On Oblivious PTASs for Nash Equilibrium

On Oblivious PT AS’s for Nash Equilibrium ∗ Constan tinos Dask alakis † EECS and CSAIL, MIT costis@mit.edu Christos P apadimitriou ‡ Computer Science, U.C. Berk eley c hristos@cs.berkele y .edu Octob er 24, 2018 Abstract If a class of games is known to have a Nash e q uilibrium with probability v alues that are either zero or Ω(1) — and thus with supp ort of bo unded siz e — then obviously this equilibrium can be found exhaustively in p olynomia l time. Somewhat surprisingly , we show that there is a PT AS for the class of g a mes whose equilibria are guar an teed to hav e smal l — O ( 1 n ) — v alues, and therefor e la rge — Ω( n ) — supp orts. W e als o point out that there is a PT A S for games with sparse pay oﬀ matrices, a family for which the exact problem is known to be PP AD-complete [6]. Both algo rithms ar e of a sp ecial kind that we call oblivious: The algo rithm just samples a ﬁxed distribution on pairs of mixed stra tegies, and the game is only used to deter mine whether th e sampled strategies co mprise an ǫ - Na sh equilibrium; the answer is “yes” with inv erse p olynomial probability (in the seco nd case, the a lgorithm is a ctually deterministic). These results bring ab out t he question: Is ther e an oblivious PT AS for ﬁnding a N ash e quilibrium in gener al games? W e a nsw er this question in the n ega tiv e; our lower b ound comes close to th e quasi-po lynomial upper b ound of [18]. Another recent PT AS for anonymous g ames [13, 14, 7] is a lso oblivious in a weak er sense appropria te for this cla ss of games (it samples from a ﬁxed distribution on unor der e d collections of mixed stra teg ies), but its r unning time is exp onential in 1 ǫ . W e prove that a n y o blivious PT AS for anonymous games with tw o str ategies and three play er types m ust hav e 1 ǫ α in the exp onen t of the running time for some α ≥ 1 3 , r e nder ing the alg orithm in [7] (which works with an y b ounded num b er o f play er types) essen tially o ptimal within oblivious algo r ithms. In contrast, we devis e a poly ( n ) · (1 /ǫ ) O (log 2 (1 /ǫ )) non-oblivious PT AS for anon ymous games with t wo strategies and any b ounded num b e r o f pla yer t ypes. The key idea of our algorithm is to search not over unordered s ets of mixed strategies , but ov er a car efully crafted s et of collec tio ns of the ﬁrst O (log 1 ǫ ) moments of the distribution of the num be r of play ers playing strateg y 1 at equilibrium. The algorithm works b ecause of a probabilistic r esult o f mor e general interest that w e prov e: the tota l v ariation distance b et ween tw o sums of indep endent indicator random v ariables dec reases expo ne ntially with the n umber of moments o f the tw o sums that are equal, independent of the num b e r o f indicators. A byproduct of our algorithm is establishing the existence of a spars e (and eﬃcien tly com- putable) ǫ -cover of the se t of all po s sible sums of n indepe ndent indicators, under the tota l v ariation dista nce. The size o f the cov er is p oly( n ) · (1 /ǫ ) O (log 2 (1 /ǫ )) . ∗ Extended version of the pap er of the same title that app eared in STOC , 2009. † This work w as d one while the author w as a p ostdoct oral researc her at Microsoft Research, N ew En gland. ‡ Supp orted by NSF grant CCF - 0635319, a gift from Y ahoo! Research, and a MICRO grant. 