Nontransitive penney game, its generalizations, applications in finance and fractal brownian motion

Nadezhda Yu. Filimoshina

Lomonosov Moscow State University

Abstract: The paper investigates a nontransitive paradox which is known as the «Penney game». An algorithm for calculating theoretical winning probabilities is given, which makes it possible to generalize the game to binary sequences of arbitrary length in which 0 and 1 are independent and uniformly distributed. Based on a numerical experiment, it is established that for a sequence constructed from the dynamics of stock prices, this nontransitive effect persists. A generalization of the game to fractal Brownian motion is provided: a model is constructed and, using the results of numerical experiment, the parameter values are found at which nontransitivity disappears in the game. The values of the parameter at which the winning probabilities in the «Penney game» for fractal Brownian motion most accurately correspond to the game for stock price dynamics are investigated.

Keywords: Penney game, nontransitivity, fractal Brownian motion, Cholesky decomposition.

UDC: 519.2
BBK: 22.17

Received: 21.05.2025
Revised: 28.07.2025
Accepted: 15.09.2025