How hard is it to find extreme Nash equilibria in network congestion games? E. Gassner J. Hatzl Graz University of Technology, Austria S.O. Krumke H. Sperber University of Kaiserslautern, Germany G. Woeginger Eindhoven University of Technology, The Netherlands 13th Combinatorial Optimization Workshop Aussois, January 2009 Hatzl (TUG) Network congestion games January 2009 1 / 29
Talk Outline 1 Problem Formulation 2 Preliminary Results 3 Complexity Results for Worst Nash Equilibria 4 Complexity Results for Best Nash Equilibria Hatzl (TUG) Network congestion games January 2009 2 / 29
The model A directed graph G ( V , E ) with multiple edges A source s and a sink t Non-decreasing latency functions ℓ e : N 0 → R + 0 N users, each routing the same amount of unsplittable flow Strategy set for all users: P — set of all simple s - t -paths Hatzl (TUG) Network congestion games January 2009 3 / 29
The model A directed graph G ( V , E ) with multiple edges A source s and a sink t Non-decreasing latency functions ℓ e : N 0 → R + 0 N users, each routing the same amount of unsplittable flow Strategy set for all users: P — set of all simple s - t -paths x 1 . 5 x s u t 2 x 2 x 1 . 5 x Hatzl (TUG) Network congestion games January 2009 3 / 29
The model A flow is a function f : P → N 0 . The latency on a path P ∈ P is the sum of the latencies on its edges, i.e., � � � � ℓ P ( f ) := ℓ e f P e ∈ P P ∈P : e ∈ P Given a flow f the social cost are given by C max ( f ) := P ∈P : f P > 0 ℓ P ( f ) . max x 1 . 5 x s u t 2 x 2 x 1 . 5 x C max ( f ) = max { 1 + 3 , 2 + 3 , 1 . 5 } = 5 Hatzl (TUG) Network congestion games January 2009 4 / 29
Nash Equilibrium Definition (Nash Equilibrium) A flow f is a Nash equilibrium, iff for all paths P 1 , P 2 with f P 1 > 0 we have f P − 1 if P = P 1 ℓ P 1 ( f ) ≤ ℓ P 2 (˜ f ) with ˜ f P = f P + 1 if P = P 2 . f P otherwise x x 1 . 5 x 1 . 5 x s u t s u t 2 x 2 x 2 x 2 x 1 . 5 x 1 . 5 x Hatzl (TUG) Network congestion games January 2009 5 / 29
Roughgarden Model Network Congestion Game Roughgarden single-commoditiy multicommodity unsplittable, unweighted splittable makespan sum Hatzl (TUG) Network congestion games January 2009 6 / 29
Existence of Nash equilibria Hatzl (TUG) Network congestion games January 2009 7 / 29
Existence of Nash equilibria Theorem (Roughgarden and Tardos (2002)) The Nash flows of an instance are precisely the optima of a non-linear convex programming problem. If f and ˜ f are Nash flows then ℓ e ( f ) = ℓ e (˜ f ) for all e ∈ E. Hence, all Nash equilibria have the same social cost. Hatzl (TUG) Network congestion games January 2009 7 / 29
Existence of Nash equilibria Theorem (Roughgarden and Tardos (2002)) The Nash flows of an instance are precisely the optima of a non-linear convex programming problem. If f and ˜ f are Nash flows then ℓ e ( f ) = ℓ e (˜ f ) for all e ∈ E. Hence, all Nash equilibria have the same social cost. Theorem (Fabrikant et al. (2004)) Given a network congestion game the optimal solution of the following min-cost flow problem MCF(G) yields a Nash equilibrium: For every edge e ∈ E we need N copies with costs c e i = ℓ e ( i ) , i = 1 , . . . , N. The capacities of all edges are 1 and we send N units of flow from s to t. Hatzl (TUG) Network congestion games January 2009 7 / 29
Extreme Nash equilibria Consider the following instance with N = 2: 2 x 2 x 2 x 2 x u 1 u 2 u 3 s t 3 x 3 x 5 x 5 x Hatzl (TUG) Network congestion games January 2009 8 / 29
Extreme Nash equilibria Consider the following instance with N = 2: 2 x 2 x 2 x 2 x u 1 u 2 u 3 s t 3 x 3 x 5 x 5 x The solution with minimum social cost of 12 is given by 2 x 2 x 2 x 2 x u 1 u 2 u 3 s t 3 x 3 x 5 x 5 x Hatzl (TUG) Network congestion games January 2009 8 / 29
