Total Latency in Singleton Congestion Games

Transcription

1 Introuction Unrestricte Restricte 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 Grauate School Dynamic Intelligent Systems, University of Paerborn, Paerborn, Germany December 13, 2007 Total Latency in Singleton Congestion Games 1 / 22

2 Introuction Unrestricte Restricte Conclusion Singleton Congestion Games Define 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 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 efine: Set of pure strategy profiles S := S 1 S n Set of mixe strategy profiles as a subset of (S) Special Cases: unweighte: For all i [n]: w i = 1 unrestricte ( symmetric): For all i [n]: S i = [m] s t Total Latency in Singleton Congestion Games 2 / 22

3 Introuction Unrestricte Restricte Conclusion Notation Loa 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 si (s)) Social cost is total latency. For a mixe profile S: SC(P) := P(s) δ e (s) f e (δ e (s)) s S e E = P(s) w i f e (δ e (s)) = w i PC i (P). s S e s i i [n] Pure Price of Anarchy for a set G of games: i [n] PoA pure (G) := sup Γ G sup p is NE of Γ SC Γ (p) OPT, Mixe price of anarachy efine analogously. Total Latency in Singleton Congestion Games 3 / 22

4 Introuction Unrestricte Restricte Conclusion Motivation Scenario: Selfish loa balancing Selfish players may choose the machine to process their job on Player s cost = time until all jobs on that machine are processe Note: Only ifference to KP-moel [Koutsoupias & Papaimitriou, 1999] is the social cost function The case of non-atomic (singleton) congestion games has long been settle [Roughgaren & Taros, 2003]. What is known for singleton congestion games? Total Latency in Singleton Congestion Games 4 / 22

5 Introuction Unrestricte Restricte Conclusion Relate Work unrestricte restricte latencies players PoA pure : LB an UB PoA mixe : LB an UB x ient /m [7] x arb. 9/8 [7] 2 1/m [7,6] ax ient. 4/3 [7] 2 1/m ax arb Φ [2] Φ [2] x ient. 1 B +1 [5] j=0 a jx j arb. B +1 Φ +1 [1] B +1 Φ +1 [1] x ient [8] [3] [8] 5/2 [4] ax ient. 5/2 [3] 5/2 [8] 5/2 [3] 5/2 [4] j=0 a jx j ient. Υ() Υ() [1] Υ() Υ() [1] ax arb. 1 + Φ [3] 1 + Φ [2] 1 + Φ [3] 1 + Φ [2] j=0 a jx j arb. Φ +1 Φ +1 [1] Φ +1 Φ +1 [1] 1. Alan, Dumrauf, Gairing, Monien, Schoppmann. STACS Awerbuch, Azar & Epstein. STOC Caragiannis, Flammini, Kaklamanis, Kanellopoulos & Moscarelli. ICALP Christooulou, Koutsoupias. ESA Gairing, Lücking, Mavronicolas, Monien, Roe. ICALP Gairing, Monien & Tiemann. SPAA Lücking, Mavronicolas, Monien & Roe. STACS Suri, Tóth & Zhou. SPAA 04 Total Latency in Singleton Congestion Games 5 / 22

6 Introuction Unrestricte Restricte Conclusion Relate Work unrestricte restricte latencies players PoA pure : LB an UB PoA mixe : LB an UB x ient /m [7] x arb. 9/8 [7] 2 1/m [7,6] ax ient. 4/3 [7] 2 1/m ax arb Φ [2] Φ [2] x ient. 1 B +1 [5] j=0 a jx j arb. B +1 Φ +1 [1] B +1 Φ +1 [1] x ient [8] [3] [8] 5/2 [4] ax ient. 5/2 [3] 5/2 [8] 5/2 [3] 5/2 [4] j=0 a jx j ient. Υ() Υ() [1] Υ() Υ() [1] ax arb. 1 + Φ [3] 1 + Φ [2] 1 + Φ [3] 1 + Φ [2] j=0 a jx j arb. Φ +1 Φ +1 [1] Φ +1 Φ +1 [1] 1. Alan, Dumrauf, Gairing, Monien, Schoppmann. STACS 2006 B2. Awerbuch, Azar & Epstein. STOC 05 := -th Bell number 3. Caragiannis, Flammini, Kaklamanis, Kanellopoulos & Moscarelli. ICALP Christooulou, Koutsoupias. ESA 2005 Φ 5. := positive real root of (x + 1) = x +1 Gairing, Lücking, Mavronicolas, Monien, Roe. ICALP Gairing, Monien (k+1) & Tiemann. kspaa (k+2) 7. Lücking, Mavronicolas, Monien & Roe. STACS Suri, Tóth & Zhou. SPAA 04 Υ() :=, where k = Φ (k+1) +1 (k+2) +(k+1) k +1 Total Latency in Singleton Congestion Games 5 / 22

