Colored Resource Allocation Games

Transcription

1 Colored Resource Allocation Games Evangelos Bampas 1, Aris Pagourtzis 1, George Pierrakos, and Vasileios Syrgkanis 1 1 School of Elec. & Comp. Engineering, ational Technical University of Athens {ebamp,pagour}@cs.ntua.gr, syrganis@corelab.ntua.gr School of Electrical Engineering & Computer Sciences, UC Berkeley georgios@cs.berkeley.edu Abstract. e introduce Colored Resource Allocation Games as a new model for selfish routing and wavelength assignment in multifiber all-optical networks. Colored Resource Allocation Games are a generalization of congestion and bottleneck games where players have their strategies in multiple copies (colors). e focus on two main subclasses of these games depending on the player cost: in Colored Congestion Games the player cost is the sum of latencies of the resources allocated to the player, while in Colored Bottleneck Games the player cost is the maximum of these latencies. e investigate the pure price of anarchy for three different social cost functions and prove tight bounds for each separate case. Our bounds for the two standard social cost functions (maximum and average player cost) generalize earlier results. The third social cost function is particularly meaningful in the setting of multifiber all-optical networks, where it captures the objective of fiber cost minimization. 1 Introduction Potential Games are a widely used tool for modeling network optimization problems under a non-cooperative perspective. Initially studied in [1] with the introduction of congestion games and further extended in [] in a more general framework, they have been successfully applied to describe selfish routing in communication networks (e.g. [3]). The advent of optical networks as the technology of choice for surface communication has introduced new aspects of networks that are not sufficiently captured by the models proposed so far. This work comes to close this gap and presents a class of potential games useful for modeling selfish routing and wavelength assignment in multifiber optical networks. In optical networking it is highly desirable that all communication be carried out transparently, that is, each signal should remain on the same wavelength from source to destination. The need for efficient access to the optical bandwidth has given rise to the study of several optimization problems in the past years. The most well-studied among them is the problem of assigning a path and a color (wavelength) to each communication request in such a way that paths of the same color are edge-disjoint and the number of colors used is minimized. onetheless, it has become clear that the number of wavelengths in commercially available fibers is rather limited and will probably remain such in the foreseeable future. Therefore, the use of multiple fibers has become inevitable in large scale networks. In the context of multifiber optical networks several optimization problems have been defined and studied, the objective usually being to minimize either

2 the maximum fiber multiplicity per edge or the sum of these maximum multiplicities over all edges of the graph. 1.1 Contribution e introduce Colored Resource Allocation Games, a class of games that can model non-cooperative versions of routing and wavelength assignment problems in multifiber all-optical networks. They can be viewed as an extension of congestion games where each player has his strategies in multiple copies (colors). hen restricted to (optical) network games, facilities correspond to edges of the network and colors to wavelengths. The number of players using an edge in the same color represents a lower bound on the number of fibers needed to implement the corresponding physical link. Having this motivation in mind, we consider both egalitarian (max) and utilitarian (sum) player costs. For our purposes it suffices to restrict our study to identity latency functions. e use the price of anarchy (PoA) introduced in [4] as a measure of the deterioration caused by lack of coordination. e estimate the PoA of our games under three different social cost functions. Two of them are standard in the literature (see e.g. [5]): the first (SC 1 ) is equal to the maximum player cost and the second (SC ) is equal to the sum of player costs (equivalently, the average player cost). The third one is specially designed for the setting of multifiber all-optical networks; it is equal to the sum over all facilities of the maximum color congestion on each facility. ote that in the optical network setting this function represents the total fiber cost needed to accommodate all players; hence, it captures the objective of a well-studied optimization problem ([6 9]. Let us also note that the SC 1 function under the egalitarian player cost captures the objective of another well known problem, namely minimizing the maximum fiber multiplicity over all edges of the network [8, 10, 11]. SC 1(A) = max C i(a) SC (A) = X C i(a) SC 3(A) = X max n f,a(a) a [] f F Colored Congestion «Games Congestion Games Θ Θ [5] q 5 p Θ F 5 [5] Table 1. The pure price of anarchy of Colored Congestion Games (utilitarian player cost). Results for classical congestion games are shown in the right column. Our main contribution is the derivation of tight bounds on the price of anarchy for Colored Resource Allocation Games. These bounds are summarized in Tables 1 and. It can be shown that the bounds for Colored Congestion Games remain tight even for network games. Observe that known bounds for classical congestion and bottleneck games can be obtained from our results by simply setting = 1. On the other hand one might

