Marc Gaudry
email: marc.gaudry@umontreal.ca

Emile Quinet
email: quinet@enpc.fr

Michel Barbier

Robert Fogel

Daniel Mcfadden

Émile Quinet

Claude Abraham

Stanley Warner

EOLE Abraham-Mcfadden Approach

Shannon's measure of information, path averages and the origins of random utility models in transport itinerary or mode choice analysis

Keywords: B23, C49, R41 Multipath assignment, Aggregation of path characteristics, Path aggregation, Inclusive values, Multinomial Logit, Shannon's measure of information, Origins of Random Utility Models (RUM), Probit, Logarithmic Logit, Abraham multipath assignment, aggregation of path characteristics, path aggregation, inclusive values, Multinomial Logit, Shannon's measure of information, origins of Random Utility Models (RUM), Probit, Logarithmic Logit, Abraham's Law of traffic assignment, Kirchhoff's distribution, non linearity B23, C49, R41

Table of contents

List of tables

Table 1. Impact of Shannon aggregation error on simulated public transit share in Île-de-France.......12 Table 2. Four steps in the development of random utility path assignment models around 1961.........16

List of figures

The necessary aggregation of paths for transport demand model specification

Passenger transport models that explain D od,m , the quantity demanded by mode m between any origin o and destination d, require, for all relevant origin-destination (OD) pairs and modes (M), the construction of level-of-service indicators LOS od,m typically comprising at least the modal Cost or Fare F od,m and Travel time T od,m characteristics. Given that other variables ETC are also required to complete specifications, a summary formulation of such demand functions might well be: In the particular case of stochastic (random utility) mode choice models commonly treated as demand models, such as the Multinomial Logit (MNL) and Probit, these variables appear in "representative utility functions" (RUF) typically specified only in terms of own-mode LOS characteristics, such as:

(0-A) , , 1 , ( ,.
(0-B) , , ( , )
od m m od m V f LOS ETC  , m= 1, …, M; o, d = 1, …, Z.
The construction of such LOS indicators resulting in unique values of modal trip characteristics poses problems of its own because, as a rule, multiple paths of varying "lengths" are in fact used between any given origin and destination: the actual multiplicity of paths has to be both modeled and adequately represented to obtain the unique LOS vector of element values for each OD pair and mode in question. The first task of assignment algorithms therefore consists in espousing traveler choices in networks consisting in sets of techniques1 or groups of small plants jointly used to simultaneously produce different (trip) outputs. Their second task is to derive the LOS elements from the results of such multiple path assignments where itineraries actually used differ with respect to those elements (time, fare, etc.): the question is then how to weigh, or aggregate, them.

In this paper, we assume that the first task has been carried out to perfection and we concentrate on the second to discuss some current weighing practices and their implications for the specification and estimation of the demand functions and their consequent use in project appraisal. The immediate problem giving rise to this concern is for us the extension of a suburban train line westward from central Paris in an urban environment where other competing suburban train lines already exist along with many other transit options (metros, buses, tramways, etc.). In the presence of a significant density of transit alternatives, it is tempting for analysts to choose in the menu of transit assignment procedures offered by commercial computer packages the "average LOS" option to generate the transit indicators needed in the demand or mode choice model: we compare such "average" options with the "Logsum" also available from Logit path assignments in some transit assignment programs.

We provide proof of a Path Aggregation THeorem (PATH) stating that, in transport networks where unique measures of the utility of multiple paths are required for demand model formulation purposes, constructs based on weighted averages of network path characteristics derived from multipath assignments always underestimate the utility of multiple paths by an amount exactly equal to [START_REF] Shannon | A Mathematical Theory of Communication[END_REF] measure of information if the true path choice model is Multinomial Logit. The aggregation of path characteristics therefore differs from the aggregation of paths in measurable ways, a number of which are also considered in passing.

We study the properties of this measure of aggregation error. First, the issue of proper service aggregation is not limited to public transit projects with closely competing more or less "common" lines in dense urban networks (our particular Paris predicament): it also arises in other modes whenever distinct lines compete within a single mode, and even in freight assignments very briefly alluded to. Notably, the validity of the PATH does not depend on the mathematical form of the path utility functions or on the identification of a common path constant in assignment models.

2. Does the aggregation of paths differ from that of their characteristics?

The Logit context and three average constructs of path characteristics

To establish the structural properties of various path weighing methods, we provisionally neglect all observational subscripts -to be reintroduced in the next section where individual observations need to be identified -and reinterpret the remaining running index as applying to paths instead of modes.

If the choice function among path alternatives i (i = 1, …, M) is assumed to be MNL, namely:

(1-A) 1 exp( ) / exp( )

M i i i i p V V    , (1-B) 1 log ln exp( ) M i i i i p V V    
, where the Logsum or Inclusive value term derived by [START_REF] Williams | On the Formation of Travel Demand Models and Economic Evaluation Measures of User Benefit[END_REF] or [START_REF] Mcfadden | Modelling the Choice of Residential Location[END_REF] is easily recognizable. And we wish to consider three ways of aggregating itinerary use by performing a calculation of mean path utility. The first two are readily found in most assignment program menus:

(1-C)

p i i i V p V    [probabilistic mean]
(1-D)

a i i i V m V   
[arithmetic mean] where, by convention2 :

 the pi denote shares or choice probabilities of the M paths (or itineraries) used ;  the mi are all equal to 1/M ;  the V i denote, as in (0-B), the representative utility functions (RUF) of the paths.

The third construct, included here for good measure, is inspired by prospect theory which introduces a rupture or "twisting" in the evaluation of choice probabilities, in this case:

(1-E)

1 * * * 1 1 1 M k k pp i i k k i i V p p V                             

   [prospect power mean]

where :

 starred values of probabilities (p*1, ..., p*i, …, p*M) and of RUF * k V signify that the latter are ordered in increasing fashion. And we imagine, in this hypothetical case of a mean construct incorporating an attitude towards path utility, that 1   stands for instance for the risk of agoraphilia (the fear of little used itineraries), 1 



for the risk of agoraphobia (the fear of heavily used itineraries), and 1   for a neutral attitude towards itinerary size or inherent attractiveness: in this latter case, (1-E) collapses back to (1-C);

 the simple power transformation p  is chosen among probability transforming functions that maintain the capacity of the distribution. Stott (2006, Table 3) lists seven current examples of such functions, obviously excluding from his ménagerie the [START_REF] Box | An Analysis of Transformations[END_REF] power transformation applied below to LOS variables but including in it the convoluted inverted S-shape animal used by [START_REF] Tversky | Advances in prospect theory: cumulative representation of uncertainty[END_REF],

  1 (1 ) p p p    
  : all seven do indeed guarantee that transformed probabilities sum to unity.

Our retained prospect power formulation of a mean (1-E) is directly borrowed from a model of Air France flight choice between Charles de Gaulle airport and two of the three airports serving London where the author [START_REF] Lapparent | La valeur du temps dans les déplacements réguliers professionnels[END_REF][START_REF] Lapparent | Optimism, Pessimism toward Risk of Time Losses in Business Air Travel Demand, and Insurance Premium[END_REF] applied the simple power "twist" to Travel time * k T , a variable included in the i V functions of different flights, not to those complete functions themselves. Expressing an attitude towards risk of realization of a RUF instead of a variable, formulation (1-E) is therefore speculative in the sense that it has never been applied as such.

Underestimation of the value of paths by constructs averaging their characteristics

We are now in a position to ask how the aggregation of paths differs from that of their characteristics by comparing the Logsum to various weighted averages. Generally speaking, consider from (1-B):

(2-A) ln ln Vi i i i V p e        ,
and, after multiplying all terms by weights i w normalized to sum to unity, sum over all paths to obtain the weighted mean w i i i V wV   , and simply derive:

(2-B) ln( ) ln[ ] j V w i i i i i i j V wV w p e       ,
from which different special cases will follow, depending on the nature of the weights

1 i i w  
. For instance, if the weights are the probabilities themselves as in (1-C), we obtain:

(2-C) ln exp( ) ln( )

p i i i i i V V p p     
which can be stated formally as the path aggregation theorem (PATH):

In transport networks where unique measures of the utility of multiple paths are required for demand model formulation purposes, constructs based on probability-weighted averages of network path characteristics derived from multipath assignments always underestimate the utility of multiple paths by an amount equal to [START_REF] Shannon | A Mathematical Theory of Communication[END_REF]



play a central role in information theory as measures of information, choice, uncertainty. The form of H will be recognized as that of entropy in certain formulations of statistical mechanics where pi is the probability of a system being in cell i of its phase space. H is then, for example, the H in Boltzmann's famous H theorem. We shall call Like the entropy measure, the measure ln( )

i i i S p p  



is null if all pi except one are zero -but this possibility is excluded by the MNL, except if all utilities except one are minus infinity. Otherwise, the properties of the measure of information S (or minus-one times the Entropy H) for our path aggregation problem duly match those of the entropy measure, mutatis mutandis in view of the sign change. In particular, we note without surprises that S:

(i) is negative: the Logsum is always larger than (1-C), the probabilistically weighted average

p i i i V p V   
, because probabilities are always positive fractions and all ln ( ) 0 i p  .

