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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?
User Equilibrium Braess paradox
What just happened?
User Equilibrium Braess paradox
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.
User Equilibrium Braess paradox
The Prisoners’ Dilemma
We can visualize these results in a matrix.
User Equilibrium Braess paradox
The Prisoners’ Dilemma
No matter what you think your friend will do, you are better offtestifying against them.
User Equilibrium Braess paradox
The Prisoners’ Dilemma
The same logic holds for your friend.
User Equilibrium Braess paradox
The Prisoners’ Dilemma
If both of you act selfishly, it leads to the worst possible outcome.
User Equilibrium Braess paradox
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.
User Equilibrium Braess paradox
Is the Braess paradox “realistic”?
User Equilibrium Braess paradox
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.
User Equilibrium Braess paradox
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?
User Equilibrium Braess paradox
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
User Equilibrium Solving for Equilibrium
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.
User Equilibrium Solving for Equilibrium
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.
User Equilibrium Solving for Equilibrium
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
User Equilibrium Solving for Equilibrium
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
User Equilibrium Fixed Point Problems
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.
User Equilibrium Fixed Point Problems
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.
User Equilibrium Fixed Point Problems
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).
User Equilibrium Fixed Point Problems
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.
User Equilibrium Fixed Point Problems
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 Fixed Point Problems