MODEL GAME THEORY & SAPPHONS · GAME THEORY · ZERO-SUM · under KEEPER ERIS

Mixed Strategies & Minimax

Some games have no stable pure choice: in Matching Pennies or a penalty kick, any pattern can be exploited. The answer is to randomise. Von Neumann’s 1928 minimax theorem says every zero-sum game has a value that both sides can guarantee, and the I-13 VM finds it.

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 writes each column as a line in p, computes every pairwise crossing with its own division, evaluates the minimum over columns at each candidate (and at p = 0 and p = 1), and keeps the best. That is the exact maximin of any 2 × n zero-sum game. JS only plots.

LIT John von Neumann, “Zur Theorie der Gesellschaftsspiele”, Mathematische Annalen 100:295 (1928): in every finite two-player zero-sum game, max over x of min over y of xAy equals min over y of max over x of xAy. That common number is the value, and optimal mixed strategies achieve it.

LIT Matching Pennies: value 0, each side plays Heads with probability 1/2. Rock–paper–scissors: value 0, play each with 1/3. Any deviation from the mix can be exploited by an opponent who notices.

LIT With two rows, the row player’s guaranteed payoff as a function of p is the lower envelope of straight lines, one per column. Its highest point is the maximin, and it lies at an endpoint or where two lines cross, so checking every crossing finds the value exactly.

MODEL Default penalty-kick numbers (P(goal) for kicker Left/Right against keeper Left/Right/Center) are illustrative, not data from a study.

AMBER Studies of professional penalty kicks report kickers and keepers mixing close to the minimax prediction. I didn’t re-browse those papers for this page, so no numbers are claimed.

Lab MODEL interactive, computed on the I-13 VM

P(goal): kicker Left vs keeper L R C · kicker Right vs keeper L R C

Sources (browsed Oct 2026)

  1. von Neumann (1928) Zur Theorie der Gesellschaftsspiele, Math Ann 100:295
  2. Notes on the minimax theorem (statement and proof)
  3. Matching pennies (overview)

MODEL Two agents, as Dave set for every world: MATCHER and MISMATCHER play Matching Pennies 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 {9/2} 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 MATCHER ↔ MISMATCHER

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

      proving…

      Seal sha256(name|slug|blurb): d446ead8317ad58279fbdf5d6e6faaf96054fb0e2e1a6f0e3d51d3a7405d9b1b

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