Mittelalterliche Würfelspiele mit Einsatz und Gewinn

Thomas Bronder's study examines the betting structures of medieval dice games, focusing on the twelve games described in Alfonso X's thirteenth-century Book of Games, together with the closely related games Azar and Guirguiesca. The central argument is that these games were two-player, zero-sum contests in which stakes were settled by mutual agreement rather than a neutral pot, and that their outcomes and risks were shaped by chance-point rules. The analysis draws on the surviving sources, which document betting amounts and timing only sparsely for several of the games.

A key finding concerns the probabilities underlying three-dice play. The distribution of three-dice sums, with 216 outcomes forming a hat-shaped curve and an 18.52% chance of ending a game in the Azar range, was likely known to Alfonso through De Vetula but was never stated in the text. Medieval players could judge single-roll odds from experience, yet they lacked the formal probability theory, not developed until the seventeenth century, needed to compute how risk shifted across successive throws. For Azar, the unequal hit probabilities of its two outcome groups, roughly 1:4.4, meant that about two-thirds of games were decided by repeated hits on the chance points.

Guirguiesca receives particular attention. Roughly 30 percent of its games were decided by the opening Azar bet and about 70 percent by the subsequent point bet. Because winnings were tied to the opponent's stake, raising one's own wager offered no rational gain, and a mutually agreed 1:1 structure approximated a near-even private wager, with the first player's win probability about 52 percent before the first throw and varying with the opponent's revealed point. The study concludes that such symmetric games balanced expected outcomes over many sessions, though high single-session stakes could leave lasting disputes, and that Guirguiesca stands as a precursor to the later games Hazard and Craps, whose asymmetric house edge contrasts with the medieval zero-expected-return design.