AGlimpseofAGT: Selfish Routing

Transcription

1 AGlimpseofAGT: Selfish Routing Guido Schäfer CWI Amsterdam / VU University Amsterdam g.schaefer@cwi.nl Course: Combinatorial Optimization VU University Amsterdam March 12 & 14, 2013

2 Motivation

3 Situations of strategic interaction Viewpoint: many real-world problems are complex and distributed in nature: 1 involve several independent decision makers (lack of coordination) 2 each decision maker attempts to achieve his own goals (strategic behavior) 3 individual outcome depends on decisions made by others (interdependent) Examples: routing in networks Internet applications auctions... Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 3

4 Network routing Network routing: (think of urban road traffic) large number of commuters want to travel from their origins to their destinations in a given network commuters are autonomous and choose their routes so as to minimize their individual travel times travel time along a network link is affected by congestion individual travel time depends on the choices made by others Applications: road traffic, public transportation, Internet routing, etc. Observation: selfish route choices lead to inefficient outcomes Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 4

5 Inefficient outcomes Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 5

6 Consequences Negative consequences: environmental pollution waste of natural resources, time and money stress on the traffic participants... In 2010, congestion caused urban Americans to travel 4.8 billion hours more and to purchase an extra 1.9 billion gallons of fuel for a congestion cost of $101 billion. [Texas Transportation Institute, 2011 Urban Mobility Report] Need: gain fundamental understanding of the effect of strategic interactions in such applications Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 6

7 Approach Game theory: provides mathematical toolbox to study situations of strategic interaction different models of games several solution concepts for the prediction of rational outcomes Algorithmic game theory: use game-theoretical models and solution concepts take computational/algorithmic issues into account Goals: study the effect of strategic behavior analyze the efficiency loss due to lack of coordination provide algorithmic means to reduce the inefficiency Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 7

8 Today s goals Goals for today: get a glimpse of the research in algorithmic game theory running example: network routing get an idea of the phenomena and questions that arise familiarize with used approaches and techniques approach the state-of-the-art of recent studies (perhaps) Algorithmic game theory is a young, challenging, interdisciplinary research field... addresses many aspects of practical relevance... can have an impact on real-world applications... unites ideas from game theory, algorithms, combinatorial optimization, complexity theory, etc.... is fun! Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 8

9 AGT book N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani (Eds.) Algorithmic Game Theory Cambridge University Press, Available online (free!) at: Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 9

10 Roadmap 1 Examples of Some Phenomena 2 Wardrop Model and the Price of Anarchy 3 Coping with the Braess Paradox 4 Stackelberg Routing 5 Network Tolls Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 10

11 Examples of Some Phenomena

12 Cost minimization games A cost minimization game G is given by set of players N = {1,...,n} set of strategies S i for every player i N cost function C i : S 1 S n R S = S 1 S n is called the set of strategy profiles. Interpretation: every player i N chooses a strategy s i S i so as to minimize his individual cost C i (s 1,...,s n ) one-shot, simultaneous-move: players choose their strategies once and at the same time full-information: every player knows the strategies and cost function of every other player Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 12

13 Example: Congestion game n = 4 S i = {e 1, e 2 } l e1 (x) =x s t l e2 (x) =4 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 13

14 Example: Congestion game n = 4 S i = {e 1, e 2 } l e1 (x) =x s 4 l e2 (x) =4 t Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 13

15 Example: Congestion game l e1 (x) =x n = 4 S i = {e 1, e 2 } s 1 3 l e2 (x) =4 t Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 13

16 Example: Congestion game l e1 (x) =x n = 4 S i = {e 1, e 2 } s 2 2 l e2 (x) =4 t Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 13

17 Example: Congestion game l e1 (x) =x n = 4 S i = {e 1, e 2 } s 3 1 l e2 (x) =4 t Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 13

18 Nash equilibrium and social cost Nash equilibrium: strategy profile s =(s 1,...,s n ) S is a pure Nash equilibrium (PNE) if no player has an incentive to unilaterally deviate i N : C i (s i, s i ) C i (s i, s i) s i S i (s i refers to (s 1,...,s i 1, s i+1,...,s n )) Social cost of strategy profile s =(s 1,...,s n ) S is C(s) = i N C i (s) Astrategyprofiles that minimizes the social cost function C( ) is called a social optimum. Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 14