1 1 In tro duction Is there a p olynomial time approximat ion sc heme (PT AS) f or computing appro ximate Nash equilib- ria? This has emerged, in th e w ak e of the intrac tabilit y results for Nash equilibria [9, 5], as th e most cen tral question in equilibr ium computation. Ove r the p ast thr ee y ears there has b een relativ ely slo w progress to w ards smal ler ǫ ’s [ 10, 11, 4, 21] — so slo w that it is hard to b eliev e that a PT AS is around the corner. On the other hand, [13, 14] p ro vide a PT AS for the imp ortant special case of anon ymous games with a b oun ded num b er of strategies (those for w hic h the utilities of th e pla y ers, although diﬀeren t, d ep end on t he numb e r of pla y ers p laying eac h strategy , not on the identities of the pla y ers that do). This PT AS pro ceeds b y discretizing the pr obabilitie s in the mixed strategies of the play ers to multiples of 1 k , for appropr iate integ er k , and works ev en in the generalizatio n in whic h the pla ye rs are d ivided into a b ounded num b er of typ e s , and u tilitie s dep end on how many pla y ers of e ach typ e c ho ose eac h strategy . In this pap er we rep ort signiﬁcan t progress on this imp ortan t algorithmic problem; we pr esen t sev eral new algorithms, but also the ﬁ r st nont rivial low er b ounds. W e start by p oin ting out tw o new interesting classes of b imatrix games 1 that h a v e PT AS ’s: • It wa s sh o wn in [6] that computing a Nash equilibrium for the sp ecial case of sp ars e games , that is, games whose pa y oﬀ matrices ha ve a b ound ed num b er of non-zero entries in eac h ro w and column [6 ], is PP AD-co mplete. W e p oin t out that there is a trivial PT AS — in particular, the pair of uniform mixed strategies w orks. This is in teresting in that this is th e ﬁrst PP AD-complete case of a p roblem that is so appr o ximable. • W e also giv e a randomized PT AS for smal l pr ob ability games, that is, games that are guaran - teed to ha ve Nash equilibria w ith small O ( 1 n ) nonzero probability v alues, and thus with linear supp ort (Th eorem 3). It is qu ite surp r ising that games with small probabilit y v alues (our second s p ecial case ab o v e) are easy , since games with lar ge (b ou n ded from b elo w b y a constan t) prob ab ility v alues are also easy . What probabilit y v alues are diﬃcult then? Our next result, a lo w er b ound, seems to suggest inverse lo garithmic pr obabilit y v a lues are h ard (c ompare with the quasi-PT AS of [18], whose equilibr ia h a v e roughly logarithmic s upp ort). T o explain our negativ e result, we ﬁrst note that b oth PT AS’s outlined ab o v e (as well as those for anonymous games discussed later) are oblivious . This means that they hav e access to a ﬁxed set of p airs of mixed str ateg ies, in the generic case by s ampling, and th ey lo ok at the game only to d etermine whether a sampled pair of strategies constitutes an appro ximate Nash equilibriu m. Is there an oblivious PT AS for th e general Nash equilibrium ? The answ er here is a deﬁnite “no” — in fact, we pro v ed our negativ e r esult after w e had b een w orking for s ome time on develo ping such an algorithm . . . W e sho w th at any oblivious algo rithm must sample at least Ω  n ( . 8 − 34 ǫ ) log 2 n  pairs in exp ectation (Theorem 4). F or comparison, [18]’s algorithm tak es time n O ( log n/ǫ 2 ) . Another imp ortant class of games for w hic h a PT AS was recen tly disco v ered is that of anony- mous games with two str ate gi e s [13, 7]. These are m ulti-pla ye r games, in wh ic h the utilit y of eac h pla y er dep end s on th e s tr ateg y (0 or 1) pla y ed, an d the numb er of other pla y ers p laying strategy 1 ( not their ident ities). In fact, the PT AS works eve n in the more sophisticated case in whic h th e pla y ers are partitioned into t yp es, and the utilities d ep en d on the n umb er of p la y ers of e ach typ e pla ying strategy 1. The PT AS in [13], and the relativ ely more eﬃcien t one in [7], b oth hav e r un- ning times that are exp onentia l in 1 ǫ . They are also oblivious in a sense appropriate f or anon ymous 1 These results can b e extended to any b ounded num b er of pla yers, bu t in what follo ws we on ly discuss the tw o-pla yer, or bimatrix, case. 2 games, in that they work b y sampling an unor der e d set of n m ixed strategies, wh ere n is the num b er of play ers, and they only lo ok at the game in order to determine if ther e is an assignment of these strategies to the pla y ers that results in an appro ximate equilibriu m. W e pro ve that any oblivious appro ximation algorithm, for anonymous games with t w o strategies and three play er types, must sample an exp onential—in 1 ǫ —sized collecti on of unord ered sets of m ixed strategies — and so our PT AS’s are n ear-optimal . Finally , we circum ve nt this negativ e result to develo p a non-oblivious P T AS which ﬁnd s an ǫ - appro ximate Nash equilibrium in anonymous games with t wo strategies an d any b oun ded n umber of pla y er t yp es in time p oly( n )  1 ǫ  O (log 2 1 ǫ ) . This algorithm is b ased (in add ition to man y other insigh ts an d techniques for anonymous games) on a new result in applied probability w h ic h is, w e b eliev e, interesting in its own r ight: S upp ose that y ou h a v e t w o sums of n indep endent Bern oulli random v ariables, which ha v e the same ﬁrst momen t, th e same second momen t, and so on u p to momen t d . Then the d istributions of the t w o sums h a v e v ariati onal distance that v a nish es exp onen tially fast with d , regardless of n . T o turn this theorem int o an algorithm, w e d iscretize the mixed strategies of the pla y ers usin g tec hniques from [7] and , in the range of parameters where the algorithm of [7] br eaks d o wn, we iterate o v er all p ossible v alues of the ﬁr st log (1 /ǫ ) m omen ts of the pla y ers’ aggregat e b ehavio r; we th en try to ident ify , via an inv olv ed d ynamic programming sc heme, mixed strategies, implemen ting the giv en moments, which co rresp ond to appro ximate Nash equilibria of the game. Our app ro ximation guaran tee f or su ms of indep endent indicators is rather strong, esp ecially when the num b er of indicators is sm all, a regime w here Berry-Ess ´ een type b ounds pro vide w eak er guarantee s and r esult in slow er algorithms [13, 7]. It is quite in triguing that a quasi- p olynomial time b ound, such as the one w e provide in this p ap er, shows up again in the analysis of algorithms f or approximat e Nash equilibr ia (cf. [18]). As a bypro duct of our r esults we establish the existence of a sparse (and eﬃcien tly compu table) ǫ -co v er of the set of all p ossible sums of n indep enden t indicators, und er the total v ariatio n distance. The s ize of our co ver is p oly( n ) · (1 /ǫ ) O (log 2 (1 /ǫ )) . W e discuss this result in Section 6. 1.1 Preliminaries A tw o-pla y er, or bimatrix , game G is pla ye d by t wo pla y ers, the r ow player and the c olumn player . Eac h pla y er has a set of n pur e str at e gies , wh ic h w ith ou t loss of generalit y we assume to b e th e set [ n ] := { 1 , . . . , n } for b oth pla y ers. The game is describ ed then b y t w o p ayoﬀ matric es R , C corresp onding to the ro w and column p lay er resp ectiv ely , so th at, if the ro w pla ye r c ho oses strategy i and the column p la y er str ategy j , the row play er gets pa y oﬀ R ij and th e column p la y er C ij . As it is customary in the literature of approxima te Nash equilibria, we assume that the matrices are normalized, in that their en tries are in [ − 1 , 1]. The pla y ers can pla y mixe d str a te gies , that is, pr obabilit y distributions