero with 5D or more on level ground [Exception: Don't bother vs. Laglor with Raelin b/c he just doesn't kill fast enough]; Stingers on level ground vs. Braxas, Zelrig, Nilfheim, or Cyprien)

ALTERNATIVELY

Stinger Drain makes sense statistically when all of the following criteria are met:

1. You have Su-Bak-Na in your army.

2. The Stingers can't get height advantage. (Exceptions: Stingers on height vs. a multi-attacking hero with 5D or more when Stingers attacking from height; Stingers on height vs. Braxas, Zelrig, Nilfheim, or Cyprien)

3. The Stingers are contending with a figure that has:
a. Sufficient defenses to withstand numerous attacks from the Stingers (i.e. a multi-lifepoint hero)
b. High offensive output per Order Marker
3. (contd') meaning that this opposing figure can kill its points worth in Stingers before it dies. (i.e. Stingers vs. a multi-attacking hero with 3D or more on level ground)



Heroscape Defensive Survival Point Economy Rankings:

I calculated the defensive point depletion rates in the face of various attack thresholds (using Sisyphus' probability tables) to determine which Heroscape figures represent the best bargain in terms of survivability and point economy.

Hero Defensive Survival Point Economy (SPE)
= Hero Defensive Point Depletion Rate [Points Lost per Attack]
= (Probabilistic Damage Inflicted) x (Points Lost per Wound)

Squad Defensive Survival Point Economy (SPE)
= Squad Defensive Point Depletion Rate [Points Lost per Attack]
= (Probability of at least 1 wound) x (Points Lost per Figure Killed)

Simulation of the Proportions of Attacks that Defending Figures will face in the Heroscape Metagame:
65% of attacks will be 3A
25% of attacks will be 4A
10% of attacks will be 5A



What does SPE tell us?

A low SPE value (and high SPE ranking) indicates:

1. Units that would do well in a battle of attrition as dictated by point economy.

2. Units that serve well as melee fodder from a point economy perspective.

3. In the event that start-zone sizes were unrestricted and that there were no locations of tactical significance on a Heroscape map (i.e. no glyphs, no terrain variation, the entire map is flat, etc.), SPE tells us what army compositions would be the best melee screens and/or attrition figures.

Heroscape Tactical Battlefield Positioning (TBP) Rankings:

I ranked the survival probabilities of defending figures for various attack thresholds (using Sisyphus' probability tables) to determine which Heroscape figures have the best likelihood of surviving attacks and retaining positions of tactical significance on the Heroscape battlefield.

Hero Survival Probability
= Hero Tactical Battlefield Positioning (TBP)
= 1 - (Hero Death Probability)
= 1 - ((Probabilistic Damage Inflicted) / Life )

Squad Figure Survival Probability
= Squad Figure Tactical Battlefield Positioning (TBP)
= 1 - Squad Figure Death Probability
= 1 - (Probability of at least 1 wound)

Simulation of the Proportions of Attacks that Defending Figures will face in the Heroscape Metagame:
65% of attacks will be 3A
25% of attacks will be 4A
10% of attacks will be 5A



What does TBP tell us?

A high TBP value (and high TBP ranking) indicates*:

1. Units that would serve well as glyph holders in the immediate short-term (i.e. vs. a single attack) from a survival standpoint.

2. Units that will be able to hold height advantage in the immediate short-term (i.e. vs. a single attack) from a survival standpoint.

3. Units that can be used to protect or provide cover for other units in the immediate short-term (i.e. vs. a single attack) due to their defensive resiliency.

4. Units that will be able to hold a choke point in the immediate short-term (i.e. vs. a single attack) from a survival standpoint.

5. In the case of squads, units that would serve well as glyph holders from a survival standpoint.

6. In the case of squads, units that would be able to successfully retain height advantage from a survival standpoint.

7. In the case of squads, units that can protect or provide cover for other units due to their defensive resiliency.

8. In the case of squads, units that would be able to hold a choke point from a survival standpoint.

[gorthan313]

Woah! +rep for that interesting post!