For the same reason, use of the arithmetic mean (1-D) yields a difference equal to ln( )

i i i m p   , clearly
always negative again. In fact, as demonstrated in (2-B), any set of path utility weights normalized to sum to unity will necessarily produce a value inferior to that of the Logsum. The difference will be more or less close to S, depending on how the weights differ from Shannon's untransformed probabilities: a case in point would be the use of the prospect power mean (1-E) leading to a difference equal to

    1 * * * 1 1 1 ln( ) M k k i i k k i i p p p              

  

, again negative and S  unless 1   ;

(ii) is at best constant but generally decreasing with path splitting [S(x,y) ≥ S(x) + S(y)]: the information error concerning a joint path is at least as important as the sum of information errors concerning the paths considered individually. If all options have equal weight, splitting them (or generally increasing their number) will decrease S eventually to 0. Formally, if the measure S(M) pertains to M alternatives of equal probability 1/M, then

( ) 0 S M  as M   . (3-A)   ( ) (1 / ) ln(1 / ) S M M M M   , (3-B) ( ) ln(1/ ) ln( ) S M M M      , (3-C) ( ) 0 S M  ;
(iii) has a minimum equal to ln(M) , for a constant number of paths M, when all the p i are equal.

i i i i i L p p p      
, the following first order conditions, for all i:

(4-C) ln( ) 1 0 i p    
implying that all i p be equal. This means that any change away from equality increases S.

(iv) does not always increase with the variance of probabilities.

Does the fact that changes away from equality of probabilities increase S mean that higher variance in probabilities always increases it as well? To see that it does not, consider firstly particular cases of changes away from equality and secondly the general case:

(a) S increasing with variance: to see that S can increase with the variance of choice3 probabilities, start with cases of null variance.

Consider first a pair of options of equal importance [p1 = (1-p1) = 0,5]. It is easy to show that we then have

1 1 1 1 / ln( ) ln(1 ) 0 0,5 S p p p iff p       
, namely a measure S that increases with the variance of the choice probabilities: for instance, S increases from -0,6931 to -0,5623 when the probabilities change from (1/2,1/2) to (1/4, 3/4).

Similarly, start with the triplet of options of equal importance (1/3, 1/3, 1/3) and increase the last to ( , ,1-2 )

 

 by drawing equally from the first two. Again, S increases with the variance of probabilities:

it equals -1,0986 when all probabilities are equal and -1,0397 if  = 0,25. But this is another case of changes away from equality. What of the more general case?

(b) Sign of changes in S unrelated to that of changes in variance: to see that there exists no general relationship between the direction of changes in the variance of probabilities and that of changes in S, consider more formally the quantities of interest and their modifications following a marginal change in probabilities4 , namely, in succession:

(5-A) ln( )

i i i S p p    and ln( ) i i i dS p dp   ; (5-B) 2 2 2 1 1 i p i i i i p p M M              and 2 2 i p i i i d pd p    .
To ask whether the sign of dS can be deduced from the sign of 2 i p d , let us interpret these two quantities as scalar products of M-dimensional vectors: the former as the product of vector A with coordinates pi by the vector C with coordinates dpi; the latter as the product of vector B with coordinates ln(pi) by the same vector C with coordinates dpi.

Consider now a plane, in Figure 1, defined by vectors A and B, as well as Γ, the projection of vector C on that same plane. It is the case that, as a scalar product, dS is of the same sign as that of the cosine of the angle formed by A and Γ; for a similar reason, Vector A

Vector B Γ Γ'

Note that, if A and B are not collinear, one readily finds a vector such as Γ' for which those signs differ and a vector such as Γ for which they are identical: in Figure 1 indeed, the two cosines are of the same sign for Γ but not for Γ'. The signs of the two cosines are identical only if vectors A and B are collinear -the signs of dS and 2 i p d are then also identical.

Clearly, with elements pi and ln(pi), these vectors are obviously not collinear except for very special values of the pi, namely all equal 1/M exemplified in the two particular cases just above in (a). In general, the direction of changes in the variance of probabilities and that of changes in S are unrelated.

(v) is independent from the mathematical form of the utility functions: although obvious from (3-D), this important practical property will shortly be explored at some length in order to search for systematic relationships between the value of S and the mathematical form of the i V functions.

This matters because Box-Cox transformations introduce asymmetry in the form of the response curves derivable from Box-Cox Logit models. In those models, the marginal utility of the LOS is not assumed to be constant any more, as it is in the popular classical Linear Logit model. The practical question is therefore whether non linearity, which implies the presence of asymmetry of responses to modifications of LOS, affects the aggregation error in analytically predictable ways.

It will turn out that 2 1 S S S    , the difference between two aggregation errors S2 and S1, where indices refer to models varying only in the form of the RUF, requires case by case study: little of an analytic nature can be said about the link between aggregation error and non linearity in the RUF.

In fact, rejection by the data of the unreasonable assumption of constant marginal utility does not imply analytical properties of 2 1 S S S    that are accessible without use of numerical methods.

Antecedents for S and consequences of obstinacy in the use of weighted averages

In view of the celebrity of Shannon's information measure, it may be asked whether the notion of path aggregation error S is in fact interesting and what benefits could arise by shifting from Mean to Logsum measures in processing results from path assignment models. Given that people know very well, if only by applying Jensen's inequality, that these measures differ in practice, why not just live with a long-noticed and familiar difference? We argue and show by simulations that recognizing and interpreting this "well known" difference as S may prompt a redirection of practice to avoid significant error in the estimation of demand and in the derivation of welfare measures.

Interpretation of a familiar difference: consequences in terms of context

What is in a name? In view of the extreme simplicity of (2-C), is it in fact well known as S to all but the present authors? Lest we break down an open door, what are then the published transport demand antecedents of measure S? There appears to be two independent streams. In the first, either Shannon's name appears but the formula S is used for purposes other than the measurement of aggregation error (2-C) or as the basis for an analogy; similarly, the entropic form -S also gives rise to another analogy where the probabilities are replaced by other terms. In the second, S is gratuitously computed ex machina in the sense of (2-C), but without demonstration and without being named or interpreted after Shannon. At the very least, some benefit may arise from the new interpretation of a "known difference" beyond that of unifying the streams, from which we consider representative cases.

In the first stream, always related to demand estimation, one finds for instance [START_REF] Anas | Discrete choice theory, information theory and the multinomial logit and gravity models[END_REF] who has shown that the MNL can be derived equivalently by minimization of Shannon's information measure or by maximization of utility. Explicit reference had also been made to Shannon in a Logit context by [START_REF] Theil | A Multinomial Extension of the Linear Logit Model[END_REF] who had defined a measure "derived from information theory" for the degree to which a vector p of N expenditure shares i p , that are each functions of income and of N prices, varies between two periods: (6-A) ( : ) log log

N i i i i I p p p p p     



where i p and i p refer to prior and posterior values, and the derived closeness to S is obvious5 .

In demand estimation contexts where due reference is made to Shannon's S, one also finds Picard (1987) who substitutes flows for probabilities in a minimization of the error between the origindestination flows ij D to be estimated under various constraints6 and some a priori ij D ~ flows. His objective function, derived from the [START_REF] Kullback | On Information and Sufficiency[END_REF] distance between distributions7 , is:

(6-B) ln ln ij ij ij i j MIN D D D         .
A closely related construct in terms of quantities of goods is found as part of the following utility function for n variants of a differentiated product and for commodity 0, a Hicksian composite good:

(6-C) 0 ln n n i i i i i i U aX X X N X      which is assumed to hold if n i i X N   (i.e.
if the individual commodities sum to the amount N of the differentiated product) and the i a and  (the Logit homoscedasticity term) are nonnegative scalars.

