Fast Convergence to Nearly Optimal Solutions in Potential Games

Transcription

1 Fast Convergence to Nearly Optmal Solutons n Potental Games Baruch Awerbuch Yoss Azar Amr Epsten Vahab S. Mrrokn Alexander Skopalk Abstract We study the speed of convergence of decentralzed dynamcs to approxmately optmal solutons n potental games. We consder α-nash dynamcs n whch a player makes a move f the mprovement n hs payoff s more than an α factor of hs own payoff. Despte the known polynomal convergence of α-nash dynamcs to approxmate Nash equlbra n symmetrc congeston games [7], t has been shown that the convergence tme to approxmate Nash equlbra n asymmetrc congeston games s exponental [23]. In contrast to ths negatve result, and as the man result of ths paper, we show that for asymmetrc congeston games wth delay functons that satsfy a bounded jump condton, the convergence tme of α-nash dynamcs to an approxmate optmal soluton s polynomal n the number of players, wth approxmaton rato that s arbtrarly close to the prce of anarchy of the game. In partcular, we show ths polynomal convergence under the mnmal lveness assumpton that each player gets at least one chance to move n every T steps. We also prove that the same polynomal convergence result does not hold for (exact) best-response dynamcs, showng the α-nash dynamcs s requred. We extend these results for congeston games to other potental games ncludng weghted congeston games wth lnear delay functons, cut games (also called party afflaton games) and market sharng games as follows. Dept. of Computer Scence, Johns Hopkns Unversty. E-Mal: baruch@cs.jhu.edu. Research supported by NSF grants ANIR and CCR School of Computer Scence, Tel-Avv Unversty. E-Mal: azar@tau.ac.l. Research supported n part by the German-Israel Foundaton. School of Computer Scence, Tel Avv Unversty. E-Mal: amrep@tau.ac.l. Research supported n part by the German-Israel Foundaton. Theory Group, Mcrosoft Research, E-Mal: mrrokn@mcrosoft.com. Dept. of Computer Scence, RWTH Aachen, E-Mal: skopalk@cs.rwth-aachen.de.

2 1 Introducton Computatonal game theory has lead already to many mportant nsghts for understandng Nash equlbra n systems under the control of self-nterested agents. Promnent results for the qualty of Nash equlbra nclude bounds on the prce of anarchy, whch s the rato between the worst Nash equlbrum and the global optmal soluton [21, 10, 24, 19], and for computatonal complexty [12, 11, 6]. Intutvely, a hgh prce of anarchy a system ndcates that t requres a coordnaton mechansm to acheve good performance. On the other hand, low prce of anarchy does not necessarly mply good performance of the system [18, 15]. One man reason for ths phenomenon s that n many games wth selfsh players actng n a decentralzed fashon, the repeated selfsh behavor of the players may not lead to a Nash equlbrum [15]. Moreover, the convergence rate mght be very slow [12]. Ths motvates the queston of whether selfsh players actng n a decentralzed fashon, converge to approxmate solutons n a reasonable amount of tme [18, 15, 8, 5]. In ths paper, we address ths queston for the general class of congeston games, whch are used to model routng, network desgn and other resource sharng scenaros n dstrbuted systems [21, 2, 14]. We also consder other potental games. In a congeston game there are n players and a set of resources. The strategy of a player conssts of a subset of these resources. Each resource possesses a delay functon d e, whch depends on the number of players usng ths resource and the delay(cost) of each player s the sum of the delays assocated wth hs selected resources. Rosenthal [20] prove that every congeston game has a pure Nash equlbrum, by showng a potental functon that s strctly decreasng after any strct mprovement of a player. Thus, ths property, shows that the natural Nash Dynamcs, n whch players teratvely play best response converges to a pure Nash Equlbrum. It has been shown that the problem of fndng pure Nash equlbra n congeston games s PLS-complete [12] even wth lnear latency functons [1]. Ths result holds even for symmetrc congeston games. These results mply examples of congeston games and ntal states from whch n the Nash dynamcs all Nash equlbra have dstance exponental n the number of players n. For ths reason, Chen and Snclar [7] study convergence to approxmate equlbra n symmetrc congeston games. They consder α-nash equlbra whch are states n whch no player can decrease hs cost by more than a factor of 1 α by unlaterally changng hs strategy. They also nvestgate α-nash dynamcs, n whch we only allow moves that mprove the cost of a player by a factor of more than 1 α. For symmetrc congeston games where each resource delay satsfes the bounded jump assumpton, they show that convergence to α-nash equlbra occurs wthn a number of steps that s polynomal n the number of players [7]. Recently, Skopalk and Vöckng [23] show examples of asymmetrc congeston games wth n players and O(n) resources and bounded jump delay functons such that there are states that have dstance exponental n the number of players n to all α-nash equlbra. Thus, the results for convergence to α-nash equlbra appear n [7] cannot be extended to asymmetrc congeston games. These negatve results motvate the study of convergence to approxmate solutons n asymmetrc congeston games. Mrrokn and Vetta [18] and Goemans et. al [15] study convergence to approxmate solutons n load balancng games, vald-utlty games, and congeston games. Chrstodoulou et al. [8] study the speed of convergence to approxmate solutons n potental games. They show that after a constant number of rounds of α-nash dynamcs the approxmaton factor of the soluton 1

