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Combinatorial games on Galton–Watson trees involving several-generation-jump moves

Dhruv Bhasin and Moumanti Podder

Vol. 13 (2024), No. 1, 1–58
Abstract

We study the k-jump normal and k-jump misère games on rooted Galton–Watson trees, expressing the probabilities of various possible outcomes of these games as specific fixed points of functions that depend on k and the offspring distribution. We discuss phase transition results pertaining to draw probabilities when the offspring distribution is Poisson (λ). We compare the probabilities of various outcomes of the 2-jump normal game with those of the 2-jump misère game, and a similar comparison is drawn between the 2-jump normal game and the 1-jump normal game, under the Poisson regime. We describe the rate of decay of the probability that the first player loses the 2-jump normal game as λ . We also discuss a sufficient condition for the average duration of the k-jump normal game to be finite.

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Keywords
two-player combinatorial games, normal games, misère games, rooted Galton–Watson trees, fixed points, Poisson offspring, generalized finite state tree automata
Mathematical Subject Classification
Primary: 05C57, 05C80, 60C05, 68Q87, 91A46
Milestones
Received: 15 March 2023
Revised: 27 December 2023
Accepted: 11 January 2024
Published: 1 February 2024
Authors
Dhruv Bhasin
Indian Institute of Science Education and Research (IISER)
Pune
India
Moumanti Podder
Indian Institute of Science Education and Research (IISER)
Pune
India