o v er their p u re strateg ies whic h are rep resen ted by p robabilit y vect ors x ∈ R n + , | x | 1 = 1. If the ro w pla y er c ho oses mixed strategy x and the column pla y er mixed strategy y , then the row play er gets exp ected pa yo ﬀ x T Ry and the column pla ye r exp ected p a y oﬀ x T C y . A pair of mixed strategies ( x, y ) is a Nash e quilibrium of the game G = ( R, C ) iﬀ x maximizes x ′ T Ry among all pr obabilit y vect ors x ′ and, simulta neously , y maximizes x T C y ′ among all y ′ . It is an ǫ -appr oximate N ash e quilibrium iﬀ x T Ry ≥ x ′ T Ry − ǫ , for all x ′ , and, sim ultaneously , x T C y ≥ x T C y ′ − ǫ , for all y ′ . In this pap er, w e will use the stronger notion of ǫ -appr oximately wel l-supp or te d Nash e quilibrium , or simply ǫ -Nash e quilibrium . This is an y pair of strategies ( x, y ) such that, for all i with x i > 0, e T i Ry ≥ e T i ′ Ry − ǫ , for all i ′ , and similarly for y . That is, eve ry strategy i in the s upp ort of x guarantee s exp ected p a y oﬀ e T i Ry w hic h is within ǫ fr om the optimal resp onse to y , and similarly ev ery strategy in the supp ort of y is within ǫ from the optimal r esp onse to x . 3 An anonymous game is in a sense the dimensional du al of a bimatrix game: T here are n play ers, eac h of whic h has t w o strategies, 0 and 1. F or eac h play er i there is a u tility function u i mapping { 0 , 1 } × [ n − 1] to [ − 1 , 1]. In tuitive ly , u i ( s, k ) is the utilit y of play er i when s/he p lays str ateg y s ∈ { 0 , 1 } , and k ≤ n − 1 of the r emaining pla y ers pla y strategy 1, while n − k − 1 play strategy 0. In other words, the utilit y of eac h p la y er dep end s, in a pla y er-sp eciﬁc wa y , on th e strategy play ed b y the play er and the n umber of other pla ye rs who pla y strategy 1 — but not their ident ities. The n otio ns of Nash equilibrium and ǫ -Nash equilibr ium are extend ed in the natural wa y to th is setting. Brieﬂy , a mixed strategy f or the i -th pla y er is a f u nction x i : { 0 , 1 } → [0 , 1] su c h that x i (0) + x i (1) = 1. A set of mixed strategies x 1 , . . . , x n is then an ǫ -Nash equilibrium if, for ev ery pla y er i and ev ery s ∈ { 0 , 1 } with x i ( s ) > 0: E x 1 ,...,x n u i ( s, k ) ≥ E x 1 ,...,x n u i (1 − s, k ) − ǫ , wh ere for the purp oses of the exp ectation k is d ra wn from { 0 , . . . , n − 1 } b y ﬂipping n − 1 coins acco rdin g to the distributions x j , j 6 = i , and counti ng the num b er of 1’s. A more sophisticated kin d of anon ymous games divides the p lay ers int o t typ es, so that the utilit y of play er i dep ends on the str ategy pla ye d by him/her, and the num b er of p la y ers of e ach typ e who p la y strategy 1. 2 PT AS for T w o Sp ecial Cases 2.1 Small Games W e sa y that a class of bimatrix games is smal l if th e sum of all en tries of the R and C matrices is o ( n 2 ). One such class of games are the k -sp arse games [6 ], in which ev ery r o w and column of b oth R and C ha ve at most k non -zero entries. The follo wing r esult by Chen, Deng and T eng shows that ﬁ nding an exact Nash equilibr ium remains hard for k -sparse games: Theorem 1 ([6 ]) . Finding a O ( n − 6 ) -Nash e quilibrium i n 10 -sp arse normalize d bi matr ix games with n str ate gie s p er player is a PP AD-c o mplete pr oblem. In con trast, it is easy to see that there is a PT AS for this class: Theorem 2. F or any k , ther e is a PT A S for the N ash e quilibrium pr oblem in k -sp arse bi matr ix games. Our origi nal pro of of this theorem consisted in showing that there alwa ys exists an ǫ -Nash equilibrium in wh ich b oth p la y ers of the game us e in their m ixed strategies probabilities that are