3 mght be a superconstant. They also show that the approxmaton factor of a state after one round of Nash dynamcs s Θ(n). Our Results. In ths paper, we study the convergence of unrestrcted α-nash dynamcs to an approxmately optmal soluton n dfferent classes of asymmetrc congeston games and other potental games. We consder the unrestrcted α-nash dynamcs wth a lveness property that no player s prevented from movng for arbtrarly many steps. We consder asymmetrc congeston games wth resources satsfyng a bounded jump condton. For γ 1, a resource e satsfes the γ-bounded jump condton, f ts delay functon satsfes d e (t + 1) γd e (t) for all t 1. Ths condton s rather weak. In partcular, a resource wth d e (t) = γ t satsfy the γ-bounded jump condton. We show that for asymmetrc congeston games wth the bounded jump condton, the unrestrcted α-nash dynamcs wth a lveness property converges to approxmate solutons wth approxmaton rato of arbtrarly close to the prce of anarchy n tme that s polynomal n the number of players (For detals, see Theorem 3.3 and Remark 3.5). Ths result mples fast convergence to good approxmate solutons for the nterestng case of polynomal latency functons of degree d. These results are n contrast to the negatve results that appear n [12, 1, 23]. We also prove that the same polynomal convergence result does not hold for (exact) best-response dynamcs, showng the α-nash dynamcs s requred 3.4. We extend ths result for other potental games. We frst extend ths result to weghted congeston games wth lnear delay functons for whch we show that any unrestrcted α-nash dynamcs satsfyng the lveness property converges to a ( ɛ)-approxmate soluton after polynomal number of α-moves. Furthermore, we extend the results to proft maxmzng potental games ncludng cut games (also called party afflaton games) and market sharng games. In these games, players maxmze ther payoff nstead of mnmzng ther cost. For these games, we need to assume that players play a best-response α-moves,.e., an α-move that has the maxmum possble payoff. For both of these games, we show that any unrestrcted α-nash best-response dynamcs satsfyng the lveness property converges to a (2 + ɛ)-approxmate soluton after polynomal number of α-moves. Ths s n contrast to the negatve result of Chrstodoulou et al [8] for cut games that shows that convergence tme of (exact) best-response dynamcs to a constant-factor soluton n ths game s exponental. Related Work. The study of convergence of Nash dynamcs s related to local search problems, and PLS-complete problems ntroduced by Johnson et. al [16]. Fabrkant et al [12] proved that fndng a pure Nash equlbrum of network congeston games s PLS-complete. Ackermann et al [1] showed that the same problem for network congeston games wth lnear latency functons s PLS-complete as well. Skopalk and Vöckng [23] showed that fndng an approxmate Nash equlbrum n congeston games s also PLS-complete. Mrrokn and Vetta [18] ntated the study of convergence to approxmate solutons n the context of load balancng games and vald-utlty games [24]. They consder coverng walks of best responses n whch each player has at least one chance to play n each round. Motvated by studyng the Nash dynamcs and convergence to approxmate solutons, Goemans et al [15] ntroduced snk equlbra, and proved that n weghted congeston games, random Nash dynamcs converges to a constant-factor approxmately optmal soluton n expected polynomal tme. However, they do not provde any bound for the convergence tme of determnstc unrestrcted Nash dynamcs. In fact, n Theorem 3.4, we show a lower bound for determnstc Nash dynamcs for these games, showng that the above result only holds for random Nash 2

4 dynamcs. Chrstodoulou et al [8] showed a tght bound of Θ(n) for the approxmaton factor of the soluton after one round of α-nash dynamcs n congeston games wth lnear latency functons. They also showed that for congeston games wth lnear latency functons, after a constant rounds of Nash dynamcs, players may not converge to an approxmate soluton. Here, we show that after a polynomal rounds of α-nash dynamcs, players converge to a constant-factor soluton. Chekur et al [5] and Charkar et al [4] studed convergence of Nash dynamcs to approxmate solutons n network cost sharng games. The study of α-moves for convergence to approxmate solutons has been also consdered by Chrstoulou [8] n the context of cut games. They show that for any constant α (and not for an α = o(1)) after one round of α-moves of players n a cut game, the value of the cut s a constant-factor approxmate soluton. Ther proof does not handle the convergence of unrestrcted dynamcs. For a more complete lst of results n these areas, see Mrrokn [17]. Cut Games (or party afflaton games) are potental games defned on an edge-weghted graph [12, 22, 8]. Nash dynamcs for these games correspond to the local search algorthm for the Max-Cut problem. Schaffer and Yannakaks [22] proved that fndng a Nash equlbrum n ths game s PLS-complete. Chrstodoulou et. al [8] showed an exponental lower bound for the convergence tme of (exact) best-response dynamcs to constant-factor approxmate solutons n these games. In contrast, we show polynomal convergence of α-nash best-response dynamcs n these games. Market sharng games are a specal case of proft maxmzng congeston games and vald-utlty games [24] that has been studed for the content dstrbuton n servce provder networks [14]. Mrrokn and Vetta [18] show that after one round of best responses n whch each player get exactly one chance to play best response, players reach an O(log n)- approxmate soluton. 2 Prelmnares 2.1 General Defntons Strategc games. A strategc game (or a normal-form game) Λ =< N, (Σ ), (u ) > has a fnte set N = {1,..., n} of players. Player N has a set Σ of actons (or strateges). We call a game symmetrc f all players share the same set of strateges, otherwse we call t asymmetrc. The jont acton set s Σ = Σ 1 Σ n and a jont acton S Σ s also called a profle or strategy profle. The payoff functon of player s u : Σ R, whch maps the jont acton S Σ to a real number. Let S = (S 1,..., S n ) denote the profle of actons taken by the players, and let S = (S 1,..., S 1, S +1,..., S n ) denote the profle of actons taken by all players other than player. Note that S = (S, S ). An mprovement move S for a player n a profle S s a move for whch u (S, S ) u (S). A best response move S for a player n a profle S s an mprovement move that has the maxmum payoff. In ths paper, we consder two types of games: cost mnmzng games and proft maxmzng games. In cost mnmzng games, each player wants to mnmze the cost c (S) = u (S) n strategy profle S. Ths type of games nclude congeston games wth polynomal latency functons. In proft maxmzng games, each player wants to maxmze the proft p (S) = u (S) n strategy profle S. Ths type of games nclude market sharng games and cut games. Nash equlbra (NE): A jont acton S Σ s a pure Nash equlbrum f no player N can beneft from unlaterally devatng from hs acton to another acton,.e., N S 3

