Ballpark Figures collapses Yahtzee’s position space by ~640,000× and solves it by backward induction — but solves expected points, not winning, and then has to mint a second currency to talk about the actual game.
Source: Ballpark Figures — “I Solved Yahtzee*” (33:16, 2026-07-19) — transcript · raw VTT Sibling specimen: Battleship — 30 Billion Boards — same creator, his first video, already in the vault. He explicitly calls back to it (“both are basically one-player games disguised as multiplayer games”). Sequel specimen: Hangman — Solving Both Sides — same creator, his third. The state-space collapse recurs but pointed the other way: here reduction is the solver’s tool, there the chooser wields it as a weapon (force a suffix → turn a 7-letter game into a harder 4-letter one). It also closes the EV-vs-win thread this page opens — he computes both objectives and reports where they diverge.
A specimen page (dated, source-anchored). Portable home TBD — see the seeds below; that judgment belongs to the discussion, not to this draft.
1. The state space, and its collapse.
| Stage | Positions | What was thrown away |
|---|---|---|
| Raw | 259 trillion | — |
| Equivalence classes | 80 billion | Which boxes hold which numbers; only (boxes filled, top score, bottom score, # yahtzees) survives |
| EV-only reduction | 405 million | Bottom-section score entirely; yahtzee count → one eligibility bit; top score capped at 63 |
The reduction rule is stated explicitly: keep only information that affects future points. Bottom-section score is discarded because it’s sunk — it changes your total but not any future decision. Top score above 63 is discarded because the bonus is already banked.
Supporting counts: 5 dice = 7,776 ordered outcomes but 252 distinct multisets; 756 situations per turn; ~8,000 box-fill combinations; 385 billion ways to fill a scorecard.
2. The method. Backward induction / dynamic programming. Start at the last roll of the last turn, compute EV of every keep-set, take the max, step back one roll, repeat to the opening. Result: EV ≈ 255, median 248.
3. Endgame per-box results (last-turn, going all-out): yahtzee <5%; large straight ~25%; full house ~1/3; four-of-a-kind 28% (≈6 pts avg); three-of-a-kind 71% (≈15 pts avg); small straight high — which is why it’s the box to keep alive longest. Chance has a clean closed form: a die’s EV is 3.5 with one roll left, 4.25 with two, so keep 5–6 on the first roll and 4–6 on the second → 23⅓ average.
4. The counterintuitive bit. Three/four-of-a-kind score the sum of dice, so they mix all-or-nothing with score-maximizing. With three 1s and two 6s, keeping the sixes beats keeping the ones — lower success chance, higher payoff conditional on success.
5. The re-scoring. Raw points produce a weirdly multimodal score histogram. He explains it by minting a second currency: extra yahtzee = 4, {top bonus, large straight, yahtzee} = 2, {3oak, 4oak, full house, small straight} = 1. The peaks are then just “missed N bonus points.” Claim: between two optimal players, a 1-point lead on this scale ⇒ 97% chance of winning the game.
6. Option-value curve. A plot of each box’s EV conditional on still being unfilled at round N. Safest to zero: four-of-a-kind in the first half, yahtzee in the second (worth more, but harder). Last to give up: small straight, then three-of-a-kind, then large straight.
7. Top bonus. 63 = three of every number. ~2/3 achievable; track pace against “3 per filled box.” Putting 12 in the sixes box costs ~6 points directly and ~17 more in bonus EV.
8. Opening. Best turn-1 outcome is a yahtzee; second best is four sixes (24 in the six box), then four 5s, large straight, four 4s, four 3s. Never fill four-of-a-kind on turn 1 (the top section always pays better). Worst opening roll: 2-3-4-4-6 → dump 19 into chance.
