Existential monadic second order logic on random rooted trees

Holroyd, Alexander E.; Levy, Avi; Podder, Moumanti; Spencer, Joel

doi:10.1016/j.disc.2018.09.012

Cited by 2 publications

(4 citation statements)

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“…For > 0 this has D =D = 0, but for sufficiently small > 0 we have p 1 > 1, and therefore E (2) > 0 by Theorem 5. This gives the last case of (8).…”

Section: Proof Of Theorem 5(iii)mentioning

confidence: 88%

“…Other work on combinatorial games in random settings includes the study of positional games (such as Maker-Breaker games) on random graphs, for example, [2,6,15], and [4] which deals with matching games played on random point sets, with an intimate connection to Gale-Shapley stable marriage. In another direction, [8] uses certain games as tools for proving statements involving second-order logic on random trees, and [16] uses a game in the analysis of optimization problems in a random setting. One striking observation from all these examples is that games, by their competitive nature, often automatically tease out and magnify some of the most interesting and subtle structural properties of random systems.…”

Section: Background and Related Workmentioning

confidence: 99%

“…S (1) < D; S (1) < E (2) ; E (1) <P;D <P; E (1) < D;D < E (2) ; (8) and that any inequality between any pair of N,Ñ, P, and S (2) is possible except for S (2) < P. We start by giving examples for the possibilities in (8) in turn.…”

Section: Proof Of Theorem 5(iii)mentioning

confidence: 99%

“…In any nontrivial case where the tree is finite with probability 1, for example, p 0 = p 1 = 1/2, we have E (1) =D = 0 andP > 0, giving the third and fourth cases of (8).…”

Section: Proof Of Theorem 5(iii)mentioning

confidence: 99%

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Galton–Watson games

Holroyd

Martin

2021

Random Struct Algorithms

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We address two-player combinatorial games whose graph of positions is a directed Galton-Watson tree. We consider normal and misère rules (where a player who cannot move loses or wins, respectively), as well as an "escape game" in which one designated player loses if either player cannot move. We study phase transitions for the probability of a draw or escape under optimal play, as the offspring distribution varies. Across a range of natural cases, we find that the transitions are continuous for the normal and misère games but discontinuous for the escape game; we also exhibit examples where these properties fail to hold. We connect the nature of the phase transitions to the length of the game under optimal play. We establish inequalities between the different games. For instance, the draw probability is no smaller in the misère game than in the normal game. KEYWORDSBranching process, combinatorial game, phase transition, random game INTRODUCTIONGame theory naturally often focuses on carefully chosen games for which interesting mathematical analysis is possible. What can be said about games in the wild? One approach to this question is to consider games whose rules are typical, that is, chosen at random, although known to the players. In this article, we consider rules arising from random trees. We consider combinatorial games whose positions and moves are described by a directed acyclic graph . A token is located at some vertex of , and the two players take turns to move it along aThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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“…For > 0 this has D =D = 0, but for sufficiently small > 0 we have p 1 > 1, and therefore E (2) > 0 by Theorem 5. This gives the last case of (8).…”

Section: Proof Of Theorem 5(iii)mentioning

confidence: 88%

Section: Background and Related Workmentioning

confidence: 99%

Section: Proof Of Theorem 5(iii)mentioning

confidence: 99%

“…In any nontrivial case where the tree is finite with probability 1, for example, p 0 = p 1 = 1/2, we have E (1) =D = 0 andP > 0, giving the third and fourth cases of (8).…”

Section: Proof Of Theorem 5(iii)mentioning

confidence: 99%

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