*This week’s Hammer of Math takes a look at abilities which allow for Command Points to be refunded, and determines the likelihood they’ll be worth the cost.*

When 8th Edition came about, players quickly realized the immense value of maximizing the number of Command Points available for Stratagems. This impulse most egregiously manifested itself as the “Loyal 32”, an Astra Militarum battalion consisting of two HQ choices and 3 squads of 10 men which, under the 8th edition army construction rules, would give a player 5 additional CP to start with (there was also a brigade variation on this). Everywhere the Imperium could be found, from the awe-inspiring assaults of the Imperial Knights to the most secretive operations of the Adeptus Custodes, there would be 32 guys running around firing lasguns. In every one of these battalions there was a Warlord who was given **Kurov’s Aquila** and **Grand Strategist **so that every time either player used a Stratagem there would be a chance that the Loyal 32 would generate more CP. Games Workshop eventually curtailed the power of this approach, first by limiting the generation of CP to one per turn and then in 9th Edition changing the rules so that detachments *cost* CP instead of generating them. 9th Edition also saw every faction generate one CP at the beginning of their turn, significantly improving equity between factions.

### The Rules of Refunds

In 9th Edition CP start at a fixed value based on the size of the engagement (3 CP for a Combat Patrol, 6 for Incursion, 12 for Strike Force, 18 for Onslaught). At the beginning of each player’s Command Phase that player generates one additional CP. Players may use abilities and powers to gain or refund additional CP, but there are four caveats:

- You cannot gain or refund more than 1 CP per battle round.
- CP used before the battle or at the end of a battle round cannot be refunded.
- The CP bonus for having a battle-forged army, or CP gained at the start of the Command Phase via mission special rules, are exempt from the limit.
- The limit of gaining or refunding 1 CP per battle round does not apply to CPs gained via Stratagems, such as
**Priority Threat Neutralised**for Officio Assassinorum.

So how do we assess the value of relics, traits, and abilities that let players refund or generate CP? Since the majority of these effects come from either Warlord Traits or Relics we can assign a ‘value’ of 1 CP for each ability, since if these abilities are able to pay for themselves and generate additional CP it becomes an automatic decision to pay additional CP to include them in most circumstances. So let’s look at the likelihood they’ll recoup their cost, and the likelihood they’ll go beyond.

Before we do that, we have to note the upper bound: For most abilities and relics, the maximum number of CP a player can gain or refund during a game is 5, and realistically they’re likely to end up with less than that.

## Types of Abilities

The majority of abilities we’ll be looking at can be broken down into a four categories.

**Chance per command point spent.**Abilities like the Ynnari**Walker of Many Paths**, the Aeldari**Path of Command**, or the**Grand Strategist**Trait of the Astra Militarum provide a chance to refund a CP for every point spent. So if you use a Stratagem that costs 3 CP, you roll 3 dice to see if you get one back. The Nexos unit for Genestealer Cults has the**Strategic Coordinator**ability which can regain a CP on a 5+ or 6+ from either player.**Chance per Stratagem.**Generally reserved for abilities that gain a CP based on the actions of the opponent, this includes Relics such as**Kurov’s Aquila**for the Astra Militarum or the**Puretide Engram Neurochip**T’au Signature System. There’s one unique ability; the Harlequin**Player of the Twilight**Warlord Trait. Every time an opponent uses a Stratagem the Harlequin player rolls a D6. If that result exactly matches the number of Command Points that that opponent spent then the Harlequin player receives that many Command Points. The 1 CP per round limit explicitly does*not*apply to this ability, but once the ability has been triggered it cannot be used again until the next battle round.**Fixed chance to gain CP each battle round.**The**Master of the Codex**Warlord Trait for Adeptus Astartes Chapter Masters can gain this on a 4+ each round, while the Thousand Sons**Perfidious Tome**can gain a CP for the player on a 4+*or*give the opponent a CP on a roll of 1. This can also include abilities that are used once per round such as the Ultramarine’s**Scryer’s Gaze**(WC 7) or the Death Guard Tallyman**The Seven-fold Chant**. Both of these abilities effectively gain 1 CP per turn on a 2D6 roll of 7+, although the Ultramarine psychic power can be denied.**Psychic powers which target a CHARACTER and gain a CP if the power beats their Leadership on a 3D6.**This includes the**Interrogation**(WC 6) power for the Inquisition and the**Mind Raid**(WC 6) ability from the Obscuration discipline of the Adeptus Astartes. Both powers include a secondary effect to make the power applicable to targets other than CHARACTERS.

The fourth type is extremely situational and has a variable probability, so we’ll focus on the other three for this analysis.

### Probabilities

Because each D6 roll is independent and with a fixed probability, we can use a binomial distribution to calculate the probability that at least N successes will occur over X trials. We had some fun with this in the past when looking at the probability that a Space Marine character would be interred in a dreadnought, and it works equally well here. To read the charts below first select the target probability; 6+, 5+, 4+, or a 7+ on 2D6. Then look at the minimum number of desired successes and see how many D6 rolls (trials) are required to achieve a given probability. For example in order for an ability that triggers on a 5+ to have a better than even chance to gain at least 2 CP (and therefore actually provide a benefit) requires at least 5 rolls. Don’t forget that the number of D6 rolls is limited by the rules of the game; once an ability has generated a CP no further rolls can be attempted until the next round.

Here are some takeaways:

- The majority of abilities generate a refund or gain a CP on a 5+. The majority point is 2 rolls for 1 success, 5 rolls for 2 successes, 8 rolls for 3 successes, and 11 rolls for 4 successes. Given the lower number of CP available in smaller games it seems like Strike Force is the smallest size game where these abilities could are likely to actually generate a net benefit.
- Abilities that occur once per Command Phase really require a 4+ to be worth purchasing. If the ability triggers on a 5+ like
**Exalted Lord of Change**or**Tyrants Banner**then the model has to survive for all five rounds to have a 54% chance of producing more CP than it cost to acquire. **The Seven-fold Chant**and**Scryer’s Gaze**are two of the most reliable CP generators in the game, and have the added bonus of not actually costing any CP to obtain. The Tallyman is particularly interesting given how powerful dice roll modifiers are in this game; being able to reliably gather CP and use**Malicious Calculation**to boost attacks of friendly CORE units is pretty nice.- I don’t know what Orks did to deserve a Warlord Trait which only refunds CP on a 6+, but I’ve
**Got a Plan, Ladz!**is the only one and it’s about as bad as you would think. You could spend all 12 CP of a Strike Force game and there would still be a 10% chance that the Trait would be less than worthless.

In the end these abilities are a question of opportunity cost. The CP spent obtaining them, the limitations of generating a maximum of one CP per turn (and the subsequent loss of additional opportunities until the next round), and the Warlord Traits and Relics that could otherwise be obtained mean that some of these can be hard to justify. Generally speaking abilities that refund CP on a 6+ will seldom be worth spending an extra CP on where those that offer a 5+ (the most common) are likely to pay for themselves, but again you’ll have to weigh taking them against the opportunity cost of not taking another valuable relic or Warlord Trait. Personally I think things would be more palatable if players were limited to generating a maximum of 5CP per game (equivalent to 1 CP per turn). That way these abilities are still capped, but at least they have a better chance of being useful.

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