Optimal Decisions Under Partial Refunds: A Reward-Earning Random Walk on a Parity Dial

Abstract

We solve an unsolved problem posed in Sarkar (2020), which proposed a reward-earning binary random walk game on a parity dial whose twelve nodes, when read clockwise, are labeled as (1, 11, 3, 9, 5, 7, 6, 8, 4, 10, 2, 0). Starting from Node 0, at each step the player tosses a fair coin and moves one step clockwise (if heads) or counterclockwise (if tails). The player pays 25c + k cents if she intends to capture c nodes and toss the coin k times. When the c non-zero nodes are captured or when the k tosses are over the game ends; and the player earns as many nickels as the sum of the labels of the captured nodes. The player’s objective is to determine (c, k) to minimize the expected percentage loss. Here we consider a more complex game in which the player is offered several options for a partial refund on each unused toss on payment of an additional upfront overhead fee. Which partial refund offer should she choose? Having chosen the refund option, how should she determine (c, k) to minimize the expected percentage loss? Under partial refund offers, the player may choose a higher c and a higher k compared to those in the no refund scenario. The optimal choice is discovered through computer simulation, leaving open the theoretical development. Lessons learned from such games empower all parties engaged in the marketplace to determine when to intervene and how to make decisions to benefit from an opportunity and/or prevent a catastrophe.