Introduction Unrestricted Restricted Conclusion Total Latency in Singleton Congestion Games Price of Anarchy Martin Gairing 1 Florian Schoppmann 2 1 International Computer Science Institute, Berkeley, CA, USA 2 International Graduate School Dynamic Intelligent Systems , University of Paderborn, Paderborn, Germany December 13, 2007 Total Latency in Singleton Congestion Games 1 / 22
Introduction Unrestricted Restricted Conclusion Singleton Congestion Games � � Defined by tuple Γ = n , m , ( w i ) i ∈ [ n ] , ( S i ) i ∈ [ n ] , ( f e ) e ∈ E where ◮ n ∈ N is number of players ◮ m ∈ N is number of resources s t … ◮ w i ∈ R > 0 is weight of player i ◮ S i ⊆ 2 [ m ] is set of strategies of i ◮ f e : R ≥ 0 → R ≥ 0 is latency function of resource e Implicitly defined: ◮ Set of pure strategy profiles S := S 1 × · · · × S n ◮ Set of mixed strategy profiles as a subset of ∆( S ) Special Cases: ◮ unweighted: For all i ∈ [ n ] : w i = 1 ◮ unrestricted ( ⇒ symmetric): For all i ∈ [ n ] : S i = [ m ] Total Latency in Singleton Congestion Games 2 / 22
Introduction Unrestricted Restricted Conclusion Notation ◮ Load on e ∈ E : δ e ( s ) := � i ∈ [ n ] | s i = e w i ◮ Private cost of i ∈ [ n ] : PC i ( P ) := � s ∈ S P ( s ) f s i ( δ s i ( s )) ◮ Social cost is total latency. For a mixed profile S : � � SC ( P ) := P ( s ) δ e ( s ) · f e ( δ e ( s )) s ∈ S e ∈ E � � � � w i · f e ( δ e ( s )) = w i · PC i ( P ) . = P ( s ) e ∈ s i s ∈ S i ∈ [ n ] i ∈ [ n ] ◮ Pure Price of Anarchy for a set G of games: SC Γ ( p ) PoA pure ( G ) := sup sup OPT , Γ ∈G p is NE of Γ Mixed price of anarachy defined analogously. Total Latency in Singleton Congestion Games 3 / 22
Introduction Unrestricted Restricted Conclusion Motivation Scenario: Selfish load balancing ◮ Selfish players may choose the machine to process their job on ◮ Player’s cost = time until all jobs on that machine are processed Note: Only difference to KP-model [Koutsoupias & Papadimitriou, 1999] is the social cost function The case of non-atomic (singleton) congestion games has long been settled [Roughgarden & Tardos, 2003]. What is known for singleton congestion games? Total Latency in Singleton Congestion Games 4 / 22
Introduction Unrestricted Restricted Conclusion Related Work latencies players PoA pure : LB and UB PoA mixed : LB and UB x ident. 1 2 − 1 / m [7] unrestricted x arb. 9 / 8 [7] 2 − 1 / m [7,6] ax ident. 4 / 3 [7] 2 − 1 / m ax arb. 2 1 + Φ [2] 2 . 036 1 + Φ [2] x d ident. 1 B d + 1 [5] � d Φ d + 1 Φ d + 1 j = 0 a j x j arb. B d + 1 [1] B d + 1 [1] d d x ident. 2 . 012 [8] 2 . 012 [3] 2 . 012 [8] 5 / 2 [4] ax ident. 