Pachinko

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 $\{1,2,4,8,16\}$ 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 $2$ 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 result: any distribution of $n$ dyadic probabilities, each specified by $k$ bits, can be constructed using $O(n k^2)$ pins, which is close to the information-theoretic lower bound of $\Omega(n k)$.