A Multi-Computational Exploration of Some Games of Pure Chance

TL;DR

This paper explores various computational methods to analyze simple chance-based games, emphasizing the importance of symbolic computation in deriving qualitative insights and providing tools for further research.

Contribution

It introduces a multi-computational approach combining simulation, numeric, symbolic, and conceptual methods, highlighting symbolic computation's role in understanding probabilistic game models.

Findings

Symbolic computation reveals rational and algebraic numbers in game analysis.

Numerical methods can be optimized using symbolic insights.

Qualitative results about sequences and numbers are derived from symbolic algorithms.

Abstract

In the spirit of "multi-culturalism", we use four kinds of computations: simulation, numeric, symbolic, and "conceptual" to explore some "games of pure chance" inspired by children board games like "Snakes and Ladders" (aka as "Chutes and Ladders") and "gambler's ruin with unlimited credit". Even more interesting than the many computer-generated actual results described in this paper and its web-site extension, is our broad-minded, ecunemical approach, not favoring, a priori, any one of the above four kinds of computation, but showing that, a posteriori, symbolic computation is the most important one, since (except for simulation, that is very inaccurate) numerics can be made more efficient with the help of symbolics (in the "downward" direction), and, (in the "upward" direction) the mere existence of certain symbolic-computational algorithms imply interesting "qualitative" results, that certain numbers are always rational, or always algebraic, and certain sequences are always polynomial, or C-recursive, or algebraic, or holonomic. This article is accompanied by four Maple packages, and numerous input and output files, that readers can use as templates for their own investigations.