Maths of the Psychic Phase, Part 1: Description

The new 7th edition psychic phase presents some interesting mechanics to model mathematically. If you're not familiar with the system, perhaps because you more routinely play a non-psychic race like Dark Eldar, Tau, or Necrons, each power has a cost to manifest, of between 1 and 3 warp charge. Warp charge dice are generated at the start of each psychic phase, at the rate of one per friendly mastery level, plus d6 because GamesWorkshop. Each warp charge dice can be converted to an actual warp charge on the roll of a 4+.

The catch is that you have to allocate dice to manifesting a power before finding out if they will convert to warp charge. The danger comes from the old standby, Perils of the Warp, that kicks in if two or more 6s are rolled on the dice for manifesting a power. The balance arises from the desire to manifest as many powers as possible per phase, for as efficient a cost, at minimal risk of inflicting Perils.

To the maths!













We'll start with some simple descriptive maths.

The chart above shows the expected outcome for a simple power with a cost of 1 warp charge, after attempting to manifest it with between 1 and 10 dice. Clearly, the extra dice are counterproductive. The highest odds of success without the burden of Perils is when 4 dice are rolled. 3 dice provide a very similar outcome however, for 1 dice less cost and a lower chance of Perils.


I should point out that for now, I'll be treating Perils as strictly undesirable. In future posts I'll assess the cost of Perils in a more nuanced way.

How does the chart look for powers that cost 2 warp charge?

Now, the number of dice required to get good odds of success start to climb. Firstly, the chance of achieving success without Perils never climbs above 62.7% at a cost of 6 dice, meaning that if you want to have a good chance of casting these powers you are going to need to accept a high rate of either failure or Perils. Similar success rates with lower rates of Perils can be seen when using 4 or 5 dice if you are risk-averse, and at 7 dice or more diminishing returns and higher chances of incurring Perils will cause you some concern.

Clearly, these powers are more difficult to manifest. But what about the biggest powers, the warp charge 3 powers?



These powers are, simply, very difficult to manifest, and even harder to pull off without Perils. Notice that at no stage is the chance of success without incurring Perils higher than 50%! A whopping 9 dice are needed to achieve the best chance of this success, but with a further 43% chance of incurring Perils, you will want to weigh this cost very carefully.

At this point it seems worthwhile to contrast the chances of success with the previous system, which was a simple Ld test to manifest a power, and the warp charge cost was purely a resource cost rather than an impediment to manifestation.



Yup, as a Dark Eldar player I am quite happy with the new system, thank you very much. Even the bottom rung of Ld 8 psykers could safely manifest powers at rates much greater than current psykers could hope to achieve for warp charge 2 or 3 powers; ld 9 was enough to safely manifest warp charge 1 as well as can be done for now at a much lower cost in resources. The pendulum has definitely swung away from ease of casting, although to balance that some of the powers are definitely stronger.

If the chance of successful manifestation seems the most important limiting factor to consider, the resource cost is the next. You will note that successfully manifesting any power with a good chance of success requires more dice than the psyker is individually contributing to the warp charge pool. Therefore, if you desire to cast many powers, you must optimise your use of resources.

Without yet considering the additional costs of Perils, or the prioritisation of powers within a turn, I'll look briefly at optimising the number of powers that can be successfully (for now, that also means without incurring Perils) manifested given their warp charge cost and dice pool availability. To do this, I imagine a hypothetical warp charge pool of 60 dice, and measure how many powers can be successfully manifested by allocating each power a constant pool of between 1 and 10 dice.



The answers are informative. Despite the fact that manifesting a 1 warp charge power with 1 die is only 50% likely, much lower than the 80.6% optimal rate, manifesting these powers with 1 die each is the best way to manifest the largest number of powers.

When manifesting 2 warp charge powers, using 3 or 4 dice are both efficient outcomes. 3 dice is slightly more efficient. Although using 4 dice is 13 percentiles more likely to manifest successfully than using 3, consistently using 3 dice to trial all of your warp charge 2 powers will give you the most psychic bang for your buck.

At a cost of 3 warp charge, the difficulty in manifesting the powers heavily influences the number of dice required even to cast efficiently. The best outcome is at 6 dice, which is only 3 percentiles below he highest possible chance of a successful and safe cast.

To better understand the optimal use of the psychic phase in 40k, I will explore the cost and benefit functions of manifesting powers more closely, and include both Perils and the Deny the Witch mechanism. For now, accept that you can cast a lot of powers at high risk of failure, or few powers at a higher but still only moderate chance of success.

And if you play a non-psychic army, don't stress.