But the authors of (6-C) merely refer to its second term as "entropy-type" (Anderson et al., 1986, p. 6;1988, p. 461) or "of an entropic form" expressing the variety-seeking behavior of the representative consumer (Anderson et al., 1992, p. 79): in contrast with Theil and Picard, they ignore Shannon. In cases (6-B) and (6-C), the analogy 8 arises from the form but the constructs in terms of goods are short of Shannon's information or of Boltzmann's entropy proper which both duly require probabilities 9 .

What is in a formula? In the second stream, apparently consisting only in articles or computer programs pertaining to public transit assignment, the S formula is again "given" -apparently never derived -but without any reference to Shannon's own S or to the related notion of entropy -S.

For instance one finds, in the appendix of an interesting paper on schedule-based transit assignment [START_REF] Daly | The use of schedule-based assignments in public transport modeling[END_REF], the exact expression for S, gratuitously presented as the difference between [

p i i i V p V   
] and the Logsum, with the emphasis put on its negativity, duly imputed to ln ( ) 0 i p  , but S does not seem to already have a meaning, pedigree, name or existence on its own.

The same restraint prevails in documented commercial computer programs performing transit assignment. The closest term-for-term match to (2-C) is found in the Cube Voyager ( 2008) manual where the three mathematical expressions belong to a menu of variables calculable from effected multi-path assignments. These output variables are respectively called (op. cit., p. 812

) Average Generalized Cost Skim [ p V ], Composite Cost Skim [ ln exp( ) i i V  ] and Value of Choice [ ln( ) i i i p p   ].
But no reference to Shannon is found in that manual or in those of other popular programs such as EMME/2 or VISUM which has an option called Utility to allow computation of output variable exp( )

i i V 
from realized multipath assignments (PTV AG, p. 464). The forthcoming Emme 3 program [START_REF] Inro | Emme 3 at www[END_REF][START_REF] Florian | Emme strategy transit assignments with variants[END_REF], should add to the deterministic 10 optimal strategy assignment implemented in EMME/2 a Logit choice option [START_REF] Florian | A note on logit choices in strategy transit assignment[END_REF].

Consequences in terms of expected benefits

Estimation of demand or mode choice functions with Logsums instead of Means should improve path choice fit as much as, in the past, similar operations have improved fits in demand models requiring some inclusive value of modal services, traditionally taken to be an average or some "optimal" path.

But that is not all: in the appendix mentioned above, Daly makes an important theoretical point concerning probabilistically weighted averages p V of type (1-C). They can reverse the direction of the effect of path improvements implied by a Logit choice because, as he demonstrates, the effect on p V of improving particular path c must obey the following positivity constraint:

(7-A) (1 ) 0 p c c c p V V p V V          
8 Another analogy based on a multiplicative form is the use of the expression "Gravity Model" for CES utility functions and, even more frequently, for economic trade and transport models based on Activities and Distance or Cost, rather than on Newton's Masses and Distance. 9 As recognized by Anderson et al. in their footnote on entropy (op. cit., p. 79). 10 The unrealistic idea of the "optimal strategy" [START_REF] Spiess | Optimal strategies: a new assignment model for transit networks[END_REF]) is that transit users always walk to the stop or station that generates the lowest generalized path cost for them. Solutions (optimal strategy assignments) are -sensitive because, in representative urban areas (e.g. Stockholm), or if large zones are used, a traveler departing at a given moment typically has more than a unique path to each destination.

namely any impact of a LOS change on the weighted average p V must be of the same sign as that on path c V . To understand this requirement, note that, starting with (1-C) and remembering (1-A), we have, if varies by

c c V d V : (7-B) p c c i i i c dV p dV V dp     ; but, as it is the case that for i c  (7-C) ( )² i c i V V i c i c c V i e e dp dV p p dV e     
and that for c

(7-D) 2 ( )² [ ] [ ] ( )² i i i i V V c c c c c V V i i e e dp dV p p dV e e       ; it follows that (7-E) ( ) ( ) p c i i c c c c c c c p c c i dV p V p p V p dV p p V V p dV        .
Daly's implied condition that 1 c p V V    for both sides of (7-E) to have the same sign will obviously not hold if c V is one or more units smaller than p V . Consequently (op. cit., p. 157):

"[…] changes in the average can be used as an approximation to changes in the 'Logsum' valuetheoretically the correct value for Logit models -only when the changes relative to the initial probabilities are small", lest changes in path c produce an incoherent result on the indicator p V , i.e. violate condition (7-A).

But small changes are precisely not what large transit projects or modal improvements are about. For this reason, not only is p V an inadequate substitute for the Logsum in general but it can lead to counterintuitive results in the demand or mode choice model precisely when the change considered is relatively important, relative to existing "reference" options, in the project evaluation context11 .

A case in point is presented in Figure 1, which pertains to Paris, where an extension of RER train line E is envisaged. Over a significant part of its itinerary in the East, this line currently competes with other train lines A and C. But E V , the current utility of line E in the area affected by the planned westerly extension, is quite low relative to Train V because it does not yet exist in the West.

RER C RER A RER E Planned extension of RER E RER C RER C
The westerly extension, including a new intersection with train line C, would naturally raise E V considerably, at least relative to the mean. In this case then, use of Train V to represent the utility of "trains" in the demand or mode choice functions is likely to violate Daly's condition (7-A).

In fact the benefit is necessarily understated, even if the direction of the effect is not reversed. It is therefore interesting to perform simulations in order to determine, for various representative situations met in urban mode choice models, the impact of using weighted averages rather than Logsums.

Consequences in terms of impact on representative minority public transit share

How much does it matter that Shannon aggregation error be avoided in the construction of LOS variables for mode choice models? To get an idea of its importance for Greater Paris, we construct an example from representative utilities assumed known for 3 alternatives: Car with a utility 1 V and Public transit (PT), composed of a first public mode with utility 21

V and a second one with utility 22 V .

For two distinct models of Car and PT market shares, index L denoting the Logsum formulation and index W the Weighted Average specification of transit utility, we need the four expressions:

(8-A)

V V V V V V e e V V p V p V L L W W V p V p V V p V p V e e
                      ,
and we purport to simulate the impact of changes in V22 on the difference between PT shares:

(8-B)

                                                   .
In Case A of Table 1, where the actual 2006 Île-de-France12 daily trip shares of the car (0,64) and transit (0,36) are first reproduced by a Logsum term based on a set of hypothesized representative utility values of [2,25; 1,00; 1,00], one finds that substitution of the Weighted Average term (smaller by S = -0,6931) underestimates the true transit share by 64%. V V S   , the ratio of PT shares obtained under each specification in (8-A) is:

(8-C) 1 1 1 1 2 2 p p p L p p p p L p V S V V V S V V V L V S V V V V V V W p e
         g
where it is clear that, for a given attractiveness of the other modes represented by 1 V , the ratio falls with increases in S. It is also clear that, if these other modes have overwhelming market shares, the relative importance of S decreases: if the other mode has about two thirds of the market, as in our Paris example, the correction matters.

But, if Shannon path aggregation error can clearly make an important practical difference to demand (and derived welfare) estimation, does the mathematical form of the utility function -important, as we shall recall, since the beginning of RUM path choice modeling in 1961 -mitigate or worsen the error?

Shannon's information measure S and the functional form of RUF

The mathematical form of random utility functions in transport demand analysis, ignored in the PATH derivation of the S measure above, is in fact a fundamental behavioral issue. Linearity is extremely rare in nature and in practice rejected almost every time it is tested in RUF, as a survey of more than 50 passenger and freight mode choice models where both Time and Fare variables were both subjected to Box-Cox transformations, has recently shown for models developed by some 30 researchers belonging to 10 countries (Gaudry, 2011, Tables 7, 8, 9 and18). As path choice does not differ fundamentally from mode choice, non linearity of Logit path utility functions is also expected and indeed allowed for by many commercial assignment packages13 : we will now recall how early path and mode choice studies, concerned with form, conscientiously retained non linear formulations.

