Passing the Buck and Firing Fibonacci: Adventures with the Stochastic Abacus
Bruce F. Torrence · American Mathematical Monthly · 2019
A “pass-the-buck” game begins with n players arranged as the vertices in a graph. One player has a dollar bill. The game is played in turns where, at random, the current bill-holder either wins outright or else passes the bill to a player on an adjacent vertex. If the vertex of the bill-holder has degree d, then each of the d + 1 outcomes is equally likely. We analyze this class of game using Arthur Engel’s stochastic abacus—a chip firing algorithm—revealing a natural relationship with the Fibonacci numbers.