Call of Cthulhu sanity as gambler's ruin — when the math says you're going to zero

The cure ratio decides whether the chain converges

Same dichotomy as the HP problem (covered in expected strikes to kill):

• E[recovery] ≥ E[loss]: the chain doesn't absorb at zero in finite expectation. The investigator's sanity wanders around or trends back up. Player can take unlimited Mythos exposure in the long run.
• E[recovery] < E[loss]: strict drift toward zero. The chain absorbs at insanity with probability 1; the only question is when. The expected time to insanity is bounded by (starting sanity) / (E[loss] − E[recovery]).

Most published CoC scenarios pace per-investigation sanity loss at 1d6 or 1d10 per scene of Mythos contact, and per-downtime recovery at +1 per week of psychotherapy. Investigation pacing typically front-loads Mythos contact and back-loads downtime. The cure ratio rarely favours the player. The walk drifts; the only design lever is how long the player has before the descent.

Variance changes the whole picture

Two investigators, same starting sanity 50, both facing 15 sanity-loss encounters with the same expected loss per encounter (3.5). One faces 1d6 per encounter; the other faces 2d4 − 1.5 for the same mean but lower variance. Same expected total loss (52.5). Same expected ending sanity (-2.5).

But the distributions are different. The high-variance investigator (1d6) has a longer tail in both directions — they might stagger out at sanity 12 if the dice run cold, or hit zero by encounter 9 if they run hot. The low-variance investigator (2d4 − 1.5) clusters tightly around the mean — they reliably hit zero between encounters 14-15, with very little chance of finishing on either side of that band.

For a published scenario, the variance picture matters more than the mean. The "this scenario is rough on sanity" warning that authors put on adventures usually translates to "the variance of per-encounter loss is high enough that some investigators will absorb early". A scenario with the same mean sanity loss but lower variance gives every investigator a more similar experience; high variance is what produces "the table of investigators ended in vastly different mental states" stories. The variance and kill probability pillar covers this for HP — the same math applies, just relabeled.

The Mythos cap and the temporary insanity carousel

Two CoC-specific wrinkles the basic gambler's-ruin model doesn't capture out of the box:

The Cthulhu Mythos cap. Maximum sanity is 99 − Cthulhu Mythos skill. As an investigator learns more about the Mythos to fight cultists, their maximum sanity drops — even healthy recovery hits a hard ceiling. So the gambler's ruin chain has a shrinking pool ceiling over time. Compose this by chunking: run the chain one segment per Mythos-acquisition milestone (each segment uses the ceiling-of-the-moment as the starting sanity for the next), then compose the per-segment survivor distributions together. Standard pattern: a sequence of independent gambler's-ruin chains with input distributions tracked across segment boundaries.

Temporary vs. indefinite vs. permanent insanity. The "absorbed at zero" event in the HP analogy isn't a single thing in CoC. Hitting 0 sanity = permanent insanity (the character is retired). But losing 5+ sanity in one scene = temporary insanity (player loses control of the character for a few rounds). Losing 1/5 of starting sanity in a single day = indefinite insanity. The basic gambler's-ruin model treats all sanity loss uniformly. For richer modelling, split the input into "shock loss" and "drain loss" expressions and run two parallel chains — temporary insanity events become a separate per-round Bernoulli on top of the cumulative-loss Markov, composed via the same multi-track pattern.

What the engine ships today

The same call that computes strikes-to-kill against a target answers the sanity question:

# A 5-scenario campaign arc, sanity 50, 1d6 loss per scenario,
# +1d4 recovery per scenario (downtime woven in):
{:ok, r} = Diceplots.Engine.rounds_to_kill(["1d6"], ["1d4"], 50, 5)
r.converged            # true if the player makes it through all 5
r.expected_rounds      # if converged is false: E[rounds until insanity]
r.cumulative           # P(insane by round N) for each N

# An arc with 2d6 loss (heavier Mythos density) and no recovery —
# strict drift to zero:
{:ok, r} = Diceplots.Engine.rounds_to_kill(["2d6"], [], 50, 10)
r.expected_rounds      # ~ 50 / 7 ≈ 7.14 by linearity, exact value via engine

Multi-pool sanity questions (e.g., the party splits into two investigation tracks; one faces Mythos contact, the other recovers; recombine sanity at the end) compose by running separate rounds_to_kill evaluations per track and joining the marginal end-state distributions — same pattern used for chunked Mythos-cap segments above.

Where this matters in practice

The gambler's-ruin frame surfaces design and play decisions that the "did my investigator survive?" binary lens hides.

Scenario length tuning. If your sanity-loss / recovery cure ratio is 2:1, expected sessions to insanity is (starting sanity) / (loss − recovery). A POW 50 character (sanity ~70) with 1d6 loss / 1 recovery per scenario hits insanity at ~28 scenarios in expectation. A 6-session arc is well-within survival territory; a 12-session campaign with the same pacing is on the cusp. Variance still matters — the 95th-percentile time to insanity is much shorter than the mean for high-variance loss distributions.

The "down for the count" cliff. Once an investigator drops below (starting sanity − 20) they're at risk of indefinite insanity from any moderate shock. So the practical "broken" threshold isn't zero — it's about 20 below starting, where the next significant Mythos contact has a ~50% chance of crossing the indefinite line. Keepers tuning the late-game should compute "P(reach 20-below) by session N" rather than "P(reach zero) by session N" — the former captures when the player effectively loses control.

Cross-system: any drift-to-absorption mechanic. The same shape applies to Eclipse Phase stress, Delta Green's sanity track, Trail of Cthulhu's stability, Vampire's Hunger die — anywhere a finite pool drains via per-encounter loss against trickle recovery, the engine's rounds_to_kill substrate computes the survivors curve. You don't need a sanity-specific module; the framing transfers.