Kombinatorische Optimierung Matroids and Polymatroids in der - PowerPoint PPT Presentation

Kombinatorische Optimierung Matroids and Polymatroids in der Logistik und im Verkehr in Congestion Games Tobias Harks Augsburg University WINE Tutorial, 8.12.2015

Outline ◮ Part I: Congestion Games ◮ Existence of Equilibria ◮ Computation of Equilibria ◮ Matroids ◮ Part II: Integral Splittable Congestion Games ◮ Existence and Computation of Equilibria ◮ Integral Polymatroids ◮ Part III: Nonatomic Congestion Games ◮ Efficiency of Equilibria ◮ The Braess Paradox ◮ Matroids are Immune to Braess Paradox

Strategic Games ◮ Strategic game G = ( N , X , π ) ◮ N = { 1 , . . . , n } set of players ◮ X = × i ∈ N X i set of pure strategies ◮ x = ( x 1 , . . . , x N ) strategy profile ◮ π i ( x ) : X → R , i ∈ N private cost/utility

Strategic Games ◮ Strategic game G = ( N , X , π ) ◮ N = { 1 , . . . , n } set of players ◮ X = × i ∈ N X i set of pure strategies ◮ x = ( x 1 , . . . , x N ) strategy profile ◮ π i ( x ) : X → R , i ∈ N private cost/utility ◮ Mixed strategy: for each player a probability distribution over pure strategies

Solution Concept Definition Pure Nash equilibrium (PNE): no player has an incentive to unilaterally deviate. Definition Mixed Nash equilibrium (MNE): no player has an incentive to unilaterally change her mixed strategy. ”Minimax Theorem” John von Neumann (1928), Nash (1950)

Motivation: Party Affiliation Games ◮ each strategy set X i = { 1 , − 1 } ◮ Weight w i , j measures relationship between i and j ◮ payoff u i ( x ) = � j ∈ N x i x j w i j → max − 1 1 u 2 ( x ) = 0 2 2 u 3 ( x ) = 0 3 1 u 1 ( x ) = 1 1 − 3 − 1 3 2 1 u 4 ( x ) = 0 u 5 ( x ) = 5 4 5

Solution Concept: Pure Nash Equilibrium Definition Pure Nash equilibrium (PNE): no player has an incentive to unilaterally change his pure strategy. Definition Mixed Nash equilibrium (MNE): no player has an incentive to unilaterally change his mixed strategy.

Motivation: Party Affiliation Games ◮ each strategy set X i = { 1 , − 1 } ◮ Weight w i , j measures relationship between i and j ◮ payoff u i ( x ) = � j ∈ N x i x j w i j → max − 1 1 u 2 ( x ) = 0 2 2 u 3 ( x ) = 0 3 1 u 1 ( x ) = 1 1 − 3 − 1 3 2 1 u 4 ( x ) = 0 u 5 ( x ) = 5 4 5

Existence of Nash Equilibria Nash (1951) Theorem Every finite game possesses a mixed Nash equilibrium. Pure Nash Equilibrium need not exist! Example: Assymmetric Party Affiliation Game 1 1 2 − 1 In the mixed Nash equilibrium, each player chooses each party with probability 1 / 2.

Part I Congestion Games

Congestion Games Modell ◮ N = { 1 , . . . , n } set of players ◮ R = { r 1 , . . . , r m } set of resources ◮ X = × i ∈ N X i set of strategy profiles with X i ⊆ 2 R ◮ Strategy profile x = ( x 1 , . . . , x n ) ∈ X ◮ Load of a resource x r = |{ i ∈ N : r ∈ x i }| for x ∈ X ◮ Cost functions c r : N → R nondecreasing (convex) ◮ private cost: π i ( x ) = � r ∈ x i c r ( x r )

Example

Example

Example

Fundamental Questions 1 When do pure Nash equilibria exist ? 2 How do players find them ? 3 How difficult is it to compute them ?

Potential Functions Definition (Exact potential function) P : X 1 × · · · × X n → R If a player changes his action, the change in the potential function value is equal to the change in her payoff. u i ( x i , x − i ) − u i ( y i , x − i ) = P ( x i , x − i ) − P ( y i , x − i )

Potential Functions Definition (Exact potential function) P : X 1 × · · · × X n → R If a player changes his action, the change in the potential function value is equal to the change in her payoff. u i ( x i , x − i ) − u i ( y i , x − i ) = P ( x i , x − i ) − P ( y i , x − i ) b –Potential u i ( x i , x − i ) − u i ( y i , x − i ) = b i ( P ( x i , x − i ) − P ( y i , x − i )) Monderer and Shapley (1996)

Merits of Potential Games Theorem Every finite exact potential game (with potential P) ◮ possesses a PNE ◮ every sequence of improving moves is finite (FIP) ◮ every local minimum of P is a PNE Monderer and Shapley (1996)

Proof ◮ path γ = ( x 0 , x 1 , . . . , ) sequence of unilateral moves ◮ improvement path γ = ( x 0 , x 1 , . . . , ) sequence of unilateral improving moves γ = x 0 , x 1 , . . . improvement path ⇒ P ( x 0 ) > P ( x 1 ) > · · · > must be finite .

