Optimization Decisions with Bounded Rationality in Customer-Intensive Services Under Gumbel Distributions

Customers’ bounded rationality significantly influences the effectiveness of advertising and related decision-making in customer-intensive services. This paper, based on the M/M/1 queuing model and incorporating the characteristics of customers’ bounded rationality, conducts an optimization analysis of the decision-making of customer-intensive service providers. In this optimization process, we use the Gumbel distribution to model the random noise in customers’ bounded rationality, where the scale parameter characterizes the degree of bounded rationality, and customers’ choices follow a multinomial logit (MNL) choice model. We adopt the symmetric Nash equilibrium to prove the uniqueness of equilibrium in the duopoly market. Furthermore, we derive the optimal service rate and corresponding advertising expenditure levels for both monopoly and duopoly markets through optimization analysis. Our findings indicate that, regardless of market structure, an increase in the scale parameter of the Gumbel distribution, which captures customers’ bounded rationality, leads service providers to adopt a lower service rate to attract customers and reduce advertising expenditures. Notably, this strategy paradoxically results in higher profits for service providers. Additionally, we mathematically prove that, in both monopoly and duopoly markets, the proportion of customers choosing to enter the system follows a unimodal function of the service rate, and a unique and stable Nash equilibrium exists. Furthermore, we find that in the duopoly market, service providers, as a competitive strategy, set a lower service rate compared to the monopoly market. However, in the duopoly market, the proportion of customers attracted by each service provider is lower than in the monopoly market. We also confirmed the results through numerical simulation analysis.