The issue of curvature

Most specifications of LOS variables used by Logit practitioners are in fact nested special cases of the Box-Cox transformation (BCT) applicable to any strictly positive variable Var v :

(8-D) ( ) ( ) 1 , 0 , ln ( ) , 0. v v v Var Var Var              
and notably to the variables of interest for transport project appraisal, primarily Time (for passengers) and Fare (for freight), present in the RUF (0-B) which can be rewritten explicitly with BCT:

(8-E) ik ( ) i i 0 i k i k k V X      
As already mentioned above, non linearity, as illustrated in Figure 3 for the binomial case, means that the reaction curve to improvements in variable X 1 associated with alternative 1 will be asymmetric with respect to its inflexion point: it would be symmetric with an inflexion point at p 1 = 0,50 only if the data supported in (8-E) the unlikely assumption of constant marginal utility 1, for , Asymmetry is therefore critically important given that, in forecasts of important changes in LOS, everything is in the curvature because there is no real disagreement on the identity of important variables and because LOS changes considered are far from marginal, consisting for instance in a division by two of travel time. In fact, the asymmetric logarithmic response, implying a curve situated above that of the linear response for [1 < X 1 < 5,5] in the case illustrated in Figure 3, prevailed in the careful empiricism of the founders of path and mode choice analysis, as we now recall before formally addressing the issue of the impact of the form of path RUF on the measure S.

ik i k    .

The secret origin of random utility models (RUM) and their functional form

We studied the foundations of random utility modeling, apparently first developed to explain road path choice, and not mode choice, in a close-knit engineering milieu of the type described by [START_REF] Ekelund | Secret Origins of Modern Microeconomics: Dupuit and the Engineers[END_REF] for the "secret origins of microeconomics": we carefully consulted written sources and queried witnesses, notably experienced transport engineers from public and private institutions.

In a nutshell, it seems that formal derivations of formulae based on the Gaussian and Rectangular distributions, published in a scientific journal for roads and airports [START_REF] Abraham | La répartition du trafic entre itinéraires concurrents: réflexions sur le comportement des usagers, application au calcul des péages[END_REF] 14 , served for the milieu of French engineers as an explicit justification of prevailing pre-existing road path choice assignment models and of their descendants, all then based on a Logit core.

Notably, a particular variant, sometimes called a "Logarithmic Logit" (e.g. [START_REF] Leurent | Calcul de deux modèles d'affectation[END_REF], was soon designated in official French documents, and remains so to this day, under the name Abraham's Law of traffic assignment, despite the fact that the Logit form itself had not been derived by Claude Abraham who claimed then and now that he is not the author of the so-called "Loi d'Abraham" used as an effective approximation of the presumably real McCoys derived and documented at length in his demonstrations. The attribution of this label by the milieu is all the less surprising that Abraham had made it crystal clear that it did not really matter15 what the underlying distribution really is.

For transit assignment problems, the preference went to a linear RUF: Barbier's binomial Probit application to bus and metro, published in 1966, may well have inspired later generalizations by [START_REF] Domencich | Urban Travel Demand: A Behavioral Analysis[END_REF], still all formulated with linear RUF. Now for the "petite histoire".

Logit practice and Abraham's 1961 derivations from Normal & Rectangular distributions

The mood on the banks of the Seine: Logit road choice applications, with non linear utilities, in search of a justification. It is useful to comment on four Parisian steps, distinguished and summarized in Table 2, to document the first RUM:

1) both Setec (1959) and [START_REF] Abraham | La répartition du trafic entre itinéraires concurrents: réflexions sur le comportement des usagers, application au calcul des péages[END_REF] provide a panorama of American and French road assignment practices. In the former case of a Channel Tunnel study, Logit forms dominated and were applied with both linear (model M1) and logarithmic (model M2) RUF, both of which contained explicit AGC and were considered "approximations" of the Probit, called "Modèle Normal";

2-3) all formal derivations from assumed distributions of random terms found in Equation (9-B) below, published or not and based on the Gaussian or on the Rectangular distribution, used linear LOS terms but insisted that they applied as well to logarithms. They were also explicitly derived as "justifications" of then-current Logit practices.

In a footnote (op. cit. p. v) of the anonymous Setec derivation, "Mr. Malcor" was credited with the idea of distributed differences in "subjective valuations" but no reference was given. And the formulation is not mentioned in his just-published article on road traffic and operations research [START_REF] Malcor | Théorie de la circulation routière et recherche opérationnelle[END_REF].

In his seminal paper, [START_REF] Abraham | La répartition du trafic entre itinéraires concurrents: réflexions sur le comportement des usagers, application au calcul des péages[END_REF] considered an individual n choosing between two road paths 1 and 2 with generalized costs (GC) composed of mean linearly weighted Length and Time elements, respectively GC 1 and GC 2 . He formulated (op. cit., p. 66) 16 the choice probability, or proportion of users, for the first path as given by:

(9-A)

1 2 1 Prob 0 n n p U U        ,
where the utilities of the paths are assumed to be (9-B)

1 1 1 1 n n n U GC    and 2 2 2 2 n n n U GC      ,
and the two errors of zero mean are respectively associated with Length (Cost) and Time elements of the representative GC term. He then considered how assumptions concerning the distribution of  and  affected the structure of the path choice model, first deriving a Probit under the assumption of Normal distributions and another model under the alternate assumption of Rectangular distributions. He performed simulations with both structures for 2-path and 3-path cases based on California expressway data and finally addressed related problems, in particular how to handle cases of paths sharing a link and how to calculate revenues from tolls applied only to subsets of links.

2 1 0 n n V V        [ ) ] T i C i Time Cost    
0 0 0, , i j i j paths          [ ln( )] i Generalized Cost  
1 The derivation uses a linear utility function but the author considers non linearity much more credible 17 . 2 With standard errors assumed equal across distributions.

Although his formulation (9-A)-(9-B) is identical to that found later in CRA (1972, Ch. 5) or in Domencich & McFadden (1975, Ch. 4, S. 4) 18 , he did not consider the Weibull and Cauchy distributions; 4) in particular, the Probit derivation was seen as a justification of Logit practice. We could not successfully date exactly the first use of the expression "Abraham's Law" which combined (1-A) with logarithms of the i GC to implement non linear path RUF without constants 19 :

(9-C) 1 exp( ) / exp( )

M i i i i p V V    , with ln i i V GC    .
In step, French road manuals have long recommended values of  as high as 8 or 10 for intercity road path choice modeled according to "Abraham's Law".

The use of a logarithmic RUF was also voluntary in [START_REF] Warner | Stochastic Choice of Mode in Urban Travel: A Study in Binary Choice[END_REF] binary urban Logit mode choice model: concerned with goodness-of-fit, he compared various LOS forms 20 and retained the logarithmic one after a very careful analysis of residuals, noticing the inferior fit obtained under a linear form and referring explicitly to traditional log-linear CES production functions. This is not to say that all path choice models then used in France were straightforward applications of Abraham's Law. But the Probit rapidly became part of the common toolbox: in an analysis of the 16 He claimed in a footnote linked to Equation 9-A that Setec (1959) had first formalized this model in Channel studies. That unpublished consulting report, easily downloadable from the referenced Ministry of Transport site, indeed contains a derivation of a Probit model, with RUF based solely on path costs, under the assumption of "Laplace-Gauss" errors. 17 As reiterated by Claude Abraham in an email to the authors, dated August 18, 2011: "In the classical [Linear] Logit, there is no difference between a two-minute gain on a 10 minute and on a 60 minute trip, a manifestly absurd hypothesis". 18 Those authors called their model derived from a Rectangular distribution the "truncated linear probability model". 19 In practice, this is compensated by correction factors called « bonuses », for instance for highways, etc. 20 Thomas (1967) applies the same careful methodology in a study of car commuter values of time.

profitability of 1961-1962 road works, the Probit curve appears, without further comment, listed among four diversion curve methods (Abraham & Thédié, 1966, p. 145).

Barbier's casual 1963 Probit application: transit choice, with linear utility. This casual state-ofthe-art use of the Probit model is also found in transit studies. In 1963, for instance, Michel Barbier, an urban planning engineer 21 working for the Paris Region Planning Institute (IAURP) studied the choice between bus and metro with a sample collected by Setec and SNCF. In his working paper, Barbier (B., 1963b): (i) carried out a full discriminant analysis to find good combinations of costs, frequency, itinerary time and number of transfers which resulted in the same proportion of travelers to a particular destination using the bus or the metro; (ii) formulated an explicit Binomial Probit model of choice between these means of transport based on a difference in generalized cost (a combination of Cost and Time) expressed in time units D, namely 22 :

"Noticing that the users' indifference point towards means of transport, such that 50% of users take the bus [rather than the metro], corresponds to a time difference between them of D=5 minutes, we assume that each user takes the bus if he estimates that 5 D  . If P(D) denotes the probability of taking the bus, we then have: Linking the Seine and the Charles rivers. The reason for pointing to this model as a potential source of their inspiration is that Barbier's working paper and its published version are to some extent summarized and referenced 23 in the of key documents resulting from these authors' work.