5 / 2 [3] 5 / 2 [8] 5 / 2 [3] 5 / 2 [4] restricted � d j = 0 a j x j ident. Υ( d ) Υ( d ) [1] Υ( d ) Υ( d ) [1] ax arb. 1 + Φ [3] 1 + Φ [2] 1 + Φ [3] 1 + Φ [2] � d Φ d + 1 Φ d + 1 Φ d + 1 Φ d + 1 j = 0 a j x j arb. [1] [1] d d d d 1. Aland, Dumrauf, Gairing, Monien, Schoppmann. STACS 2006 2. Awerbuch, Azar & Epstein. STOC’05 3. Caragiannis, Flammini, Kaklamanis, Kanellopoulos & Moscardelli. ICALP’06 4. Christodoulou, Koutsoupias. ESA 2005 5. Gairing, Lücking, Mavronicolas, Monien, Rode. ICALP’04 6. Gairing, Monien & Tiemann. SPAA‘05 7. Lücking, Mavronicolas, Monien & Rode. STACS‘04 8. Suri, Tóth & Zhou. SPAA’04 Total Latency in Singleton Congestion Games 5 / 22
Introduction Unrestricted Restricted Conclusion Related Work latencies players PoA pure : LB and UB PoA mixed : LB and UB x ident. 1 2 − 1 / m [7] unrestricted x arb. 9 / 8 [7] 2 − 1 / m [7,6] ax ident. 4 / 3 [7] 2 − 1 / m ax arb. 2 1 + Φ [2] 2 . 036 1 + Φ [2] x d ident. 1 B d + 1 [5] � d Φ d + 1 Φ d + 1 j = 0 a j x j arb. B d + 1 [1] B d + 1 [1] d d x ident. 2 . 012 [8] 2 . 012 [3] 2 . 012 [8] 5 / 2 [4] ax ident. 5 / 2 [3] 5 / 2 [8] 5 / 2 [3] 5 / 2 [4] restricted � d j = 0 a j x j ident. Υ( d ) Υ( d ) [1] Υ( d ) Υ( d ) [1] ax arb. 1 + Φ [3] 1 + Φ [2] 1 + Φ [3] 1 + Φ [2] � d Φ d + 1 Φ d + 1 Φ d + 1 Φ d + 1 j = 0 a j x j arb. [1] [1] d d d d 1. Aland, Dumrauf, Gairing, Monien, Schoppmann. STACS 2006 B d := d -th Bell number 2. Awerbuch, Azar & Epstein. STOC’05 3. Caragiannis, Flammini, Kaklamanis, Kanellopoulos & Moscardelli. ICALP’06 Φ d := positive real root of ( x + 1 ) d = x d + 1 4. Christodoulou, Koutsoupias. ESA 2005 5. Gairing, Lücking, Mavronicolas, Monien, Rode. ICALP’04 ( k + 1 ) 2 d + 1 − k d + 1 ( k + 2 ) d 6. Gairing, Monien & Tiemann. SPAA‘05 Υ( d ) := ( k + 1 ) d + 1 − ( k + 2 ) d +( k + 1 ) d − k d + 1 , where k = ⌊ Φ d ⌋ 7. Lücking, Mavronicolas, Monien & Rode. STACS‘04 8. Suri, Tóth & Zhou. SPAA’04 Total Latency in Singleton Congestion Games 5 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Affine, Weighted: Bounding all NE Lemma Let P be NE in an unrestricted, affine, weighted game. Then, for all subsets of resources M ⊆ [ m ] : b j W + ( |M| − 1 ) w i + � j ∈M a j � SC ( P ) ≤ w i · 1 � j ∈M a j i ∈ [ n ] Proof omitted here. Total Latency in Singleton Congestion Games 6 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Affine, Weighted: Bounding OPT (1/2) Lemma Let s ∈ S be optimal and let M := { e | δ e ( s ) > 0 } . Define X := { x ∈ R M > 0 | � j ∈M x j = W } and let x ∗ ∈ arg min x ∈ X { � j ∈M x j · f j ( x j ) } . Denote M ∗ = { j ∈ M | x ∗ j > 0 } . Then, W 2 + W b j 2 · � j ∈M ∗ a j SC ( s ) ≥ . j ∈M ∗ 1 � a j Proof. � � f j ( x ∗ j ) · x ∗ � f j ( x ∗ j ) · x ∗ SC ( s ) = f j ( δ j ( s )) · δ j ( s ) ≥ j = j j ∈M j ∈M j ∈M ∗ j + b j x ∗ a j � a j · x ∗ · x ∗ � · x ∗ � � = j + b j j = j 1 a j j ∈M ∗ j ∈M ∗ Total Latency in Singleton Congestion Games 7 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Affine, Weighted: Bounding OPT (2/2) x is an equilibrium in the nonatomic game where each f e ( x ) is d replaced by dx ( x · f e ( x )) = 2 a e x + b e . Hence, for all resources j ∈ M ∗ , 2 · b j x ∗ j + 1 2 · b k k ∈M ∗ b k k ∈M ∗ ( x ∗ k + 1 W + 1 � a k ) 2 · � a j a k = = . 