Effect of time length for decision-making in the ultimatum game

Mobility systems featuring shared vehicles are often unable to serve all potential customers, as the distribution of demand does not coincide with the positions of vehicles at any given time. System operators often choose to reposition these shared vehicles (such as bikes, cars, or scooters) actively during the course of the day to improve service rate. They face a complex dynamic optimization problem in which many integer-valued decisions must be made, using real-time state and forecast information, and within the tight computation time constraints inherent to real-time decision-making. We first present a novel nested-flow formulation of the problem, and demonstrate that its linear relaxation is significantly tighter than one from existing literature. We then adapt a two-stage stochastic approximation scheme from the generic SPAR algorithm due to Powell et al., in which rebalancing plans are optimized against a value function representing the expected cost (in terms of fulfilled and unfulfilled customer demand) of the future evolution of the system. The true value function is approximated by a separable function of contributions due to the rebalancing actions carried out at each station and each time step of the planning horizon. The new algorithm requires surprisingly few iterations to yield high-quality solutions, and is suited to real-time use as it can be terminated early if required. We provide insight into this good performance by examining the mathematical properties of our new flow formulation, and perform rigorous tests on standardized benchmark networks to explore the effect of system size. We then use data from Philadelphia’s public bike sharing scheme to demonstrate that the approach also yields performance gains for real systems.