The Paradox Severity Linear Latency General Latency Extensions Conclusion. Braess Paradox. Julian Romero. January 22, 2008.

Transcription

1 Julian Romero January 22, 2008 Romero 1 / 20

2 Outline The Paradox Severity Linear Latency General Latency Extensions Conclusion Romero 2 / 20

3 Introduced by Dietrich Braess in Adding costless edges doesn t necessarily decrease traffic costs. v v l (x) = x l (x) = 1 l (x) = x l (x) = 1 s l (x) = 0 t s t l (x) = 1 l (x) = x l (x) = 1 l (x) = x w w Romero 3 / 20

4 Cost of (a) is 2, Cost of (b) is 3 /2 PoA= 4 /3. Worst Case scenario. (PoA is bounded by 4 /3 with linear cost functions. v v l (x) = x l (x) = 1 l (x) = x l (x) = 1 s l (x) = 0 t s t l (x) = 1 l (x) = x l (x) = 1 l (x) = x w w Romero 4 / 20

5 Examples Stuttgart, Germany invested in new route in city center to ease traffic. Traffic got worse, didn t improve traffic until it was blocked from traffic. Earth Day NYC closed 42nd street. Many were worried, actually improved. Seoul, South Korea - Invested $380m to tear down one of three main bridges into town. Santa Cruz added 2 lanes to highway. Carpool Lanes? Romero 5 / 20

6 Other Work Steinberg and Zangwill (1983) - Show that occurs about half the time in random graphs. Pas and Principio (1997) - Show that only occurs demand for travel fall in an intermediate range. Tumer and Wolpert (2000) - Show that can be avoided with COIN. In experimental lab, Braess paradox has been shown, but not for larger Braess graphs. Romero 6 / 20

7 Application to Other Models Loss Networks Queuing Networks Used to explain Downs-Thomson Paradox Electric Networks Simple model of water pipes exhibits no Braess paradox, but more complicated model does (not sure about experimental data). Resolving Newcomb s Problem String and Springs network Romero 7 / 20

8 Game Not uncommon in game theory to give more options and make worse off. L M U 4,4 5,1 M 1,5 3,3 Romero 8 / 20

9 Game Not uncommon in game theory to give more options and make worse off. L M U 4,4 5,1 M 1,5 3,3 = L M R U 4,4 5,1-10,10 M 1,5 3,3-10,10 D 10,-10 10,-10-9,-9 Romero 8 / 20

10 Game Not uncommon in game theory to give more options and make worse off. L M U 4,4 5,1 M 1,5 3,3 = L M R U 4,4 5,1-10,10 M 1,5 3,3-10,10 D 10,-10 10,-10-9,-9 This is not a real paradox but only a situation which is counterintuitive. Romero 8 / 20

11 Paper On the Severity of : Designing Networks for Selfish Users is Hard Tim Roughgarden Typically network designers look to find smallest or cheapest network satisfying conditions. This paper takes a given network, and tries to determine if cost can be improved by removing links. Can we determine what networks are susceptible to Braess Paradox, and would benefit by eliminating links? Romero 9 / 20

12 Model Directed Network G (V, E). Unique source s and sink t. P is the set of s-t paths. f : P R + Flow on edges f e = P:e P f P. Traffic flow r. Feasible flow, P P f P = r. Each edge has latency function l e ( ). l e is non-negative, continuous, and non-decreasing. Latency on path, l P (f) = e P l e (f e ) An instance is the triple (G, r, l). Romero 10 / 20

13 Definitions A flow f feasible for (G, R, l) is at Nash equilibrium, or is a Nash flow, if for all P 1, P 2 P with f P1 > 0 and δ (0, f P1 ], we have, ) l P1 (f) l P2 ( f Cost of flow, and f P = C(f) = P P f P δ if P = P 1 f P + δ if P = P 2 f P otherwise l P (f)f P A c-approximation run in polynomial time, and returns no more than c times the optimal solution. Romero 11 / 20

14 Properties of Nash Flows In equilibrium l P1 (f) = l P2 (f) (latency equal on all paths). Call this common latency L (G, R, l). Equilibrium exists for all instances (G, r, l). Equilibrium unique, i.e. l P (f) = l P (f ). At equilibrium C(f) = r L (G, r, l) For every instance, L (G, r, l) is non-decreasing in r. Romero 12 / 20

15 Linear Latency Functions l e (x) = a e x + b e with a e, b e 0. Prop: f feasible flow, f Nash flow. For any instance (G, r, l) with linear latency functions, C(f) 4 3 C (f ) Trivial algorithm - network will all candidate edges. The trivial algorithm is a 4 /3-approximation algorithm for linear latency network design. Theorem - Assuming P NP, for every ε > 0, there is no ( 4 3 ε) -approximation algorithm for Linear Latency Network Design Romero 13 / 20

16 Linear Latency Call instance (G, r, l) paradox-free if, L (G, r, l) L (H, r, l) for all sub graphs H of G paradox-ridden if, L (G, r, l) = 4 L (H, r, l) for some subgraph H of H 3 Corollary - Given an instance (G, r, l) that has linear latency functions and is either paradox-free or paradox-ridden, it is NP-hard to decide whether or not (G, r, l) is paradox-ridden. Romero 14 / 20

17 General Latency Functions Examine continuous and non-decreasing latency functions. PoA is unbounded for these functions. 1 s t Compare only Nash-flows rather than all feasible flows. Theorem - For every instance (G, r, l) with n vertices, the trivial algorithm returns a solution of value at most n/2 times that of an optimal solution. p x Romero 15 / 20

18 Braess Graph Define class of Braess Graphs Vertices -V k = {s, v 1,..., v k, w 1, w k, t} Edges - E k - (s, v i ), (v i, w i ), (w i, t), (v i, w i 1 ), (v 1, t), (s, w k ) Latency Functions A - e = (v i, w i ) l k e(x) = 0 B - e = (vi, w i 1 ), (s, w k ), (v 1, t) C - e = (wi, t), (s, v k i+1 ) l e (x) = l k e(x) = 1 0 x = k k+1 i x = 1 cont., non-decr. otherwise Romero 16 / 20

19 Braess Graph w 2 w 3 s B C v 2 A B w 1 C C t s B C C v 3 A B A w 2 C C C t C A B C v 2 B A w 1 B v 1 v 1 Proposition - For every integer n 2, there is an instance (G, r, l) in which G has n vertices and a subgraph H with n L (G, r, l) = L (H, r, l) 2 Romero 17 / 20

20 Hardness Theorem - Assuming P NP, for every ε > 0 there is no ( n/2 ε)-approximation algorithm for General Latency Network Design. Corollary - Given an instance (G, r, l) with general latency functions that is paradox-free or paradox-ridden, it is NP-hard to decide whether or not (G, r, l) is paradox-ridden. Romero 18 / 20

21 Polynomial Latency Now consider polynomial latency functions l(x) = a p x p + a p 1 x p a 0 with a i 0. Compare Nash flow to any feasible flow. Proposition - There is a constant c 1 > 0 such that for every p 2 and every instance (G, r, l) with polynomial latency functions of degree p that admits a Nash flow f and a feasible flow f, p C(f) c 1 ln p C(f ) Romero 19 / 20

22 Conclusions Braess paradox is impossible to recognize efficiently. The effect is bounded with linear or polynomial latency functions. Effect can become unbounded for general continuous non-decreasing latency functions. Important to recognize that effect of is unbounded. Romero 20 / 20