3 SC 1(A) = max C i(a) SC (A) = X C i(a) SC 3(A) = X f F max n f,a (A) a [] Colored Bottleneck Games Bottleneck Games Θ ` Θ() [1] Θ ` Θ() [1] E A E OPT Table. The pure price of anarchy of Colored Bottleneck Games (egalitarian player cost). Results for classical bottleneck games are shown in the right column. notice that our games can be casted as classical congestion or bottleneck games with F facilities. However we are able to derive better upper bounds for most cases by exploiting the special structure of the players strategies. In addition to the above, we provide a potential function for Colored Bottleneck Games in order to prove the existence and convergence to pure equilibria and we show that the price of stability [13] is Related ork One of the most important notions in the theory of non-cooperative games is the ash Equilibrium (E) [14], a stable state of the game in which no player has incentive to change strategy unilaterally. A fundamental question in this theory concerns the existence of pure ash Equilibria (PE). For congestion and bottleneck games [1,, 15] it has been shown with the use of potential functions that they converge to a PE. Bottleneck games have been studied in [1, 15 17]. In [1] the authors study atomic routing games on networks, where each player chooses a path to route her traffic from an origin to a destination node, with the objective of minimizing the maximum congestion on any edge of her path. They show that these games always possess at least one optimal PE (hence PoS = 1) and that the PoA of the game is determined by topological properties of the network. A further generalization is the model of Banner and Orda [15], where they introduce the notion of bottleneck games. In this model they allow arbitrary latency functions on the edges and consider both splittable and unsplittable flows. They show existence, convergence and non-uniqueness of equilibria and they prove that the PoA for these games is exponential in the users demand. Selfish path coloring in single fiber all-optical networks has been studied in [18 1]. Bilò and Moscardelli [18] consider the convergence to ash Equilibria of selfish routing and path coloring games. Bilò et al. [19] consider several information levels of local knowledge that players may have and give bounds for the PoA in chains, rings and trees. The existence of ash Equilibria and the complexity of recognizing and computing a ash Equilibrium for selfish routing and path coloring games under several payment functions are considered by Georgakopoulos et al. [0]. In [1] upper and lower bounds of the PoA for selfish path coloring with and without routing are presented under functions that charge a player only according to her own strategy. 3

4 Selfish path multicoloring games are introduced in [] where it is proved that the pure price of anarchy is bounded by the number of available colors and by the length of the longest path; constant bounds for the PoA in specific topologies are also provided. In those games, in contrast to the ones studied here, routing is given in advance and players choose only colors. Model Definition Definition 1 (Colored Resource Allocation Games). A Colored Resource Allocation Game is defined as a tuple F,,, {E i } such that: 1. F is a set of facilities f i. [] is a set of colors 3. [] is a set of players 4. E i is a set of possible facility combinations for player i such that: a. i [] : E i F b. S i = E i [] is the set of possible strategies of player i c. A i = (E i,a i ) S i is the notation of a strategy for player i with E i E i denoting the set of facilities used and a i [] the corresponding color 5. A = (A 1,...,A ) is a strategy profile for the game 6. For a strategy profile A, f F, c [], n f,c (A) is the number of players that use facility f in color c in strategy profile A Depending on the player cost function we define two subclasses of Colored Resource Allocation Games: Colored Congestion Games (CCG), where the player cost is C i (A) = e E i n e,ci (A) Colored Bottleneck Games (CBG), where the player cost is C i (A) = max e E i n e,ci (A) For each of the above variations we will consider three different social cost functions: SC 1 (A) = max C i (A) SC (A) = SC 3 (A) = f F C i (A) = f F max n f,a(a) a [] a [] n f,a (A) From the definition of pure ash Equilibrium we can derive the following two facts that hold in Colored Congestion and Bottleneck Games respectively: Fact 1. For a PE strategy profile A of a CCG it holds: E i E i, c [] : C i (A) e E i (n e,c (A) + 1) (1) 4

