December 2022: newer quickstart, older rules
In December 2022, 2LM released a new version of the Household quickstart on DriveThruRPG. It seems they elected to go back to the older version of the system as used in their previous title, Broken Compass. The basic mechanic works as follows:
- Roll a pool of between two and nine d6s.
- Count matching sets.
- 2-of-a-kind: Basic success (valued at 1 point).
- 3-of-a-kind: Critical success (valued at 3 points).
- 4-of-a-kind: Extreme success (valued at 9 points).
- 5-of-a-kind: Impossible success (valued at 27 points).
- 6-of-a-kind: What a Hero! (valued at 81 points). (As of December 2022 this is only listed in Broken Compass; it remains to be seen whether Household adopts this in the final version.)
- Total up the points. (For purposes of plotting only. The point values reflect the "3X1" rule, though by the rules the exchange only goes one way.)
The first quickstart had a 2:1 ratio between success levels; this may reflect the fact that the blanks made it difficult to ensure Basic success while at the same the Jokers made Impossible success more achievable.
Broken Compass also has a core reroll mechanism.
- The character can potentially reroll up to two times.
- Only numbers that are not part of a success (i.e. only one of that number was rolled) can be rerolled.
- If the character rolled at least one success in the initial roll, they make take the first reroll ("Risk").
- If the first reroll does not add or improve any successes, they lose one of their existing successes (of their choice) and the roll is over.
- If the first reroll adds or improves at least one success, the character may either accept the result or make a second reroll ("All or nothing"). (The Household quickstart suggests that there will be a similar "All In" mechanic in the final version, but there are no explicit rules yet.)
- If the second reroll does not add or improve any successes, all successes are lost.
We'll analyze two strategies:
- "Known-difficulty": The difficulty is known; stop when that success level is reached.
- "Always-reroll": Take every possible reroll, even at the risk of busting.
For known-difficulty, for any decision point, the highest difficulty reached is the max of the current roll (since the player can always choose to exit if the successes are sufficient) and the potential reroll (which the player will surely take otherwise). If the result of the roll is a binary pass/fail, knowing the target difficulty is as good as knowing exactly what numbers will come up on your rerolls!
For always-reroll, we just compute the result of the second reroll (if the reroll is available).
The dotted lines represent the results of always-reroll, which can be thought of as the maximum penalty for taking risks.
Interestingly, there are a few places where extra dice appears to hurt the always-reroll strategy. My guess is that the extra dice can increase the chance of rolling well enough that it would be silly to continue, but always-reroll does so anyways.
If the character has Expertise in the task:
- They can make the first reroll even without successes in the initial roll.
- They don't lose any successes if the first reroll doesn't add or improve any success.
- They can make the second reroll ("All or nothing") even if the first reroll doesn't add or improve any success.
For large pools and low target successes, always-reroll can perform worse than in the non-Expertise case! This is due to the increased possibility of bad bets on "All or nothing".
You can run the script in this JupyterLite notebook. There are a lot of subtleties in the rerolling process, so let me know if you spot any discrepancies.