Multifractal Aspects of an Efficient Change-Making Process

Abstract

In this paper, we focus on the "minimal payment" which minimizes the number of coins left after the payment. Two kinds of multifractal properties of the minimal payment are studied. The first one is a frequency distribution of change amounts, and the second one is a visiting probability on the delay plot of successive change amounts. When the face values of coins are power of two (1, 2, 4, 8, …), we find that these two distributions are related to well-known multifractal models, and derive analytical expressions of multifractal spectra.

Fractal patterns related to dividing coins

“…Even with a canonical coin system, which is easy to study, the change-making process possesses a rich mathematical structure. If we repeatedly pay money so that the number of coins in the purse after each payment is minimized, a fractal pattern is obtained from a sequence of change amounts [12,13].…”