Strangely, however, this connection appears in the literature review (CRA, 1972, p. 3-9;Domencich & McFadden, 1975, p. 25), rather than in the theory development section, and no mention is made of the Probit analysis (9-D). All that is said is:

"A French study [Institut d'aménagement et d'urbanisme de la Région Parisienne (1963)] also carried out a full discriminant analysis to find good combinations of times, costs, and number of transfers which resulted in the same proportion of travelers to a particular destination using transit."

for which the provided reference in both documents is simply: (Barbier, 1966) is identical, as may be verified in Appendix 1 below where Figure 4 is captured from it. 23 The only reference in French among the 126 listed in the CRA version, or in the list augmented to 243 for the book. 24 Only the author's initials are used on the front page. Barbier, M. (1966) A "Gang of Four". In any case, some 11 years after Abraham's publication, a consulting study by Charles River Associates Incorporated presented in its theoretical section entitled "stochastic specification and estimation techniques" (CRA, 1972, Ch. 5) four derivations from (9-A)-(9-B), effectively adding to previous Gaussian and Rectangular branches new ones based on Weibull and Cauchy error distribution assumptions, yielding the Logit and Arctan form determinations of binomial choice probabilities. This "gang of four", doubling Abraham's twin achievement, even produced un embarras de richesse: multinomial derivations were also presented for the Gaussian and Weibull cases27 and the computational simplicity of Multinomial Logit estimation naturally emphasized.

Randomness in earlier, less general, approaches. Interestingly, the revised version of the consulting report 28 , published as a book [START_REF] Domencich | Urban Travel Demand: A Behavioral Analysis[END_REF], starts with a laudatory foreword by Richard E. Quandt who had already noted elsewhere (Quandt, circa 1974) the extent to which the authors' work constituted a "more general and sophisticated development of" some models by Anthony J. Blackburn and himself. These developments, treating trade-offs between modal cost and time as random variables29 in models estimated from aggregate data, are very globally referenced in Ch. 4 ("A theory of Population Travel Demand Behavior") of the book as forming one of two earlier approaches yielding functional forms for the choice probabilities, the other being attributed to [START_REF] Thurstone | A law of comparative judgement[END_REF] "suggested particular case" (op. cit., p. 53). This under-identified nameless particular case30 turns out to be a Probit application, just like Barbier's model.

RUF forms in Abraham-McFadden (A-M) derivations and in their environment.

Published RUM derivations forming "the Abraham-McFadden approach" [START_REF] Abraham | La répartition du trafic entre itinéraires concurrents: réflexions sur le comportement des usagers, application au calcul des péages[END_REF][START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF] all assumed linear forms for the RUF, as did applications of this A-M approach by Barbier (1963bBarbier ( , 1966)), CRA (1973) and [START_REF] Domencich | Urban Travel Demand: A Behavioral Analysis[END_REF].

But the environment was changing: if [START_REF] Blackburn | A nonlinear model of passenger demand[END_REF] formulates a linear RUF, many, including [START_REF] Warner | Stochastic Choice of Mode in Urban Travel: A Study in Binary Choice[END_REF], practitioners of Abraham's Law, and Quandt (1968), formulate or test and willfully select multiplicative RUF.

Fogel's unresolved cliometric question of 1964

By contrast with these practitioners of Abraham's Law or with Warner and Quandt, many at the time indeed avoided hard form test work and relied on fast and easy linearity. For instance, [START_REF] Fogel | Railroads and American Economic Growth: Essays in Econometric History[END_REF] was interested in defining a measure of changes in generalized transport costs following the introduction of rail in a 19 th Century world where water and rail modes competed and maintained positive market shares. Instead of due changes in a Logsum, a notion available only later [START_REF] Small | Applied Welfare Economics with Discrete Choice Models[END_REF], he used the difference between water and rail costs (extreme cases of 0 and 100% market shares of rail) to build his principal measure, called alpha31 :  

rail water GC GC   .
As noted in the survey of 50 models referred to above, such a priori linear specifications have since been rejected in numerous Logit freight mode choice cases (op. cit., Table 9) where Box-Cox transformations, e.g.  

( ) ( ) C C rail water GC GC    
, were tested for rail, road and water, and notably in the very large European canal project Seine-Nord Europe linking the Paris area to Belgium (Setec International et al., 2006).

Fogel found that, as an explanatory variable of "social savings", the difference between linear costs (alpha) was not very significant [START_REF] Davis | Review Essay[END_REF]. The cliometric question is then whether a more adequate Logsum measure, or even appropriately non linear forms of alpha derived with Warner's due care, would have given less disappointing results than those obtained under unlikely linearity.

The impact of RUF form on S

Problem formulation. Non linearity of path choice RUF specifications might be established by trial and error, as in applications of Abraham's Law where the coefficient  is conventionally changed manually according to the road type, or by Warner's ad hoc method where simple powers of variables are manipulated to improve fit. To reach beyond such special cases, we consider the more general specification of the RUF formally based on determination by the Box-Cox transformation32 (8-D). The question in practice is then whether there exists a systematic link between the value of the BCT on the key LOS variables found in path RUF and the S measure.

To answer it, we reintroduce Abraham's observation subscripts, useful here to isolate specific values of LOS variables with particular properties, and compare two choice models differing only in RUF form. By assumption, we have 1 2 and in in p p , the estimated individual shares or choice probabilities for models 1 and 2, formulated with the same number of variables (to simplify notation without loss of generality) but different constraints on the  BCT parameters applied to the independent variables of each model, namely:

( 10-A)

1 1 1 exp exp in in mn m V p V   , with 1 ( ) 1 1 1 1 ik in io ik ikn k V X      
, and

(10-B) 2 2 2 exp exp in in mn m V p V   , with 2 ( ) 2 2 2 2 ik in io ik ikn k V X       ,
i i k i k    and 2 0 2 2 ( , , ) i i k i k   
summarize the parameters associated to models 1 and 2, respectively.

Note again that use of BCT in path choice models assumes that an alternative-generic constant (AGC) is used on all M paths of each origin-destination pair because invariance of BCT form estimates to changes in the units of measurement of the variables requires the presence of a regression constant [START_REF] Schlesselman | Power families: a note on the Box and Cox transformation[END_REF]. In path choice however, none of the M constants can be set at zero for a reference path (as in the case of modes) because paths have no natural labels. But the required AGC constant, which modifies all path shares by adding a common amount to all RUF and consequently has a modeling role of its own, still has to be estimated33 if one is to stay away from simple power functions which have uncorrectable problems34 .

Path constants are typically ignored in applications of Abraham's (non linear) Law, as they were in (linear) Logit applications to road tracé choice [START_REF] Mcfadden | The Revealed Preferences of a Government Bureaucracy[END_REF](McFadden, , also 1975[START_REF] Mcfadden | The Revealed Preferences of a Government Bureaucracy: Empirical Evidence[END_REF] and to road path choice [START_REF] Dial | A Probabilistic Multipath Traffic Assignment Model which Obviates Path Enumeration[END_REF]: in the latter case, vehicles are assigned, between an origin and a destination separated by the shortest length L*, to each path of length L "proportionately to (op. cit., p. 91).

  * exp L L       "
This said about (10), the difference of interest for our purposes can be defined compactly as follows:

(11)     2 1 2 2 1 1 2 2 1 1 2 1 2 1 ln ln ln ln i i i i j j j j j i S S S p p p p p p p p D D C C          ,
where the terms D 2 and D 1 collected in the first parenthesis are direct (own) terms and the others found in the second parenthesis, C 2 and C 1 , are cross terms in the sense that we are interested in effects of changes in a particular variable 1 2 iqn iqn iqn X X X   (such as Time or Fare) that is common to both models but, in (10), appears only in the own RUF of alternative i.

A first question concerning S

 might be whether there exists a value of 1 2 iqn iqn iqn X X X   for which errors of aggregation S2 and S1 are equal: to find such a crossing point, we could solve 0 n S   for iqn X  , but the result would be of limited interest.

Much more interesting for our purposes should be the existence of a maximum or minimum difference between the two measures obtained by first solving / 0 n i q n S X    for the turning point iqn X  and then by determining whether it is a maximum or a minimum by considering the second derivative

2 2 / n i q n S X   
, evaluating it at the critical point iqn X  , and finding out how it changes signs when iqn X passes through it.