5 Σ : u (S, S ) u (S). We can also defne α-nash equlbra as follows. For 1 > α > 0, a state S s an α-nash equlbrum f for every player, c (S, S ) (1 α)c (S) for all S Σ. State graph. Gven any game Λ, the state graph G(Λ) s an arc-labelled drected graph as follows. Each vertex n the graph represents a jont acton S. There s an arc from state S to state S wth label ff there exsts player and acton S Σ such that S = (S, S ),.e., S s obtaned from S by a move of a sngle player that mproves hs payoff from S to S. Exact potental games. A game s called an exact potental game f there s a functon φ such that for any edge of the state graph (S, S ) wth devaton of player, we have φ(s ) φ(s) = u (S ) u (S). We denote the mnmal potental of the game by φ. Socal functon. Gven any game Λ, n order to measure the performance of strategy profles of players, we defne a socal functon for any strategy profle S. Ths socal functon for mnmzng cost games s denoted by cost(s) and we denote by OP T (Λ) the mnmal socal cost of a game Λ..e., OP T (Λ) = mn S Σ cost Λ (S). We denote by cost Z (S), the sum of the payoffs of the players n the set Z, when the game Λ s clear from the context,.e., cost Z (S) = Z c (S). For proft maxmzng games, the socal functon s denoted by proft(s) and we denote by OP T (Λ) the maxmal socal cost of a game Λ..e., OP T (Λ) = max S Σ proft Λ (S). We denote by proft Z (S), the sum of the payoffs of the players n the set Z, when the game Λ s clear from the context,.e., proft Z (S) = Z p (S). α-nash dynamcs. For 0 < α 1, ths dynamcs allows only α-moves of the players, where α-move of a player s a move that mproves hs cost by a factor more than 1 α,.e., f player moves from acton S to acton S then c (S, S ) < (1 α)c (S). We consder the unrestrcted α-nash dynamcs wth lveness property, whch allows an adversary to order the players moves n each round as long as every player has at least one chance to move n each round. The lveness property requrement s that n each nterval of length T every player appears at least once. For proft maxmzng games, an α-move s a move that ncreases the payoff by a factor more than 1+α. In these games, we study α-nash dynamcs under the assumpton that players play a best response when they get a chance. We call ths dynamcs, the α-nash best-response dynamcs. Also, an α-nash best-response move s a best response α-move. α-nash best-response dynamcs s also consdered by [17, 8] (called 1 + α-greedy players). The lveness property have been consdered by [7] and [18]. Mrrokn and Vetta [18] call a round n whch each player gets at least a chance to move, a coverng walk. Nce Potental Games. Consder a potental game Λ. Let S be a profle of the players and let S be the best response for any player. For each player, let (S) = c (S) c (S, S ) and let (S) = (S). Also, for any set of players Z, let Z (S) = Z (S). We may drop the (S) part of the terms and denote these terms by and Z, f the profle s determned clearly n the context. Defnton 2.1 An exact potental game Λ wth potental functon φ s β-nce ff for any state S, t holds that () cost(s) βop T (Λ) + 2 (S), and () φ(s) cost(s). We consder exact potental games, whch are β-nce, where β s the prce of anarchy of the game. We show that the α-nash dynamcs converges n polynomal tme to a state S wth (S) that s arbtrarly close to zero. Therefore the approxmaton rato of the soluton S s arbtrarly close to the prce of anarchy. Bounded Jump Property. 4

6 Defnton 2.2 (γ-bounded Jump). For any value γ 1, a game Λ satsfes the γ-bounded jump condton f for every profle S and every player wth mprovement move S, t holds that 1. c j (S) c j (S, S ) c (S). 2. for every mprovement acton S j of player j, t holds c j(s {,j}, S, S j ) c j(s j, S j ) γ c (S, S ). Lemma 4.6 shows that congeston games wth resources that satsfy the γ-bounded jump condton, studed n [7, 23], satsfy the γ-bounded jump property accordng to defnton 2.2. Therefore t s suffcent to assume the bounded jump property accordng to defnton 2.2 for ths class of games. ε-approxmate α-equlbra. Gven a strategy profle S, we call the set of players that cannot make an α-move, α-equlbrum players. Defnton 2.3 A state S s an ε-approxmate α-equlbrum f O (S) ε cost(s) where O s the set of players that can play an α-move. 2.2 Cost Mnmzng Congeston Games In ths part, we defne cost mnmzng congeston games. Snce the focus of ths paper s on these games, and for brevty, we call these games, congeston games. Unweghted Congeston Games. An unweghted congeston game s defned by a tuple < N, E, (Σ ) N, (d e ) e E > where E s a set of facltes, Σ 2 E the strategy space of player, and d e : N Z a delay functon assocated wth resource e. For a jont acton S, we defne the congeston n e (S) on resource e by n e (S) = { e S }, that s n e (S) s the number of players that selected an acton contanng resource e n S. The cost c (S) of player n a jont acton S s c (S) = u (S) = e S d e (n e (S)). [20] showed that every congeston game possesses at ne(s) =1 d e (). least one pure Nash equlbrum by consderng the potental functon φ(s) = e Weghted Congeston Games. In weghted congeston games, player has weghted demand w. We denote by l e (S), the congeston(load) on resource e n a state S,.e., l e (S) = e S w. The cost of a player n a state S s c (S) = e S d e (l e (S)). The total cost s the weghted sum cost(s) = N w c (S) = e E l ed e (l e (S)). Note that congeston games s a specal case of weghted congeston games wth w = 1 for every player. [13] showed that every weghted congeston game wth lnear latency functons possesses at least one pure Nash equlbrum by consderng a potental functon equvalent to φ(s) = 1 ( 2 e l e(s)d e (l e (S)) + e S w d e (w ) ). We use the fact that ths potental functon s an exact potental functon f the cost of a player n a state S s w c (S). To smplfy the presentaton of the results we assume that the cost of any player n a state S s c (S) = w c (S). 2.3 Proft Maxmzng Congeston Games Cut Games. Cut game s a proft maxmzng congeston game that s defned on an edgeweghted undrected graph G(V, E), wth n vertces and edge weghts w : E(G) Q +. We assume that G s connected, smple, and does not contan loops. For each v V (G), let deg(v) 5

