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I've recently learnt about CthulhuTech, which has a unique dice pool system:

CthulhuTech uses a proprietary game system known as "Framewerk" that focuses on a dice pool system. An unusual feature is the selection of numbers from the d10 rolls – a player can pick a single highest number, all like numbers, or straights of consecutive numbers, and use these to generate the result. The result is then added to a "base" consisting of a related attribute score bought by the player at character creation. The sum of the base and the result is then compared to target numbers or to the results of an opponent's rolls to determine success or failure.

So for instance a roll of 1,2,3,4,5,5,10 the player could pick either 1,2,3,4, 5,5 or 10 to add to their base score for a dice roll, before comparing it to the target number.

As a future game designer, I'd like to understand what sort of probability curve this creates for a given set of dice pool sizes, say 1-10 dice. This will hopefully let me see if I want to incorporate a similar dice pool system into a game I am planning.

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  • \$\begingroup\$ If I recall the CthulhuTech rules correctly, wouldn't the example roll allow taking '1, 2, 3, 4, 5' as a straight? I must confess that I mostly bought CthulhuTech in order to do a write-up of the statistics of the system. It also has a slight paradoxical effect, as half of the "add one more to the pool" actually makes fumbles more probable, and half makes a fumble roughly 10% as likely as the previous level. \$\endgroup\$
    – Vatine
    May 18 at 15:02
  • \$\begingroup\$ @Vatine I've never read the rules so I couldn't say. What you said makes sense though, feel free to edit that in if you're confident? \$\endgroup\$ May 18 at 16:20
  • \$\begingroup\$ I would have to go find the rulebook and double-check. Unfortunately, I am not sure exactly where it is, and there's enough demands on my time that "a comment" works, but "try to locate a rulebook I last saw within 6 months of the game being released" may be out of scope. \$\endgroup\$
    – Vatine
    May 19 at 8:39
  • \$\begingroup\$ @Vatine it was probably devoured by a great old one anyway, thanks regardless! \$\endgroup\$ May 19 at 10:46
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    \$\begingroup\$ I have had a look in the place where I last recall seeing the book, but it was not there. I will, over time, look in more places (and take a more careful look). \$\endgroup\$
    – Vatine
    May 21 at 18:50
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The result of having multiple ways to get choose numbers from the dice pool are quite chaotic.

The creator of anydice tweeted this anydice function for CthulhuTech dice pool rolls.

The graph of these results, seems to follow a fairly regular outline, despite some values being wildly less likely. For instance with a pool of 5d10, a result of 25 is 0.29% likely, but 24 and 27 are 4.47% and 5.85% likely respectively.

graph of two CthulhuTech dice rolls in 'at least' and 'graph' mode. 5d10 hits 10 at least 75% of the time, and 6d10 hits 10 at least 87.5% of the time.

It's unclear what the benefit of such a dice pool system is.

The anydice function time out for numbers over 7d10 so a full range of graphs isn't currently possible.

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    \$\begingroup\$ @HeyICanChan: The writing in CthulhuTech is ... confusing. The dice mechanic seems to have been designed by someone who found oWoD cool, but overly scientific. \$\endgroup\$ Mar 30 at 17:22
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    \$\begingroup\$ The original tweeted code has a bug in that repeated values break straights. I've submitted an edit to this answer with a corrected version: anydice.com/program/2509e \$\endgroup\$ Oct 31 at 7:29
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    \$\begingroup\$ That dice mechani is... Chaothic. How fitting for Cthulhutech. \$\endgroup\$
    – Trish
    Oct 31 at 10:41
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    \$\begingroup\$ @nick012000 I don't think so. Follow the anydice link and have a look \$\endgroup\$ Oct 31 at 11:14
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    \$\begingroup\$ @nick012000: Straights are required to be of at least length 3. \$\endgroup\$ Nov 1 at 1:26
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All pools up to 10 dice

Chance of rolling at least a given result.

(Technically the results go up to 100, but the chances above 60 are so small as to be practically invisible so I omitted them to give more space to the actually-visible part of the graph.)

Tabular form (Google Sheets).

Algorithm notes

How long did it take to compute a 10-die pool? 26 milliseconds. While I'm sure there's more micro-optimization possible, the important thing is that this can be computed using a polynomial-time algorithm rather than using exponential-time enumeration of all possible rolls.

Here's a general strategy for computing dice pools (where the individual dice are identical) in polynomial time:

  • Instead of making the outer loop go over dice, make the outer loop go over faces.
  • Formulate the problem as a time-varying Markov chain where timesteps = faces.
  • The states of the Markov chain are defined by the following:
    • The number of dice that have been consumed so far.
    • The best score among the dice seen so far.
    • Any other information you need to compute future scoring. In this case, this is the running straight length.
  • The initial state (with 100% probability) is:
    • Zero dice consumed.
    • Zero (or minimum) score.
    • Zero running straight length.
  • The transitions out of each state at a particular timestep (face) are defined by:
    • The chance of a particular number of the non-consumed dice rolling the current face. This is just a binomial distribution, which crucially can be computed in polynomial time rather than considering all \$2^N\$ ordered possibilities.
    • The new number of consumed dice = previous number of consumed dice + dice that rolled the current face.
    • The score produced by the dice so far. In this case it's the score of the just-rolled matching set, the current straight, or the previous score, whichever is highest.
    • The effect of those dice on the running straight length, i.e. if at least one die rolled the current face, the straight length increases by one, otherwise it resets to zero.

As long as the state space and score updates are polynomial, the algorithm as a whole will be polynomial-time.

Example code. Note though that there are some differences to the above, and this code and API may change in the future.

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