in teger m ultiples of ǫ/ 2 k . Hence, we can eﬃcien tly enumerate o v er all p ossib le pairs of mixed strategies of this form, as long as k is ﬁxed. Shang-Hua T eng p ointe d out to us a m uch s impler algorithm: The pair of uniform mixed strategies is alwa ys an ǫ -Nash equilibriu m in a sparse game! The diﬀerence with our algorithm is this: the unif orm equilibrium giv es to b oth play ers pa y oﬀ of at most k/n ; our algorithm can b e used instead to appro ximate the pa y oﬀs of the pla y ers in the Nash equilibrium with the optimal so cial we lfare (or more generally the Nash equilibrium that op timizes some other smo oth fu n ction of the p lay ers’ pay oﬀs). 2.2 Small Probabilit y Games If all games in a class are guarante ed to hav e a Nash equilibr ium ( x, y ), where the nonzero prob- abilit y v alues in b oth x and y are larger than some constan t δ > 0, th en it is trivial to ﬁnd this equilibrium in time n O ( 1 δ ) b y exhaustive searc h ov er all p ossible sup p orts and linear programming. But, what if a class of games is known to h av e smal l , sa y O ( 1 n ), prob ab ility v a lues? Clearly , exhaus- tiv e searc h o ve r sup p orts is not eﬃcien t an ymore, since those ha v e n o w linear size. Sur prisingly , we sho w that an y such class of games h as a (randomized) PT AS, b y exploiting the tec hnique of [18]. 4 Deﬁnition 1 (Small Probabilit y Games) . A bi matrix game G = ( R, C ) is of δ -small p robabilities , for some c onstant δ ∈ (0 , 1] , i f it has a Nash e quilibrium ( x, y ) such that al l the entries of x and y ar e at most 1 δn . Remark 1. Observe that a game of δ -smal l pr o b abilities has an e quilibrium ( x, y ) , in which b oth x and y have supp or t of size at le ast δ n . M or e over, ther e exi sts a subset of size at le ast δn 2 fr o m the supp ort of x , al l the str a te gies of which have pr ob ability at le ast 1 2 n , and similarly for y ; that is the pr ob ability mass of the distributions x and y spr e ads non-trivial ly over a subset of size Ω( n ) of the str ate gies. Henc e, smal l pr ob ability games c omp rise a sub class of lin ear supp ort games , games with an e quilibrium ( x, y ) in which b oth x and y assign to a c onstant fr action αn of the str ate gies pr ob ability at le a st 1 /β n , for some c onstants α and β . However, this br o ader class of games is essential ly as har d as the gener al: take any Nash e quilibrium ( x, y ) of a normalize d game and deﬁne the p air ( x ′ , y ′ ) , wher e x ′ := (1 − ǫ 5 ) · x + ǫ 5 ·  1 n , 1 n , . . . , 1 n  and similarly for y ′ . It is not har d to se e that the new p air is an ǫ -N ash e quilibrium; stil l, r e gar d less of what ( x, y ) is, b oth x ′ and y ′ assign to αn str ate gies pr ob ability at le ast 1 /β n , for an appr opriate sele ction of α and β . Theorem 3. F or any δ ∈ (0 , 1] , ther e is a r ando mize d PT AS for normalize d bimatrix games of δ -smal l pr ob abilities. Pr o of. W e show ﬁr st the follo wing (stronger in terms of the type of approximati on) v arian t of the theorem of Lipton, Mark akis and Meh ta [18]. Th e pro of is p ro vided in App endix A.1. Lemma 1. L et G = ( R, C ) b e a normalize d bimatrix game and ( x, y ) a Nash e quilibrium of G . L et X b e the distribution forme d by taking t = ⌈ 16 log n/ǫ 2 ⌉ indep endent r andom samples fr om x and deﬁning the u niform distribution on the samples, and similarly deﬁne Y by taking samples fr om y . Then with pr ob ability at le ast 1 − 4 n the fol lowing ar e satisﬁe d 1. the p air ( X , Y ) is an ǫ -N ash e quilibrium of G ; 2. | e T i R Y − e T i Ry | ≤ ǫ/ 2 , for al l i ∈ [ n ] ; 3. |X T C e j − x T C e j | ≤ ǫ/ 2 , f or al l j ∈ [ n ] ; Supp