Having found the expression for the second derivative, one could envisage solving

2 2 / 0 n i q n S X     to
determine the inflexion point iqn X  , but this value would also be of very marginal interest for the central issue, that of the existence of a general impact of distinct RUF forms on S, and in particular of the difference between linear symmetric and non linear asymmetric cases.

A focus on the turning point. We therefore focus the analysis on the derivation and properties of the turning point iqn X  . We present here the short form of the first and second derivatives of S  with respect to iqn X and reserve for appendices, using a longer form wherein probabilities are made explicit, iqn X is isolated, and solutions for it can be readily considered -at least qualitatively-, the demonstration that there is no analytical solution for this turning point iqn X  , which must be found and also signed by numerical methods.

Starting with the first derivative of (11) with respect to iqn X , which may be written in short form:

(12)         2 1 2 1 2 1 2 1 1 ln 1 ln 1 ln 1 ln , n i n i n in in iqn iqn iqn jn jn jn jn j i iqn iqn S p p p p X X X p p p p X X                                                                            
it is made clear in Appendix B, using the long form of ( 12), that finding the critical value iqn iqn X X   that equalizes it to zero is not analytically feasible and requires numerical methods.

But of course this value iqn iqn X X   will be a maximum or a minimum depending on whether the second derivative evaluated at that point is negative or positive. For this sign determination, the second derivative of (11) with respect to iqn X may in turn be written in short form from (12) as:

(13)       2 2 2 2 2 2 2 1 1 2 1 2 2 2 2 1 2 2 2 2 2 2 2 1 1 1 ln 1 ln 1 1 ln n i n i n i n i n in in iqn iqn in iqn iqn in iqn jn jn jn iqn jn iqn S p p p p p p X X p X X p X p p p X p X                                                                                                  2 2 1 1 1 2 1 1 1 ln jn jn jn j i iqn jn iqn p p p X p X                                               
and it is shown in Appendix C, using the long form of (13) with terms arranged to again put the iqn X in evidence, that such sign determination also requires a numerical exercise.

The maximum or minimum difference between error measures S 2 and S 1 therefore bears a systematic link with the functional form of the RUF but that linkage can only be evaluated by numerical methods, on a case by case basis: it always depends on many variables and parameters in non linear ways unfortunately not amenable to analytic treatment.

This conclusion remains if the comparison between the two models distinguished solely by functional form of the RUF is effected between any sophisticated nonlinear (asymmetric response) Box-Cox Logit and a pedestrian Linear (symmetric response) Logit.

How many Logsums for dense transit networks?

The clear superiority of Logsum over weighted average LOS measures implies that the choice among paths within transit and car networks should be addressed by Logit models. In many actual modal networks, this could mean accounting for subtle differences among transit modes or road types by dummy variables associated to the relevant links of the transit or road networks and concentrating the effort on the explanation of path shares between origin-destination points. This includes a role for path constants, notably when the long-demonstrated non linearity in LOS is handled by Box-Cox transformations rather than by simple power functions of generalized costs 35 or of individual LOS service variables. But modes have specificities.

A baker's combinatorial dozen?

In public transit networks, should a compromise layer of branches be considered between the traditional modes (car, transit, on foot, etc.) and the transit path access means, especially when the number of transit modes is plethoric? Our Paris predicament is that there are at least 4 different types of buses 36 (Ordinary, Bus Rapid Transit (BRT), T-Zen 37 , Local minibuses), 2 kinds of tramways (large ones on rails, with high windows; smaller ones on tires) and of metros (ordinary and automatic) and regional trains of quite different characteristics, "feel" and comfort. If a hierarchy is considered, which of these 10+ means are the high modes and which the low modes merely serving as access to the higher modes and requiring a path access model of their own?

And should the transit hierarchy depend on the direction of a return trip? Conceivable hierarchies 38 are many but are not nested in a statistical sense. It might well be easier to obtain specific cross-effects (including complementarity) by including in the RUF of a MNL the characteristics of some other modes (with due constraints on LOS forms), as in classical microeconomic demand systems, than to decide on the most credible hierarchy of higher and lower (access) modes among 10 transit modes.

Is Wardrop equilibrium in palliative care? Path costs are always generalized costs. So, if equilibrium methods are used to model road path choice, two acute problems arise. First, even in the simplistic case where time and cost intervene linearly, user equilibrium is unique only if users have a single value of time or if cost and time change with flow on each link in identical manner [START_REF] Dafermos | A Multicriteria Route-Mode Choice Traffic Equilibrium Model[END_REF]. Moreover, as in Wardrop's equilibrium link flows are unique but the number of itineraries used is unknown and their identity is not analytically derivable from the optimal solution 39 , their due aggregation by Logsums is problematic. The necessity of identifying all itineraries effectively used, in conformity with the path aggregation theorem (PATH), should prompt a movement of analysts and commercial programs away from equilibrium assignment and towards the use of Logit based methods.

This hold also for extensions of Wardrop equilibrium to public transit passenger flows, as in Pirandello [START_REF] Piron | Pirandello, un nouvel outil d'aide à la décision[END_REF] where volume-delay curves are replaced by volume-discomfort curves -with comfort defined by the number of users per square meter of vehicle space. The same objection applies to replacement of comfort by other notions, such as wait-time at stops.

Are freight assignment models exceptions? It might be thought that freight, where choice of mode and path is often combined within an extended mode-and-path abstract choice formulation, would avoid the multipath assignment and aggregation problems discussed above for passengers and still 35 As noted above, the Kirchhoff formula (PTV AG, op. cit., p. 511) is indistinguishable from Abraham's Law. 36 Among the 1 433 bus lines covering 24 660 km of routes, many are complementary with the rail system but many are in competition with it. The metro part of the rail system currently has 300 stations on 215 km of lines and the Grand Paris Automated Metro would add 175 km of lines and about 60 stations. 37 T-Zen buses in service since 2011 in the Paris area benefit from dedicated Bus Rapid Transit (BRT) lanes but have tramway-type doors and windows. Fish or fowl? 38 A Bus-Train layer was developed in the first version of the SAMPERS regional model for Sweden [START_REF] Transek | The SAMPERS System: Overview[END_REF] and abandoned when the model was revised and updated in 2003 [START_REF] Transek | Utveckling av Sampers 2.1. Del 1 Estimeringeninkl tilläggstester[END_REF][START_REF] Transek | Utveckling av Sampers 2.1. Del 1 Estimeringeninkl tilläggstester[END_REF]. Other models, such as PRISM, developed for Birmingham (Rand Europe, 2004), had many fewer transit modes than Paris. 39 Sometimes authors use very astute patches (e.g. [START_REF] Bar-Gera | Primal Method for Determining the Most Likely Route Flows in Large Networks[END_REF][START_REF] Bar-Gera | User-equilibrium route flows and the condition of proportionality[END_REF] to compensate for this lack.

obtain relatively good fits by simple shortest path methods. But that would underestimate the refinement of freight assignment procedures. For instance NODUS, originally conceived in this way [START_REF] Jourquin | Un outil d'analyse économique des transports de marchandises sur des réseaux multimodaux et multi-produit. Le réseau virtuel, concept, méthodes et applications[END_REF][START_REF] Jourquin | A decade of freight transport modelling with virtual networks: Acquired experiences and new challenges. GTM-paper[END_REF] has recently added a procedure which retains the least cost mode-path itinerary and assigns the origin-destination flow to competing itineraries in proportion to costs [START_REF] Jourquin | Une procédure d'affectation multi-flux et multimodale appliquée aux réseaux transeuropéens de transport de fret[END_REF]: but proportions bring us back to Abraham's Law... It would seem that, between the opposite excesses of non nested hierarchies of dubious meaning and simple-minded shortest path and equilibrium assignments, multiple path realism combined to the Logsum can go a long way, notably in plethoric networks, towards making sense of observed multiple path use and reflect the value added by enriching already dense modal networks, urban or not.

Conclusion

We have interpreted the often mentioned difference between Logsum and average utility in terms of [START_REF] Shannon | A Mathematical Theory of Communication[END_REF] 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 have studied 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 or mode choice model formulation 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 have showed by simulation that averaging modes or sub-modes -a frequent feature of traffic modeling studies -can lead to important errors in terms of level of traffic and of welfare measurement.