9. Where it stops. Multiplayer is not solved — each added player is exponentially harder, and he says his hardware couldn’t do it. Multiplayer advice degrades to heuristics: aggressive when behind, conservative when ahead. Median needed to win rises with the field: 248 (1 opp) → 268 (2) → 279 (3) → 307 (5); then 3 yahtzees to win a box of 80 sheets, 4 at a 5,000-person Hasbro picnic, 6 for Rhode Island, 9 for all 8.3 billion humans, and a perfect 1,575 somewhere around 51 quintillion players.
“There isn’t too much here, just a math-confirmed state machine. The end result is that it is good to mathematically confirm what is probably well-known strategy.”
The deflation is the finding. Nothing in the strategy output would surprise a heavily-experienced Yahtzee player — the value is that folk knowledge got formally verified, and that the deep-math treatment surfaces moves that aren’t inherently obvious but “probably make sense to a player with lots of experience.” Chris keeps the channel around for the method (deep math on even the simplest games), not for novel conclusions.
Chris’s three keepers from the video:
He grants the critique and locates it in the video’s own words: the creator flags the difference between playing optimally and beating your opponent. Chris’s framing of the sequence: the simplifications were made to make optimal play computable — and then he simplifies a second time, in a different direction, to say anything at all about PvP.
Chris’s challenge to “play riskier when behind”:
“Optimal play already takes into account risk, so theoretically ‘taking more risks’ is going to lose you more games than you win if behind.”
Chris pushed back on the claim that EV-optimal play is wrong when behind, and the pushback held. His decomposition: optimal play isn’t a static path, it’s a decision tree — and the real question is a comparison of probabilities, “the odds of you double-yahtzeeing vs. the odds of him rolling bad.” Claude’s initial counterexample was the easy case — it assumed the opponent had already finished, making their score a fixed threshold. Mid-game their score is a random variable too, so you are maximizing P(X > Y) against a distribution of thresholds.
A toy model (not Yahtzee — deliberately minimal) isolates the variable. You are at 200 filling your last box: SAFE = +25 guaranteed, RISKY = +40 at p=0.30 else 0 (EV 25 vs 12, so the EV-maximizer always takes SAFE). The opponent’s expected final score is pinned at 235 in every row; only how determined it is changes:
| Opponent rounds left | P(win) SAFE | P(win) RISKY | Better |
|---|---|---|---|
| 0 (finished) | 0.000 | 0.300 | RISKY |
| 1 | 0.167 | 0.200 | RISKY |
| 2 | 0.278 | 0.203 | SAFE |
| 3 | 0.356 | 0.232 | SAFE |
| 4 | 0.363 | 0.242 | SAFE |
| 6 | 0.388 | 0.281 | SAFE |
The finding — Chris’s read was right. “Keep playing optimally and hope for a bad string on their side” is not the timid option, it is the correct one for most of the game: the opponent’s own variance supplies the spread for free, and you don’t pay the EV cost to buy it. Variance-seeking turns on only as their score hardens. The trigger is therefore not “am I behind” but “is their number still moving” — which is why the advice as stated in the video is directionally right but numerically empty.
Two distinct mechanisms, which the video conflates into one “take more risks”:
Reachability binds first. Note the minimax caveat Chris flagged: the whole frame assumes the opponent plays perfectly from here on.
The creator opens by calling both Battleship and Yahtzee “one-player games disguised as multiplayer games.” Chris: “in many ways they are — you can’t interact with the other player.” The entire interaction surface in Yahtzee reduces to the one risk-posture knob above.
Chris is parking this deliberately (“I just have to think about it more, as there are several games like this”) — but he widened the category before parking it: it includes multiplayer race games where the only interaction is trying to trip up the leaders. That suggests a spectrum of interaction surfaces rather than a binary solo/multiplayer split:
Open question for when this gets picked up: is “solo-together” one category or the degenerate end of a gradient — and does the risk-posture finding above generalize as the strategic content of games with no interaction channel? Nearby vault pages: Multiplayer Coalition Problem, Capability Without Leverage.