19 Example: Congestion game n = 4 x s t 4 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 15

20 Example: Congestion game n = 4 s x 4 t 4 Nash equilibrium: C(s) =4 4 = 16 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 15

21 Example: Congestion game n = 4 x s t social optimum: C(s )=4 + 8 = 12 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 15

22 Example: Congestion game n = 4 s x 2 2 t inefficiency: 4 C(s) C(s ) = = 4 3 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 15

23 Example: Congestion game n = 4 x s t 4 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 15

24 Example: Congestion game n = 4 x s t Nash equilibrium: C(s) =9 + 4 = 13 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 15

25 Example: Congestion game n = 4 s x 3 1 t inefficiency: 4 C(s) C(s ) = Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 15

26 Inefficiency of equilibria Price of anarchy: worst-case inefficiency of equilibria POA(G) = max s PNE(G) C(s) C(s ) [Koutsoupias, Papadimitriou, STACS 99] Price of stability: best-case inefficiency of equilibria POS(G) = min s PNE(G) C(s) C(s ) [Schulz, Moses, SODA 03] Define the price of anarchy/stability of a class of games G as POA(G) =max POA(G) and POS(G) =max POS(G) G G G G Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 16

27 Example: Braess Paradox n = 4 x 4 s t 4 x Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 17

28 Example: Braess Paradox n = 4 s x 2 4 t 4 2 x Nash equilibrium: C(s) =24 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 17

29 Example: Braess Paradox n = 4 x 4 s t 4 x Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 17

30 Example: Braess Paradox n = 4 x 4 s 0 4 x t Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 17

31 Example: Braess Paradox n = 4 x 4 s 0 4 t 4 x Nash equilibrium: C(s) =32 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 17

32 Example: Braess Paradox n = 4 x 4 s 0 4 t 4 x cost increase: = 4 3 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 17

33 [New York Times Article]

34 Example: Network tolls n = 4 x s t 4 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 19

35 Example: Network tolls n = 4 x +2 s t 4 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 19

36 Example: Network tolls n = 4 x +2 2 s 2 4 t Nash equilibrium = social optimum Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 19

37 Wardrop Model and the Price of Anarchy

38 Wardrop model Given: directed network G =(V, A) k commodities (s 1, t 1 ),...,(s k, t k ) demand r i for commodity i [k] l a(x) flow-dependent latency function l a ( ) for every arc a A (non-decreasing, differentiable, semi-convex) x Interpretation: send r i units of flow from s i to t i,correspondingtoan infinitely large population of non-atomic players latency experienced on arc a A is l a (x), wherex is the amount of flow on arc a selfishness: playerschooseminimum-latencypaths one-shot, simultaneous-move, full information game Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 21

39 Optimal flows Notation: P i = set of all (simple) directed s i, t i -paths P = i [k] P i = set of all relevant paths flow f is a function f : P R + f is a feasible flow if P P i f P = r i for all i flow on arc a: f a := P P:a P f P latency of path P: l P (f ):= a P l a(f a ) Social cost of flow: C(f )= f P l P (f )= f a l a (f a ) P P i i [k] a A Optimal flow: feasible flow f that minimizes C( ) Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 22

40 Nash flows Nash flow: feasible flow f that satisfies for every i [k] P 1, P 2 P i with f P1 > 0, δ (0, f P1 ]: l P1 (f ) l P2 ( f ), where f P δ if P = P 1 fp := f P + δ if P = P 2 otherwise. f P Exploiting continuity of l a ( ) and letting δ 0, we obtain the following equivalent definition: Wardrop flow: feasible flow f that satisfies for every i [k] P 1, P 2 P i with f P1 > 0 : l P1 (f ) l P2 (f ) Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 23