Extreme Nash equilibria Consider the following instance with N = 2: 2 x 2 x 2 x 2 x u 1 u 2 u 3 s t 3 x 3 x 5 x 5 x A Nash equlibirum with social cost of 13 is given by 2 x 2 x 2 x 2 x u 1 u 2 u 3 s t 3 x 3 x 5 x 5 x Hatzl (TUG) Network congestion games January 2009 8 / 29
Extreme Nash equilibria Consider the following instance with N = 2: 2 x 2 x 2 x 2 x u 1 u 2 u 3 s t 3 x 3 x 5 x 5 x A Nash equlibirum with social cost of 14 is given by 2 x 2 x 2 x 2 x u 1 u 2 u 3 s t 3 x 3 x 5 x 5 x Hatzl (TUG) Network congestion games January 2009 8 / 29
Extreme Nash equilibria Worst Nash Equilibrium (W-NE for short): Given: Network congestion game ( G = ( V , E ) , ( ℓ e ) e ∈ E , s ∈ V , t ∈ V , N ) and a number K > 0 Question: Does there exist a Nash equilibrium f such that C max ( f ) ≥ K ? Best Nash Equilibrium (B-NE for short): Given: Network congestion game ( G = ( V , E ) , ( ℓ e ) e ∈ E , s ∈ V , t ∈ V , N ) and a number K > 0 Question: Does there exist a Nash equilibrium f such that C max ( f ) ≤ K ? Unfortunately, it can be shown that in general neither a best nor a worst Nash equilibrium is an optimal solution of MCF( G ). Hatzl (TUG) Network congestion games January 2009 9 / 29
Extreme Nash equilibria Theorem (Fotakis(2002), Gairing(2005)) If the users have different weights and the graph G has only parallel links W-NE and B-NE are N P-hard even for linear latency functions. Hatzl (TUG) Network congestion games January 2009 10 / 29
Nash equilibria in series-parallel graphs The series composition G = S ( G 1 , G 2 ) : P 1 Q 1 P 2 Q 2 P 3 Q 3 Lemma Let f i be a flow in G i (i = 1 , 2 ). Let f ∈ f 1 ⊗ f 2 then f is a Nash equilibrium in S ( G 1 , G 2 ) if and only if f i are Nash equilibria in G i (i = 1 , 2 ). Hatzl (TUG) Network congestion games January 2009 11 / 29
Nash equilibria in series-parallel graphs The parallel composition G = S ( G 1 , G 2 ) : Lemma Let f i be a flow in G i (i = 1 , 2 ). Then f = f 1 ∪ f 2 is a Nash equilibrium in P ( G 1 , G 2 ) if and only if f i are Nash equilibria in G i (i = 1 , 2 ) and C max ( f 2 ) ≤ L + G 1 ( f 1 ) and C max ( f 1 ) ≤ L + G 2 ( f 2 ) . Hatzl (TUG) Network congestion games January 2009 12 / 29
Worst Nash equilibrium Worst Nash Equilibrium (W-NE for short): Given: Network congestion game ( G = ( V , E ) , ( ℓ e ) e ∈ E , s ∈ V , t ∈ V , N ) and a number K > 0 Question: Does there exist a Nash equilibrium f such that C max ( f ) ≥ K ? Hatzl (TUG) Network congestion games January 2009 13 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; 4 x ( 0 ) x ( 0 ) 6 x ( 0 ) 6 x ( 0 ) Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; 4 x ( 0 ) 4 x ( 0 ) 1 6 x ( 0 ) 6 6 x ( 0 ) 6 current makespan of user 1 = 5 Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; 4 x ( 1 ) 4 x ( 1 ) 1 6 x ( 0 ) 6 6 x ( 0 ) 6 current makespan of user 1 = 5 Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; 4 x ( 1 ) 8 x ( 1 ) 2 6 x ( 0 ) 6 6 x ( 0 ) 6 current makespan of user 1 = 5 current makespan of user 2 = 6 Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; 4 x ( 1 ) 8 x ( 1 ) 2 6 x ( 0 ) 6 6 x ( 1 ) 6 current makespan of user 1 = 5 current makespan of user 2 = 6 Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; 4 x ( 1 ) 8 x ( 1 ) 2 6 x ( 0 ) 6 6 x ( 1 ) 12 current makespan of user 1 = 6 current makespan of user 2 = 6 current makespan of user 3 = 8 Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1. end do; 4 x ( 1 ) 8 x ( 2 ) 2 6 x ( 1 ) 6 6 x ( 1 ) 12 current makespan of user 1 = 6 The last user yields the current makespan of user 2 = 6 maximum makespan! current makespan of user 3 = 8 Hatzl (TUG) Network congestion games January 2009 14 / 29
Worst Nash equilibria in SP-graphs Greedy Best Response (GBR): For i = 1 to N do User i chooses a path with minimal latency with respect to load = current flow +1 end do; Hatzl (TUG) Network congestion games January 2009 15 / 29
Recommend
More recommend
