User Equilibrium CE 392C. September 1, User Equilibrium

Transcription

1 CE 392C September 1, 2016

2 REVIEW

3 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 Review

4 OUTLINE

5 1 Braess paradox 2 User equilibrium vs. system optimum 3 Techniques for small networks 4 Fixed point problems Outline

6 BRAESS PARADOX

7 Consider the following network, with 6 vehicles traveling from node 1 to node 4 50+x 3 10x 1 10x 2 50+x 4 What s the equilibrium solution? Braess paradox

8 Now, a third link is added to the network. 50+x 3 10x 1 10x 2 10+x 50+x 4 What happens now? Braess paradox

9 What just happened? Braess paradox

10 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 free but they go to jail for 15 years. If you both testify against each other, you both to to jail for 14 years. Braess paradox

11 The Prisoners Dilemma We can visualize these results in a matrix. Braess paradox

12 The Prisoners Dilemma No matter what you think your friend will do, you are better off testifying against them. Braess paradox

13 The Prisoners Dilemma The same logic holds for your friend. Braess paradox

14 The Prisoners Dilemma If both of you act selfishly, it leads to the worst possible outcome. Braess paradox

15 In the Braess paradox, adding a new network link actually increased travel times for all travelers. Why? As we moved from the original equilibrium state to the new one, whenever someone switched routes, travel times increased for others. This is an example of an externality: when users choose routes, they do not consider the impact of their choice on other users. Braess paradox

16 Is the Braess paradox realistic? Braess paradox

17 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. Braess paradox

18 This suggests two possible traffic assignment rules: User equilibrium (UE): Find a feasible assignment in which all used paths have equal and minimal travel times. System optimum (SO): Find a feasible assignment which minimizes the total system travel time TSTT = x ij t ij When might each of these rules be used? (i,j) A Braess paradox

19 SOLVING FOR EQUILIBRIUM

20 How many vehicles will choose each link? In two-link networks, a graphical approach can be used. Solving for Equilibrium

21 Route 1 Route 2 Solving for Equilibrium

22 This method can be generalized in any network with a single OD pair (r, s): 1 Select a set of paths ˆΠ rs which you think will be used. 2 Write equations for the travel times of each path in ˆΠ rs as a function of the path demands. 3 Solve the system of equations enforcing equal travel times on all of these paths, together with the requirement that the total path demands must equal the total demand d rs. 4 Verify that this set of paths is correct; if not, refine ˆΠ rs and return to step 2. Solving for Equilibrium

23 The trial and error method doesn t work well for realistic-sized networks: The Chicago regional network has nodes, links, and over 3 million OD pairs The Philadelphia network has nodes, links, and over 2 million OD pairs The Austin network has 7388 nodes, links, and around 1 million 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 the class we ll discuss methods which scale better. Solving for Equilibrium

24 Next week we ll take a detour into optimization and other mathematical techniques which help us formulate and solve traffic assignment on large networks. If your multivariable calculus is a bit rusty, I d advise reviewing the 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 Solving for Equilibrium

25 FIXED POINT PROBLEMS

26 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 will cover some basic results from fixed point problems, variational inequalities, and optimization. Fixed Point Problems

27 In the last class, we interpreted user equilibrium as a consistent solution to this loop. Path demands h Assignment rule Path travel times c x = h c = t Link performance functions Link demands x Link travel times t For example, if there was some function R(C) which gives the path flows (route choice) as a function of path travel times Fixed Point Problems

28 This is an example of a fixed point problem. The more general definition is given below: Consider some set X and a function f whose domain is X and whose range is 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 that a fixed point exists for us, this will tell us when we known an equilibrium solution exists. Fixed Point Problems

29 Brouwer s Theorem If X is a compact convex set and f is a continuous function, then f has at least one fixed point. This theorem is a bit frustrating in that it does give us any clue as to how to find this equilibrium! But it must exist somewhere. Fixed Point Problems

30 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 within the set as well (x X and y X imply λx + (1 λ)y X for all λ [0, 1]) A function is continuous if at all points y X, lim x y f (x) exists and is equal to f (y). Fixed Point Problems

31 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 to exist. Fixed Point Problems

32 Application to traffic assignment Does the traffic assignment problem satisfy the conditions of Brouwer s theorem? Let H be the set of all feasible path flows. H is closed, bounded, and convex. But what should f : H H be? If paths are tied in travel time, then R(C) can take infinitely many values. If we stick with the fixed point approach, we can still make things work but we need to appeal to Kakutani s theorem instead. Another approach, which is more useful for visualizing equilibrium problems, leads us to the variational inequality. Next week, we ll see what f should be to prove equilibrium existence using Brouwer s theorem. Fixed Point Problems