41 Inefficiency of equilibria Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 24

42 Inefficiency of equilibria Nash flow f : C(f )=1 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 24

43 Inefficiency of equilibria Optimal flow f : C(f )= = 3 4 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 24

44 Existence and computation of optimal flows min l a (f a )f a a A s.t. f P = r i i [k] P P i f a = a A P P:a P f P f P 0 P P Observations: minimize a continuous function (social cost) over a compact set (feasible flows) optimal flow must exist (extreme value theorem) optimal flow can be computed efficiently (convex program) Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 25

45 KKT optimality conditions min a A h a (f a ) s.t. f P = r i i [k] P P i f a = a A P P:a P f P f P 0 P P (CP) Theorem (KKT optimality conditions) Suppose (h a ) a A are continuously differentiable and convex. f is an optimal solution for (CP) if and only if for every i [k] P 1, P 2 P i, f P1 > 0 : h a(f a ) h a(f a ) a P 1 a P 2 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 26

46 Existence and computation of Nash flows Idea: Can we determine functions (h a ) a A such that KKT optimality conditions reduce to Nash flow conditions? min s.t. fa a A 0 l a (x)dx f P = r i i [k] P P i f a = a A P P:a P f P f P 0 P P Observations: optimal solutions of (CP) above coincides with Nash flows Nash flow must exist (extreme value theorem) similar argument: cost of Nash flow is unique Nash flow can be computed efficiently (convex program) Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 27

47 Characterization of optimal flows min l a (f a )f a a A s.t. f P = r i i [k] P P i f a = a A P P:a P f P f P 0 P P Applying KKT optimality conditions yields: Theorem fisanoptimalflowwithrespectto(l a ) a A if and only if f is a Nash flow with respect to (l a) a A,wherel a(x) =(x l a (x)). Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 28

48 Price of anarchy in selfish routing Price of anarchy: given an instance I =(G, (s i, t i ), (r i ), (l a )) POA(I) = max f Nash flow C(f ) C(f ) POA for selfish routing is independent of the network topology, but depends on the class of latency functions: POA = 4 3 for linear latency functions POA =Θ( p ln p ) for polynomial latency functions of degree p POA is unbounded in general [Roughgarden, Tardos, JACM 02] Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 29

49 [Upper bound on POA for linear latency functions]

50 POA for polynomial latency functions p POA Table: POA for polynomial latency functions of degree p Theorem Let (l a ) a A be polynomial latency functions of degree at most p. POA ( ) p 1 ( ) p 1 (p + 1) (p+1)/p =Θ ln p [Roughgarden, Tardos, JACM 02] Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 31

51 Example: Unbounded POA Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 32

52 Example: Unbounded POA Nash flow f : C(f )=1 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 32

53 Example: Unbounded POA Optimal flow f : C(f )=(1 ɛ) 1+p + ɛ 1 0 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 32

54 Example: Unbounded POA POA as p Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 32

55 Reducing inefficiency Means to reduce inefficiency: infrastructure improvement (expensive, Braess paradox) centralized routing (difficult to implement) Stackelberg routing (control part of the traffic) traffic regulation (taxes on fuel, pay-as-you-go tax) network tolls Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 33

56 Coping with the Braess Paradox

57 Braess subgraph problem Consider a single-commodity instance I =(G, (s, t), r, (l a )) with linear latency functions. Define d(h) =common latency of flow carrying paths in a Nash flow for H G Braess subgraph problem: Given (G, (s, t), r, (l a )), finda subgraph H G that minimizes d(h). Algorithm TRIVIAL: simply return the original graph G Theorem TRIVIAL is a 4 3-approximation algorithm for the Braess subgraph problem. Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 35

58 Braess subgraph problem Theorem TRIVIAL is a 4 3-approximation algorithm for the Braess subgraph problem. Proof: 1 Algorithm computes a feasible solution in polynomial time. 2 Consider an arbitrary subgraph H of G. Let f = Nash flow for G and h = Nash flow for H The bound of 4 3 on the price of anarchy yields C(f ) 4 3 C(f ) 4 3 C(h) r d(g) = C(f ) 4 3 C(f ) 4 3 C(h) =4 3 r d(h) Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 36

59 As good as it gets... Theorem Assuming P NP, for every ε>0 there is no approximation algorithm for the Braess subgraph problem achieving an approximation guarantee of ( 4 3 ε). Proof idea: A ( 4 3 ε)-approximation algorithm can be used to determine whether there exist two disjoint paths between (s 1, t 1 ) and (s 2, t 2 ) in a directed graph. The latter problem is NP-complete. Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 37

