MODEL GAME THEORY & SAPPHONS · SAPPHON · WIN–LOSS NEEDLE · under KEEPER ERIS

The JIT Radial · ±1° per Turn as a Random Walk

Dave’s Just-In-Time radial: a 360° ring whose needle moves +1° when a turn worked and −1° when it failed, with two tracks: what was tried and what happened, and what has happened and why. Mathematically, the needle is a random walk. The I-13 VM computes its exact distribution and the gambler’s-ruin odds.

Dave (t252): "i love that set up can you build some pages looking into game theory and those sappons you just built, see if you can build some folded stars and stuff"

Dave’s I-13 engine on this page

booting I-13 v0.3…

LIT What the VM computes here: The VM builds log-factorials with its own ln series and evaluates every binomial probability as ex(ln C(n, k) + k ln p + (n − k) ln q), chained through runChain. It computes the gambler’s-ruin probability and expected duration with its exp and ln. JS draws 4,000 Monte Carlo walks for comparison and plots.

Dave (t246, NES Agents app): "how about a {{ Just In Time engine , Radial 360x-+1+- degree per turn, so two tracks , what was tried , what happened, and what has happened , why }}"

MODEL Sapphon notes are read from PATRICIA v43 (nes_agents.js in the NES Agents app): Dave’s words are quoted exactly as he typed them, and every game-theory mapping on this page is my reading, not his.

MODEL In v39–v43, Track A (outer) logs what was tried → what happened, and Track B (inner) what has happened → why; each lane moves its needle ±1° per turn. If a turn works with probability p, each lane’s needle is a simple random walk.

LIT After n turns, a walk with up-probability p has made k up-steps with binomial probability C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ, so it sits at 2k − n degrees. Mean n(2p − 1), variance 4np(1 − p).

LIT Gambler’s ruin: started b steps above a lower wall and a below an upper one, with r = (1 − p)/p ≠ 1, the walk hits the top first with probability (1 − rᵇ)/(1 − rᵃ⁺ᵇ). The expected time is b/(q − p) − (a + b)/(q − p) × that probability, or a·b when p = 1/2. Even a 55% needle reaches +180° before −180° almost surely.

MODEL The agents play repeat-or-explore against an opponent who keeps or changes its pattern; the needle in the scene steps with the agents’ current mix.

Lab MODEL interactive, computed on the I-13 VM

p(turn works) · turns n · walls ±a°

Sources (browsed Oct 2026)

  1. Gambler’s ruin (formulas)
  2. Random walk (overview)
  3. Binomial distribution

MODEL Two agents, as Dave set for every world: PATRICIA and OPPONENT play Repeat-or-Explore (zero-sum, MODEL) over and over by regret matching (Hart & Mas-Colell 2000), each shifting toward the moves it regrets not having played. Their score is 1 − NashConv/2 of their empirical average play (1 = an exact Nash equilibrium), and the I-13 VM re-computes NashConv from the raw averages every time the score improves. The centrepiece is a folded star {16/7} whose fold rises with the score. Three.js r160 from a pinned CDN, devicePixelRatio capped at 2. Drag to orbit, scroll to zoom.

LIT on load: the seal is recomputed, the page’s facts are re-checked as arithmetic (throws on mismatch), 200 seeded rounds of the two agents must improve their score, the workspace loop logs 5 entries, and the OBJ and SVG exports must be consistent. Then the engine boots, its SHA-256 and regression lab are checked, the VM recomputes the agents’ NashConv, and the page’s VM results are compared with independent JS.

Gallery of kept moments (WebGL snapshots):

Workspace improvement loop: instructions for agents MODEL

  1. Measure: every 40 rounds, record NashConv of the average play, the score, the rounds since the last improvement and whether regret-matching+ is on.
  2. Try next: after a 40-round plateau, switch both agents to regret-matching+; if still stuck, halve the exploration ε; if still stuck, forget half the old history. While NashConv falls, keep course (ε × 0.9); if it rises, explore more (ε × 1.25).
  3. Log back: append {entry, round, measured, tried, ε} to the workspace log (in the JSON export and the list below), and have the I-13 VM recompute NashConv.
  4. Repeat, forever: never stop improving. Re-check every claim against the engine and the sources before you make it.

Live workspace log:

    Shared board MODEL PATRICIA ↔ OPPONENT

      Background art: MODEL a generative field seeded from this page’s seal.

      proving…

      Seal sha256(name|slug|blurb): 0b4b17952bdb888211bf11841b64822f344e8cb1bacf9fd82b0487b9121c0a6b

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