Congestion Games are Potential Games Theorem (Rosenthal ’73) Every congestion game ◮ admits an exact potential function ◮ possesses a PNE ◮ possesses the Finite Improvement Property, that is, every sequence of improving moves is finite.

Proof Rosenthal’s exact potential function P : X 1 × · · · × X n → R is defined as x r � � P ( x ) := c r ( k ) . (1) r ∈ R k =1 Let x ∈ X and y i � = x i be a unilateral deviation of i . � � u i ( x − i , y i ) − u i ( x ) = c r ( x r + 1) − c r ( x r ) . r ∈ y i r ∈ x i r / ∈ x i r / ∈ y i The potential of ( x − i , y i ) is given by: x r � � � � P ( x − i , y i ) = c r ( k ) + c r ( x r + 1) − c r ( x r ) k =1 r ∈ y i r ∈ x i r ∈ R r / r / ∈ x i ∈ y i � �� � = u i ( x − i , y i ) − u i ( x ) = P ( x ) + u i ( x − i , y i ) − u i ( x ) .

Complexity of Computing PNE Theorem (Fabrikant et al. ’04, Ackermann et al. ’08) It is PLS-complete to compute a PNE even for symmetric congestion games with affine costs.

Complexity of Computing PNE Theorem (Fabrikant et al. ’04, Ackermann et al. ’08) It is PLS-complete to compute a PNE even for symmetric congestion games with affine costs. Theorem (Fabrikant et al. ’04) For symmetric network congestion games, there is a polynomial time algorithm to compute a PNE. Subdivide each arc e into n parallel arcs with capacity 1 each and assign costs c e i = c e ( i ) for i ∈ { 1 , . . . , n } .

Complexity of Computing PNE Theorem (Fabrikant et al. ’04, Ackermann et al. ’08) It is PLS-complete to compute a PNE even for symmetric congestion games with affine costs. Theorem (Fabrikant et al. ’04) For symmetric network congestion games, there is a polynomial time algorithm to compute a PNE. Subdivide each arc e into n parallel arcs with capacity 1 each and assign costs c e i = c e ( i ) for i ∈ { 1 , . . . , n } . Remark (Ackermann et al. ’08) There are instances on which every best response dynamic needs exponential convergence time.

Complexity of Computing PNE Theorem (Fabrikant et al. ’04, Ackermann et al. ’08) It is PLS-complete to compute a PNE even for symmetric congestion games with affine costs. Theorem (Fabrikant et al. ’04) For symmetric network congestion games, there is a polynomial time algorithm to compute a PNE. Subdivide each arc e into n parallel arcs with capacity 1 each and assign costs c e i = c e ( i ) for i ∈ { 1 , . . . , n } . Remark (Ackermann et al. ’08) There are instances on which every best response dynamic needs exponential convergence time. Are there other set systems X i with efficiently comp. PNE?

Introduction Matroids Definition (Matroid) A matroid is a pair M = ( R , I ) where R is a set of resources, and I is a family of subsets of S such that: 1 ∅ ∈ I . 2 If I ⊂ J and J ∈ I , then I ∈ I . 3 Let I , J ∈ I and | I | < | J | , then there exists an x ∈ J \ I such that I + x ∈ I . A set system R , I that only satisfies (1) and (2) is called an independence system.

Introduction Matroids Definition (Matroid) A matroid is a pair M = ( R , I ) where R is a set of resources, and I is a family of subsets of S such that: 1 ∅ ∈ I . 2 If I ⊂ J and J ∈ I , then I ∈ I . 3 Let I , J ∈ I and | I | < | J | , then there exists an x ∈ J \ I such that I + x ∈ I . A set system R , I that only satisfies (1) and (2) is called an independence system. Bases are sets in I of maximal cardinality, denoted by B

Example: Uniform Matroids The independent sets of a k -uniform matroid are the sets that contain at most k elements. Example 4 resources: { 1 , 2 , 3 , 4 }

Example: Uniform Matroids The independent sets of a k -uniform matroid are the sets that contain at most k elements. Example 4 resources: { 1 , 2 , 3 , 4 } Independent sets of the 3-uniform matroid: I = {∅ , { 1 } , { 2 } , { 3 } , { 4 } , { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 }}

Example: Uniform Matroids The independent sets of a k -uniform matroid are the sets that contain at most k elements. Example 4 resources: { 1 , 2 , 3 , 4 } Independent sets of the 3-uniform matroid: I = {∅ , { 1 } , { 2 } , { 3 } , { 4 } , { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 }} Bases: B = {{ 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 }}

Graphic Matroid (Cycle Matroid) 6 3 2 4 5 1 Figure : K 4 with two bases B 1 (red), B 2 (blue).