Pachinko
Hugo A. Akitaya, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg,, Ferran Hurtado, Jason S. Ku, and Jayson Lynch

TL;DR
This paper explores the probabilistic outcomes of a Pachinko-inspired model with inelastic collisions, demonstrating how various distributions can be realized through specific pin arrangements and proving a universality result for constructing dyadic distributions.
Contribution
It introduces a novel Pachinko-based model for probability distributions and proves that any dyadic distribution can be constructed efficiently with a near-optimal number of pins.
Findings
Uniform distribution is achievable for certain drop counts.
Any dyadic distribution can be approximated arbitrarily closely.
Constructing any dyadic distribution requires O(n k^2) pins, close to the information-theoretic lower bound.
Abstract
Inspired by the Japanese game Pachinko, we study simple (perfectly "inelastic" collisions) dynamics of a unit ball falling amidst point obstacles (pins) in the plane. A classic example is that a checkerboard grid of pins produces the binomial distribution, but what probability distributions result from different pin placements? In the 50-50 model, where the pins form a subset of this grid, not all probability distributions are possible, but surprisingly the uniform distribution is possible for possible drop locations. Furthermore, every probability distribution can be approximated arbitrarily closely, and every dyadic probability distribution can be divided by a suitable power of and then constructed exactly (along with extra "junk" outputs). In a more general model, if a ball hits a pin off center, it falls left or right accordingly. Then we prove a universality…
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Taxonomy
TopicsAlgorithms and Data Compression · Sports Dynamics and Biomechanics · Stochastic processes and statistical mechanics
