# Hammer of Math: Sources of Imbalance

This week’s Hammer of Math takes a look back at some of the bigger balance issues that have popped up over the last few years, and why seemingly minor modifications can lead to big issues.

One of my favorite articles in the history of Hammer of Math is the first one, written nearly two and a half years ago, and still relevant. The article discusses the basic math behind the attack sequence and focuses on the two types of values. Driving values are those that determine the magnitude of the outcome; specifically the number of attacks and the value of the damage. Gatekeeping rolls are thresholds which determine if a particular attack sequence is successful; hit, wound, save, and feel no pain. The number of variables is immense and provides a significant amount of flexibility for the designer. The equation below shows the attack sequence; P(X) is the probability of X occurring.

P(Damage) = P(Hit) x P(Wound) x (1 – P(Save)) x (1 – P(FNP))

Expected Damage = Attacks x Damage x P(Damage)

With this many variables, it seems reasonable to think that small changes can be applied as minor tweaks. While this is usually the case, sometimes a seemingly small change can lead to unintended consequences. This article will take a look at some examples, and discuss the math behind why it happens.

### Stacking Damage Reduction

Fans from 8th Edition will recall the saga of the Iron Hands Dreadnought and how the combination of a particular Relic and Stratagem made a unit so hard to kill that even a Warlord Titan could fail to do so. The reason for this was because the rules allowed a player to stack an effect that halved damage with another that reduced damage by 1. Due to the mathematical order of operations the division occurred first and then the reduced number had one subtracted from it. This meant that something with a Damage characteristic of 4 would only do a single point of damage. The presence of a multiplicative effect meant that it scaled with incoming damage, while also effectively scaling the subtractive damage reduction (plus any healing). GW corrected this by eliminating the halved damage effect from Iron Hands, reserving the ability for only marquee units such as the Avatar of Khaine and Morvenn Vahl, and preventing halving damage from being combined with other damage mitigation abilities such as subtraction or healing.

### Saves and Feel No Pain

The armour save has been with 40k since the beginning, and when combined with armour penetration it allows for units and weapons to have a lot of character. Often weapons with the ability to penetrate armour and deal a lot of damage will have a low number of shots, while weapons designed to deal with hordes will lack the punch needed to get past the armour save gatekeeper. For units that are particularly resilient Games Workshop also applies an ability to ignore wounds (previously called “Feel No Pain” back when 40k had universal special rules). Effectively a second save that cannot be modified, this provides another layer of flexibility for units.

The challenge with saves and feel no pain is that every roll that a model passes is another attempt that the attacker will have to make. Those attempts can also be saved and so forth. The end result is that a model effective has more wounds than if it had no save, and the number of effective wounds depends on the probability of success. We previously covered this when GW released Codex: Death Guard and changed howÂ Disgustingly Resilient worked from ignoring wounds through a die roll to direct damage reduction.

Effective Wounds = 1 / (1 – P(Save))

This is a perfect example of where minor adjustments to the roll can lead to significant impact. Going from a 4+ save (particularly an invulnerable one) to a 3+ is twice as effective as going from a 5+ to a 4+, and abilities which bestowed a permanent 3+ invulnerable save such as the old Auric Aquilas for Custodes were practically mandatory selections. This also was evident in Adeptus Mechanicus and the save bonus fromÂ Solar Blessing dogma in the Lucius Forgeworld; when combined with Light Cover it provided a +2 bonus against D1 attacks and made infantry significantly more resilient to the kind of attacks designed to kill them. GW corrected this by preventing the dogma from stacking with Light Cover back in July.

GW corrected the Auric Aquilas in the latest iteration of Codex: Adeptus Custodes, but made a similar error in terms of one of the abilities of The Emperor’s Chosen and the modification that Shield Host applies to Aegis of the Emperor.Â For most Custodes,Â Aegis of the Emperor allows them to ignore mortal wounds on a 6+. The Emperor’s Chosen, on the other hand, receive a major buff and ignore those same mortal wounds on a 4+. This means that instead of needing roughly 20% more mortal wounds to do the same job, attacking players need to effectively double their output to achieve the same number of mortal wounds that they would expect from an unprotected army. The challenge is that, by design, mortal wounds are fairly limited and simply doubling your output is not terribly feasible. This is one of the reasons why the consensus opinion on our Competitive Roundtable was that at the very least The Emperor’s Chosen should have their bonus reduced to a 5+.

### Bypassing Rolls

The traditional attack sequence allows weapons and units to be well defined by their intended roles. Something like a Tyranid monster will be large, with a high Toughness characteristic and a lot of wounds. Typically it will need a high Strength weapon to hurt as weaker weapons simply fail the wound roll. When this sequence is bypassed through something like the auto-wound effect of a radium carbine (hit rolls of 6 automatically wound non-VEHICLE targets) it can produce unintended consequences. When Codex: Adeptus Mechanicus dropped players immediately noticed the Enriched Rounds Stratagem, which cost only 1 CP but changed the roll from a 6+ to a 4+. This effectively eliminated the influence of Toughness and allowed units of 20 Skitarii Vanguard to hit much harder than they should have. The same FAQ that corrected the Lucius dogma also tookÂ Enriched Rounds behind the shed and put it out of our misery; the auto-wound was changed to trigger on a 5+ and the Stratagem cost 2CP for larger units.

### Bonus Attacks

8th Edition was the era of the Smash Captain, a quick flying, hard hitting Space Marine Captain who generally used a thunder hammer and a variety of faction-specific bonuses to rocket into the biggest threat on the board and break them in pieces. No Smash Captain was better than Wolf Lord Wolfsmash Smashwolf the 2nd, a product of the Saga of the Beast Psychic Awakening Supplement and unquestionably the most brutal unit I have ever seen during my time writing Hammer of Math. Fully boosted it could theoretically deal anÂ average of at least 85 wounds to a Warhound Titan, and even without every possible enhancement it was practically guaranteed to kill a Knight in one turn.

A key part of this was due toÂ Touch of the Wild, a 1 CP Stratagem which allowed every hit roll of 4+ to count as an additional hit, meaning the unit effectively got 50% more attacks (which is actually an underestimate when you consider that the Wolf Lord also re-rolled hit rolls of 1, the actual boost is closer to 55%). Since the effect is multiplicative it meant that things could easily escalate by boosting the Attack characteristic as much as possible (which was trivial for a Space Wolf). Normally abilities like this (such as Savage Fury orÂ Exhortation of Rage) only provide a double hits on a hit roll of 6+, but in this case GW decided to be more generous. The bonus was short lived as Touch of the Wild was completely deleted from their updated supplement released half a year later.

### Conclude on a 4+

Finding an equilibrium between novelty, effectiveness, and game balance is an extremely nontrivial task. By design many of the most powerful abilities in the game will only trigger on a very high roll such as a 6+. We’ve seen multiple examples of where a particular ability or Stratagem breaks this tradition and allows those abilities to go off 50% of the time. Historically this has led to an unbalancing result for both offensive and defensive effects. Fortunately when it does happen Games Workshop has been willing to make changes, and with the new update cycle those changes will ideally happen faster.Â

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