The Gini index of travel demand imbalances
Along a public transport line operators serve multiple spatio temporally differentiated markets with the same second best capacity. Horcher and Graham (2016) showed in the simplest back haul setting that the asymmetry in demand between jointly served markets may have crucial impact on (1) the optimal capacity, (2) the equilibrium occupancy rate of vehicles and thus the crowding experience of passengers, and (3) the optimal pricing decisions. In this research the authors extend the analysis of the back haul problem to a more realistic urban public transport setting: a transit line along which capacity is still indivisible due to operational constraints, but more than two origin destination pairs have to be served. We investigate what may be a suitable measure of demand imbalances in this setting that could replace the share of main haul demand in total ridership in the back haul problem (Horcher and Graham, 2016) In a network not only the spread of demand between line section matters, but also the spread of operational costs between them, as it may be increasingly expensive to handle excess demand on relatively long line section. To tackle these challenges we propose the Gini coefficient of demand imbalances, a statistical index frequently used in macroeconomics as a measure of income inequality. The Gini coefficient measures statistical dispersion between two frequency distributions. In a public transport context we can adopt this concept by plotting the cumulative share of operational costs on line sections against the cumulative share of demand, after sorting both variables in increasing order. The resulting Lorenz curve is the diagonal of the graph in case of perfect equality, while if all demand is concentrated on a negligably short line section, then the Lorenz curve moves along the two sides of the graph (see Figure 1) The Gini index is the share of the area between the actual Lorenz curve and the one belonging to perfect equality