7 Introuction Unrestricte Restricte Conclusion Unrestricte, Affine, Weighte: Bouning all NE Lemma Let P be NE in an unrestricte, affine, weighte game. Then, for all subsets of resources M [m]: SC(P) w i W + ( M 1)w i + j M Proof omitte here. i [n] j M 1 a j b j a j Total Latency in Singleton Congestion Games 6 / 22

8 Introuction Unrestricte Restricte Conclusion Unrestricte, Affine, Weighte: Bouning OPT (1/2) Lemma Let s S be optimal an let M := {e δ e (s) > 0}. Define X := {x R M >0 j M x j = W } an let x arg min x X { j M x j f j (x j )}. Denote M = {j M xj > 0}. Then, SC(s) W 2 + W 2 b j j M a j j M 1. a j Proof. SC(s) = j M = f j (δ j (s)) δ j (s) f j (xj ) xj j M ( aj x j + b j ) x j = x j + b j a j 1 = j M f j (x j ) x j j M j M a j Total Latency in Singleton Congestion Games 7 / 22 x j

9 Introuction Unrestricte Restricte Conclusion Unrestricte, Affine, Weighte: Bouning OPT (2/2) x is an equilibrium in the nonatomic game where each f e (x) is replace by x (x f e(x)) = 2a e x + b e. Hence, for all resources j M, xj bj a j k M = (x k bk a k ) 1 a j k M 1 = W a k We get SC(s) xj + b j a j x 1 j j M a j = W b k M k a k k M 1 a k j M x j bj a j 1 a j j M x j = b k M k a k k M 1. a k x j W 2 + W 2 k M b k a k k M 1. a k Total Latency in Singleton Congestion Games 8 / 22

10 Introuction Unrestricte Restricte Conclusion Unrestricte, Affine, Weighte: Upper Boun Theorem Let G be set of unrestricte, affine, unweighte games. Then, PoA(G) < 2. Proof. Using M n, we get for any NE P: SC(P) OPT n2 + n ( M b 1) + n j j M a j n 2 + n 2 b j j M a j 1 + n2 M 1 M + n 2 b j j M a j n 2 + n 2 j M b j a j < 2 Total Latency in Singleton Congestion Games 9 / 22

11 Introuction Unrestricte Restricte Conclusion Unrestricte, Polynomial, Weighte: Lower Boun (1/3) Theorem Let G be class of unrestricte, polynomial (with max egree ), weighte games. Then, PoA pure (G) B +1. Proof. Construction with parameter k N: Resources: k + 1 isjoint sets M 0,..., M k of resources M k = 1 an M j = 2(j + 1) M j+1 for j [k 1] 0 For all j [k] 0 an for all e M j : f e (x) = x 2 j Players: k isjoint 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

12 Introuction Unrestricte Restricte Conclusion Unrestricte, Polynomial, Weighte: Lower Boun (2/3) Proof (continue). Example for k = 2: M 0 M 1 M 2 # resources: ( x ) ( x ) ( x ) latency: This profile s is a Nash equilibrium with SC(s) = M j j 2 j j = 2 k k! j [k] j [k] j +1 j! Total Latency in Singleton Congestion Games 11 / 22

