Public transport transfers assessment via transferable utility games and Shapley value approximation

Abstract

The importance of transfer points in public transport networks is estimated by exploiting an approach based on transferable utility cooperative games, which integrates the network topology and the demands. Transfer points are defined as clusters of nearby stops, from which it is easily possible to switch between routes. The methodology is based on a solution concept from cooperative game theory, known as Shapley value. A special formulation of the game is developed for public transport networks with an emphasis on transfers. Based on such a game, the Shapley value is evaluated as an attribute of each transfer point to measure its relative importance: the greater the associated value, the larger the relevance. Due to the computational requirements of the Shapley value calculation for large-size networks, a Monte Carlo approximation is investigated and adopted. A case study of a real-world network is presented to demonstrate the model’s viability.

KEYWORDS:

• Network analysis
• public transport
• transfers
• centrality measures
• cooperative games
• transferable utility
• Shapley value
• Monte Carlo methods

Acknowledgements

G. Gnecco and M. Sanguineti are members of GNAMPA-INdAM (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni – Istituto Nazionale di Alta Matematica). M. Sanguineti is Research Associate at INM (Institute for Marine Engineering) of CNR (National Research Council of Italy) under the Project PDGP 2018/20 DIT.AD016.001 ‘Technologies for Smart Communities’.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Chebyshev’s inequality states that, for any random variable $X$ with finite mean $\mu$ and finite standard variation $\sigma$, and for any real number $k>0$, one has $Pr(|X-\mu |>k\sigma )\le \frac{1}{k^2}$.

2 Boole’s inequality, also known as ‘union bound’, states that, given any countable set of events $A_1$, $A_2$, $A_3$, … , one has $Pr\left(\bigcup _i{A}_i\right)\le \sum _{i}Pr(A_i)$.