Indexing the archive…
Your Universe of Digital Possibilities
Four dice on the felt, every face printed in the open. You pick first; Vega picks second; the higher roll takes the round. There is no best die — “beats” runs in a circle — so whoever chooses first hands the choice of weapon to the house.
“Beats more often than not” is not an order: it can run in a circle. Efron’s four dice each beat the next two times in three, and the last beats the first — so whatever is chosen first, there is an answer that wins. The arrow refuses to be a straight line.
The exact central interval for K, the count of wins in n independent plays at win probability p — computed from the rules alone, before a single play. A record inside it is the rules working; a record outside it happens one time in twenty, and the band says so in advance. When the chance of each play is fixed before it is played but differs from play to play, the same band comes from the Poisson-binomial sum.
Pairwise “better” can run in a circle, and intuition insists it cannot. Each of Efron’s dice beats the next two times in three, and the last beats the first — so the first pick is a handicap, and the chair hands the same 2/3 to whoever picks second. The Call is the same circle drawn in time, on eight coin sequences; The Lineup shares the grammar of an object arranged all along; and The Wager owns what a bankroll does with an edge like this one.