Shannon's measure of information, path averages and the origins of random utility models in transport itinerary or mode choice analysis
We interpret the often mentioned difference between Logsum and average utility in terms of Shannon's (1948) information measure S, leading to a Path Aggregation THeorem (PATH). It states that, in transport networks where unique measures of the utility of multiple paths are required for demand model formulation purposes and the true path choice model is Multinomial Logit (MNL), constructs based on weighted averages of path characteristics derived from multipath assignments always underestimate the utility of multiple paths, a deficit exactly equal to S (corresponding to minus-one times entropy) if the weights are the path choice probabilities. We study the properties of this S measure of aggregation error, along with those arising from other types of averages of path characteristics, outlining some implications for demand estimation and project appraisal. Notably, the validity of the PATH does not depend on the specific contents of the representative utility functions (RUF) associated to paths, such as their mathematical form or their eventual inclusion of alternative-generic constants (AGC). We show by simulation that averaging modes or sub-modes ― a frequent feature of traffic modeling studies ― can lead to important error in terms of level of traffic and welfare measurement. Concerning the mathematical form of the RUF, we recall that, after the publication of Abraham's 1961 random utility model (RUM) of road path choice deriving the Probit specification based on the Gaussian error distribution (and another specification based on the Rectangular error distribution), French engineers used this seminal approach as justification of road path choice formulae then in current use and assigned the name "Abraham's Law" to a particular standard one, effectively a "Logarithmic Logit" close to the logarithmic RUF carefully specified for Logit mode choice by Warner in 1962. For transit problems, the preference went to a linear RUF, as evidenced in Barbier's casual binomial Probit application to bus and metro, published in 1966, which may have inspired the later generalizations by Domencich and McFadden. In view of many founders' conscientiously crafted nonlinear Logit formulations, and more generally of the repeatedly demonstrated presence of nonlinearity in RUF path and mode specifications since their careful work 50 years ago, we analyze the impact of such nonlinearity on S. This impact is tractable through a comparison of measures S2 and S1 associated with two path choice models differing only in RUF form, as determined by Box-Cox transformations applied to their level-of-service (LOS) variables. We show that, although the difference between measures S2 and S1 may reach a minimum or a maximum with changes in LOS, the solution for such a turning point cannot be established analytically but requires numerical methods: the demonstrable impact on S of nonlinearity, or asymmetry of Logit curve response, is tractable, but only at non trivial computational cost. We point out that the path aggregation issue, whereby aggregation of paths by Logsums differs from aggregation of their characteristics by averages, is not limited to public transit (PT) projects with more or less "common" lines competing in dense urban transit networks (our particular Paris predicament motivating the analysis) but also arises in other modes whenever distinct itineraries or lines compete within a single mode. Concerning dense urban PT networks, we hypothesize that Logsums based on multiple path assignments treating all transit means (about 10 in our problem) as one modal network should, using Ockham's razor, be simpler than the insertion of a layer of choice hierarchies among such urban means based on non nested specifications embodying assumptions on the identity of "higher" and "lower" means, the latter reasserting the multiple path access problems the hierarchies were designed to solve in the first place. Concerning road networks, the proper accounting of multiple path use to avoid Shannon aggregation error points to an abandonment of Wardrop's equilibrium in favor of Logit choice. This completed shift should favor transit when it is the minority mode.
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