user equilibrium - github pages · 3 solve the system of equations enforcing equal travel times on...

User Equilibrium

CE 392C

September 1, 2016

User Equilibrium

REVIEW

1 Network definitions

2 How to calculate path travel times from path flows?

3 Principle of user equilibrium

4 Pigou-Knight Downs paradox

5 Smith paradox

User Equilibrium Review

OUTLINE

1 Braess paradox

2 User equilibrium vs. system optimum

3 Techniques for small networks

4 Fixed point problems

User Equilibrium Outline

BRAESS PARADOX

Consider the following network, with 6 vehicles traveling from node 1 tonode 4

1

2

3

4

50+x

50+x10x

10x

What’s the equilibrium solution?

User Equilibrium Braess paradox

Now, a third link is added to the network.

1

2

3

4

50+x

50+x

10+x

10x

10x

What happens now?


What just happened?


The Prisoners’ Dilemma

You and a friend are arrested committing a crime!

If you both stay silent, you both go to jail for 1 year.

If you testify against your friend but they stay silent, you get off freebut they go to jail for 15 years.

If you both testify against each other, you both to to jail for 14 years.



We can visualize these results in a matrix.



No matter what you think your friend will do, you are better offtestifying against them.



The same logic holds for your friend.



If both of you act selfishly, it leads to the worst possible outcome.


In the Braess paradox, adding a new network link actually increased traveltimes for all travelers. Why?

As we moved from the original equilibrium state to the new one, wheneversomeone switched routes, travel times increased for others.

This is an example of an externality: when users choose routes, they donot consider the impact of their choice on other users.


Is the Braess paradox “realistic”?


A few implications:

User equilibrium does not minimize congestion.

The “invisible hand” does not always function well in traffic networks.

There may be room for engineers and policy makers to “improve”route choices.


This suggests two possible traffic assignment rules:

User equilibrium (UE): Find a feasible assignment in which all used pathshave equal and minimal travel times.

System optimum (SO): Find a feasible assignment which minimizes thetotal system travel time

TSTT =∑

(i ,j)∈A

xij tij

When might each of these rules be used?


SOLVING FOREQUILIBRIUM

How many vehicles will choose each link?

1 27000 7000

In two-link networks, a graphical approach can be used.

User Equilibrium Solving for Equilibrium

Route 1

Route 2


This method can be generalized in any network with a single OD pair(r , s):

1 Select a set of paths Π̂rs which you think will be used.

2 Write equations for the travel times of each path in Π̂rs as a functionof the path demands.

3 Solve the system of equations enforcing equal travel times on all ofthese paths, together with the requirement that the total pathdemands must equal the total demand d rs .

4 Verify that this set of paths is correct; if not, refine Π̂rs and return tostep 2.


The “trial and error” method doesn’t work well for realistic-sized networks:

The Chicago regional network has 12982 nodes, 39018 links, and over3 million OD pairs

The Philadelphia network has 13389 nodes, 40003 links, and over 2million OD pairs

The Austin network has 7388 nodes, 18961 links, and around 1million OD pairs.

Further, the number of paths in these networks is much, much larger.

You do not want a trial-and-error method for these networks. Later in theclass we’ll discuss methods which scale better.


Next week we’ll take a detour into optimization and other mathematicaltechniques which help us formulate and solve traffic assignment on largenetworks. If your multivariable calculus is a bit rusty, I’d advise reviewingthe following concepts (see Section 4.1 of the notes):

Dot products and their geometric interpretation

First and second partial derivatives

The gradient vector

The Hessian matrix

Multivariate chain rule


FIXED POINT PROBLEMS

There are three important questions you should be asking at this point:

Does a user equilibrium solution always exist?

If so, is the user equilibrium solution unique?

Is there any practical way to find an equilibrium in large networks?

To answer these questions, we’ll need some math. Today and next week willcover some basic results from fixed point problems, variational inequalities,and optimization.

User Equilibrium Fixed Point Problems

In the last class, we interpreted user equilibrium as a “consistent” solutionto this loop.

Path demands h Link demands xx = h

Link travel times tPath travel times cc = t

Link performance functions

Assignmentrule

For example, if there was some function R(C) which gives the path flows(route choice) as a function of path travel times


This is an example of a fixed point problem. The more general definition isgiven below:

Consider some set X and a function f whose domain is X and whose rangeis contained in X . A fixed point of f is a value x ∈ X such that x = f (x).

Fixed point theorems give us conditions on X and f which guarantee thata fixed point exists — for us, this will tell us when we known anequilibrium solution exists.


Brouwer’s Theorem

If X is a compact convex set and f is a continuous function, then f has atleast one fixed point.

This theorem is a bit frustrating in that it does give us any clue as to howto find this equilibrium! But it must exist somewhere.


Mathematical definitions...

A set is compact if it is closed and bounded.

A set is closed if it contains all of its boundary points.

A set is bounded if it can be contained by a sufficiently large ball.

A set is convex if the line connecting any two points in the set lies withinthe set as well (x ∈ X and y ∈ X imply λx + (1− λ)y ∈ X for allλ ∈ [0, 1])

A function is continuous if at all points y ∈ X , limx→y f (x) exists and isequal to f (y).


To visualize the concept of fixed points, assume that X = [0, 1].

A fixed point is anywhere f (x) crosses the diagonal line y = x

One of the homework problems asks you to show that all of the conditions(closed, bounded, convex, continuous) are necessary for a fixed point toexist.


Application to traffic assignment

Does the traffic assignment problem satisfy the conditions of Brouwer’stheorem?

Let H be the set of all feasible path flows. H is closed, bounded, andconvex.

But what should f : H → H be? If paths are “tied” in travel time, thenR(C ) can take infinitely many values.

If we stick with the fixed point approach, we can still make things workbut we need to appeal to Kakutani’s theorem instead.

Another approach, which is more useful for visualizing equilibriumproblems, leads us to the variational inequality. Next week, we’ll see whatf should be to prove equilibrium existence using Brouwer’s theorem.


user equilibrium - github pages · 3 solve the system of equations enforcing equal travel times on...

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