Concerning the mathematical form of the RUF, we have recalled 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 assignment problems, the preference went to a linear RUF: Barbier's casual binomial Probit application to bus and metro, published in 1966, 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 have analyzed the impact of such nonlinearity on S. This impact is tractable through a comparison of measures S 2 and S 1 associated with two path choice models differing only in RUF form, as determined by Box-Cox transformations applied to their level-ofservice (LOS) variables. We have showed that, although the difference between measures S 2 and S 1 may reach a minimum or a maximum with changes in LOS, the solution for such a turning point cannot be established analytically but requires the use of numerical methods. The demonstrable impact on S of nonlinearity, or asymmetry of Logit curve response, is consequently tractable only at non trivial computational cost.

We have pointed 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) but also arises in other modes whenever distinct itineraries or lines compete within a single mode. Concerning dense urban PT networks, we have hypothesized 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, as it does for any application of Wardrop to transit path choice.

This shift should favor transit when it is the minority mode.

Appendix A. Barbier's Probit transit mode choice model

Figure 4 presents a pasted extract from Barbier (1966, Annexe 3, p. 55), repeating the title of the appendix in which it is found.

Appendix B. The long form of the first derivative

We wish to document more explicitly, but still partially40 for brevity, the steps taken to isolate iqn X in (B-2), the long form of ( 12), the first derivative of S  with respect to iqn X .

Making probabilities in (12) explicit and first isolating iqn X only in all partial derivatives, one obtains:

(B-1) 2 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 ln
X S V V V X V V V X V V V V V V V                                                                    g g         2 1 1 2 2 2 2 2 2 2 1 1 1 1 1 2 1 1 exp ln exp 1 exp / exp exp 1 ln exp 1 exp / exp iq iq jn jn mn iq iqn m mn in m i j i jn jn mn iq iqn m mn in m i V V X V V V V V X V V                                                               g g     However, as 1 exp in V = 1 1 1 1 exp exp( / ) iq in iq iqn iq V X     and 2 exp in V = 2 2 2 2 exp exp( / ) iq in iq iqn iq V X     , where 1 ( ) 1 10 1 1 1 1 ( / ) ik in i ik ikn iq iq k q V X            and 2 ( ) 2 20 2 2 2 2 ( / ) ik in i ik ikn iq iq k q V X            , equation (B-
m i S V X V X V X V V X X V V X                                     g                 1 1 1 1 1 * 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 * 2 1 ( / )
V X V X V V V X X V V X V                                      g                       2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 2 * 1 1 1 1 1 1 ln exp exp( / )
V X V V X X V V V V X V V                                                 g g                1 1 1 1 2 1 1 1 1 , exp / exp . iq iq j i iq iqn iq iqn iq mn in m i X X V V                                                           where 2 1 ( ) ( ) 1 2 2 20 2 2 2 2 1 10 1 1 1 1 , 0, ( / )and ( / ). ik ik iq iq in i ik ikn iq iq in i ik ikn iq iq k q k q V X V X                          10. Appendix C.

The long form of the second derivative

We wish to document more explicitly, but still partially41 for brevity, the intermediate steps taken to isolate iqn X in (C-6), the long form of the second derivatives of S  with respect to iqn X .

For this, we focus on the first and third components of ( 13

m i in in p V X X V V V V p V X V V X V p V V V V X V p                                                     2 2 1 2 2 2 1 2 2 2 xp exp 1 iq iq iq iqn in mn m i in in iq iqn X V V p p X                    and (C-2)           2 2 2 2 2 1 2 2 2 2 2 2 1 1 2 2 2 2
                                                                      2 2 2 2 2 1 2 1 2 1 1 2 2 2 2
p p X p X p X X X                                                  
where replacing 2 / in iqn p X   with its value given in (C-1) allows for the following formulation:

(C-3)                   1 1 1 1 1 1
                                                    2 2 2 2 2 1 1 1 1 2 2 2 2 2
                                                            2 2 2 2 1
                                              2 2 2(
p V j i X X V V X V V V V X V V V X V V                                                                   
p V X p p j i X X X X V X p V X p X X p p V X X                                                      2 2 2 2 2 2 1 2 1 2 2 2 2
X V p X V X X p p V X V X p V X V X                                               2
p V X p V X X p V X p X                                                1
p V V p V V V X m i V X m i V X X V X                                                1
p V V j i p V V j i X m i X m i V V X X m i                                                     1 1 2 1 ( ) ( ) 1 1 ( ) (
X X m i V X j i V X j i                          
an expression that clearly cannot be signed analytically for given turning point values iqn iqn X X   .

Figure 1 .

 1 Figure 1. Representation of total differentials of Equations (5-A) and (5-B)..........................................8 Figure 2. Extension of Regional Express train line E in the densely served centre of Paris..................11 Figure 3. Classical Linear-Logit vs Standard Box-Cox-Logit Responses.............................................14 Figure 4. Formulation théorique des résultats observés pour le choix entre autobus et métropolitain. .28

  the set of probabilities p1, …, pn.»

  the same sign as that of the cosine of the angle formed by B and Γ.

Figure 1 .

 1 Figure 1. Representation of total differentials of Equations (5-A) and (5-B)

Figure 2 .

 2 Figure 2. Extension of Regional Express train line E in the densely served centre of Paris West Paris Downtown East

Figure 3 .

 3 Figure 3. Classical Linear-Logit vs Standard Box-Cox-Logit Responses

  be written…[…]." a formulation, linear in the generalized cost variable, that may have influenced comparable work carried out in Boston in 1970-1971 byThomas A. Domencich and Daniel McFadden.

Figure 4 .

 4 Figure 4. Formulation théorique des résultats observés pour le choix entre autobus et métropolitain

Table 1 . Impact of Shannon aggregation error on simulated public transit share in Île-de-France

 1 Moreover, if the utility of the second public mode is successively increased by 25% in Case B and by 50% in Case C due to assumed service level improvements, PT share forecasts derived from the Weighted Average measure of inclusive value continue to underestimate the same shares as forecasted by the Logsum measure. But we note that, as S increases, the relative superiority of the Logsum measure decreases, a property that can be analyzed more formally.

	Case A

Table 2 . Four steps in the development of random utility path assignment models around 1961

 2

	Given name	Constant in the RUF		Form and variables in the RUF	Source
	1. Existing Binomial Logit practice circa 1958 M1 model 02 01 0 [ ]      M2 model			[   [ ln( Cost i Cost ] i  	)]		Setec, 1959 paragraph 2.220 p. iii paragraph 2.221 p. iv
	2. Anonymous derivation of the Multinomial Probit, based on the Gaussian distribution, in 1959	Setec, 1959
	Modèle Normal	0     0 , , i j i j paths  	 	[ 	], Cost or i	[ ln( 	i Cost	)]	paragraph 2.60, p. ix
	3. Binomial derivations 1 and tests based on Gaussian and Rectangular distributions 2 in 1961	Abraham, 1961
	Gaussian

  . Choix entre autobus et métropolitain, pp. 27-40, 51-56. Ch. II et Annexes 1-3 in Barbier, M. et Merlin, P., Choix du moyen de transport par les usagers. Cahiers 25 de l'Institut d'aménagement et d'urbanisme de la région parisienne (Cahiers de l'IAURP) 26 Vol. 4-5, 57 pages, avril.

  ) corresponding to direct term D 2n and cross term C 2n in (11). The second component, D 1n , may be obtained from the first by replacing 2in

																				p with
	1in p and the fourth component, C 2n , from the third by replacing 2 jn p with 1 jn p .
	In component 2n D , the first and second derivatives of 2in p with respect to iqn X are, in turn:
	2	in					exp		2	in
	iqn	iqn	exp	2	in	m i	exp	2	mn
		2	in	1 exp	2	in	exp iqn	2	in	exp	2	in	1	m i	exp	2	mn	exp iqn	2	in
	(C-1)	2	in	1 exp	2	in	exp		2	in	1		exp	2	mn	exp	2	in	2 iqn in
		2		1 e			exp			2

The application of this Hicksian terminology to transport networks was proposed by Åke Andersson at the International Symposium on Travel Supply Models,Montreal, November 17-19, 1977.

The use of these weighted averages appears to be based on intuition or on long-established practice reflected in computer packages: we are unaware of any derivation of (1-A) from utility that would mandate (1-C) or (1-D) as path aggregators.

