A Reward-Earning Quaternary Random Walk on a Parity Dial

Abstract

A casino offers a game which involves a symmetric quaternary random walk on a parity dial with twelve nodes labeled as (1, 11, 3, 9, 5, 7, 6, 8, 4, 10, 2, 0), reading clockwise. A player begins at Node 0; she tosses a copper coin to decide whether to move clockwise (if heads) or counterclockwise (if tails); simultaneously she tosses a silver coin to decide whether she will move one step (if tails) or two steps (if heads) in the direction determined by the copper coin. Whenever she lands at a new node she is said to have ‘captured’ it. If a player intends to capture c nodes and she wishes to toss the coins k times, then her admission fee is (25 + 25c + k) cents (one quarter to play, one quarter per node to capture and one penny per toss). The game ends as soon as either c nodes (other than Node 0) are captured or k tosses are over, whichever event happens earlier; and the player earns as many nickels as the sum of the labels of the captured nodes. How should the player determine c and k? The player’s optimal choices can be derived from the theory of stochastic processes. Alternatively, optimal choices can be anticipated through a computer simulation. Lessons learned from the game empower entrepreneurs and consumers behave optimally to determine when and how to intervene to benefit from an opportunity and/or to prevent a catastrophe.