13 Introuction Unrestricte Restricte Conclusion Unrestricte, Polynomial, Weighte: Lower Boun (2/3) Proof (continue). Example for k = 2: M 0 M 1 M 2 # resources: ( x ) ( x ) ( x ) latency: This profile s is strategy profile with SC(s ) = M j 2 j = 2 k 1 k! j! j [k 1] 0 j [k 1] 0 Total Latency in Singleton Congestion Games 11 / 22

14 Introuction Unrestricte Restricte Conclusion Unrestricte, Polynomial, Weighte: Lower Boun (3/3) Proof (continue). Hence, PoA pure (G) j=1 j +1 j! j=0 1 j! = 1 e j=1 j +1 j! = B +1. Total Latency in Singleton Congestion Games 12 / 22

15 Introuction Unrestricte Restricte Conclusion Improve Lower Boun for Linear Latencies Instance: 5 resources, 5 jobs, for parameters p [0, 1], w R >0 Nash equilibrium: p w x w p 4 w 1 p 4 1 p 4 1 p 4 x x x x With p maximal, w = 3.258, an profile on the right: PoA mixe > w 1 = w, w 2 = = w 5 = 1 Nash equation for jobs 2,..., 5: w (4 + pw) 1 p w + 1 w p w 2 8w w w Total Latency in Singleton Congestion Games 13 / 22

16 Introuction Unrestricte Restricte Conclusion Improve Lower Boun for Linear Latencies Instance: 5 resources, 5 jobs, for parameters p [0, 1], w R >0 Nash equilibrium: p w x w p 4 w 1 p 4 1 p 4 1 p 4 x x x x With p maximal, w = 3.258, an profile on the right: PoA mixe > w 1 = w, w 2 = = w 5 = 1 Nash equation for jobs 2,..., 5: w (4 + pw) 1 p w + 1 w p w 2 8w w w w w x w x x x x Total Latency in Singleton Congestion Games 13 / 22

17 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Weighte: Lower Boun (1/2) Theorem Let N an G be the set of restricte, polynomial (of max egree ), weighte games. Then, PoA pure (G) Φ +1. Proof. Consier game with n N players, n + 1 resources: Φ Φ 2 Φ 3 Φ n 1 Φ n x Φ +1 x Φ 2(+1) x Φ 3(+1) x Φ n (+1) x Φ n (+1) Let s := (i) n i=1 an s := (i + 1) n i=1. Total Latency in Singleton Congestion Games 14 / 22

18 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Weighte: Lower Boun (1/2) Theorem Let N an G be the set of restricte, polynomial (of max egree ), weighte games. Then, PoA pure (G) Φ +1. Proof. Consier game with n N players, n + 1 resources: Φ Φ 2 Φ 3 Φ n 1 Φ n x Φ +1 x Φ 2(+1) x Φ 3(+1) x Φ n (+1) x Φ n (+1) Let s := (i) n i=1 an s := (i + 1) n i=1. Total Latency in Singleton Congestion Games 14 / 22

19 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Weighte: Lower Boun (1/2) Theorem Let N an G be the set of restricte, polynomial (of max egree ), weighte games. Then, PoA pure (G) Φ +1. Proof. Consier game with n N players, n + 1 resources: Φ Φ 2 Φ 3 Φ n 1 Φ n x Φ +1 x Φ 2(+1) x Φ 3(+1) x Φ n (+1) x Φ n (+1) Let s := (i) n i=1 an s := (i + 1) n i=1. Total Latency in Singleton Congestion Games 14 / 22

20 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Weighte: Lower Boun (2/2) Proof (continue). Φ Φ 2 Φ 3 Φ n 1 Φ n x Φ +1 x Φ 2(+1) x Φ 3(+1) x Φ n (+1) x Φ n (+1) s := (i) n i=1 is a NE as: PC i (s i, i + 1) = (Φi + Φi+1 ) Φ (+1) (i+1) = (Φi (Φ + 1)) Φ (+1) (i+1) = Φi Φ (+1) i = PC i (s) The theorem follows as: n SC(s) = Φ i Φ i i=1 Φ i(+1) = n SC(s ) = (n 1) 1 Φ Total Latency in Singleton Congestion Games 15 / 22

