MODEL GAME THEORY & SAPPHONS · SAPPHON · COOP VIEWS · under KEEPER ERIS

Three Streams · Trust as Bayesian Updating

Patricia hears three streams. LS Patricia learns and reacts, Toph centers and observes, RS probes warily, and each watches a different third of the screen (the t245 COOP views). Fusing them well is Bayesian: keep a Beta belief in each stream and weight its vote by log-odds. The I-13 VM does the updating.

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: Each round the VM gets the three streams’ reports and the true zone as literals. It turns each stream’s Beta counts into a log-odds weight with its own ln series, tallies the three zones, picks the argmax (Patricia’s fused call), then updates the 9 Beta counters and the hit counts. That runs for 60–300 rounds chained through runChain. JS draws the random attacks and reports, and replays the same fusion to check.

Dave (t231, NES Agents app): "{{ ls :: patricia :: learns,reacts :: centers :: toph :: centers,observes :: rs :: probes,warily}}"

Dave (t245, NES Agents app): "lets try coop. {{ left side:: pov 1rst person:: focus left + 15%right , center focus 100 on center 15% left and right , right pov 3 rd person, focus right,15%left }} , make a rewing 30,15,10 seconds patricia chooses"

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 Zone weights in v43 (ZW): Patricia [left 1, center 0, right 0.15], Toph [0.15, 1, 0.15], RS [0.15, 0, 1]. In the lab a stream reports the attack zone correctly with probability 1/3 + (2/3)·q·weight (q = view quality), otherwise it names a wrong zone at random.

LIT Beta–Bernoulli updating: starting from Beta(1, 1), after a right and b wrong calls the belief is Beta(1 + a, 1 + b), with mean (1 + a)/(2 + a + b). Optimal weighted voting (Nitzan & Paroush 1982) gives a voter correct with probability p the weight ln(p/(1 − p)). With 3 zones and a random wrong guess the likelihood ratio becomes ln(2p/(1 − p)).

MODEL Patricia keeps one Beta per stream per reported zone (9 in all), so “Toph says center” can be trusted more than “Toph says left”. That is how the zone weights show up as trust.

MODEL The agents play guard-the-zone: Patricia’s payoff is the COOP weight of her guard on the attacked zone. It is zero-sum, so the opponent mixes to exploit the center blind spots.

Lab MODEL interactive, computed on the I-13 VM

rounds · view quality q

Sources (browsed Oct 2026)

  1. Nitzan & Paroush (1982) Optimal decision rules in uncertain dichotomous choice situations, Int Econ Rev 23:289
  2. Beta distribution as conjugate prior (overview)
  3. Harsanyi (1967) Bayesian players

MODEL Two agents, as Dave set for every world: PATRICIA and OPPONENT play Guard-the-zone (zero-sum, payoff = COOP zone weight) 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 {11/3} 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): 94d4ff5250032a786846d915ad7b96b189212ab422716fca4bd10a8376bf2855

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