60 Network Tolls

61 Reducing inefficiency Means to reduce inefficiency: infrastructure improvement (expensive, Braess paradox) centralized routing (difficult to implement) Stackelberg routing (control part of the traffic) traffic regulation (taxes on fuel, pay-as-you-go tax) network tolls Advantages of network tolls: fine-grained means: impose local taxes new technologies facilitate electronic collection of tolls dynamic pricing schemes have potential to change participants behavior... Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 59

62 Dynamic pricing: ERP in Singapore Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 60

63 Road pricing Network tolls: impose non-negative tolls τ := (τ a ) a A on arcs dynamic: flow-dependent function τ a (x) (non-decreasing, continuous) static: flow-independent constant τ a Interpretation: by traversing an arc a A with flow value x, a player experiences a delay of l a (x) and has to pay a toll of τ a (x) Assumption: players are homogeneous φ a (x) :=l a (x)+α τ a (x) α specifies how players value time over money (w.l.o.g. α = 1) Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 61

64 Marginal cost tolls Marginal cost tolls: τ a (x) :=x l a(x) a A Theorem fisanoptimalflowwithrespecttol iff f is a Nash flow with respect to l + τ. Drawbacks: 1 potentially every arc of the network is tolled 2 imposed tolls can be arbitrarily large [Beckman, McGuire and Winsten 1956] Most previous studies: no restrictions on tolls that can be imposed on the arcs of the network Question: What can be done if such restrictions are given exogenously? Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 62

65 Effective road pricing in Singapore Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 63

66 Restricted Network Toll Problem

67 Restricted Network Toll Problem Our setting: suppose we are given threshold functions θ := (θ a ) a A (dynamic or static) Tolls τ =(τ a ) a A are θ-restricted if Special cases: a A : 0 τ a (x) θ a (x) x 0 1 taxing subnetworks: allow tolls only on arcs T A 2 restrict the toll on each arc a A by a (flow-independent) threshold value θ a 3 toll on each arc a A does not exceed a certain fraction of the latency of that arc, e.g., θ a (x) =εl a (x) for some ε>0 Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 65

68 Efficiency measures Given θ-restricted tolls τ, letf τ denote a Nash flow that is induced by τ, i.e.,f τ is a Nash flow with respect to l + τ. θ-restricted tolls τ are ρ-approximate if for all Nash flows f τ induced by θ-restricted tolls τ C(f τ ) ρ C(f τ ) The efficiency of θ-restricted tolls is C(f τ ) min θ-restricted tolls τ C(f ) Note: interested in the cost of the induced Nash flow (system performance) rather than the total disutility of the players Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 66

69 Results in a nutshell 1 Derive an algorithm to compute optimal θ-restricted tolls for parallel-arc networks and affine latency/threshold functions algorithm works for general latency/threshold functions can guarantee polynomial running time only for affine case key: characterize inducible flows for single-commodity networks 2 Provide bounds on the efficiency of θ-restricted tolls for multi-commodity networks and polynomial latency functions derive a pricing scheme that realizes respective efficiency bounds are tight, even for parallel-arc networks approach based on (λ, µ)-smoothness technique insight: impose tolls on arcs that are sensitive to flow changes Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 67

70 Related work 1 special case: taxing subnetworks NP-hard to compute optimal tolls for two-commodity networks and affine latency functions algorithm to compute optimal tolls for parallel-arc networks and affine latency functions [Hoefer, Olbrich, Skopalik, WINE 08] 2 existence/computation of optimal-inducing tolls for heterogenous players single-commodity, non-atomic [Cole, Dodis, Roughgarden, STOC 03] multi-commodity, non-atomic [Fleischer, Jain, Mahdian, FOCS 04] [Karakostas, Kolliopoulos, FOCS 04] multi-commodity, atomic [Swamy, SODA 07] 3 price of anarchy and price of stability of ε-nash flows [Christodoulou, Koutsoupias, Spirakis, ESA 09] Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 68

