Elsevier

Theoretical Computer Science

Volume 839, 2 November 2020, Pages 103-121
Theoretical Computer Science

Uniform distribution for Pachinko

https://doi.org/10.1016/j.tcs.2020.05.032Get rights and content

Abstract

Pachinko is a Japanese mechanical gambling game similar to pinball. Recently, several mathematical models of Pachinko have been proposed. A number of pins are spiked in a field. A ball drops from the top of the playfield and the ball falls down. In the 50-50 model, if the ball hits a pin, it moves to the left or right passage of the pin with an equal probability. An arrangement of pins generates a distribution of the drop probability for all of the columns. This problem was considered by generating uniform distributions. Previous studies have demonstrated that the (1/2a)-uniform distribution is possible for a{0,1,2,3,4} and is conjectured so that it is possible for any positive integer a. This study describes the constructive proof for this conjecture. This study also formalizes a natural decision problem yielded by this model while investigating its computational complexity. More precisely, given any drop-probability distribution A and any partial drop-probability distribution B, this study uses non-deterministic polynomial-time (NP) hardness to determine if there exists a pin arrangement that transforms A into B.

Introduction

Pachinko is a Japanese mechanical gambling game similar to Pinball (Fig. 1) [1], [2], [3]. The machine stands vertically and the player shoots a metal ball into the playfield. Many pins are spiked in the playfield and the ball drops from the top of the field. If the ball goes into a pocket in the field, then the player earns a reward. Recently, Pachinko was analyzed in the context of discrete mathematics. The origin of mathematical Pachinko is based on a book written by Akiyama in 2008 [4]. Recently, Akitaya et al. [5] studied an idealized geometry of a simple form of Pachinko [5]. This study considers one of the mathematical models presented, which is called the 50-50 model.

The 50-50 model consists of three entities: the field, pins, and a ball. The field is a half-plane triangle grid with the top-side end. A pin can be placed at any grid point. A row is a horizontal line where the grid points exist, and a column is a vertical line where the grid points exist. Since a triangle grid was considered for this investigation, the intersection points of rows and columns do not necessarily have a grid point (see Fig. 2). The ball drops from the center of the top-end and falls down vertically. If the ball hits a pin, then it moves to the left or right passage of the pin with an equal probability. Immediately afterwards, the ball continues to fall down vertically. Once the pin arrangement is fixed under the 50-50 model, the probability of dropping the ball in each column can be calculated. In other words, a pin arrangement defines the drop probability distribution for all of the columns. Then, the inverse problem of “deciding if there exists a pin arrangement that generates a given distribution or not” can be considered.

In [5], it was shown that any probability distribution p1,p2,...,pn in the 50-50 model can be constructed within an arbitrarily small additive error; thus, the main theoretical challenge is the generation of the given distribution. The (1/2a)-uniform distribution in the 50-50 model is the probability distribution. When the ball drops in the center, the probability is 0 and the probability at the 2a closest coordinates from the center is 12a (see Fig. 3). Akitaya et al. [5] showed that the (1/2a)-uniform distribution for a{0,1,2,3,4} can be constructed. This can also be conjectured such that the (1/2a)-uniform distribution for any positive integer a can be constructed. The first contribution of this study is to show that this conjecture is true. In other words, for any a1, this study provides the pin arrangement that generates a (1/2a)-uniform distribution. The number of pins used in the construction is bounded by a polynomial of 2a. To show the result, a new formulation of the problem is introduced following the notions and terminology of language theory. Even though the language theory is simple, it is substantially useful for the analysis of the 50-50 model.

As the second contribution, a computational-complexity aspect of the 50-50 model was also considered. Since the pin arrangement in the 50-50 model corresponds to a transformation from a given probability distribution (for all x-coordinates (i.e. Z1)) to another one, the matter of its design naturally yields one decision problem. For any two input distributions A and B, is there a pin arrangement that transforms A to B? This study focuses on the computational complexity of this decision problem. Unfortunately, this study does not determine any hardness results for this problem. Instead, for a slight variant of it, where B can be partial in the sense that B specifies the drop probability only for a subset of all columns, this study uses non-deterministic polynomial-time (NP) hardness to decide the transformability from A to B.

Section 2 first presents the formalization of the 50-50 model as well as the definition of the problem. Sections 3 and 4 respectively provide an explicit construction of the (1/2a)-uniform distribution for any a1 and its analysis for the number of pins. The NP-hardness results for the transformability of the distributions is presented in Section 5. Finally, the conclusion of this study is presented in Section 6.

Section snippets

Configuration and rewriting rule

The problem was formulated in the 50-50 model using the notion of formal grammar. A Pachinko machine is represented by a triangle grid on a half plane with an infinite horizontal length and an infinite downward vertical length. Each horizontal line contains grid points and is called a row. From the top end, each row is assigned a y-coordinate 1,2,. Since the field is a triangle grid, the grid points on an odd row are half-shifted from those on an even row. To fit them into the standard

Generating uniform distribution

Section 3 is devoted to the proof of the Theorem 1. The proof consists of the following three parts:

  • 1.

    4k0$(440)k2$.

  • 2.

    (440)k2$42k302k14$.

  • 3.

    42k302k+14$22k0$.

The combination of these transformations results in the Theorem 1. The following subsections looks at the details of each part.

Analysis of the number of pins for generating a uniform distribution

This section provides the asymptotic bound for the number of pins used in the construction. The numbers of pins used in the transformations shown in the presented lemmas and corollaries are summarized in Table 1. Most of the analyses in the table are easy to check. There are some exceptions that includes the Corollary 1, Lemma 9, and Lemma 10. These analyses are presented below.

In the transformation of the Corollary 1, the Lemma 1 is repeatedly applied until it becomes inapplicable for any 4s

Problem definition

As mentioned in the introduction, a pin arrangement for the 50-50 model can be regarded as a transformer with drop probability distributions. Then, it is a natural question to ask if there exists a pin arrangement that corresponds to the transformation between two given distributions or not. The problem is formally defined as follows.

Problem 1

Let A=(pn,,p0,,pn) and B=(qm,,q0,,qm) be two configurations (pi,qiQ for any i). Does AB hold?

Note that the minimum granularity 1/2g is not assumed in the

Conclusions and discussion

This investigation proved that (1/2a)-uniform distributions in the 50-50 model can be generated for any a0. This is a positive answer for the open problem posed in [5]. This construction consumes O(23a) pins for generating (1/2a)-uniform distribution. It is still open if more compact generation (with respect to the number of pins) is possible or not. This study did not pay much attention to the number of rows in the generation process. It is crucial to abandon the simplification assumption

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We would like to thank Editage (http://www.editage.com) for English language editing.

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Cited by (0)

A preliminary version of this paper was presented in the 19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG3 2016), and the 9th International Conference on Fun with Algorithms (FUN 2018).

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