ose now that we are give n a normalized bimatrix game G = ( R , C ) of δ -small probabilities, and let ( x, y ) b e an equilibrium of G in wh ic h x i ≤ 1 δn , for all i ∈ [ n ], an d s imilarly for y . Lemma 1 asserts that, if a multiset 2 A of s ize t = ⌈ 16 log n/ǫ 2 ⌉ is formed by taking t ind ep enden t samples from x and, similarly , a multiset B is formed by taking samp les f rom y , th en ( X , Y ), wher e X is the un iform d istribution o v er A and Y the u n iform distrib u tion ov er B , is an ǫ -Nash equilibrium with probability at least 1 − 4 /n . Of cour s e, we do not kn o w ( x, y ) so w e cann ot do the sampling pro cedure describ ed ab o v e. Instead we are going to tak e a uniformly random m ultiset A ′ and a uniformly random m ultiset B ′ and f orm the u niform distribu tions X ′ , Y ′ o v er A ′ and B ′ ; we will argue that th ere is an in v erse p olynomial c hance that ( X ′ , Y ′ ) is actually an ǫ -Nash equilibriu m. F or this w e d eﬁne the set A of go o d multisets for the r ow player as A := ( A A is a m ultiset, A ⊆ [ n ], | A | = t , the uniform distribution X o v er A satisﬁes Assertion 3 of Lemma 1 ) , 2 F or our d iscussion, a multiset of size t is an or der e d collection h i 1 , i 2 , . . . , i t i of t elements from some u niv erse (rep etitions are allo w ed). 5 and, s im ilarly , the set B of go o d multisets of the c olumn player as B := ( B B is a multiset, B ⊆ [ n ], | B | = t , the uniform d istribution Y o v er B satisﬁes Assertion 2 of Lemma 1 ) . The reason for deﬁn ing A and B in this wa y is that, giv en t w o multisets A ∈ A , B ∈ B , the un iform distributions X o v er A and Y o v er B comprise an ǫ -Nash equilibrium (see the pro of of L emma 1 for a ju stiﬁcatio n). What r emains to sho w is th at, with inv erse p olynomial probabilit y , a rand om m ultiset A ′ b elongs to A and a random multiset B ′ b elongs to B . T o show th is we lo w er b oun d th e cardinalities of the sets A and B via the follo wing claim, pro ve n in App end ix A.1. W e argue that the s u bset of A con taining elemen ts that could arise by samplin g x is large: ind eed, w ith pr obabilit y at least 1 − 4 n , a multiset sampled fr om x b elongs to A an d , moreov er, eac h individu al multiset has small probabilit y of b eing sampled, since x spreads the pr obabilit y mass appr o ximately ev enly on its supp ort. Claim 1. The sets A and B satisfy |A| ≥  1 − 4 n  ( δ n ) t and |B | ≥  1 − 4 n  ( δ n ) t . Giv en Claim 1, we can sho w Claim 2; the pro of is giv en in App endix A.1. Equation (1) implies that the algorithm that samples tw o un iformly random multisets A ′ , B ′ and forms the un iform probabilit y distributions X ′ and Y ′ o v er A ′ and B ′ resp ectiv ely , s ucceeds in ﬁndin g an ǫ -Nash equilibrium with probabilit y inv erse p olynomial in n . This completes the pro of of T h eorem 3. Claim 2. If X ′ , Y ′ ar e the u ni f orm distributions over r and om multisets A ′ and B ′ then Pr  ( X ′ , Y ′ ) is an ǫ -N ash e quilibrium  = Ω  δ 2 · n − 32 log(1 /δ ) /ǫ 2  . (1) 3 A Lo w er B ound for Bimatrix Games The tw o PT AS’s presente d in the previous section are oblivious . An oblivious algorithm lo oks at the game only to chec k if the v arious pairs of mixed strategies it h as come u p with (by enumeration or, more generally , by rand om sampling) are ǫ -app ro ximate, and is guaran teed to come up with one that is with probabilit y at least in v erse p olynomial in the game description. More formally , an oblivious algorithm for the Nash equilibrium p roblem is a distribution o v er pairs of m ixed strategies, indexed by the game size n and th e desired app ro ximation ǫ . It is a PT AS if for