Should competing firms offer 'Buy n times, get one free' loyalty programs? A game-theoretic analysis

This study investigates the profitability of 'Buy n, get one free' loyalty programs (LPs) for competing firms deciding on whether to offer such LP. We use a game-theoretic approach and obtain the equilibrium by considering four subgames; two symmetric ones where both firms either offer or do not offer a LP, and two asymmetric ones where only one firm offers a LP. We derive the demand using an analytical choice model. Then, we obtain the firms' optimal decision variables (prices and reward periods) and profits for each subgame under different combinations of behavioral parameters of time and reward valuations for both symmetric and asymmetric production cost for the firms. The results show that when customers highly value reward but not time, offering loyalty programs is a dominant strategy. However, under different market conditions, firms face a prisoner dilemma or find that not offering a LP is a dominant equilibrium.