71 Computing Optimal Tolls for Parallel-Arc Networks

72 Characterization of inducible flows Question: Given an arbitrary flow f,canwedeterminewhether f is inducible by θ-restricted tolls? Note: If f is given, then l a (f a ) and θ a (f a ) are constants. Nash conditions: f is a Nash flow with respect to l + τ if P 1, P 2 P with f P1 > 0 : (l + τ) P1 (f ) (l + τ) P2 (f ) Can describe the set of θ-restricted tolls that induce f as { (τ a ) a A δ v δ u + l a + τ a a =(u, v) A \ A + δ v = δ u + l a + τ a a =(u, v) A + τ a 0 a A τ a θ a a A }, where A + := {a A : f a > 0}. Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 70

73 Characterization of inducible flows Auxiliary graph: Ĝ(f )=(V, Â) with arc-costs c : Â R: for every arc (u, v) A: introduceaforward arc (u, v) Â with c (u,v) := l (u,v) + θ (u,v) for every arc (u, v) A + :introduceabackward arc (v, u) Â with c (v,u) := l (u,v) Theorem fisinduciblebyθ-restricted tolls if and only if Ĝ(f ) does not contain a cycle of negative cost. [Bonifaci, Mahyar, Schäfer, SAGT 11] Remark: can also extract respective θ-restricted tolls from Ĝ(f ) Computing optimal tolls: compute a minimum cost flow f such that Ĝ(f ) does not contain a negative cycle Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 71

74 Algorithm for parallel-arc networks Goal: compute a minimum cost flow f such that a A, f a > 0 : l a (f a ) l a (f a )+θ a (f a ) a A. s. t Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 72

75 Algorithm for parallel-arc networks Goal: compute a minimum cost flow f such that a A, f a > 0 : l a (f a ) l a (f a )+θ a (f a ) a A. s. t ordered by decreasing l a (0) +θ a (0) Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 72

76 Algorithm for parallel-arc networks Goal: compute a minimum cost flow f such that a A, f a > 0 : l a (f a ) l a (f a )+θ a (f a ) a A. a s. t ordered by decreasing l a (0) +θ a (0) suppose we knew the arc a with f a = 0 and l a (0) +θ a (0) minimum Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 72

77 Algorithm for parallel-arc networks Goal: compute a minimum cost flow f such that a A, f a > 0 : l a (f a ) l a (f a )+θ a (f a ) a A. 0 a s. t ordered by decreasing l a (0) +θ a (0) suppose we knew the arc a with f a = 0 and l a (0) +θ a (0) minimum Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 72

78 Algorithm for parallel-arc networks Goal: compute a minimum cost flow f such that a A, f a > 0 : l a (f a ) l a (f a )+θ a (f a ) a A. 0 a s. t ordered by decreasing l a (0) +θ a (0) suppose we knew the arc a with f a = 0 and l a (0) +θ a (0) minimum Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 72

79 Algorithm for parallel-arc networks Goal: compute a minimum cost flow f such that a A, f a > 0 : l a (f a ) l a (f a )+θ a (f a ) a A. (1) Algorithm: 1 Guess the minimum value z = l a (0)+θ a (0) of a zero-flow arc a in a minimum cost flow satisfying (1). 2 Set f a = 0foreveryarca A with l a (0) > z. 3 Let A = {a A l a (0) z} be the remaining arcs and solve min a A f al a (f a ) s.t. a A f a = r f a 0 a A l a (f a ) z a A l a (f a ) l a (f a )+θ a (f a ) a, a A Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 73

80 Main result Theorem The algorithm computes optimal θ-restricted tolls for parallel-arc networks in polynomial time if all latency and threshold functions are affine. [Bonifaci, Mahyar, Schäfer, SAGT 11] Remark: [Hoefer et al., WINE 08] derived a similar result for the special case that restrictions are of the form θ a {0, } for every arc a A. Recall: problem is NP-hard for two-commodity networks and affine latency functions Open problem: Is the restricted network toll problem NP-hard for single-commodity networks and affine latency functions? Guido Schäfer A Glimpse of Algorithmic Game Theory: Selfish Routing 74