1 k ∈M ∗ 1 k ∈M ∗ 1 � � a j a k a k We get j + b j 2 · b j x ∗ x ∗ j + 1 a j a j � · x ∗ � · x ∗ SC ( s ) ≥ j ≥ j 1 1 a j a j j ∈M ∗ j ∈M ∗ W 2 + W W + 1 k ∈M ∗ b k k ∈M ∗ b k 2 · � 2 · � a k a k � x ∗ · = j = . k ∈M ∗ 1 k ∈M ∗ 1 � � a k a k j ∈M ∗ Total Latency in Singleton Congestion Games 8 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Affine, Weighted: Upper Bound Theorem Let G be set of unrestricted, affine, unweighted games. Then, PoA ( G ) < 2 . Proof. Using |M ∗ | ≤ n , we get for any NE P : n 2 + n · ( |M ∗ | − 1 ) + n · � b j SC ( P ) j ∈M ∗ a j OPT ≤ n 2 + n b j 2 · � j ∈M ∗ a j n 2 · |M ∗ |− 1 b j + n 2 · � |M ∗ | j ∈M ∗ a j ≤ 1 + < 2 n 2 + n b j 2 · � j ∈M ∗ a j Total Latency in Singleton Congestion Games 9 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Polynomial, Weighted: Lower Bound (1/3) Theorem Let G be class of unrestricted, polynomial (with max degree d), weighted games. Then, PoA pure ( G ) ≥ B d + 1 . Proof. Construction with parameter k ∈ N : Resources: ◮ k + 1 disjoint sets M 0 , . . . , M k of resources ◮ |M k | = 1 and |M j | = 2 ( j + 1 ) · M j + 1 for j ∈ [ k − 1 ] 0 2 ◮ For all j ∈ [ k ] 0 and for all e ∈ M j : f e ( x ) = x d 2 2 jd 1 1 1 1 2 … 1 1 1 1 2 Players: ◮ k disjoint sets of players N 1 , . . . , N k ◮ |N j | = |M j − 1 | for j ∈ [ k ] ◮ All players in N j have weight w i = 2 j − 1 Total Latency in Singleton Congestion Games 10 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Polynomial, Weighted: Lower Bound (2/3) Proof (continued). Example for k = 2: 2 2 1 1 1 1 2 1 1 1 1 2 … M 1 M 2 M 0 4 1 # resources: 8 � x � x � x � d � d � d latency: 2 0 2 1 2 2 This profile s is a Nash equilibrium with j d + 1 |M j | · j · 2 j · j d = 2 k · k ! � � SC ( s ) = j ! j ∈ [ k ] j ∈ [ k ] Total Latency in Singleton Congestion Games 11 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Polynomial, Weighted: Lower Bound (2/3) Proof (continued). Example for k = 2: 1 1 1 1 2 2 2 2 … M 1 M 2 M 0 4 1 # resources: 8 � x � x � x � d � d � d latency: 2 0 2 1 2 2 This profile s ∗ is strategy profile with 1 |M j | · 2 j = 2 k · k ! � � SC ( s ∗ ) = j ! j ∈ [ k − 1 ] 0 j ∈ [ k − 1 ] 0 Total Latency in Singleton Congestion Games 11 / 22
Introduction Unrestricted Restricted Conclusion Unrestricted, Polynomial, Weighted: Lower Bound (3/3) Proof (continued). Hence, j d + 1 � ∞ ∞ j d + 1 = 1 j = 1 j ! � PoA pure ( G ) ≥ = B d + 1 . � ∞ 1 e j ! j = 0 j ! j = 1 Total Latency in Singleton Congestion Games 12 / 22
Recommend
More recommend
