Editing Mode choice (section)

==Econometric formulation==
Economists deal with utility rather than physical weights, and say that

:observed utility = mean utility + random term.

The characteristics of the object, x, must be considered, so we have

:''u''(''x'') = ''v''(''x'') + ''e''(''x'').

If we follow Thurston's assumption, we again have a [[probit]] model.

An alternative is to assume that the [[errors and residuals in statistics|error term]]s are [[Independent identically-distributed random variables|independently and identically distributed]] with a [[Fisher–Tippett distribution|Weibull]], [[Fisher–Tippett distribution|Gumbel Type I]], or [[Gumbel distribution|double exponential distribution]].  (They are much the same, and differ slightly in their tails (thicker) from the [[normal distribution]]).  This yields the multinomial [[logit model]] (MNL).  Daniel McFadden argued that the Weibull had desirable properties compared to other distributions that might be used.  Among other things, the error terms are normally and identically distributed. The logit model is simply a [[log ratio]] of the probability of choosing a mode to the probability of not choosing a mode.

:<math>
\log \left( \frac{P_i }
{1 - P_i } \right) = v(x_i )
</math>

Observe the mathematical similarity between the logit model and the S-curves we estimated earlier, although here share increases with utility rather than time. With a choice model we are explaining the share of travelers using a mode (or the probability that an individual traveler uses a mode multiplied by the number of travelers).

The comparison with S-curves is suggestive that modes (or technologies) get adopted as their utility increases, which happens over time for several reasons.  First, because the utility itself is a function of [[network effect]]s, the more users, the more valuable the service, higher the utility associated with joining the network. Second because utility increases as user costs drop, which happens when fixed costs can be spread over more users (another network effect).  Third technological advances, which occur over time and as the number of users increases, drive down relative cost.

An illustration of a utility expression is given:

:<math>
\log \left( \frac{P_A }
{1 - P_A } \right) = \beta _0  + \beta _1 \left( c_A  - c_T \right) + \beta _2 \left( t_A  - t_T \right) + \beta _3 I + \beta _4 N = v_A 
</math>

where
:''P<sub>i</sub>'' = Probability of choosing mode i.
:''P<sub>A</sub>'' = Probability of taking auto
:''c<sub>A</sub>,c<sub>T</sub>'' = cost of auto, transit
:''t<sub>A</sub>,t<sub>T</sub>'' = travel time of auto, transit
:''I'' = income
:''N'' = Number of travelers

With algebra, the model can be translated to its most widely used form:
:<math>
\frac{P_A } {1 - P_A } = e^{v_A }
</math>
:<math>
P_A  = e^{v_A }  - P_A e^{v_A }  
</math>
:<math>
P_A \left( 1 + e^{v_A } \right) = e^{v_A }  
</math>
:<math>
P_A  = \frac{e^{v_A } } {1 + e^{v_A } } 
</math>

It is fair to make two conflicting statements about the estimation and use of this model:

#it's a "house of cards", and
#used by a technically competent and thoughtful analyst, it's useful.

The "house of cards" problem largely arises from the utility theory basis of the model specification.  Broadly, utility theory assumes that (1) users and suppliers have perfect information about the market; (2) they have deterministic functions (faced with the same options, they will always make the same choices); and (3) switching between alternatives is costless.  These assumptions don’t fit very well with what is known about behavior.  Furthermore, the aggregation of utility across the population is impossible since there is no universal utility scale.

Suppose an option has a net utility ''u<sub>jk</sub>'' (option ''k'', person ''j'').  We can imagine that having a systematic part ''v<sub>jk</sub>'' that is a function of the characteristics of an object and person ''j'', plus a random part ''e<sub>jk</sub>'', which represents tastes, observational errors and a bunch of other things (it gets murky here).  (An object such as a vehicle does not have utility, it is characteristics of a vehicle that have utility.)  The introduction of ''e'' lets us do some aggregation.  As noted above, we think of observable utility as being a function:

:<math>
v_A  = \beta _0  + \beta _1 \left( c_A  - c_T \right) + \beta _2 \left( t_A  - t_T \right) + \beta _3 I + \beta _4 N
</math>

where each variable represents a characteristic of the auto trip.  The value ''β<sub>0</sub>'' is termed an alternative specific constant.  Most modelers say it represents characteristics left out of the equation (e.g., the political correctness of a mode, if I take transit I feel morally righteous, so ''β''<sub>0</sub> may be negative for the automobile), but it includes whatever is needed to make error terms NID.