The arrangement of last-mile services is playing an increasingly important role in making public transport more accessible. We study the use of ridesharing in satisfying last-mile demands, with the assumption that demands are uncertain and come in batches. The most important contribution of our paper is a two-level MDP framework that is capable of generating a vehicle-dispatching policy for the aforementioned service. We introduce state summarization, representative states, and sample-based cost estimation as major approximation techniques in making our approach scalable. We show that our approach converges and solution quality improves as sample size increases. We also apply our approach to a series of case studies derived from a realworld public transport dataset in Singapore. By examining three distinctive demand profiles, we show that our approach performs best when the distribution is less uniform and the planning area is large. We also demonstrate that a parallel implementation can further improve the performance of our solution approach.
Game theory has gained popularity as an approach to analysing and understanding distributed systems with selfinterested agents. Central to game theory is the concept of Nash equilibrium as a stable state (solution) of the system, which comes with a price − the loss in efficiency. The quantification of the efficiency loss is one of the main research concerns. In this paper, we study the quality and computational characteristics of the best Nash equilibrium in two selfish scheduling models: the congestion model and the sequencing model. In particular, we present the following results: (1) In the congestion model: first, the best Nash equilibrium is socially optimum and consequently, computing the best Nash is NP-hard; second, any -approximation algorithm for finding the optimum can be transformed into an -approximation algorithm for the best Nash. (2) In sequencing model: for identical machines, we show that the best Nash is no better than the worst Nash and it is easy to compute; for related machines, we show that there is a gap between the worst and the best Nash equilibrium, and leave the analytical bound of this gap for future work.
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