Evaluation of Predicted Probabilities from Multinomial Logistic Regresion I'm currently working to develop a model which predict a traveler's choice of transportation mode (transit, auto, bike, walk) using data from the U.S. Census which has been aggregated to custom origin/destination zones. Thus, the data available to me is not traveler specific but instead contains variables representing the characteristics specific to each zonal pair (pop density, modal travel time, etc.) as well as the total number of traveler between each pair and the modal split for that pair. See below for a toy example of the data,
Zone1 Zone2 Zone1_Var1 Zone2_Var1 TT_Mode1 TT_Mode2 Tot_Trips Auto_Trips Transit_Trips
1     2     123        456        55       45       100       75         25

I'm specifically interested in estimating the proportions of each trip (e.g. from above .75 and .25). I had considered using a multinomial (for all four modes) logistic regression to estimate the proportions of each mode. To do so, I expanded the data so that there was an observation for each trip in each zonal pair with a variable indicating the mode chosen (taken from the percentage of total trips) to pass into the multinom function from the nnet package in R. For example (using data from above),
ID Zone1 Zone2 Zone1_Var1 Zone2_Var1 TT_Mode1 TT_Mode2 Mode 
1  1     2     123        456        55       45       Auto  
2  1     2     123        456        55       45       Auto
3  1     2     123        456        55       45       Auto
...
76 1     2     123        456        55       45       Transit
77 1     2     123        456        55       45       Transit
...
100 1     2     123        456        55       45       Transit

Model call:
mn_fit <- multinom(as.formula(paste("mode", "~", paste(predictors, collapse = " + "))), mode_choice_list$train_mode_choice, max_iter = 200, trace= T)

However, this produces the interesting scenario where validating the model on a test set of the data leads it to predict the same mode (when asked to classify) for all zonal pairs since the variables do not vary within a zonal pair. Alternatively, I can have the model return the predicted probabilities for each mode which should be analogous to the observed modal splits. 
This leads to my question: what is the best way to evaluate a set of predicted probabilities vs. a set of observed probabilities (ie modal splits). I've looked into scoring rules (e.g. brier, logarithmic, etc. ) and they seem promising. However, I'd like to confirm that utilizing these rules in evaluating various specifications of the mode choice model (e.g. different variables, variable transformations, etc.) is a valid route to take. If they are, are they able to account for the frequency weights of each zonal pair (i.e. there are a different number of trips between each zonal pair) which I would assume should be accounted for when evaluating performance.
Additionally, I'd welcome any feedback regarding my structuring of the data or use of the multinom package if it is incorrect or on another class of model that might be more suited to the data/problem at hand. 
 A: "I'm specifically interested in estimating the proportions of each trip (e.g. from above .75 and .25)"  
This does not seem clear to me. Why are you interested in estimating the proportion of trips taken when you already have this data? It would seem to me that the real question is "what is the impact of pop. density, modal travel time, etc. on the probability of a given transportation option being chosen." To solve this you can use multinomial logistic regression.  
Do you have information on the distribution of your independent variables? If you do, my suggestion would be to generate a large random sample using this information, and generating set of target outcomes using the proportions that you already know. You can then train a cross validated model on this information to estimate the effect each of your variables has on the probability of each mode of transportation being chosen.
It is important to note that mlogit models utilize relative risk ratios (the exponentated coefficients) and not odds ratios. That is, your coefficients will tell you the change in probability that a unit change in your variable has on one outcome occurring over the base category you specify. I highly recommend reading the ats example here.