The variance of the choice probabilities is not independent from that of the random term associated with each RUF: a large variance of this random term implies that the systematic part of the utility function, to be formulated explicitly below in our discussion of the seminal[START_REF] Abraham | La répartition du trafic entre itinéraires concurrents: réflexions sur le comportement des usagers, application au calcul des péages[END_REF] paper, has a relatively smaller role to play as compared to that of the random term. This implies heteroskedasticity of random errors, or some systematic departures from the common homoskedastic variance { 2 /6}. When the ratio between the mean of the systematic part of the utilities and the random term is extremely high, say infinite, any mean will adequately reproduce the utility of the full set of options: the case is analogous to that of the red bus blue bus paradox.

Such that the probabilities sum to one and the sum of their variations equals zero.

[START_REF] Theil | The information approach to demand analysis[END_REF][START_REF] Theil | Economics and Information Theory[END_REF] had a long-standing interest in applications of information theory to economics.

This thesis is summarized in Picard &Gaudry (1998). The problem was to find, for each of 64 categories of freight transported in Canada, the optimal flows among the 67 principal cities (as origins or destinations), with the flows satisfying row and column total constraints of regional matrices for the cities situated in 8 regions (large provinces or groupings of small provinces). The objective function guarantees that the 64 matrices of estimated flows of dimension 67x67 are "near" exogenously provided a priori or observed values.

[START_REF] Hagen-Zanker | A New Method of Adaptive Zoning for Spatial Interaction Models[END_REF] use the same distance measure applied to the normalized flows of trip distribution matrices and duly refer to Shannon.

The same problem arises in the hypothetical situation of the addition of a new very bad service s with a very low Vs. In this case, the Logsum will, by MNL logic, increase despite the fact that p V falls.

The Île-de-France region, comprising the Greater Paris area, had a population of 11,7 million in 2008. The information on shares was graciously provided by Syndicat des transports d'Île-de-France (STIF). We neglect here other modes.

For instance, VISUM 11.5 (PTV AG, op. cit., p. 507) allows for various options including two where the RUM, consisting solely of a generalized cost, is either transformed logarithmically -in which case Kirchhoff's distribution formula(Fellendorf & Vortisch, 2010, Equation 2.5) is equivalent to Abraham's Law described below -, or subjected to a Box-Cox transformation. If values for the latter are supplied from outside, the invariance problems raised by the absence of intercepts[START_REF] Schlesselman | Power families: a note on the Box and Cox transformation[END_REF] in the path utilities are dodged.

The article, which contains a short presentation by Coquand, Abraham's boss and Director of Roads and Traffic of the French Ministry of transport, is sometimes referenced with this second name, but we follow Frank Haight's usage in his extensive annotated "bibliography in road traffic"(1964) and neglect the author of the administrative imprimatur.

To quote him at the end of the section providing the derivation based on the Rectangular distribution: "Que la distribution réelle des estimations des usagers soit ou non gaussienne, nous n'en savons rien, et cela n'a, au demeurant, pas grande importance" (op. cit., p. 68).

In the MIT Libraries'catalog listing all volumes from 1 to 41 (except 28), the long form is used. In these Cahiers (ISSN 0020-2207), authors' names are found on the back of the cover page of each article or chapter. As Chapter II of Vol. 4-5 pooledBarbier's working paper (1963b) on the choice between bus and metro with other working papers by Merlin on choices between other modal pairs, the reference "Barbier & Merlin (1966)"would also be an adequate reference.

The conventional short form is used on all IAURP Cahiers document covers. The long form is «Cahiers de l'Institut d'aménagement et d'urbanisme de la région parisienne». The sub-headings on the front page of Volume 4-5 [Transports urbains. Les transports urbains et leurs usagers en région de Paris : 2] refer to Cahier theme series and their subdivisions.

27McFadden (1975, Footnote 14) later attributed the Logit derivation from the Weibull distribution solely to[START_REF] Mcfadden | Conditional logit analysis of qualitative choice behaviour[END_REF] where empirical estimates are reproduced with permission from CRA (

1972).28 Without the interesting Appendix A attempt to derive transport demand functions from specific Quadratic, Log-Linear and Stone-Geary utility

functions.29 Blackburn's Ph.D. thesis, listed among the Abstracts of theses for 1966-1967 at M.I.T., and the paper clearly derived from it[START_REF] Blackburn | A nonlinear model of passenger demand[END_REF], as well as Quandt's article[START_REF] Quandt | Estimation of Modal Splits[END_REF], are contemporaneous with early work on random coefficients in linear regression[START_REF] Hildreth | Some estimators for a linear model with random coefficients[END_REF][START_REF] Swamy | Efficient Inference in a Random Coefficient Regression Model[END_REF] and might well have their most appropriate place in that literature recently blooming again in Mixed Logit garb in transportation, after a long dormition (e.g.Hensher &

[START_REF] Hensher | A Random Coefficient Model of the Determinants of Frequency of Shopping Trips[END_REF].30 This contrasts with the claims of the consulting report where (on page 5-6)[START_REF] Thurstone | A law of comparative judgement[END_REF] had been credited with the first multinomial formulation of Equation (9), also described (on page 4-4) as a model that is "well known (see, for example,Luce, Individual Choice Behavior, Wiley, 1959) and need not be elaborated here".

To estimate alpha, Fogel focused on a sample of 30 of the 825 potential routes between pairs of cities in the West and East of the United States.

There are of course other ways to introduce non linearity of the RUF. For instance,[START_REF] Palma | Route choice decisions under uncertainty[END_REF] successfully use a cubic form on travel time in a Probit model for the Île-de-France region.

An approach to the estimation of alternative-generic path constants for all paths is discussed in[START_REF] Gaudry | Identifying all alternative-specific constants in Multinomial Logit models by Inverse Power Transformation Capture[END_REF].

For instance, in contrast to BCT, simple powers do not maintain the order of the data(Johnston, 1984, p. 63).

The detailed steps taken from (12) to (B-2) for all first derivatives are described in full in[START_REF] Gaudry | Choix modal transpyrénéen ferroviaire, intermodal et routier: un modèle Logit Universel de forme Box-Cox généralisée[END_REF].

Further documentation of the detailed steps of these manipulations from (13) to (6-C) is available from the authors.

Acknowledgements

The authors thank Réseau Ferré de France (RFF) for supporting this work carried out for the Scientific Economic Evaluation Committee of the EOLE project on the westerly extension of the existing E line of the Paris regional RER (Réseau Express Régional) train network: the derivation of the Shannon measure was presented at the July 6, 2011, meeting of the full scientific committee. They are also grateful to (i) Claude Abraham for discussions on his random utility path choice models of 1961; (ii) Staffan Algers and Andrew Daly for information on the SAMPERS and PRISM urban passenger demand model structures; (iii) Cong-Liem Tran for verification and redress of some derivations from an earlier draft; (iv) Jean-Dominique Blardone and Dany Nguyen-Luong for facilitated access to French national ministry and Île-de-France regional IAURP archives; (v) Syndicat des transports d'Île-de-France (STIF) for reference area-wide modal share information. The first author benefitted from financial support by Le Fonds National de la Recherche Scientifique de Belgique through research stays at Group Transport & Mobility (GTM), Université Catholique de Louvain-Mons in 2011 and 2012, where findings were presented in seminars, as they were at the 10 èmes journées du BETA, Université de Strasbourg, May 24, 2012.

The authors thank Réseau Ferré de France (RFF) for supporting this work carried out for the Scientific Economic Evaluation Committee of the EOLE project on the westerly extension of the existing E line of the Paris regional RER (Réseau Express Régional) train network: the derivation of the Shannon measure was presented at the July 6, 2011, meeting of the full scientific committee. They are also grateful to (i) Claude Abraham for discussions on his random utility path choice models of 1961; (ii) Staffan Algers and Andrew Daly for information on the SAMPERS and PRISM urban passenger demand model structures; (iii) Cong-Liem Tran for verification and redress of some derivations from an earlier draft; (iv) Jean-Dominique Blardone and Dany Nguyen-Luong for facilitated access to French national ministry and Île-de-France regional IAURP archives; (v) Syndicat des transports d'Île-de-France (STIF) for reference area-wide modal share information. The first author benefitted from financial support by Le Fonds National de la Recherche Scientifique de Belgique through research stays at Group Transport & Mobility (GTM), Université Catholique de Louvain-Mons in 2011 and 2012, where findings were presented in seminars, as they were at the 10 èmes journées du BETA, May 24, 2012. Agora Jules Dupuit, Publication AJD-142 ---------------

And the long form may now be written with terms arranged to again put the iqn X in evidence:

where the four probabilities and their respective RUF are, successively for own terms: