Hello. This last video is about the limits of the equilibrium approximation.
The Equilibrium approximation is now largely used, both in theory and in practice and we
sometimes forget that many factors influence the quality of this approximation.
This video does not aim at making a full list of these factors, it only highlights a few
that I think are especially important and often forgotten. Schematically, the system that we
consider involves some demand and some supply, that is a congestion mechanism,
and together, they lead to some traffic conditions that may or may not be at equilibrium.
In order to predict well the traffic conditions, we need to have good models of the demand,
of the supply, and we need the equilibrium approximation to work well.
These are quite strong requirements.
In practice, there are three main issues.
First, the approximation may not be reasonable.
Second, it's very difficult to find the right level of details for our models.
And third, we need to estimate the parameters of our models.
These problems are quite general, but I would like to highlight a few specific issues related
to the cases that we have explored in the previous videos.
So let's start with the equilibrium approximation.
One important issue is that the conditions vary from day to day.
This applies to the demand, which usually varies over the week and over the seasons,
and to the supply, which varies with the weather and with road works for instance.
Another important issue is stability.
Indeed, we have seen that equilibria may exist, but nothing guarantees that these equilibria are
actually attracting and not repelling but we will cover this in more details in the next slides.
Then, users may not be fully informed, or they might not be fully rational.
Although this is changing as nowadays, people start using navigation apps now and more often.
And with this, they become better informed, and perhaps also more rational.
Let's now discuss a bit the level of details.
The models we have described in the previous videos were very simple.
Of course, some simplifications are always necessary to keep some intuition.
But we should not omit important phenomena.
One important thing that we have mostly left aside is the cost of uncertainty.
If we have variability, we have some uncertainty, for instance, in travel times, and this
is known to be one of the key factors influencing mode choice for example.
And by focusing on a deterministic equilibrium, we somehow ignore this which is a quite questionable
choice.
And the supply side, the congestion mechanism that we considered were all very simple.
Actually, they were all unidirectional that is that the congestion only occurred between
users that traveled on the same link and in the same direction.
In practice, different directions often interact, for instance, there are traffic signals or
roundabouts.
Finally, even if he could find perfect models, we would still have great difficulties to
estimate them.
Congestion mechanisms are perhaps simpler, because the described almost physical phenomena.
But the parameters of these phenomena still depend on some behavioral parameters like
the driving aggressiveness of reaction time.
To estimate the demand, we use two types of methods which are called stated preferences
and revealed preferences.
With stated preferences, we basically ask people about what they would do in different
situations while with revealed preferences, we try to infer people's preferences from
the choice that actually make in real life.
Of course, when they're available, revealed preferences are much more reliable.
With stated preferences, people might have difficulties to imagine the situation.
And even if the situation is relatively simple, and well known from them, users may reply
in one way and act in another.
The problem with revealed preferences is that we can only observe existing alternatives.
And these typically differ in more than one aspect.
The only exception is perhaps HOT-lane, where you can pay a small toll to use a lane that
goes a bit faster and the traffic sign at the beginning of the road indicates the expected
travel times on the normal road and under HOT lane.
Let's now discuss about stability, it is difficult to obtain general results because the stability
of a user equilibrium depends not only on the condition mechanism and on the schedule
preferences, but also in how users adjust their decisions.
It should remember from the first video of this week, I made this drawing where this
is the process that people use to update their decision.
This is also called the adjustment mechanism.
So the stability of a user equilibrium depends on the congestion mechanism, on the adjustment
mechanism and on the utility function.
If you want to be rigorous, stability can be quite technical.
Here, I will just give you the main intuition.
In congestion games, the equilibrium is likely to be stable, because the cost of an alternative
increases with the number of people that chooses this alternative.
For instance, if we are in a situation like this, and the travel time is a bit larger
on Route one, then people are likely to switch from route one to route two which will increase
a little bit the travel time on route two, decrease the one on route one, and both will get
closer to each other.
So we will get closer to equilibrium.
With the budget time choice however, this is a completely different story.
If we consider that this whole time period is congested, and we add one user that arrives
at the bottleneck at time zero, then the effect on the others is null for the users that arrived
at the bottleneck before that user because the queue, they will experience will be the
same.
But all the users that arrived afterwards will see a queue that is one user longer.
So if we consider a system that is near equilibrium, and that we change it a little bit by taking
some users from this range of departure time to that range of departure time, the consequence
in the delay is as follows.
For users departing before T1 or after T2 present there is no change.
For users departing between T1+deltaT and T2, they see that there is this many users just before
them, so it increases their delay.
And in between it varies linearly.
Intuitively, this is problematic as users have the same incentive to switch where users
were added and where users were removed.
Let's make a simple simulation experiment.
So we consider a single origin destination pair with the unidirectional bottleneck and
schedule preferences that have this shape.
So the marginal utility rate at home is constant, and the one at the destination increases like
this smoothly.
And this is the case with homogeneous users.
And with heterogeneous users, we just consider a small offset in the marginal utility rate
at the destination.
So we translate a little bit this curve in different directions.
And in order to go from one day to the next, the adjustment mechanism that we use is known
as Smith’s adjustment mechanism and it is such that the proportion of the population
that changes from alternative I, to alternative J is proportional to the difference, [UJ-UI]
if the difference is positive, and it's zero, if it's negative.
Let's now see some simulations.
We start with the departures that uniformly distributed between 6:30 in the morning and
9:30 in the morning, then people will adjust based on the conditions they observe.
So if we start a simulation, that's what we observe.
So the departure rate adjusts as a function of the observed travel time and the dashed
line that you see here is the equilibrium travel time and as we see, the travel time in simulation
is always quite different and it exhibits some oscillations that are not so close to the
equilibrium travel time.
Intuitively, this is perfectly consistent with what we described just before for the
bottleneck model.
Fortunately, with heterogeneous users, we will see that there is a bit more stability.
So again, we start with departures that are uniformly distributed between 6:30 and 9:30
but now let's see what it gives once we run through the simulation.
So the different curves represent the departure rate for different families of people with
different offset in their marginal utility rate and we see that as time goes by these
different families segregate themselves, the one that have t* that it's relatively late
tend to depart quite late and altogether this confers some stability to the system and the
oscillation that we observe are much closer to the equilibrium travel time.
Thus, static code choice problems are expected to be rather stable while for departure time
choice, the stability really depends on the amount of heterogeneity and probably also
on other factors.
We are reaching the end of this lecture and it is now time to conclude.
We have seen that equilibria are very useful to predict the adjustments of travelers to
any type of perturbation.
To ensure tractability and intuition, We have only considered two decisions: route and departure
time and we considered them separately. But nothing prevents us from combining these or
even other decisions in simulations.
Simulations also allow for greater level of details and they are largely used in practice.
We should however, keep in mind that some limitations remain and in particular, that
there is no point in using models that we cannot accurately estimate.
Thank you and goodbye.