[ZBeeblebrox]

Very interesting. I like the Anti Q9 and Stealth dodge stats.

[tom]

Interesting and useful post. I'd +rep you, but I don't think I have enough myself to make it count!

[dok]

"You must spread some Reputation around before giving it to mccombju again."

Nice work. Is that a Matlab plot?

[mccombju]

Originally Posted by dok

Is that a Matlab plot?

I generated the plot using MSExcel, and an adhoc recalculation of Sisyphus' probability tables (the original numbers would work a bit better i.e. this is a work-in-process).

The trendlines and whatnot are my interpretations, and they are either linear regressions or smoothed lines fit to scatter-plot points.

*EDIT* the figures are now based strictly on Sisyphus' work i.e. no recalculations were harmed in the creation of these figures

[Mr Migraine]

Nice. That will come in handy, especially when trying to kill Q9.

[1Mmirg]

I've added this to the Index, grouped with Sisyphus' and Kroc's work. Thanks, mccombju--I look forward to digesting this further.

[mccombju]

Updated first post:

+ Updated "Probabilisitc Damage Inflicted" figure
>>>>> Added Anti-Q9 Magic Number
>>>>> Added attack dice 5A through 10A
+ Removed trendline equation
+ Added "Probability of Inflicting Damage" figure comparing the my results with those proposed by Kroc
+ Added "Gift of the Empress Aura" probability figure
+ Added "Shield of Valour" probability figure
+ Added "Tough" probability figure
+ Took out predictions and summaries relating Kroc's Rules of Thumb to Heroscape Matchup Calculator
+ Made distinction between Kroc's squad-based analysis and my multi-lifepoint hero analysis

[Jexik]

It's a bit late for me to dive into the tables and really understand them, but I will point out that 3A/4D are the stats of the Knights of Weston, and that 2A/3D are probably the 'thresholds/magic numbers' for ranged (6 range or more) squads, as evidenced by the 4th Mass, mostly because of height advantage.

[mccombju]

Updated first post:

+Added Sisyphus acknowledgement section
>>>>> Added link to Sisyphus' Probability Tables
>>>>> Added "Probability of Inflicting Damage" figure
>>>>> Added "Probabilisitc Damage Inflicted" figures (Scale: 1 wound; 3.5 wounds; 5 wounds)
+ Added median effective attack and defense values for melee and ranged figures (credited to Jexik)
+ Updated Magic Number for Attack Dice
+ Updated Defense Dice Magic Number
+ Updated "Probabilisitc Damage Inflicted" figure
>>>>> Indicated visually (via red trendlines) the instances for which the attack dice magic number holds true
>>>>> Indicated visually (via pink trendlines) the instances for which the defense dice magic number holds true
>>>>> Indicated visually (via orange trendline) the instances for which the anti-Q9 attack dice magic number holds true
+ Added "Probabilisitc Damage Inflicted" figures (Scale: 1 wound; 3.5 wounds)
+ Updated magic number proposal/explanation
>>>>> Magic numbers represent probabilistically winning combinations against the attack and defense of opposing figures
>>>>> Aside presented: Concept of lowest cost option for optimal rate of dealing damage
>>>>> Demonstrate the bias by designers of Heroscape for low numbers of attack and defense dice
>>>>> State that the distribution of attack and defense dice is tightly clustered around the median values for tournament-viable figures
>>>>> Provided definition for attack dice magic number in Heroscape
>>>>> Provided definition for defense dice magic number in Heroscape

[dragonite]

Originally Posted by mccombju

I generated the plot using MSExcel, and an adhoc recalculation of Sisyphus' probability tables (the original numbers would work a bit better i.e. this is a work-in-process).

The trendlines and whatnot are my interpretations, and they are either linear regressions or smoothed lines fit to scatter-plot points.

Somehow I think I understood that.