5 Fact. For a PE strategy profile A of CBG it holds: Equivalently: E i E i, c [] : C i (A) max(n e,c (A) + 1) () e E i E i E i, c [], e E i : C i (A) n e,c (A) + 1 (3) 3 Colored Congestion Games In this section we compute the pure price of anarchy of colored congestion games for three different social cost functions. 3.1 Pure PoA for Social Cost SC 1 Theorem 1. The price of anarchy of any Colored Congestion Game F,,, {E i } ) with social cost SC 1 is O. ( Proof. Let A be a ash Equilibrium and let OPT be an optimal strategy profile. ithout loss of generality we consider the first player to have the maximum cost, SC 1 (A) = C 1 (A). Thus we need to bound C 1 (A) with respect to the optimum social cost SC 1 (OPT) = max j [] C j(opt). Since A is a ash Equilibrium every player has no benefit of changing either her color or her choice of facilities. e denote with OPT 1 = (E 1,a 1 ) the strategy of player P 1 in OPT. Since A is a.e. it must hold: a [] : C 1 (A) (n e,a (A) + 1) n e,a (A) + C 1 (OPT) (4) e E 1 The second inequality holds since any strategy profile cannot lead to a cost for a player that is less than the size of her facility combination. Let I [] the set of players that, in A, use some facility e E1. The sum of their costs is: C i (A) n e,a(a) ( e E1 a [] n e,a(a)) E1 i I e E1 a [] e E 1 ( min a [] e E1 n e,a(a)) E1 (min a [] e E 1 n e,a(a)) E1 The first inequality holds since a player in I might use facilities (e,a) not in E1 and the second inequality holds from the Cauchy-Schwarz inequality. Let a min = arg min n e,a (A). Thus we have: n e,amin (A) e E 1 E 1 a [] e Ei (5) C i (A) (6) i I 5

6 k 1 Ô Ø ÓÚ Ö h i n 0 h 1 n 1 h n n k 1 h k n k Fig.1. A worst-case instance that proves the tightness of the upper bound, depicted as network game. From [5] we have: e E 1 C i (A) 5 C i (OPT) (7) Combining the above two inequalities we have: n e,amin (A) E 1 C i (A) E 1 i I Combining with (4) for a min, we get C 1 (A) C 1 (OPT) + 5 E 1 C i (A) 5 E1 C i (OPT) (8) C i (OPT) (9) Since E1 C 1(OPT) and C i (OPT) SC 1 (OPT), we get ( ) 5 C 1 (A) 1 + SC 1 (OPT) (10) Theorem. There exists an infinite set of Colored Congestion Games F,,, {E i } ( ) with social cost SC 1, that have pure price of anarchy Ω. Proof. e will describe the lower bound instance as a network game. The underlying network is illustrated in Figure 1. In that network major players want to send traffic from n 0 to n k+1. In every node n 0,...,n k we have (k 1) minor players that want to send traffic from n i to n i+1. In the worst-case equilibrium A all players choose the short central edge, leading to social cost SC 1 (A) = k. In the optimum the minor players are equally divided to all long paths and the major players choose the central edge. This leads to SC 1 (OPT) = k, and the price of anarchy is therefore: ( ) PoA = k = Θ (11) 6

7 3. Pure PoA for Social Cost SC The price of anarchy for Social Cost SC is upper-bounded by 5/, as proved in [5]. For the lower bound, we use a slight modification of the instance described in [5]. e have players and facilities. The facilities are separated into two groups: {h 1,...,h } and {g 1,...,g }. Players are divided into groups of players. Each group i has strategies {h i,g i } and {g i+1,h i 1,h i+1 }. The optimal allocation is for all players in the i-th group to select their first strategy and be equally divided in the colors, leading to SC (OPT) =. In the worst-case E players choose their second strategy and are equally divided in the colors, leading to SC (A) = 5. Thus, the PoA of this instance is 5/ and the upper bound remains tight in our model too. 3.3 Pure PoA for Social Cost SC 3 Theorem ( 3. The price of anarchy of colored congestion games with social cost SC 3 is ) O F. Proof. e denote by n e (S) the vector [n e,a1 (S),...,n e,a (S)]. In terms of the above vector we can write: SC 3 (S) = e F From norm inequalities we have: hence: SC 3 (S) = e F max n e,a(s) = n e (S) (1) a [] e F n e (S) n e (S) n e (S) (13) n e (S) e F n e,a(s) F n e,a(s), (14) a where the last inequality is a manifestation of the norm inequality x 1 n x, where x is a vector of dimension n. ow, from the first inequality of (13) we have: SC 3 (S) 1 n e,a(s) 1 n e,a(s) (15) a e F a Combining (15) and (14) gives: e F 1 SC (S) SC 3 (S) F SC (S) (16) From [5] we know that the price of anarchy with social cost SC (S) is 5/. Let A be a worst-case ash Equilibrium in the case of SC 3 and let OPT be an optimal strategy profile. From (16) we know that SC 3 (A) F SC (A) and SC 3 (OPT) 1 SC (OPT). Thus: PoA = SC 3(A) SC 3 (OPT) SC (A) F SC (OPT) 5 F (17) 7 e F a

