Dynamic pricing with (extra) seat reservations under the nested logit model

We suggest a dynamic pricing model for selling extra seats - seat reservations for unoccupied seats that provide additional space alongside regular reservations. Such extra space tickets share the resources of the main product and leverage the unused capacity, offering significant revenue-generation opportunities when coaches, trains, or airplanes frequently depart with empty seats. We formulate a Markov decision process (MDP) representing the ticket sales problem of a transportation company that sells tickets for a single leg in a single compartment, offering three options: (1) without seat reservation, (2) with seat reservation, and (3) with seat reservation and extra space. In this framework, seat reservations are integrated into the state space, making the problem a special case of the network dynamic pricing problem. To solve this problem, we draw from established network dynamic pricing methods to derive upper bounds and policies for pricing the three ticket types. These approaches include deterministic approximation, approximate linear programming, and a decomposition method based on seat values provided by the deterministic approximation. Under the nested logit demand model, we demonstrate that the ALP subproblem features a convex objective function in actions and linearity in the state components, enabling efficient solutions. An extensive numerical study highlights the efficiency of the decomposition approach, which delivers a superior revenue-to-runtime trade-off compared to more complex methods. This makes it a practical choice for real-world applications. Additionally, our results quantify the significant revenue potential of offering extra seats, particularly in low-demand scenarios.