21 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Unweighte: Lower Boun (1/5) Theorem Let N an G be the set of restricte, polynomial (of max egree ), unweighte games. Then, PoA pure (G) Υ(). Proof. Recursive construction with parameter k N: Level: 0 1 k k + 1 2k ( + 1)k Again, we let s be profile where each player uses strategy closer to root an s be profile where each player uses her other strategy. Total Latency in Singleton Congestion Games 16 / 22

22 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Unweighte: Lower Boun (2/5) Proof (continue): For any resource on level ( + 1 i) k + j, where i [ + 1] an j [k 1] 0, let the latency function be f i,j : R 0 R 0, f i,j (x) := [ +1 l=i+1 ] (k 1) ( ) l i j x. l + 1 i + 1 Resources on level ( + 1) k have the same latency function f 0,0 := f 1,k 1 as those on level ( + 1) k 1. Note that s is Nash equilibrium: f i+1,k 1 = f i,0 for i [ + 1] 0 f i,j (i) = f i,j+1 (i + 1) for all i [ + 1] an j [k 2] 0 Level: 0 1 k k + 1 2k ( + 1)k Total Latency in Singleton Congestion Games 17 / 22

23 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Unweighte: Lower Boun (3/5) Social cost of Nash equilibrium s: [ +1 k 1 +1 SC(s) = i=1 j=0 ( +1 = i +1 i=1 l=i+1 + (i 1) +1 + ( + 1) +1 l k ] [ +1 l=i+1 i j i f i,j (i) l [ +1 l=i ] l ] i +1 i +1 (i + 1) ) [ i +1 ] k 1 l +1 i (i 1) +1 (l + 1) l=i ( + 2) ( + 2) ( + 1) +1 (by rearranging terms an simplifying geometric series) Total Latency in Singleton Congestion Games 18 / 22

24 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Unweighte: Lower Boun (4/5) Repeat for SC(s ). Then, SC(s) SC(s ) +1 is of the form: i=0 β i α k 1 i +1 i=0 γ i α k 1 i where i [ + 1]: β i, γ i Q, α i = i +1 (+2) Hence, for fining limit: Fin i [ + 1] 0 fow which α i is max Note that α i+1 > α i is equivalent to (i + 1) l=i+2 l > i +1 i.e., (i + 1) > i +1. Moreover, α 1 = (+1)! (+2) +1 l=i+1 +1 l=i+1 l an α 0 = 1 l = i +1 (i + 1) < 1 an α +1 = (+1)+1 (+2) > 1 +1 l=i+2 Total Latency in Singleton Congestion Games 19 / 22 l,

25 Introuction Unrestricte Restricte Conclusion Restricte, Polynomial, Unweighte: Lower Boun (5/5) Let λ := Φ. Then, (λ + 1) > λ +1 but (λ + 2) < (λ + 1) +1 Thus, λ [] an α λ+1 is maximal Using stanar calculus we get lim k +1 i=0 β i α k 1 i +1 i=0 γ i α k 1 i Inserting gives the esire boun. = β λ+1 γ λ+1. Total Latency in Singleton Congestion Games 20 / 22

26 Introuction Unrestricte Restricte Conclusion Exponential Growth Even for singleton congestion games with polynomial latency functions, price of anarchy for is Θ() : Φ Υ() Φ +1 B ,514 1, ,345 14, , ,926 4, ,603 1,101,126 21, ,540,286 11,079, , ,562, ,180, ,570 Total Latency in Singleton Congestion Games 21 / 22

27 Introuction Unrestricte Restricte Conclusion Conclusion Motivation: Unerstaning the epenence of the PoA on network topology Results presente in this talk: Collection of upper an lower bouns on PoA For the unrestricte case, closing the gaps between upper an lower bouns seems challenging Surprisingly, both upper bouns on the PoA for general congestion games with polynomial latency functions are alreay exact for singleton games an pure NE Total Latency in Singleton Congestion Games 22 / 22