7 be the degree of v, and let Adj(v) be the set of neghbors of v. Let also w v = u Adj(v) w uv. A cut n G s a partton of V (G) nto two sets, T and T = V (G) T, and s denoted by (T, T ). The value of a cut s the sum of edges between the two sets T and T,.e proft(s) = v T,u T w uv. The cut game on a graph G(V, E), s defned as follows: each vertex v V (G) s a player, and the strategy of v s to choose one sde of the cut,.e. v can choose S v = 1 or S v = 1. A strategy profle S = (S 1, S 2,..., S n ), corresponds to a cut (T, T ), where T = { S = 1}. The payoff of player v n a strategy profle S, denoted by p v (S), s equal to the contrbuton of v v V p v(s). n the cut,.e. p v (S) = :S S v w v. It follows that the cut value s equal to 1 2 If S s clear from the context, we use p v nstead of p v (S) to denote the payoff of v. We denote the maxmum value of a cut n G, by c(g). These games are exact potental games, and the potental functon s φ(s) = proft(s) = v T,u T w uv. Market Sharng Games. A market sharng game s defned by a tuple < N, M, (Σ ) N, (v j ) j M > where M s a set of markets, Σ 2 M the strategy space of player, and v j the value of market j. For a jont acton S, we defne the congeston n j (S) on market j by n j (S) = { j S }, that s n j (S) s the number of players that selected an acton contanng market j n S. The payoff v j n j (S) p (S) of player n a jont acton S s p (S) = u (S) = j S. Market sharng games are maxmzaton congeston games wth potental functon φ(s) = 1 nj (S) v j log n j M =1. The socal functon s the sum of payoff of players or the total value of the market satsfed,.e., proft(s) = N p (S) = j N S v j. 3 Convergence of the α-nash Dynamcs In ths secton, we consder the unrestrcted α-nash dynamcs wth a lveness property for nce exact potental games satsfyng the bounded jump property. Throughout ths secton, let C be the set of α-equlbrum players and let O be the set of all other players,.e., the players that can make an α-move. Frst we observe the followng smple lemma. Lemma 3.1 If a state S s n an ε-approxmate α-equlbrum, then (S) (α + ε)cost(s). Proof: Snce C s the set of α-equlbrum players, C (S) α cost C (S). Thus, (S) = C (S) + O (S) (α + ε)cost(s). As a warmup example, we consder a (restrcted) basc dynamcs, where n each step, among all players that can play an α-move, we choose the player wth the maxmum absolute mprovement, and let hm move. Lemma 3.2 Let 1 8 > δ α. Consder an exact potental game Λ that satsfes the nce property and any ntal state S nt (. The basc dynamcs ) generates a profle S wth cost(s) β(1 + O(δ))OP T (Λ) n at most O n δ log( φ(s nt) φ ) steps. Proof: Consder a step that starts wth profle S. Let ε O = O (S)/cost(S). By defnton 2.3 the state S s an ε O -approxmate α-equlbrum. Now, there are two cases: Case 1: ε O δ. It follows from Lemma 3.1 that (S) (α + ε O )cost(s) (α + δ)cost(s). Hence, by defnton 2.1, the dynamcs reached β(1 + 4(α + δ))-approxmaton of the optmal cost. 6

8 Case 2: ε O > δ. It follows that O (S) > δ cost(s). Hence, there exsts a player j O such that j (S) > δ n cost(s). Thus, j(s) > δ nφ(s), snce φ(s) cost(s). Therefore the potental gan s at least δ n φ(s). Let φ(t) denote the potental n step t. Then, φ(t) φ(s nt)(1 δ n )t. Snce φ(t) φ, the upper bound on the number of steps follows. The above basc Nash dynamcs requres some coordnaton that chooses the player wth the maxmum gan at each step. Now we show smlar results for unrestrcted Nash dynamcs. Theorem 3.3 Let 1 8 > δ 4α. Consder an exact potental game Λ that satsfes the nce property and the γ-bounded jump condton. For any ntal state S nt, the unrestrcted ( α-nash ) dynamcs generates a profle S wth cost(s) β(1+o(δ))op T (Λ) n at most O γn αδ log( φ(s nt) φ ) T steps. Before provng Theorem 3.3 we pont out that the α-nash dynamcs s necessary for polynomal tme convergence to nearly optmal solutons for nce exact potental games satsfyng the bounded jump property, that s, we show that even after exponentally many steps, the unrestrcted exact Nash dynamcs wth a lveness property for asymmetrc congeston games wth lnear delay functons may generate strategy profles whose socal cost s far from the optmal soluton. Theorem 3.4 There exsts an exact potental game Λ that satsfes the nce property and the γ-bounded jump condton, and an ntal state S nt from whch the unrestrcted exact bestresponse dynamcs generates a profle S wth cost(s) Ω( n log n )OP T after an exponental number of steps. In partcular, ths holds for a congeston game wth lnear latency functons. The proof of ths theorem s based on constructng a long nvolved example wth several components, and s left to Secton A of the appendx. We now present the proof of Theorem Proof: (of Theorem 3.3) Let α = 4α. It s suffcent to consder the case that the players are not n a δ-approxmate α -equlbrum, snce otherwse t follows from Lemma 3.1 and Defnton 2.1 that the dynamcs reached a β(1 + 4(α + δ))-approxmaton of the optmal cost. We show that n each nterval of T steps the potental decreases by a factor of at least αδ 4γn. Let S 0, S 1,..., S T denote the jont actons of the players n tmes 0, 1,..., T of ths nterval respectvely. Snce S 0 s not a δ-approxmate α -equlbrum, there exsts a player wth an mprovement α -move. Consder player j wth the maxmum absolute mprovement α -move and let S j be hs best response. Recall that j(s 0 ) = c j (S 0 ) c j (S j 0, S j ). Let j = j(s 0 ) and let t be the frst tme n ths nterval that player j s allowed to move. We denote by U the set of tmes before tme t, where players made α-moves and we denote by w(t) the player that moved at tme t for each t U. Let A = t U c w(t)(s t ) be the sum of the costs of the movng players when they make ther moves. Now, we consder two cases: Case 1: A j 4γ. By the frst condton of the bounded jump property, we have for each t U Summng over all tmes t U, we obtan: c j (S t ) c j (S t+1 ) c w(t) (S t ). (1) c j (S 0 ) c j (S t ) t U c w(t) (S t ) = A j 4γ j 4. (2) 7

9 Where the frst nequalty follows snce the sum of the left hand sde of equaton (1) telescopes. Smlarly, by the second property of the bounded jump assumpton, we obtan By summng nequaltes (2) and (3), we get c j (S t j, S j) c j (S 0 j, S j) γ A γ j 4γ j 4. (3) c j (S t ) c j (S t j, S j) c j (S 0 ) c j (S 0 j, S j) j 2 = j j 2 = j 2. (4) By the second property of the bounded jump assumpton we also get Hence, c j (S t ) c j (S 0 ) + γ A c j (S 0 ) + γ j 4γ = c j(s 0 ) + j 4. (5) c j (S t ) c j (S 0 ) + j 4 < j α + j 4 < 2 j 4α = j 2α. Where the second nequalty follows from the fact that j s the mprovement of player j when makng hs best response, whch s an α -move n step 0. Thus, α c j (S t ) < j. As a result, usng ths nequalty and nequalty (4), we get α c j (S t ) < c j (S t ) c j (St 2 j, S j ). Therefore, player j can make an α-move at tme t and decrease the potental φ by at least α c j (S t ) α j 2 αδ 2n φ(s0 ). Case 2: A > j 4γ. Snce A s the sum of the costs of players makng an α-move when makng the move, these players decrease the potental φ by at least αa > α j 4γ αδ 4γn φ(s0 ). Let φ() denote the potental n round. Then, n both cases φ() φ(s nt )(1 αδ 4γn ). Snce φ() φ, the upper bound on the number of steps follows. Remark 3.5 The above theorem shows that we reach a state wth cost at most β(1 + O(δ)) of the optmum after polynomal number of α-moves. Eventhough after ths state the cost of solutons can ncrease, t follows from the proof of the theorem that the number of states n whch the cost of the soluton s more than a β(1 + O(δ))-approxmaton s at most O( γn αδ log( φ(s nt) φ )T ). In addton, snce the potental functon s always decreasng after any α-move, the cost can ncrease by a factor of at most cost(s) φ(s) rato cost(s) φ(s). It s not hard to show that the for any strategy profle n congeston games wth polynomal delay functons of degree d s at most O(d) and for weghted congeston games wth lnear functons s at most O(1). As a result, for both type of congeston games that we consder n Secton 4, the cost of any state after a polynomal number of steps reach a constant-factor approxmate soluton and remans wthn a constant factor of the optmal soluton. 4 Congeston Games In ths secton we consder weghted congeston games wth lnear latency functons and congeston games wth lnear and polynomal latency functons. 8

10 4.1 Lnear Latency Functons In ths secton we consder weghed and unweghted congeston games wth lnear latency functons. Specfcally d e (x) = a e x + b e for each resources e E, where a e and b e are nonnegatve reals. For smplcty we only consder the dentty functon d e (x) = x. It s easy to verfy that all the proofs work for the general case as well Weghted Congeston Games We frst show that weghted congeston games wth lnear latency functons are β-nce accordng to defnton 2.1 wth β = Lemma 4.1 Congeston games wth lnear latency functons are β-nce potental games wth β = Next we show that weghted congeston game wth lnear delay functons satsfy the 1-bounded jump condton. Lemma 4.2 Let Λ be a weghted congeston game wth lnear delay functons. Then, the game Λ satsfes the 1-bounded jump condton accordng to defnton 2.2. The proof of the above two lemmas can be found n the appendx. Theorem 3.3 and Lemmas 4.1, 4.2 yeld the followng corollary. Corollary 4.3 Let 1 8 > δ α. Consder a weghted congeston game Λ wth lnear latency functons and any ntal state S nt. The unrestrcted α-nash dynamcs wth ( lveness property ) generates a profle S wth cost(s) (1+O(δ))OP T (Λ) n at most O n αδ log( φ(s nt) φ ) T steps Unweghted Congeston Games We frst show that congeston games wth lnear latency functons are β-nce accordng to defnton 2.1 wth β = 2.5. In the proof of ths lemma, we use two Lemmas whch appear n [9], and are stated n the appendx. Lemma 4.4 Congeston games wth lnear latency functons are β-nce potental games wth β = 2.5. Next we show that unweghted congeston games wth resources that satsfy the γ-bounded jump condton, satsfy the γ-bounded jump condton accordng to defnton 2.2. Defnton 4.5 (resource γ-bounded jump). Resource e satsfes the γ-bounded jump condton f ts delay functon satsfes d e (x + 1) γ d e (x) for every x 1, for γ 1. Lemma 4.6 Let Λ be a congeston game wth nonnegatve, non-decreasng delay functons n whch every resource has γ-bounded jump. Then, the game Λ satsfes the γ-bounded jump condton accordng to defnton

11 The proof of the above two lemmas can be found n the appendx. Theorem 3.3, Lemmas 4.4, 4.6 and the fact that resource wth lnear latency functon has 2-bounded jump, yeld the followng corollary. Corollary 4.7 Let 1 8 > δ α. Consder a congeston game Λ wth lnear latency functons and any ntal state S nt. The unrestrcted α-nash dynamcs ( wth lveness property ) generates a profle S wth cost(s) 2.5(1 + O(δ))OP T (Λ) n at mosto n αδ log( φ(s nt) φ ) T steps. 4.2 Polynomal Latency Functons In ths secton, we consder congeston games wth polynomal latency functons of degree d. We show that congeston games wth polynomal latency functons are β-nce accordng to defnton 2.1 wth β = d d(1 o(1)). Prce of anarchy results whch appear n [9] mply that for β = d d(1 o(1)) and for every profle S equaton (??) n defnton 2.1 holds. Lemma 4.8 Congeston games wth polynomal latency functons of degree d are β-nce potental games wth β = d d(1 o(1)). Theorem 3.3, Lemmas 4.8, 4.6 and the fact that resource wth polynomal of degree d latency functon has 2 d -bounded jump, yeld the followng corollary. Corollary 4.9 Let 1 8 > δ α. Consder a congeston game Λ wth polynomal latency functons of degree d and any ntal state S nt. The unrestrcted ( dynamcs generates ) a profle S wth cost(s) d d(1 o(1)) (1 + O(δ))OP T (Λ) n at most O 2d n αδ log( φ(s nt) φ ) T steps. 5 Proft Maxmzng Congeston Games In ths secton, we extend the results for cost mnmzng congeston games to proft maxmzng congeston games. We frst defne some prelmnares for these games. Consder an exact potental game Λ. Let S be a profle of the players and let S be a best response strategy for player n strategy profle S. The payoff of player n strategy profle S s denoted by p (S) and each player wants to maxmze ts payoff. In ths settng, for each player, let (S) = p (S, S ) p (S) and let (S) = (S). Defnton 5.1 An exact potental game Λ wth potental functon φ s β-nce ff for any state S t holds that () β (proft(s) + (S)) OP T (Λ), and () φ(s) proft(s). Defnton 5.2 (γ-bounded Jump). Consder any profle S and any player wth mprovement move S. Then, for every player j the followng propertes hold: 1. p j (S, S ) p j(s) p (S, S ) 2. for every mprovement acton S j of player j, t holds p j(s j, S j ) p j(s {,j}, S, S j ) γ p (S, S ) 10

12 5.1 Convergence of proft maxmzng games Smlar to the proof of Theorem 3.3 for convergence of unrestrcted α-nash dynamcs n cost mnmzng games, we can prove the followng general theorem for convergence tme of the α-nash best-response dynamcs proft maxmzng games. Theorem 5.3 Let 1 8 > δ 4α. Consder an exact potental game Λ that satsfes the nce property and the bounded jump condton. For any ntal state S nt the unrestrcted α Nash best-response dynamcs( wth lveness property ) generates a profle S wth β(1+o(δ))prof t(s) OP T (Λ) n at most O γn αδ log( φ φ(s nt ) ) T steps. The proof of ths theorem s very smlar to that of Theorem 3.3 and s left to Secton C n the appendx. 6 Cut Games and Market Sharng Games Usng Theorem 5.3, n order to prove polynomal convergence of α-nash best-response dynamcs n cut games and market sharng games, we can show that both of these games satsfy the 2-nce and 1-bounded jump propertes. The proofs of these propertes can be found n Sectons D and E of the appendx. Corollary 6.1 Let 1 8 > δ 4α. Consder a cut game or a market sharng game Λ wth and any ntal state S nt. The unrestrcted α Nash best-response dynamcs ( wth lveness ) property 1 generates a profle S wth proft (2+O(δ)) OP T (Λ) n at most O n αδ log( φ φ(s nt ) ) T steps. References [1] H. Ackermann, H. Rögln, and B. Vöckng. On the mpact of combnatoral structure on congeston games. In Proceedngs of the 47th Annual IEEE Symposum on Foundatons of Computer Scence (FOCS), pages , [2] Ellot Anshelevch, Anrban Dasgupta, Jon M. Klenberg, Éva Tardos, Tom Wexler, and Tm Roughgarden. The prce of stablty for network desgn wth far cost allocaton. In FOCS, pages , [3] B. Awerbuch, Y. Azar, and A. Epsten. The prce of routng unsplttable flow. In Proc. 37th ACM Symp. on Theory of Computng, pages 57 66, [4] M. Charkar, C. Matteu, H. Karloff, S. Naor, and M. Saks. Best response dynamcs n multcast cost sharng Unpublshed Manuscrpt. [5] C. Chekur, J. Chuzhoy, L. Lewn, S. Naor, and A. Orda. Non-cooperatve multcast and faclty locaton games. In ACM EC, pages 72 81, [6] X Chen and Xaote Deng. Settlng the complexty of two-player nash equlbrum. In Proc. 47nd IEEE Symp. on Found. of Comp. Scence, pages ,

13 [7] Steve Chen and Alstar Snclar. Convergence to approxmate nash equlbra n congeston games. In Proc. 18th ACM-SIAM Symp. on Dscrete Algorthms (SODA), [8] G. Chrstodolou, V. S. Mrrokn, and A. Sdropolous. Convergence and approxmaton n potental games. In Proceedngs of the 18th Annual Symposum on Theoretcal Aspects of Computer Scence (STACS), [9] G. Chrstodoulou and E. Koutsoupas. The prce of anarchy of fnte congeston games. In 37th ACM Symposum on Theory of Computng, pages 67 73, Baltmore, MD, USA, May [10] Artur Czumaj and Berthold Vöckng. Tght bounds for worst-case equlbra. In SODA, pages , [11] Constantnos Daskalaks, Paul W. Goldberg, and Chrstos H. Papadmtrou. The complexty of computng a nash equlbrum. In Proceedngs of the thrty-eghth annual ACM symposum on Theory of computng, pages 71 78, [12] Alex Fabrkant, Chrstos Papadmtrou, and Kunal Talwar. The complexty of pure nash equlbra. In Proceedngs of the thrty-sxth annual ACM symposum on Theory of computng, pages , [13] D. Fotaks, S. Kontoganns, and P. Spraks. Selfsh unsplttable flows. Theoretcal Computer Scence, Specal Issue on ICALP 2004, 348: , [14] M. Goemans, L. L, V.S.Mrrokn, and M. Thottan. Market sharng games appled to content dstrbuton n ad-hoc networks. In MOBIHOC, [15] M. Goemans, V. S. Mrrokn, and A. Vetta. Snk equlbra and convergence. In FOCS, [16] D. Johnson, C.H. Papadmtrou, and M. Yannakaks. How easy s local search? Journal of Computer and System Scences, 37:79 100, [17] V. S. Mrrokn. Approxmaton Algorthms for Dstrbuted and Selfsh Agents. Massachusetts Insttute of Technology, [18] V.S. Mrrokn and A. Vetta. Convergence ssues n compettve games. In RANDOM- APPROX, pages , [19] Chrstos Papadmtrou. Algorthms, Games, and the Internet. In Proceedngs of 33rd STOC, pages , [20] R. W. Rosenthal. A class of games possesng pure-strategy nash equlbra. Internatonal Journal of Game Theory, 2:65 67, [21] T. Roughgarden and Eva Tardos. How bad s selfsh routng? Journal of the ACM, 49(2): , [22] A. Schaffer and M. Yannakaks. Smple local search problems that are hard to solve. SIAM journal on Computng, 20(1):56 87,

14 [23] A. Skopalk and B. Vöckng. Inapproxmablty of convergence n congeston games Unpublshed Manuscrp. [24] A. Vetta. Nash equlbra n compettve socetes, wth applcatons to faclty locaton, traffc routng and auctons. In Proceedngs of 43rd Symposum on Foundatons of Computer Scence (FOCS), pages , A Convergence of the exact Nash Dynamcs Here, we consder the unrestrcted exact Nash dynamcs wth a lveness property for nce exact potental games satsfyng the bounded jump property. We show that even after exponentally many steps, the exact Nash dynamcs for lnear congeston games may generate strategy profles whose socal cost s far from the optmal soluton. Theorem A.1 After exponentally many steps of the exact Nash dynamcs of congeston games wth lnear latency functons, we may reach solutons whose cost s more than Ω( n log n )OP T. Proof: We construct a congeston game Γ n consstng of four types of players: () counter players representng a bnary counter, () congeston players consstng of a group of A-players and one C-player, () bt players denoted by B-players, and (v) trgger players consstng of T -players, R-players, and a Q-player. For a large nteger number M, we show that there exsts a best-response sequence of exponental length n whch the the counter players have delays of at least M. However, each of the counter players has a strategy wth one unque extra resource wth the delay functon l(x) = 5M n x. But each tme a counter player has the chance to change hs strategy, hs extra resource s congested by n congeston players. In each step of the counter, the congeston players successvely allocate all extra resources of the counter players. Thus, after each step, every counter player gets a chance to devate but t does not have an ncentve to change to ts extra resource. However, n the optmal soluton, the counter players can devate to ther extra strategy that has delay of M n. The cost of the optmal soluton s domnated by the cost of the n congeston players that have delay of O(log n)m. Thus, even after an exponentally long sequence of best response, the socal cost n s at least Ω( n log n )OP T = Ω( n log n )OP T. Frst we descrbe the hgh-level dea of the constructon of the congeston game. A man component of ths game s an n-bt counter consstng of 4n counter players. Ths counter s smlar to exstng examples, e.g., [?]. The best response sequence of the counter players count downwards from 2 n 1 to 0. In each countng step of the counter, we gve all players a chance to move. For each player, there exsts an extra a-resource and an addtonal Alt strategy that conssts of only ths resource. If a player changes to that strategy and no other player s on that resource, he can decrease hs delay by a factor of k = n. However, there are k = n congeston players, denoted by A-players, that are on that resource makng a devaton of a counter player not favorable In addton, n each countng step of the counter, we let the C-player successvely occupy the c-resources of the bts of the counter that are 1. Ths prevents these bts from swtchng to 0. Thus, we can gve all counter players a chance 13

15 Player Strateges Resources Delays Int One r 1 4M + 8δ Zero r 2 M/M + 2δ Alt r 3 c r 5 a 1 M/M + 4δ M/2M M/M + 9δ 5M k x Change One r 2 M/M + 2δ r M Zero r 4 M/M + 3δ q 100n(n + 1)x r 1 6 M + 10δ 1 Alt a 2 5M k x Player Strateges Resources Delays Done One r 3 M/M + 4δ Zero r 4 M/M + 3δ Alt a 3 5M k x Reset One r 6 M + 10δ r M Zero r 5 M + 9δ Alt r 1 6 a 4 Fgure 1: The strateges of the four counter players for the -th bt. For delay functons, f they have x n the descrpton, the delay functon f(x) s gven n terms of the congeston x. If the resource s used only by one or two players, the delay functon s denoted by r 1 /r 2 where r 1 and r 2 are delays for congeston 1 and 2 respectvely. The scalng factor δ s at least 200n(n + 1). M + 10δ 1 5M k x to move. To ensure that all the A-players ple up on the same a-resources, we have log4n B- players that encode bnary numbers correspondng to the a-resources whch are to be allocated by the A-players. Ths way, we make sure that the only proftable devaton of the A-players s to allocate ths partcular a-resource, snce the delay of the other strateges s hgher due to the B-players. Fnally, the delay of the trgger players ncreases by the counter players n each step of the counter. Ther best responses ncrease the delays of the bt and congeston players such that the all the aforementoned strategy changes are best responses. We now descrbe the constructon of the game. We say a player s actvated or we actvate a player f we let hm play hs best response. We frst descrbe the detals of the n-bt counter. For each bt wth 1 n we have 4 players, see Fgure A for the complete descrpton of ther strateges and of the resources. We say the -th bt of the counter s 1 f player Int plays hs One-strategy and 0 otherwse. We construct a best-response sequence that consst of exponentally many rounds. We start the sequence wth all counter players playng One and we ensure that no counter player changes to Alt untl the end of the process. In each round, all players are actvated at least once. In order to prove the result, we make certan assumptons on the usage of certan resources by other players. We wll later show, that these assumptons hold throughout the process. Let x be the value of the counter and be the bt that flps from 1 to 0 when changng to x 1. Throughout the process, we make sure that the followng three condtons hold: 1. (A1) Each tme we actvate a player, the a-resource n hs Alt strategy s congested by k A-players. 2. (A2) If we actvate a player Int wth and Int plays One, then the resource b s allocated by another player. 14

16 3. (A3) The resource q s allocated by at most one other player. actvated player best response 1 Int +1,...,Int n unchanged 2 Done +1,...,Done n One f hs Int player plays One, Zero otherwse. 3 Change +1,...,Change n One 4 Reset,...,Reset n One 5 Int Zero 6 Trgger Zero 7 Reset 1,...,Reset 1 Zero 8 Int 1,...,Int 1 One 9 Done 1,...,Done 1 One 10 Trgger 1,...,Trgger 1 One 11 Done Zero 12 Trgger One Fgure 2: Sequence of actvatons and best responses of one round four counter players. Ths round corresponds to one step of the counter n whch the -th bt swtches from 1 to 0. The actvaton sequence of counter players s descrbed n Fgure A. Essentally, the players correspondng to bts greater than do not change ther strateges. The player Int changes to Zero. Ths results n a sequence of best responses n whch the less sgnfcant bts change to 1. Thus, under the above assumptons, the value of the counter decreases by exactly one n each step. Note, that durng each step one trgger player allocates the resource T and leaves t agan. We wll now descrbe the next component, a set of trgger players that make use of ths fact. The trgger component conssts of a large set of trgger players. We use these player to repeatedly ncrease and decrease the delay on some resources of the components that we descrbe later. Trgger players consst of one player Q and several T -players and R-players. Each trgger player has two strateges Wat and Trgger. See Fgure A for a detaled descrpton of the strateges and resources. Only player Q s nterested n resource q. Hence, condton A3 s satsfed. The best response for Q s Wat, f none of the players of the counter allocates the resource q, otherwse hs best response s Trgger. The best response for a R-player s Wat, f Q s on Wat, otherwse hs best response s Trgger. The best response for a T -player s Wat f Q s on Wat or the R-player wth the same ndex s on Trgger, otherwse hs best response s Trgger. Assume all players play Wat except player Q who plays Trgger. Then there s a bestresponse sequence that can be dvded to n + 1 segments. Each segment contans contans strategy profles S 1,..., S 6 wth S occurs before S +1. These strategy profles have the followng propertes: (S 1 ) For an arbtrary set I {0,..., n}, each resource n {t C } I s allocated by one T -player. Each resource n {t C } I s not allocated by any T -player. (S 2 ) No resource t Aj s allocated by any T -player. 15

17 Player Strateges Resources Delays Q Wat q 100n(n + 1)x Trgger r T 100n(n + 1) q j C 5x p j C 7x q A 5kx p A 7kx p B 0 7mx p B 1 7mx q B 0 5mx 5mx T j C for each 1 n and 1 j n + 1 Wat q j C 5x Trgger t C 2x r j C 5x R j C for each 1 n and 1 j n + 1 Wat p j C 7x Reset r j C 5x 3 T A for each 1 n + 1 Wat q A 5kx Trgger t Aj for all 1 j k 2x r A 5kx R A for each 1 n + 1 Wat p A 7kx Reset r A 5kx 3k T B 0for each 1 n + 1 Wat q B 0 5mx Trgger t B 0 for all 1 j m j 2x q B 1 r B 0 5mx R B 0for each 1 n + 1 Wat p B 0 7mx Reset r B 0 5mx 3k T B 1for each 1 n + 1 Wat q B 1 5mx Trgger t B 1 for all 1 j m j 2x r B 1 5mx R B 1for each 1 n + 1 Wat p B 1 7mx Reset r B 1 5mx 3k Fgure 3: Descrpton of all trgger players. The descrpton of delay functons f(x) s gven n terms of the congeston x. 16

18 (S 3 ) Each resource t B 0 j s allocated by one T -player. No resource t B 1 j s allocated by any T -player. (S 4 ) Each resource t B 1 j s allocated by one T -player. No resource t B 0 j s allocated by any T -player. (S 5 ) Each resource t Aj s allocated by one T -player. (S 6 ) No resource t C s allocated by any T -player. Note that durng each step of the counter, the resource q s allocated by a player of the counter. At the end of each step, t s no longer allocated by any player of the counter. We therefore assume that durng each step of the counter, there exst a best-response sequence of the trgger players consstng of n + 1 of the aforementoned segments. We are now ready to descrbe the remanng component consstng of bt players and congeston players. In ths part, we make use of sequence of strategy profles of the trgger players and show that the desred assumptons for the counter are met. There are m = log 4n bt players, denoted by B-players, that encode whch of the a-resources are to be allocated by k = n A-players. Furthermore, there s one C-player that allocates one of the c-resources. The complete descrpton of these players and ther strateges can be found n Fgure A. Player Strateges Resources Delays A j Zero r A 6mM + 5M k j for each 1 j k t Aj 2x (, l) a j 5M k x for each 1 n b 0 y f the y-th bt of 4 + l s 0 6Mx and 1 l 4 b 1 y f the y-th bt of 4 + l s 1 6Mx r 3 B j Zero b 0 j 3Mx for each 1 j m t B 0 j 2x One b 1 j 3Mx t B 1 j 2x C Zero r C M t C0 2x c Mx for each 1 n t C 2x Fgure 4: The A-, B-, and C-players and ther strateges. The descrpton of delay functons f(x) s gven n terms of the congeston x. Now we descrbe another condton nvolvng B-players that s met through the process. Condton C1 No other player except the B-players allocate any b-resource. Consder a strategy profle correspondng to S 3. In ths case, the best response of every B-player s One. Consder a strategy profle correspondng to S 4. Then the best response of every B-player s Zero. Thus, n each of the n + 1 segments of the best-response sequence of 17

19 the Trgger players, there s a best response sequence of the B-players that leads to a strategy profle n whch the B-player encode an arbtrary number. We now show, that gven a segment of best-responses by the Trgger players there s a best response sequence such that all k A-players allocate an arbtrary resource a j We start wth a strategy profle S 2 and actvate the A players n ascendng order startng wth A 1. Each player s best response s Zero. Ths satsfes condton C1. In the profle S 3 and S 4 the B- players encode the number 4 + j. In S 5 we actve the A players n ascendng order startng wth A 1. Each player s best response s (, j). Thus, n each step of the counter, the A-players successvely allocate n+1 arbtrary a-resources. We choose the a-resources such that condton A1 s satsfed. Fnally, there s exsts C-player that s used to make sure that condton A2 s satsfed. Gven a strategy profle S 1 n whch exactly one resource t C s not allocated by a trgger player, hs best response s the strategy. Thus, there s a best-response sequence n whch n every step of the counter the C-player successvely allocate n + 1 arbtrary c-resources. We choose the latency functons of c-resources n a way to make sure that condton A2 s satsfed. B Mssng Proofs of Secton 4 Proof of Lemma 4.2 The proof requre the followng two Lemmas. The frst lemma appears n [3] and the second lemma s a smple fact. Lemma B.1 Consder a weghted congeston game Λ wth lnear delay functons. Let S be any profle and S be a profle of the optmal soluton, then c (S, S ) cost(s) cost(s ) + cost(s ). Lemma B.2 For every par of nonnegatve ntegers x, y, f x 2 x+1+y, then x y. Proof: Let S be a profle of the optmal soluton and let S be any profle. Applyng Lemma B.1, we get c (S, S ) cost(s) cost(s ) + cost(s ). Note that cost(s) c (S, S ) (S), snce for any player wth best response S, c (S, S ) c (S, S ). Thus, by addng (S) to both sdes of the nequalty, we get cost(s) cost(s) cost(s ) + cost(s ) + (S). Let x = cost(s) cost(s ) and let y = (S) cost(s ). Now, we dvde the above nequalty by cost(s ) and express the result n terms of x and y. Thus, x 2 x+1+y. Applyng Lemma B.2, we get x y. Ths completes the proof of the Lemma. Proof of Lemma

20 Proof: Consder any profle S and any player wth mprovng acton S. We frst show the frst property n defnton 2.2. Consder any player j. Then, c j (S) c j (S, S ) w j l e (S) (l e (S) w ) e (S \S ) S j = w j w = w w j e (S \S ) S j w e (S \S ) S j e (S \S ) S j l e (S) w e S l e (S) = c (S). For the second property n defnton 2.2. Consder any player j wth acton S j. Then, c j (S {,j}, S, S j) c j (S j, S j) w j (l e (S j, S j) + w ) l e (S j, S j) e (S \S ) S j = w j w = w w j e (S \S ) S j w e (S \S ) S j l e (S, S ) w l e (S, S ) e (S \S ) S j = c (S, S ). e S Proof of Lemma 4.4 To present the proof of Lemma 4.4, we need requres the followng two Lemmas whch appear n [9]. Lemma B.3 Consder a congeston game Λ wth nonnegatve, non-decreasng delay functons. Let S be any profle and let S be a profle of the optmal soluton, then c (S, S ) n e (S )d e (n e (S) + 1). e E Lemma B.4 For every par of nonnegatve ntegers x, y, t holds x(y + 1) 5 3 x y2. Proof: Let S be a profle of the optmal soluton and let S be any profle. Applyng Lemma B.3, we obtan c (S, S ) e E n e(s )d e (n e (S) + 1). Applyng Lemma B.4, we get c (S, S ) ( 5 3 n e(s ) ) 3 n e(s) 2 = 5 n e (S ) n e (S) e E e E e E = 5 3 cost(s ) cost(s). Recall that cost(s) c (S, S ) (S), where S s the best response of any player. Thus, by multplyng the nequalty by 3/2, addng (S) to both sdes and rearrangng the terms, we get c (S, S ) 2.5 cost(s )+ (S) 2. Therefore, cost(s) 2.5 cost(s )+ 3 2 (S). 19