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Designing competitive loyalty programs: a stochastic game-theoretic model to guide the choice of reward structure

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Abstract

We develop a game-theoretic model to guide the choice of the reward structure in customer loyalty programs. We model a duopoly market in which one firm adopts a loyalty program. Firms independently and simultaneously set the prices and rewards. Heterogeneous customers buy homogeneous products in a multi-period setting. Customers are segmented into three groups based on their level of strategic behavior, which is expressed in terms of their degree of forward-lookingness. We use two exogenous parameters to represent the size of each segment. A third parameter captures the point pressure effect, which refers to the increase in customer spending as they approach a reward threshold. In each period, customers choose the firm that maximizes their utility, which is a function of offered prices, rewards, and the distance to the next reward. We use the logit model to model the customer choice behavior. Customers’ accumulated purchases evolve as a Markov chain. We derive the limiting distribution of accumulated purchases, which is subsequently used to formulate the firm’s expected revenue functions. We develop two algorithms to find the Nash equilibrium for both the linear and nonlinear rewards in term of the three parameters. Using a thorough numerical analysis, we show that the choice of the structure becomes more critical as the size of the strategic segment increases. The nonlinear scheme is superior when the size of the highly-strategic segment is very small. The linear rewards is superior in markets where the size of the highly-strategic segment and the sensitivity to distance are simultaneously not small.

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Authors and Affiliations

  1. Frank G. Zarb School of Business, Hofstra University, 1000 Hempstead Turnpike, Hempstead, NY, 11549, USA

    Amir Gandomi

  2. Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, M5B 2K3, Canada

    Amirhossein Bazargan & Saeed Zolfaghari

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  1. Amir Gandomi
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  2. Amirhossein Bazargan
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Correspondence to Amir Gandomi.

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Appendices

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Table 3 List of symbols
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The effect of parameters on the equilibrium values of decision variables

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Fig. 14
figure 14

The effect of parameters on equilibrium prices and rewards in the nonlinear rewards case

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Fig. 15
figure 15

The effect of parameters on equilibrium prices and rewards in the linear rewards case

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Gandomi, A., Bazargan, A. & Zolfaghari, S. Designing competitive loyalty programs: a stochastic game-theoretic model to guide the choice of reward structure. Ann Oper Res 280, 267–298 (2019). https://doi.org/10.1007/s10479-019-03179-1

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