8 Theorem 4. There exists an infinite set of Colored Congestion Games with social cost SC 3 that have PoA = F. Proof. Consider a colored congestion game with players, F = facilities and = colors. Each player has strategies the singleton sets consisting of one facility. In other words E i = {{f 1 }, {f },..., {f }}. The above instance has a worst-case equilibrium with social cost when all players choose a different facility in an arbitrary color. On the other hand in the optimum strategy profile players fill all colors of the necessary facilities. This needs facilities with maximum capacity over their colors 1. Thus the optimum social cost is leading to a PoA = F. 4 Colored Bottleneck Games 4.1 Convergence to Equilibrium Definition. (Player Congestion Vector). A player congestion vector for a strategy profile A of a CBG is a vector [b,...,b 1 ] where b i = {P k [] : c k (A) = i} (18) Theorem 5. For a CBG F,,, {E i } any ash dynamics converges to a ash Equilibrium in finitely many steps. Proof. Consider an arbitrary initial strategy profile A 0 and its corresponding Player Congestion Vector CV 0. At every step m of a ash dynamics one player k must make an improving move. Let C k (A m ) = j. Then, b j of CV m must decrease at least by 1, since no other player s cost can be increased to b j and no player with higher cost is affected. Thus, the quantity i (b i (A m ) i ) decreases at every step and must converge to a PE in a finite number of steps. Corollary 1. For any CBG F,,, {E i } the price of stability is Pure PoA for Social Cost SC 1 Theorem 6. The price of anarchy of any CBG game with social cost SC 1 (A) is at most. Proof. It is obvious that SC 1 (OPT) 1. Let SC 1 (A) + 1. From Fact at least players must play each of the other colors. This needs at least +1 players. Theorem 7. There exist instances of a CBG game with pure PoA =. 8

9 Proof. Consider the following class of CBG games. e have players and facilities. Each player P i has two possible strategies: E i = {{f i }, {f 1,...,f }}. In a worst-case E all players choose the second strategy and they are equally divided in the colors. This leads to player cost for each player and thus to a social cost. In the optimal strategy all players would choose their first strategy leading to player and social cost 1. Thus the PoA for this instance is. 4.3 Pure PoA for Social Cost SC Theorem 8. The price of anarchy of any CBG game with social cost SC (A) is at most. Proof. By Theorem 4. we know that C i (A). Moreover it is obvious that SC (OPT). Thus PoA = C i(a) SC (OPT). The instance used in the previous section can also be used here to prove that the above inequality is tight for a class of CBG games. 4.4 Pure PoA for Social Cost SC 3 To state the following theorem we have to define a set. Definition 3. e define E S to be the set of facilities used by at least one player in the strategy profile S = (A 1,...,A ), i.e. E S = E 1... E (19) Theorem 9. The price of anarchy of any CBG game with social cost SC 3 (A) is at E most A E OPT. Proof. e exclude from the sum over the facilities, those facilities that are not used by any player since they do not contribute to the social cost. Thus we focus on facilities with max a n e,a > 0. Let a max (e) denote the color with the maximum multiplicity at facility e. Let P i be a player that uses the facility copy (e,a max (e)). Since C i = max e Ei n e,ai (A) it must hold that n e,amax(e)(a) C i (A). In fact we can state the following general property: e F, i [] : n e,amax(e) C i (A) (0) From the above sections we know that C i (A). Moreover it is obvious that SC 3(OPT) E OPT. From the above we can conclude: SC 3 (A) SC 3 (OPT) E A E OPT (1) E OPT. Theorem 10. There exists a class of CBGs with PoA = E A 9

10 Proof. e use a slight modification of the instances used in the above two sections. Each player i has strategies E i = {{f i }, {f 1,...,f M }} for any M. In the worst-case E all players will play the second strategy leading to SC 3 (A) = M and E A = M. On the other hand in the optimal outcome all players will play the first strategy leading to SC 3 (OPT) = and E OPT =. Thus the price of anarchy for this instance is PoA = M = E A E OPT 5 Discussion. In this paper we introduced Colored Resource Allocation Games, a class of games which generalize both congestion and bottleneck games. The main feature of these games is that players have their strategies in multiple copies (colors). Therefore, these games can serve as a framework to describe routing and wavelength assignment games in multifiber all-optical networks. Although we could cast such games as classical congestion games, it turns out that the proliferation of resources together with the structure imposed on the players strategies allows us to prove better upper bounds. Regarding open questions, it would be interesting to consider more general latency functions. This would make sense both in the case where fiber pricing is not linear in the number of fibers, and also in the case where the network operator seeks to determine an appropriate pricing policy so as to reduce the price of anarchy. Another interesting direction is to examine which network topologies result in better system behavior. References 1. Rosenthal, R..: A class of games possessing pure-strategy nash equilibria. Int. J. Game Theory (1973) Monderer, D., Shapley, L.S.: Potential games. Games and Economic Behavior 14 (1996) Roughgarden, T.: Selfish Routing and the Price of Anarchy. The MIT Press (005) 4. Koutsoupias, E., Papadimitriou, C.H.: orst-case equilibria. In Meinel, C., Tison, S., eds.: STACS. Volume 1563 of Lecture otes in Computer Science. (1999) Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC. (005) omikos, C., Pagourtzis, A., Zachos, S.: Routing and path multicoloring. Inf. Process. Lett. 80(5) (001) omikos, C., Pagourtzis, A., Potika, K., Zachos, S.: Routing and wavelength assignment in multifiber wdm networks with non-uniform fiber cost. Computer etworks 50(1) (006) Andrews, M., Zhang, L.: Complexity of wavelength assignment in optical network optimization. In: IFOCOM, IEEE (006) 9. Andrews, M., Zhang, L.: Minimizing maximum fiber requirement in optical networks. J. Comput. Syst. Sci. 7(1) (006) Li, G., Simha, R.: On the wavelength assignment problem in multifiber DM star and ring networks. IEEE/ACM Trans. etw. 9(1) (001) Margara, L., Simon, J.: avelength assignment problem on all-optical networks with fibres per link. In: ICALP. (000) Busch, C., Magdon-Ismail, M.: Atomic routing games on maximum congestion. In Cheng, S.., Poon, C.K., eds.: AAIM. Volume 4041 of Lecture otes in Computer Science. (006)

11 13. Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., exler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. Foundations of Computer Science, 004. Proceedings. 45th Annual IEEE Symposium on (Oct. 004) ash, J.: on-cooperative games. The Annals of Mathematics 54() (1951) Banner, R., Orda, A.: Bottleneck routing games in communication networks. In: IFOCOM, IEEE (006) 16. Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of nash equilibria for a selfish routing game. In: ICALP. (00) Libman, L., Orda, A.: Atomic resource sharing in noncooperative networks. Telecommunication Systems 17(4) (001) Bilò, V., Moscardelli, L.: The price of anarchy in all-optical networks. In Kralovic, R., Sýkora, O., eds.: SIROCCO. Volume 3104 of Lecture otes in Computer Science. (004) Bilò, V., Flammini, M., Moscardelli, L.: On nash equilibria in non-cooperative all-optical networks. In Diekert, V., Durand, B., eds.: STACS. Volume 3404 of Lecture otes in Computer Science. (005) Georgakopoulos, G.F., Kavvadias, D.J., Sioutis, L.G.: ash equilibria in all-optical networks. In Deng, X., Ye, Y., eds.: IE. Volume 388 of Lecture otes in Computer Science. (005) Milis, I., Pagourtzis, A., Potika, K.: Selfish routing and path coloring in all-optical networks. In: CAA. Volume 485 of Lecture otes in Computer Science. (007) Bampas, E., Pagourtzis, A., Pierrakos, G., Potika, K.: On a non-cooperative model for wavelength assignment in multifiber optical networks. In: ISAAC. (008)