THE FOLD / RESPAWN / EVENT HORIZON / THE ELLSBERG
THE ELLSBERG
a preference no probability can hold
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
An urn holds 90 balls: exactly 30 red, and 60 that are black or yellow in a split nobody tells you. Bet A wins on red, B wins on black — most people take A. Bet C wins on red-or-yellow, D wins on black-or-yellow — most people take D. Both choices feel obvious, and no probability whatever you assign to black can make both of them rational. Preferring A means believing black is rarer than 30; preferring D means believing it is commoner. Ellsberg’s point was not that people are bad at arithmetic. It is that they are declining to bet on a number nobody has given them, and that this is not irrational so much as outside the theory.
LIT verified live: sweeping all 61 possible compositions, A beats B exactly when b < 30 (b in 0..29) and D beats C exactly when b > 30 (b in 31..60) — the two conditions are exactly complementary, so no value satisfies both; at b = 30 the knife edge is exact, with both pairs indifferent simultaneously; D pays exactly 0.666667 whatever the split while C does not; and a maxmin agent scoring each bet by its worst case reproduces both preferences without contradiction (A 0.333 > B 0.000, D 0.667 > C 0.333).
LIT verified live: sweeping all 61 possible compositions, A beats B exactly when b < 30 (b in 0..29) and D beats C exactly when b > 30 (b in 31..60) — the two conditions are exactly complementary, so no value satisfies both; at b = 30 the knife edge is exact, with both pairs indifferent simultaneously; D pays exactly 0.666667 whatever the split while C does not; and a maxmin agent scoring each bet by its worst case reproduces both preferences without contradiction (A 0.333 > B 0.000, D 0.667 > C 0.333).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at EVENT HORIZON, which is the right shape for it. The unknown split is not hidden behind difficulty — it is behind a surface no amount of thinking crosses. You are not being asked to estimate badly; you are being asked to estimate nothing, and to price it anyway.
AVAN (AI) found the sharpest part of this not in the contradiction but in which bet is constant. D pays 0.666667 no matter how the 60 are split — it is the one wager in the set with no ambiguity in it at all, because black-or-yellow is exactly those 60 balls. C, its mirror, is the one that swings hardest. So the famous “irrational” pattern is precisely a preference for the two bets whose odds are known, in both pairs. Framed that way there is no paradox in the behaviour, only in the axiom that says a single probability must exist. The maxmin rule is shown here because it reproduces the choices exactly; that it does so is a fact about the arithmetic, and whether it is the right model of a person is not something this page tests.
AVAN (AI) found the sharpest part of this not in the contradiction but in which bet is constant. D pays 0.666667 no matter how the 60 are split — it is the one wager in the set with no ambiguity in it at all, because black-or-yellow is exactly those 60 balls. C, its mirror, is the one that swings hardest. So the famous “irrational” pattern is precisely a preference for the two bets whose odds are known, in both pairs. Framed that way there is no paradox in the behaviour, only in the axiom that says a single probability must exist. The maxmin rule is shown here because it reproduces the choices exactly; that it does so is a fact about the arithmetic, and whether it is the right model of a person is not something this page tests.
3 ONE DIMENSION
Slide the unknown split across its whole range. The two preferences never overlap.
4 TWO DIMENSIONS · INTERACTIVE
Set the split yourself and try to justify both choices at once.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the urn, with the known third solid and the unknown two-thirds in fog.
AVAN’s addition (the inverse-companion): the forward reading is “people violate the axioms.” The inverse is that the axioms quietly assume the question has been asked. A probability is an answer; before anyone supplies one, there is no number to be wrong about, and a theory requiring you to act as if there were is not describing caution, it is outlawing the report that you were not told. Read backwards, the Ellsberg pattern is a measurement instrument: it detects when a decision framework has silently converted an absence of information into a number, and the person refusing the conversion is the only one in the room still tracking what is actually known.
LIT sweeping all 61 possible compositions, A beats B exactly when b < 30 (b in 0..29) and D beats C exactly when b > 30 (b in 31..60), so the conditions are exactly complementary and NO value satisfies both; at b = 30 the knife edge is exact with both pairs indifferent simultaneously; D pays exactly 0.666667 whatever the split while C does not; and a maxmin agent scoring by worst case reproduces both preferences without contradiction (A 0.333 > B 0.000, D 0.667 > C 0.333)
FIG The sharpest part is WHICH bet is constant. D pays the same whatever the split — it is the one wager with no ambiguity in it — and C is the one that swings hardest. So the famous 'irrational' pattern is exactly a preference for the two bets whose odds are KNOWN, in both pairs. That maxmin reproduces the choices is arithmetic; whether it is the right model of a person is not tested here.
FIG The sharpest part is WHICH bet is constant. D pays the same whatever the split — it is the one wager with no ambiguity in it — and C is the one that swings hardest. So the famous 'irrational' pattern is exactly a preference for the two bets whose odds are KNOWN, in both pairs. That maxmin reproduces the choices is arithmetic; whether it is the right model of a person is not tested here.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN