A Simple Graph-Theoretic Model for Selfish Restricted Scheduling

Transcription

1 A Simple Graph-Theoretic Moel for Selfish Restricte Scheuling Roert Elsässer 1, Martin Gairing 1, Thomas Lücking 1, Marios Mavronicolas 2, an Burkhar Monien 1 1 Faculty of Computer Science, Electrical Engineering an Mathematics, University of Paerorn, Fürstenallee 11, Paerorn, Germany {elsa,gairing,luck,m}@uni-paerorne 2 Department of Computer Science, University of Cyprus, P O Box 20537, Nicosia CY-1678, Cyprus mavronic@ucyaccy Astract In this work, we introuce an stuy a simple, graph-theoretic moel for selfish scheuling among m noncooperative users over a collection of n machines; however, each user is restricte to assign its unsplittale loa to one from a pair of machines that are allowe for the user We moel these oune interactions using an interaction graph, whose vertices an eges are the machines an the users, respectively We stuy the impact of our moeling assumptions on the properties of Nash equiliria in this new moel The main finings of our stuy are outline as follows: We prove, as our main result, that the parallel links graph is the est-case interaction graph the one that minimizes expecte makespan of the stanar fully mixe Nash equilirium among all 3-regular interaction graphs The proof employs a graph-theoretic lemma aout orientations in 3-regular graphs, which may e of inepenent interest We prove a lower oun on Coorination Ratio [16] a measure of the cost incurre to the system ue to the selfish ehavior of the users In particular, we prove that there is an interaction graph incurring Coorination Ratio Ω log n This oun is shown for pure Nash equiliria We present counterexample interaction graphs to prove that a fully mixe Nash equilirium may sometimes not exist at all Moreover, we prove properties of the fully mixe Nash equilirium for complete ipartite graphs an hypercue graphs 1 Introuction Motivation an Framework Consier a group of m non-cooperative users, each wishing to assign its unsplittale unit jo onto a collection of n processing (ientical) machines The users seek to arrive at a stale assignment of their jos for their joint interaction As usual, such stale assignments are moele as Nash equiliria [21], where no user can unilaterally improve its ojective y switching to a ifferent strategy We use a structure an sparse representation of the relation etween the users an the machines that exploits the locality of their interaction; such locality almost always exists in complex scheuling systems More specifically, we assume that each user has access (that is, finite cost) to only two machines; its cost on other machines is infinitely large, giving it no incentive to switch there The (expecte) cost of a user is the (expecte) loa of the machine it chooses Interaction with just a few neighors is a asic esign principle to guarantee efficient use of resources in a istriute system Restricting the numer of interacting neighors to just two is then a natural starting point for the theoretical stuy of the impact of selfish ehavior in a istriute system with local interactions Our representation is ase on the interaction graph, whose vertices an (unirecte) eges represent the machines an the users, respectively Multiple eges are allowe; however, for simplicity, our interaction multigraphs will e calle interaction graphs The moel of interaction graphs is interesting ecause it is the simplest, non-trivial moel for selfish scheuling on restricte parallel links In this moel, any assignment of users to machines naturally correspons to an orientation of the interaction graph (Each ege is irecte to the machine where the user is assigne) We will consier pure Nash equiliria, where each user assigns its loa to exactly one of its two allowe machines with proaility one; we will also consier mixe Nash equiliria, where each user employs a proaility istriution to choose etween its two allowe machines Of particular interest to us is the fully mixe Nash equilirium [20] where every user has strictly positive proaility to choose each of its two machines In the stanar fully mixe Nash equilirium, all proailities are equal to 1 2 It is easy to see that the stanar fully mixe Nash equilirium exists if an only if the (multi)graph is regular With each (mixe) Nash equilirium, we associate a Social Cost [16] which is the expecte makespan - the expectation of the maximum, over all machines, total loa on the machine Best-case an worst-case Nash equiliria minimize an maximize Social Cost, respectively For a given type of Nash equilirium such as the stanar fully mixe Nash equilirium, est-case an worst-case graphs among a graph class minimize an maximize Social Cost of Nash equiliria of the This work has een partially supporte y the DFG-Sonerforschungsereich 376 Massive Parallelität: Algorithmen, Entwurfsmethoen, Anwenungen, y the VEGA grant No 2/3164/23, y the IST Program of the European Union uner projects FLAGS (contract numer IST ) an DELIS (contract numer ), an y research funs at University of Cyprus

2 given type, respectively The assignment of users to machines that minimizes Social Cost might not necessarily e a Nash equilirium; call Optimum this least possile Social Cost We will investigate Coorination Ratio [16] - the worst-case ratio over all Nash equiliria, of Social Cost over Optimum We are intereste in unerstaning the interplay etween the topology of the unerlying interaction graph an the various existence, algorithmic, cominatorial, structural an optimality properties of Nash equiliria in this new moel of selfish restricte scheuling with oune interaction Contriution an Significance We partition our results into three major groups: 3-regular interaction graphs (Section 3) It is easy to prove that the Social Cost of the stanar fully mixe Nash equilirium for any -regular graph is f(, n), where f(, n) is a function that goes to 0 as n goes to infinity This gives a general ut rather rough estimation of Social Cost for -regular graphs; moreover, it oes not say how the specific structure of each particular 3-regular graph affects the Social Cost of the stanar fully mixe Nash equilirium We continue to prove much sharper estimations for the special class of 3-regular graphs Restricting our moel of interaction graphs to 3-regular graphs le us to iscover some nice structural properties of orientations in 3-regular graphs, which were motivate y Nash equiliria However, we have so far een unale to generalize these properties to regular graphs of egree higher than 3 We pursue a thorough stuy of 3-regular interaction graphs; these graphs further restrict the oune interaction y insisting that each machine is accessile to just three users Specifically, we focus on the stanar fully mixe Nash equilirium where all proailities of assigning users to machines are 1 2 We ask which the est 3-regular interaction graph is in this case This question rings into context the prolem of comparing against each other the expecte numer of 2-orientations an 3-orientations - those with makespan 2 an 3, respectively The manner in which these numers outweigh each other rings Social Cost closer to either 2 or 3 We evelop some eep graph-theoretic lemmas aout 2- an 3-orientations in 3-regular graphs to prove, as our main result, that the simplest 3-regular parallel links graph is the est-case 3-regular graph in this setting The proof ecomposes any 3-regular graph own to the parallel links graph in a way that Social Cost of the stanar fully mixe Nash equilirium oes not increase The graph theoretic lemmas aout 2- an 3-orientations are prove using oth counting an mapping techniques; oth the lemmas an their proof techniques are, we elieve, of more general interest an applicaility Boun on Coorination Ratio (Section 4) For the more general moel of restricte parallel links, a tight oun of Θ( log n ) on Coorination ratio restricte to pure Nash equiliria was shown in [9, Theorem 52] an inepenently in [1, Theorem 1] This implies an upper oun of O( log n ) on the Coorination Ratio for pure Nash equiliria in our moel as well We construct an interaction graph incurring Coorination Ratio Ω( log n ) to prove that this oun is tight for the moel of interaction graphs as well The construction extens an approach followe in [9, Lemma 51] that prove the same lower oun for the more general moel of restricte parallel links The Fully Mixe Nash Equilirium (Section 5) We pursue a thorough stuy of fully mixe Nash equiliria across interaction graphs Our finings are outline as follows: There exist counterexample interaction graphs for which fully mixe Nash equiliria may not exist Among them are all trees an meshes These counterexamples provie some insight aout a possile graph-theoretic characterization of interaction graphs amitting a fully mixe Nash equilirium 4-cycles an 1-connectivity are factors expecte to play a role in this characterization We next consier the case where infinitely many fully mixe Nash equiliria may exist In this case, the fully mixe Nash imension is efine to e the imension of the smallest -imensional space that can contain all fully mixe Nash equiliria For complete ipartite graphs, we prove a ichotomy theorem that characterizes unique existence The proof employs arguments from Linear Algera For hypercues, we have only een ale to prove that the fully mixe Nash imension is the hypercue imension for hypercues of imension 2 or 3 We conjecture that this is true for all hypercues, ut we have only een ale to oserve that the hypercue imension is a lower oun on the fully mixe Nash imension (for all hypercues) We are finally intereste in unerstaning whether (or when) the fully mixe Nash equilirium is the worst-case one in this setting We present counterexample interaction graphs to show that the fully mixe Nash equilirium is sometimes the worst-case Nash equilirium, ut sometimes not For the hypercue, there is a pure Nash equilirium that is worse (with respect to Social Cost) than the fully mixe one On the other han, for the 3-cycle the fully mixe Nash equilirium has worst Social Cost Relate Work an Comparison Our moel of interaction graphs is the special case of the moel of restricte parallel links introuce an stuie in [9], where each user is now further restricte to have access to only two machines The work in [9] focuse on the prolem of computing pure Nash equiliria for that more general moel Aweruch et al [1] log n also consiere the moel of restricte parallel links, an prove a tight upper oun of Θ( log ) on Coorination Ratio for all (mixe) Nash equiliria This implies a corresponing upper oun for our moel of interaction graphs It log n log is an open prolem whether this oun of O( ) is tight for the moel of interaction graphs, or whether a etter upper oun on Coorination Ratio for all (mixe) Nash equiliria can e prove The moel of restricte parallel links is, in turn, a generalization of the so calle KP-moel for selfish routing [16], 2

3 which has een extensively stuie in the last five years; see eg [3 5, 7 11, 19, 20] Social Cost an Coorination Ratio were originally introuce in [16] Bouns on Coorination Ratio are prove in [3, 8 10, 20] The fully mixe Nash equilirium was introuce an stuie in [20], where its unique existence was prove for the original KP-moel The Fully Mixe Nash Equilirium Conjecture, stating that the fully mixe Nash equilirium maximizes Social Cost, was first explicitly state in [11] It was prove to hol for special cases of the KP-moel [11, 19] an for variants of this moel [9, 10] Recently the Fully Mixe Nash Equilirium Conjecture was isprove for the original KP-moel an the case that jo sizes are non-ientical [6] This stans in sharp contrast to the moel consiere in this paper where jo sizes are ientical The moel of interaction graphs is an alternative to graphical games [14] stuie in the Artificial Intelligence community The essential ifference is that in graphical games, users an resources are moele as vertices an eges, respectively The prolem of computing Nash equiliria for graphical games has een stuie in [13, 14, 18] Other stuie variants of graphical games inclue the network games stuie in [12], multi-agent influence iagrams [15] an game networks [17] 2 Framework an Preliminaries For all integers k 1, enote [k] = {1,,k} Interaction Graphs We consier a graph G = (V, E) where eges an vertices correspon to users an machines, respectively Assume there are m users an n machines, respectively, where m > 1 an n > 1 Each user has a unit jo From here on, we shall refer to users an eges (respectively, machines an vertices) interchangealy So, an ege connects two vertices if an only if the user can place his jo onto the two machines Strategies an Assignments A pure strategy for a user is one of the two machines it connects; so, a pure strategy represents an assignment of the user s jo to a machine A mixe strategy for a user is a proaility istriution over its pure strategies A pure assignment L = l 1,, l m is a collection of pure strategies, one for each user A pure assignment inuces an orientation of the graph G in the natural way A mixe assignment P = (p ij ) i [n],j [m] is a collection of mixe strategies, one for each user A mixe assignment F is fully mixe [20, Section 22] if all proailities are strictly positive The stanar fully mixe assignment F is the fully mixe one where all proailities are equal to 1 2 The fully mixe imension of a graph G is the imension of the smallest -imensional space that contains all fully mixe Nash equiliria for this graph Cost Measures For a pure assignment L, the loa of a machine j [n] is the numer of users assigne to j The Iniviual Cost for user i [m] is λ i = {k : l k = l i }, the loa of the machine it chooses For a mixe assignment P = (p ij ) i [m],j [n], the expecte loa of a machine j [n] is the expecte numer of users assigne to j The Expecte Iniviual Cost for user i [m] on machine j [n] is the expectation, accoring to P, of the Iniviual Cost for user i on machine j, then, λ ij = 1 + k [m],k i p kj The Expecte Iniviual Cost for user i [m] is λ i = j [n] p ijλ ij ( Associate with a mixe assignment P is the Social Cost SC(G,P) = E P maxv [n] {k : l k = v} ), that is, Social Cost is the expectation, accoring to P, of makespan (that is, maximum loa) The Optimum OPT(G) is efine as the least possile, over all pure assignments L = l 1,, l n [n] m, makespan; that is, OPT(G) = min L [n] m max v [n] {k : l k = v} Nash Equiliria an Coorination Ratio We are intereste in a special class of (pure or) mixe assignments calle Nash equiliria [21] that we escrie here The mixe assignment P is a Nash equilirium [9, 16] if for each user i [m], it minimizes λ i (P) over all mixe assignments that iffer from P only with respect to the mixe strategy of user i Thus, in a Nash equilirium, there is no incentive for a user to unilaterally eviate from its own mixe strategy in orer to ecrease its Expecte Iniviual Cost Clearly, this implies that λ ij = λ i if p ij > 0 whereas λ ij λ i otherwise We refer to these conitions as Nash equations an Nash inequalities, respectively The Coorination Ratio CR G for a graph G is the maximum, over all Nash equiliria P, of the ratio SC(G,P) OPT(G) ; thus, SC(G,P) CR G = max P The Coorination Ratio CR is the maximum, over all graphs G an Nash equiliria P, of the ratio OPT(G) SC(G,P) OPT(G) ; thus, CR = max G,P SC(G,P) OPT(G) Our efinitions for CR G an CR exten the original efinition of Coorination Ratio y Koutsoupias an Papaimitriou [16] to encompass interaction graphs Graphs an Orientations Some special classes of graphs we shall consier inclue the cycle C r on r vertices; the complete ipartite graph (or iclique) K r,s which is a simple ipartite graph with partite sets of size r an s respectively, such that two vertices are ajacent if an only if they are in ifferent partite sets; the hypercue H r of imension r whose vertices are inary wors of length r connecte if an only if their Hamming istance is 1 For a graph G, enote G the maximum egree of G A graph is -regular if all vertices have the same egree The graph consisting of 2 vertices an 3 parallel eges will e calle necklace Also, for even n, G (n) will enote the parallel links graph, ie, the graph consisting of n 2 necklaces An orientation of an unirecte graph G results when assigning irections to its eges The makespan of a vertex in an orientation α is the in-egree it has in α The makespan of an orientation is the maximum vertex makespan For any integer, a -orientation is an orientation with makespan in a graph G; enote -or(g) the set of -orientations of G 3

4 3 3-Regular Graphs In this section, we consier the prolem of etermining the est-case -regular graph among the class of all -regular graphs with a given numer of vertices (an, therefore, with the same numer of eges), with respect to the Social Cost of the stanar fully mixe Nash equilirium, where all proailities are equal to 1/2 A Rough Estimation We start with a rough estimation of the Social Cost of any -regular graph G, where 2 We first prove a technical lemma aout the proaility that such a ranom orientation has makespan at most 1 Denote this proaility q (G) Lemma 1 Let I e an inepenent set of G Then, q (G) ( ) I We are now reay to prove: Theorem 1 For a -regular graph G with n vertices, SC( F, G) = f(n, ), where f(n, ) 0 as n Proof Since every maximal inepenent set of G has size at least n +1, Lemma 1 implies that q (G) ( ) 1 1 n +1 2 Thus, SC( F, G) q (G) + (1 q (G)) = ( 1)q (G), so that SC( F, G) = f(n, ), where f(n, ) tens to 0 as n, as neee Cactois an the Two-Sisters Lemma The rest of our analysis will eal with 3-regular graphs We will e ale to significantly strengthen an improve Theorem 1 for the special case of 3-regular graphs We efine a structure that we will use in our proofs Definition 1 (Cactois) A cactoi is a pair Ĝ = V, Ê, where V is a set of vertices an Ê is a set consisting of unirecte eges etween vertices, an pointers to vertices, ie, loose eges incient to one single vertex A cactoi is calle 3-regular if each vertex is incient to three elements from Ê A cactoi may e consiere as a stanar multigraph if we a a special vertex an we replace each pointer y an ege which connects the special vertex with the vertex the pointer is incient to Consier now any aritrary ut fixe orientation σ of Ĝ Call it stanar orientation We will now efine variales x α (e) for each e Ê, which take values from {0, 1} in each possile orientation α of Ĝ The values are efine with reference to the stanar orientation σ So, take any aritrary orientation α of Ĝ For each e Ê, x α(e) = 1 if e has the same irection in α an σ, an 0 otherwise Note that x σ (e) = 1 for all e Ê We now continue with a lemma that estimates the proaility that a ranom orientation is a 2-orientation in a 3-regular cactoi Ĝ Consier two vertices u an v calle the two sisters with incient pointers π u an π v Assume that in the stanar orientation σ, π u an π v point away from u an v, respectively Denote P G (i, j) the proaility that a ranom orientation α with x α (u) = i an x α (v) = j, where i, j {0, 1}, is a 2-orientation Clearly, y our assumption on the stanar orientation σ, P G (1, 1) is not smaller than each of P G (0, 0), P G (0, 1) an P G (1, 0) However, we prove that P G (1, 1) is upper oune y their sum Lemma 2 (Two Sisters Lemma) For any 3-regular cactoi Ĝ = V, Ê an any two sisters u, v V, it hols that P G (0, 0) + P G (0, 1) + P G (1, 0) P G (1, 1) Proof Denote 1, 2 an 3, 4 the other eges or pointers incient to the two sisters u an v, respectively Define the stanar orientation σ so that these eges or pointers point towars u or v, respectively Denote Ĝ the cactoi otaine from Ĝ y eleting the two sisters u an v an their pointers π u an π v Define P G (x 1, x 2, x 3, x 4 ) the proaility that a ranom orientation α of the cactoi Ĝ with x α ( i ) = x i for 1 i 4 is a 2-orientation Then, P G (1, 1) = 1 16 P G (0, 1) = 1 16 x 1,x 2,x 3,x 4 {0,1} x 1,x 2 {0,1},x 3 x 4=0 P G (x 1, x 2, x 3, x 4 ), P G (0, 0) = 1 16 P G (x 1, x 2, x 3, x 4 ), an P G (1, 0) = 1 16 x 1 x 2=0,x 3 x 4=0 x 1 x 2=0,x 3,x 4 {0,1} Set now D = 16 (P G (0, 0) + P G (0, 1) + P G (1, 0) P G (1, 1) ) It suffices to prove that D 0 Clearly, D = 2 x 1 x 2=0,x 3 x 4=0 P G (x 1, x 2, x 3, x 4 ) P(1, 1, 1, 1) P G (x 1, x 2, x 3, x 4 ), P G (x 1, x 2, x 3, x 4 ) 4

5 Use now the cactoi Ĝ to efine the proailities Q(i, j) an R(i, j) where i, j {0, 1} as follows: Q(i, j) is the proaility that a ranom orientation α of the cactoi Ĝ with x α ( 1 ) = i an x α ( 2 ) = j is a 2-orientation; R(i, j) is the proaility that a ranom orientation α of the cactoi Ĝ with x α ( 3 ) = i an x α ( 4 ) = j is a 2-orientation Clearly, Q G (i, j) = P G (i, j, x 3, x 4 ) an R G (i, j) = P G (x 1, x 2, i, j) x 3,x 4 {0,1} x 1,x 2 {0,1} We procee y inuction on the numer of vertices of Ĝ So, it suffices to assume the claim for the cactoi Ĝ an prove the claim for the cactoi Ĝ Assume inuctively that Q G (0, 0) + Q G (0, 1) + Q G (1, 0) Q G (1, 1) an R G (0, 0) + R G (0, 1) + R G (1, 0) R G (1, 1) These inuctive assumptions an the efinitions of Q G an R G imply that x 3,x 4 {0,1} x 1 x 2 =0 x 1,x 2 {0,1} x 3 x 4 =0 From the first inequality we otain, P G (x 1, x 2, x 3, x 4 ) x 3 x 4 =0 x 1 x 2 =0 From the secon inequality we get, P G (x 1, x 2, x 3, x 4 ) x 1 x 2 =0 x 3 x 4 =0 P G (x 1, x 2, x 3, x 4 ) P G (x 1, x 2, x 3, x 4 ) x 3,x 4 {0,1} x 1,x 2 {0,1} x 3,x 4 {0,1} x 1,x 2 {0,1} P G (1, 1, x 3, x 4 ) P G (x 1, x 2 1, 1) P G (1, 1, x 3, x 4 ), P G (x 1, x 2, 1, 1) x 1 x 2=0 x 3 x 4=0 P G (x 1, x 2, 1, 1) P G (1, 1, x 3, x 4 ) Aing up the last two inequalities yiels that 2 x 1 x 2 =0 P G (x 1, x 2, x 3, x 4 ) 2P G (1, 1, 1, 1), which implies D 0, x 3 x 4 =0 an the claim follows Orientations an Social Cost In this section, we prove a graph-theoretic result, namely that the regular parallel links graph minimizes the numer of 3-orientations among all 3-regular graphs with the same numer of vertices Theorem 2 For every 3-regular graph G with n vertices it hols that 3-or(G) 3-or(G (n)) Proof In orer to prove the claim, we start from the graph G 0 = G = (V, E 0 ) an iteratively efine graphs G i = (V, E i ), 1 i r, for some r n, in a way that G r equals G (n) an 3-or(G i ) 3-or(G i+1 ) hols for all 1 i < r Note that in each 3-regular graph, each connecte component is either isomorphic to a necklace or it contains a path of length 3 connecting four ifferent vertices, such that only the mile ege of this path can e a parallel ege If in G i all connecte components are necklaces, than G i is equal to G (n), otherwise some connecte component of G i contains a path c, a,, with 4 ifferent vertices a,, c, In the latter case, construct a new graph G i+1 = (V, E i+1 ) y eleting the eges {a, c}, {, } from E i an aing the eges {a, }, {c, } to the graph as escrie in the following paragraph e 4 a e 2 c e 6 e 4 a c e 6 G i e 1 Gi+1 e 1 e 2 e 3 e 3 Fig 1 Constructing the graph G i+1 from G i As illustrate in Figure 1, the eges incient to vertices a,, c, are numere y some j, where 1 j 9 In this figure, all the eges are ifferent This oes not necessarily have to e the case It may happen that e 4 = resulting in two parallel eges etween a an in G i an three parallel eges etween a an in G i+1 It may also happen that e 6 or is equal to or It is not possile that e 6 or is equal to e 2 (or that or is equal to e 3 ) since we assume that in the path c, a,, only the mile ege may e a parallel ege It may e also possile that e 4 is equal to or, an that is equal to e 6 or Note also that in each iteration step, the numer of single eges is ecrease y at least 1 So the numer of iteration steps is oune y n 5

6 First, we will show that 3-or(G i ) 3-or(G i+1 ) hols if all the eges e 1,, are ifferent We will consier the more general case in which some of the e j s are equal at the en of the proof To make the notation simpler, we set i = 1, ie, we consier the graphs G 1 an G 2 Note that there is a one-to-one corresponence etween eges in G 1 an eges in G 2 This implies that any aritrary orientation in G 1 can e interprete as an orientation in G 2 an vice versa Take the stanar orientation of G 1 to e the one consistent with the arrows in Figure 2 The interpretation of this orientation for G 1 yiels the stanar orientation for G 2 (also shown in Figure 2) e 4 a e 2 c e 6 e 4 a c e 6 G 1 e 1 G 2 e 1 e 2 e 3 e 3 Fig 2 The stanar orientations in G 1 an G 2 We will prove our claim y efining an injective mapping F : 3-or(G 2 ) 3-or(G 1 ) We want to use the ientity mapping as far as possile We set C 2 = {α ; α 3-or(G 2 ), α / 3-or(G 1 )} an C 1 = {α ; α 3-or(G 1 ), α / 3-or(G 2 )}, an we will efine F such that F(α) = α for α 3-or(G 2 ) \ C 2 an that F : C 2 C 1 is injective Note that a mapping F : 3-or(G 2 ) 3-or(G 1 ) efine this way is injective, since if β C 1, then β / 3-or(G 2 ) an therefore β is not an image when using the ientity function Let α e an aritrary orientation Note that all vertices u / {a,, c, } have the same makespan in G 1 an in G 2 with respect to α We ientify first the class C 2 an consier the vertices a,, c, We oserve: a has makespan 3 in G 2 x 1 = x 2 = x 4 = 1 a has makespan 3 in G 1 has makespan 3 in G 2 x 3 = 0, x 8 = x 9 = 1 has makespan 3 in G 1 has makespan 3 in G 2 x 1 = x 2 = 0, x 5 = 1 x 3 = 1 has makespan 3 in G 1 x 3 = 0, x 8 = x 9 = 1 has makespan 3 in G 1 x 6 = x 7 = 1 c has makespan 3 in G 1 c has makespan 3 in G 2 x 3 = x 6 = x 7 = 1 x 2 = 0 c has makespan 3 in G 1 x 1 = 0 x 5 = 1 has makespan 3 in G 1 x 2 = 1 x 1 = x 4 = 1 a has makespan 3 in G 1 Collecting this characterization, we construct the class C 2 as C 2 = {α / 3-or(G 1 ) ; x 1 = x 2 = x 3 = 0 x 5 = 1 x 6 x 7 = x 8 x 9 = 0} {α / 3-or(G 1 ) ; x 2 = x 3 = x 6 = x 7 = 1 x 1 x 4 = 0 (x 1 = 1 x 5 = 0)} In a similar way, we construct the class C 1 as C 1 = {α / 3-or(G 2 ) ; x 1 = 0 x 2 = x 3 = x 5 = 1 x 6 x 7 = 0} {α / 3-or(G 2 ) ; x 2 = x 3 = 0 x 6 = x 7 = 1 x 8 x 9 = 0 (x 1 = 1 x 5 = 0)} Now, to efine F, we consier four cases aout orientations α C 2 : (1) Consier α C 2 with x 2 = x 3 = x 6 = x 7 = 1 x 1 x 4 = 0 x 8 x 9 = 0 (x 1 = 1 x 5 = 0) Set F(x 1, 1, 1, x 4, x 5, 1, 1, x 8, x 9, ) = (x 1, 0, 0, x 4, x 5, 1, 1, x 8, x 9, ) α in G 2 e 4 e 6 e 2 e 6 a c e 4 a c e 1 e 2 e 3 F( α) in G 1 e 1 e 3 Fig 3 The mapping F Note that vertices from {a,, c, } have the same connections to vertices outsie {a,, c, }; therefore, α / 3-or(G 1 ) implies that F(α) / 3-or(G 2 ) This implies that F(α) C 1 6

7 (2) Consier α C 2 with x 1 = x 2 = x 3 = 0 x 5 = 1 x 6 x 7 = 0 x 8 x 9 = 0 Set F(0, 0, 0, x 4, 1, x 6, x 7, x 8, x 9, ) = (0, 1, 1, x 4, 1, x 6, x 7, x 8, x 9, ) In a way similar to case (1), we conclue that F(α) C 1 After these two cases, any orientation α H 2 with H 2 = {α C 2 x 2 = x 3 = x 6 = x 7 = 1 x 1 x 4 = 0 x 8 = x 9 = 1 (x 1 = 1 x 5 = 0)} has not een mappe y F, an orientations β H 1 with H 1 = {β C 1 x 2 = x 3 = 0 x 6 = x 7 = 1 x 1 = x 4 = 1 x 8 x 9 = 0} {β C 1 x 1 = 0, x 2 = x 3 = x 5 = 1 x 6 x 7 = 0 x 8 = x 9 = 1} are not images uner F We continue with these orientations (3) Set H 21 = {α C 2 ; x 2 = x 3 = x 6 = x 7 = x 8 = x 9 = 1 x 1 = 1 x 4 = 0} H 11 = {β C 1 ; x 2 = x 3 = 0 x 1 = x 4 = x 6 = x 7 = 1 x 8 x 9 = 0} We will show that H 21 H 11 hols Consier the cactois T 21 an T 11 otaine y omitting the vertices a,, c, from H 21 an H 11, respectively T 21 an T 11 consist of eges an 6 pointers e j, 4 j 9 Fixing the irections of the pointers in the same way as in the efinitions of H 21 an H 11, respectively, the numer of 2-orientations of T 21 is equal to H 21 an the numer of 2-orientations of T 11 is equal to H 11 See Figure 4 for an illustration e 4 a c e 6 e 2 e 6 e 4 a c H 21 e 1 e 2 e 3 H 11 e 1 e 3 Fig 4 Orientations from the sets H 21 an H 11 The pointers e 6 an have the same irections in T 21 an T 11 an has no specifie irection in oth cases Ege e 4 has ifferent irections in T 21 an T 11 Directing ege e 4 in T 21 towars vertex a woul lea to an increase numer of 2-orientations since the other vertex incient to e 4 has in this case makespan 2 with a larger proaility Let T 21 e the cactoi otaine from T 21 y irecting ege e 4 towars a Then T 21 an T 11 iffer only in the irections given to eges an Let P(i, j) e the proaility of a 2-orientation in G 2 if x 8 = i an x 9 = j Set m = 3 H11 2n Then, 2 = P(0, 0) + m 3 P(0, 1) + P(1, 0) P(1, 1) H21 2 m 3, ecause of Lemma 2 It follows that H 21 H 11 (4) To finish the first part of the proof, set H 22 = {β C 2 ; x 2 = x 3 = x 6 = x 7 = x 8 = x 9 = 1 x 1 = x 5 = 0} H 12 = {β C 1 ; x 1 = 0 x 2 = x 3 = x 5 = x 8 = x 9 = 1 x 6 x 7 = 0} See Figur elow for an illustration In the same way as in case (3), we show that H 22 H 12 e 4 a c e 6 e 2 e 6 e 4 a c H 22 e 1 e 2 e 3 H 12 e 1 e 3 Fig 5 Orientations from the sets H 22 an H 12 7

8 Since H 2 = H 21 H 22 an H 1 = H 11 H 12, there exists an injective mapping F : 3-or(G 2 ) 3-or(G 1 ) in the case that all eges e 4,, are ifferent Now we consier the case that some of these eges are equal If e i = e j then in each orientation α the variales x i an x j get opposite values Recall that the construction an proof of injectivity of the mapping F, which we escrie aove, was one in 3 steps: (i) We efine F(α) = α for all α 3-or(G 2 ) \ C 2 (ii) In cases (1) an (2) for some well efine α = (x 1,,x 9, ), the value F(α) is otaine y negating x 2 an x 3 an leaving the other irections unchange (iii) H 2 H 1 is shown for the remaining cases Steps (i) an (ii) are not influence y setting x i = x j for some i, j {4,,9}, i j So it remains to consier step (iii) If e i = e j for i {6, 7}, j {8, 9}, then x i = x j hols an this implies that H 2 =, since for all α H 2 it hols x 6 = x 7 = x 8 = x 9 = 1 Clearly, this implies H 2 H 1 So we can assume now that e i e j for i {6, 7}, j {8, 9} an we consier the case e 4 = We will show first that H 21 H 11 hols also in this case Consier the cactois T 21 an T 11 otaine y eleting the vertices a,, c, from H 21 an H 11 Since ege e 4 = connects vertices a an, it is also elete when the cactois are forme Each of the cactois T 11 an T 21 has now only the 4 pointers e j, 6 j 9 A simple inspection of the proof given aove shows that H 21 H 11 hols also in this case Furthermore, H 22 H 21 can e shown in the same way The cases e 4 = an = e 6 can e hanle in a very similar way This completes the proof of the claim Our main result follows now as an immeiate consequence of Theorem 2 Corollary 1 For a 3-regular graph G with n vertices, SC(G, F) SC(G (n), F) = 3 ( 3 4) n/2 We can also show that equality oes not hol in Corollary 1 Example 1 There is a 3-regular graph for which the Social Cost of the stanar fully mixe Nash equilirium is larger than for the corresponing parallel links graph 4 Coorination Ratio In this section, we present a oun on the Coorination Ratio for pure Nash equiliria ( ) Theorem 3 Restricte to pure Nash equiliria, CR = Θ log n ( ) Proof Upper oun: Since our moel is a special case of the restricte parallel links moel, the upper oun O log n in [9] also hols for our moel Lower oun: Let G e the complete tree of height k, where each vertex in layer l, 0 l k has k l chilren Denote y k l = k(k 1) (k l) the lth falling factorial of k Then, the numer of vertices is n = 0 l k kl < (k+1)! = Γ(k+2) This implies k > Γ 1 (n) 2 (1) Denote y L 1 the pure assignment in which all users are assigne towar the root Clearly, the Iniviual Cost of a user assigne to a vertex in layer l is k l Moreover, such a user can not improve y moving to its vertex in layer (l + 1) Thus, L 1 is a pure Nash equilirium with Social Cost k (2) Denote y L 2 the pure assignment in which all users are assigne towar the leaves Clearly, the Iniviual Cost of all users is 1 Thus, the Social Cost of L 2 is 1 It follows that max G,L SC(G,L) OPT(G) SC(G,L1) SC(G,L 2) = k > Γ 1 (n) 2 = Ω ( ) log n, as neee Oservation 1 Restricte to pure Nash equiliria, for any interaction graph G, CR G G, an this oun is tight 8

9 5 The Fully Mixe Nash Equilirium In this section, we stuy the fully mixe Nash equilirium For a graph G = (V, E), for each ege jk E, enote jk the user corresponing to the ege jk Denote p jk an p kj the proailities (accoring to P) that user jk chooses machines j an k, respectively For each machine j V, the expecte loa of machine j excluing a set of eges Ẽ, enote πp(j)\ẽ, is the sum kj E\ E p kj As a useful cominatorial tool for the analysis of our counterexamples, we prove: Lemma 3 (The 4-Cycle Lemma) Take any 4-cycle C 4 in a graph G, an any two vertices u, v C 4 that are non-ajacent in C 4 Consier a Nash equilirium P for G Then, π P (u) \ C 4 = π P (v) \ C 4 Non-Existence Results We first oserve: Counterexample 1 There is no fully mixe Nash equilirium for trees an meshes We remark that the crucial property of trees that was use in the proof of Counterexample 1 is that each tree contains at least one leaf Thus, Counterexample 1 actually applies to the more general class of graphs with no vertex of egree 1 We continue to prove: Counterexample 2 For each graph in Figure 1, there is no fully mixe Nash equilirium Our six counterexample graphs suggest that the existence of 4-cycles across the ounary of a graph or 1-connectivity may e crucial factors that isallow the existence of fully mixe Nash equiliria Of course, this remains yet to e proven Uniqueness an Dimension Results For Complete Bipartite Graphs, we prove: Theorem 4 Consier the complete ipartite graph K r,s, where s r 2 an s 3 Then, the fully mixe Nash equilirium F for K r,s exists uniquely if an only if r > 2 Moreover, in case r = 2, the fully mixe Nash imension of K r,s is s 1 Hypercue Graphs Oserve first that, in general, any point in (0, 1) r is mappe to a fully mixe Nash equilirium with equal Nash proailities on all eges of the same imension (an pointing to the same irection) This implies: Oservation 2 Consier the hypercue H r, for any r 2 Then, the fully mixe Nash imension of H r is at least r To show that r is also an upper oun, we nee to prove that no other fully mixe Nash equiliria exist We manage to o this only for r {2, 3} Theorem 5 Consier the hypercue H r, for r {2, 3} Then, the fully mixe Nash imension is r Worst-Case Equiliria We present two counterexamples to show that a fully mixe Nash equilirium is not necessarily the worst-case Nash equilirium, ut it can e Counterexample 3 There is an interaction graph for which no fully mixe Nash equilirium has worst Social Cost Counterexample 4 There is an interaction graph for which there exists a fully mixe Nash equilirium with worst Social Cost 6 Epilogue We introuce a simple graph-theoretic moel, calle interaction graphs, to aress the effect of structure an sparse interactions among users an machines in complex multischeuling systems Within our new moel, we stuie the impact of selfish ehavior of the users reaching a stale state of the system moele as a Nash equilirium [21] In this setting, we investigate the amount of performance loss uner various topological assumptions on interaction graphs As our main result, we etermine that the simplest parallel links graph is the est among all 3-regular graphs with respect to expecte makespan in the stanar fully mixe Nash equilirium The proof of our main result has require a lot of non-stanar structural graph theory to e proven Our work presents a new genre of mathematical prolems in relation to the moel of interaction graphs that remain tantalizingly open We conclue y listing a few of them here: Exten our analysis on the optimality of the parallel links graph to all -regular graphs, for any fixe > 3 Is the stanar fully mixe Nash equilirium essential for the optimality of the parallel links graph? Or oes the optimality hol for all fully mixe Nash equiliria? Characterize in graph-theoretic terms the graphs for which a fully mixe Nash equilirium exists, an those for which a fully mixe Nash equilirium is (respectively, is not) the worst Nash equilirium Is Θ( log n it is Ω( log n ) the right oun on Coorination Ratio for all mixe Nash equiliria? Or is it Θ( log ) an O( log n log ) n log )? We know 9

10 Exten our moel to encompass the more realistic assumptions of non-unit weights for the users an capacities for the links (cf [16]), or the capaility of users to place their jos on more then two machines (that is, the interaction graph ecomes a hypergraph) It will e very interesting to stuy the impact of these aitional imensions In conclusion, our work eals with a currently treny topic, namely the (impact of) selfish ehavior of users, in a simple graph-theoretic moel for restricte scheuling, namely the interaction graphs Numerous open prolems an issues remain, an we elieve that our work will stimulate further research on the topic Acknowlegments: We thank Paul Spirakis an Karsten Tiemann for helpful iscussions on the topic of our work References 1 B Aweruch, Y Azar, Y Richter an D Tsur, Traeoffs in Worst-Case Equiliria, Proceeings of the First International Workshop on Approximation an Online Algorithms, K Jansen an R Solis-Oa es, pp 41 52, Vol 2909, LNCS, Springer-Verlag, Septemer G Christooulou, E Koutsoupias an A Nanavati, Coorination Mechanisms, Proceeings of the 31st International Colloquium on Automata, Languages an Programming, J Diaz, J Karhumäki, A Lepistö et al, pp , Vol 3142, LNCS, Springer- Verlag, July A Czumaj an B Vöcking, Tight Bouns for Worst-Case Equiliria, Proceeings of the 13th Annual ACM Symposium on Discrete Algorithms, pp , January E Even-Dar, A Kesselman an Y Mansour, Convergence Time to Nash Equiliria, Proceeings of the 30th International Colloquium on Automata, Languages an Programming, J C M Baeten, J K Lenstra, J Parrow an G J Woeginger es, pp , Vol 2719, LNCS, Springer-Verlag, June/July R Felmann, M Gairing, T Lücking, B Monien an M Roe, Nashification an the Coorination Ratio for a Selfish Routing Game, Proceeings of the 30th International Colloquium on Automata, Languages an Programming, J C M Baeten, J K Lenstra, J Parrow an G J Woeginger es, pp , Vol 2719, LNCS, Springer-Verlag, June/July S Fischer an B Vöcking, A Counterexample to the Fully Mixe Nash Equilirium Conjecture, Technical Report, RWTH Aachen, May D Fotakis, S Kontogiannis, E Koutsoupias, M Mavronicolas an P Spirakis, The Structure an Complexity of Nash Equiliria for a Selfish Routing Game, Proceeings of the 29th International Colloquium on Automata, Languages an Programming, P Wimayer, F Triguero, R Morales, M Hennessy, S Eienenz an R Conejo es, pp , Vol 2380, LNCS, Springer-Verlag, July M Gairing, T Lücking, M Mavronicolas an B Monien, Computing Nash Equiliria for Scheuling on Restricte Parallel Links, Proceeings of the 36th Annual ACM Symposium on Theory of Computing, pp , June M Gairing, T Lücking, M Mavronicolas an B Monien, The Price of Anarchy for Polynomial Social Cost, Proceeings of the 29th International Symposium on Mathematical Founations of Computer Science, J Fiala, V Kouek an J Kratochvil es, pp , LNCS, Springer-Verlag, August M Gairing, T Lücking, M Mavronicolas, B Monien an M Roe, Nash Equiliria in Discrete Routing Games with Convex Latency Functions, Proceeings of the 31st International Colloquium on Automata, Languages an Programming, J Diaz, J Karhumäki, A Lepistö an D Sannella es, pp , Vol 3142, LNCS, Springer-Verlag, July M Gairing, T Lücking, M Mavronicolas, B Monien an P Spirakis, Extreme Nash Equiliria, Proceeings of thth Italian Conference on Theoretical Computer Science, C Bluno an C Laneve es, pp 1 20, Vol 2841, LNCS, Springer-Verlag, Octoer 2003 Full version: The Structure an Complexity of Extreme Nash Equiliria, accepte to Theoretical Computer Science 12 G Gottlo, G Greco an F Scarcello, Pure Nash Equiliria: Har an Easy Games, Proceeings of thth Conference on Theoretical Aspects of Rationality an Knowlege, pp , June S Kakae, M Kearns, J Langfor an L Ortiz, Correlate Equiliria in Graphical Games, Proceeings of the 4th ACM Conference on Electronic Commerce, pp 42 47, June M Kearns, M Littman an S Singh, Graphical Moels for Game Theory, Proceeings of the 17th Conference on Uncertainty in Artificial Intelligence, pp , August D Koller an B Milch, Multi-Agent Influence Diagrams for Representing an Solving Games, Games an Economic Behavior, Vol 45, No 1, pp , Octoer E Koutsoupias an C H Papaimitriou, Worst-Case Equiliria, Proceeings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, G Meinel an S Tison es, pp , Vol 1563, LNCS, Springer-Verlag, March P La Mura, Game Networks, Proceeings of the 16th Conference on Uncertainty in Artificial Intelligence, pp , M Littman, M Kearns an S Singh, An Efficient Exact Algorithm for Solving Tree-Structure Graphical Games, Proceeings of the 15th Conference on Neural Information Processing Systems Natural an Synthetic, pp , Decemer T Lücking, M Mavronicolas, B Monien, M Roe, P Spirakis an I Vrto, Which is the Worst-Case Nash equilirium?, Proceeings of the 28th International Symposium on Mathematical Founations of Computer Science, B Rovan an P Vojtas es, pp , Vol 2747, LNCS, Springer-Verlag, August M Mavronicolas an P Spirakis, 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11 Appenix Fig 6 Six counterexample graphs Proof of Lemma 1 Consier an inepenent set I = {v 1,, v r } of G For each j, 1 j r, enote E j the set of eges incient to j Clearly, E j = for all j Moreover, for all inices j an h, j h, E j an E h are isjoint since I is an inepenent set Choose any aritrary vertex v I The proaility that v has makespan 1 is 1 1 Since all sets E 2 j, 1 j r, are isjoint, the proaility that all vertices from I have makespan 1 is ( ) I The proaility that all vertices from V have makespan 1 is no larger, an the claim follows Proof of Corollary 1 For each orientation of a 3-regular graph, the makespan is at least 2 So SC(G, F) = 2 q 3 (G)+3(1 q 3 (G)) = 3 q 3 (G) Because of Theorem 33 q 3 (G) q 3 (G (n)) = ( 3 4) n/2, as neee Example 1 Let G e the complete graph with 4 vertices an G e the corresponing parallel links graph Denote F an F the stanar fully mixe Nash equiliria for the graph G an G, respectively Each graph has exactly 6 eges, so there are 2 6 = 64 possile assignments of the users Since we consier the stanar fully mixe Nash equilirium, each of these assignments is equiproale Enumerating all possile assignments an counting the numer of 3-orientations (the ones with makespan 3) we get 32 an 28 for G an G, respectively Thus, SC(G, F) > SC(G, F ) i

12 Proof of the 4-Cycle Lemma Denote C 4 = u, x, v, y, u We will write own the Nash equations for users ux, xv, vy an yu These are (i) π P (u) \ C 4 + p yu = π P (x) \ C 4 + p vx, (ii) π P (x) \ C 4 + p ux = π P (v) \ C 4 + p yv, (iii) π P (y) \ C 4 + p uy = π P (v) \ C 4 + p xv, (iv) π P (u) \ C 4 + p xu = π P (y) \ C 4 + p vy Aing all these equations an using the fact that for any user a we have p a = 1 p a, it follows that 2π P (u) \ C 4 + π P (x) \ C 4 + π P (y) \ C 4 = 2π P (v) \ C 4 + π P (x) \ C 4 + π P (y) \ C 4 This implies that π P (u) \ C 4 = π P (v) \ C 4, as neee Counterexample 1 Assume, y way of contraiction, that a fully mixe Nash equilirium F exists for a tree T Take any ege uv for a leaf v in T The Nash equation for user uv is π P (u) f vu = π P (v) f uv or π P (u) f vu = 0 (since v is a leaf) Since u is not a leaf, π P (u) f vu > 0 A contraiction The non-existence of fully mixe Nash equiliria for meshes is an immeiate consequence of the 4-Cycle Lemma Counterexample 2 Consier the top left graph in Figure 6 Assume, y way of contraiction, that there is a fully mixe Nash equilirium for it Name the machines x, y, z, z, x, y from top to ottom The Nash equations ecome (i) fzx = f zy (ii) fyx = f yz + f z z (iii) fxy = f xz + f z z (iv) fxz + f yz = f x z + f y z (v) fy x = f y z + f zz (vi) fx y = f x z + f zz (vii) fz x = f z y Recall, that for any user a we have f a = 1 f a It follows from (ii) an (iii) with (i) that f xy = 1 2 By symmetry, f x y = 1 2 Now aing (ii) an (v) yiels f yx + f y x = f yz + f y z +1 which implies that f yz + f y z = 0, a contraiction to the assumption that there is a fully mixe Nash equilirium The 4-Cycle Lemma immeiately implies that there is no fully mixe Nash equilirium for the three graphs at the ottom The non-existence of the fully mixe Nash equilirium for the two remaining graphs follows with arguments similar to those we use for the top left graph Proof of Theorem 4 For any integer k 2, enote I k k an J k k the ientity matrix an the complementary ientity matrix, respectively; that is, I k k = an J k k = Recall that s r 2 an s 3 We show in (1) that there exists a unique fully mixe Nash equilirium if an only if r > 2 In (2), we prove that the fully mixe imension is s 1 if r = 2 ii

13 (1) Define vectors f 1,f 2,,f r so that for each inex l, 1 l r, f l contains the s proailities for each of s users attache to machine l in the right ipartition to assign its loa to machine l So, each vector f l correspons to a vertex in the left partite set (of size r); each such vector has s components, each corresponing to a vertex in the right partite set It is immeiate to erive that the fully mixe Nash equations ecome J s s I s s I s s I s s f 1 1 I s s J s s I s s I s s f 2 1 = (r 1) I s s I s s I s s J s s 1 Take any two ajacent lock rows in the Nash equations For example, take the first lock row an the secon lock row; these are J s s f 1 + f f r = (r 1)1 s 1 an f 1 + J s s f f r = (r 1)1 s 1 By sutraction, it follows that J s s (f 1 f 2 ) = f 1 f 2 Since 1 is not an eigenvalue of J s s, it follows that f 1 = f 2 In this way, it is prove that f 1 = f 2 = = f r ; set this common value to f Then, each lock row may e written as J s s f + (r 1)f = (r 1)1 s 1, or r r f = (r 1)1 s r 1 This linear system has the solution r 1 r+s 2 1 s 1, which is unique if an only if the system matrix is non-singular; thus, the fully mixe Nash equilirium F exists uniquely if an only if r > 2, as neee (2) Assume now that r = 2 Similar to the previous case (y swapping r an s), we can express the Nash equations with help of the matrix J 2 2 I 2 2 I 2 2 I 2 2 I 2 2 I 2 2 J 2 2 I 2 2 I 2 2 I 2 2 M = I 2 2 I 2 2 I 2 2 J 2 2 I 2 2 I 2 2 I 2 2 I 2 2 I 2 2 J 2 2 We now procee y eriving the imension of the solution space with help of matrix manipulation From i = 2 to s, sutract the ith row lock from the (i 1)th row lock This yiels J 2 2 I 2 2 I 2 2 J J 2 2 I 2 2 I 2 2 J M = J 2 2 I 2 2 I 2 2 J 2 2 I 2 2 I 2 2 I 2 2 I 2 2 J 2 2 Then, from i = 1 to s 1, a the ith row lock of M to the (i + 1)the column lock This yiels J 2 2 I J 2 2 I M = J 2 2 I I 2 2 2I 2 2 3I 2 2 (s 1)I 2 2 J (s 1)I 2 2 Since M is a lower triangular matrix, it suffices to erive the rank of the matrices on the iagonal On the one han, the eterminant of J 2 2 I 2 2 is ( ) et(j 2 2 I 2 2 ) = 1 1 = f r iii

14 Thus, the rank of J 2 2 I 2 2 is 1 On the other han, the eterminant of J (s 1)I 2 2 is ( ) et(j (s 1)I 2 2 ) = s 1 1 = (s 1) 1 s s 3 > 0 Thus, the rank of J (s 1)I 2 2 is 2 Comining these results, we get that the rank of M is s + 1 This implies that the kernel has imension 2s (s + 1) = s 1, proving the claim Proof of Theorem 5 The lower ouns follow from Oservation 2 For r = 2, note that H 2 = C 4 = u, x, v, y, u, the 4-cycle The Nash equations for users ux an xv are f yu = f vx an f ux = f yv, which implies that im H2 (F) 2 Consier now the case r = 3, where im H3 (F) = 12 Using the Nash equations an the 4-Cycle Lemma, we prove that the Nash proailities on eges of the same imension (an pointing to the same irection) are necessarily equal, which implies that im H3 (F) 3 Counterexample 3 Let G e the 4-cycle s, t, u, v, s For G there exists a pure Nash equilirium with social cost 2: user st an tu are assigne to machine t, user uv is assigne to machine u, an user vs is assigne to machine s Since the social cost of any pure assignment is at most 2 an there exist pure assignments with social cost 1 which contriute to the social cost of any fully mixe Nash equilirium, the social cost of any fully mixe Nash equilirium is strictly less than 2, proving the claim Counterexample 4 Let g e the 3-cycle For G there are two symmetric pure Nash equiliria where there is exactly one user assigne to each machine Let L e such a pure Nash equiliria It is, SC(G,L) = 1 Clearly, there is only one further Nash equilirium for G, which is the stanar fully mixe Nash equilirium F In F each of the three users chooses each of its two possile links with proaility 1 2 This implies SC(G,F) = 175 > SC(G,L) iv