an y game the probabilit y that a pair dr a wn from the d istribution is an ǫ -Nash equilibrium is inv ersely p olynomial in n . Notice that, since we are ab out to prov e lo w er b ounds, we are opting for the generalit y of randomized oblivious algorithms—a d etermin istic algorithm that en umerates o v er a ﬁxed set of pairs of mixed strategies can b e seen as a (randomized) oblivious algorithm b y considering the uniform d istribution ov er th e set it enumerates ov er. The rather s urprising simplicit y and success of these algorithms (as w ell as th eir cousin s for anon ymous games, see the next s ection) raises the question: Is ther e an oblivious PT AS f or the general Nash equ ilibrium problem? W e show that the answer is negativ e. 6 Theorem 4. Ther e i s no oblivious PT A S for the N ash e quilibrium i n bimatrix games. Pr o of. W e construct a sup er-p olynomially large family of n × n games with the p rop ert y that ev ery t w o games in the family do not share an ǫ -Nash equilibrium. This quic kly leads to the pro of of th e theorem. Our construction is based on a construction by Alth¨ ofer [1], except th at we need to pay more atten tion to ens ure th at the ǫ -Nash equ ilibria of the games w e construct are “close” to the exact Nash equilibria. F or ℓ ev en and n =  ℓ ℓ/ 2  , w e deﬁ ne a family of n × n t wo-pla y er games G S = ( R S , C S ), indexed by all subsets S ⊆ [ n ] with | S | = ℓ . Letting { S 1 , S 2 , . . . , S n } b e the set of all subsets of S with cardinalit y ℓ/ 2, w e imagine that column j of the game G S corresp onds to sub set S j . Then, for ev ery j , w e ﬁll column j of the pa yo ﬀ matrices R S and C S as follo ws: • for all i / ∈ S , R S,ij = − 1 and C S,ij = 1; • for all i ∈ S j , R S,ij = 1 and C S,ij = 0; and • for all i ∈ S \ S j , R S,ij = 0 and C S,ij = 1. In other w ords, our construction has tw o comp onen ts: In the ﬁrst comp onen t (deﬁn ed by the ro ws lab eled with the elemen ts of S ), th e game is 1-sum, whereas in the second (corresp ond ing to th e complemen t of S ) the game is 0-sum w ith th e ro w p la y er alwa ys getting p a y oﬀ of − 1 and the column p la y er alwa ys getting pay oﬀ of 1. The pa yo ﬀs of the ﬁrst comp onent are more balanced in the follo wing w a y: as we said, ev ery co lumn corresp onds to a subset S 1 . . . , S n of S of ca rd inalit y ℓ/ 2; if the column pla y er c ho oses column j , then the ro w pla y er gets 1 for c ho osing a ro w corresp onding to an element of S j and 0 for a ro w corresp onding to an elemen t in S \ S j . See Figure 1 of App endix A.2 for an illustration of R S for the case n = 6, ℓ = 4, S = { 1 , 2 , 3 , 4 } . Lemma 2 pro vides th e follo wing c haracterization of the approxima te equ ilibria of the game G S : in an y ǫ -Nash equilibriu m ( x, y ), the strategy x of the row pla yer must h a v e ℓ 1 distance at most 8 ǫ from u S —the uniform distribution o v er the set S . That is, in all approximat e equilibria of the game the strategy of the row pla y er m ust b e close to th e unif orm distribu tion o v er S . F ormally , Lemma 2. L et ǫ < 1 . If ( x, y ) is an ǫ -Nash e quilibrium of the g ame G S , wher e x is the mixe d str ate gy of the r ow player and y that of the c olumn player, then the fol lowing pr o p erties ar e satisﬁe d by x : 1. x i = 0 , for al l i / ∈ S ; 2. ℓ 1 ( x, u S ) ≤ 8 ǫ , wher e u S is the u ni f orm distribution over S , and ℓ 1 ( x, u S ) r epr esents the ℓ 1 distanc e b etwe en distributions x , u S . The pro of of the ﬁrst assertion is straightfo rward: the row pla y er will not assign an y probability mass to the rows which give her − 1, sin ce an y r o w in S w ill guaran tee h er at least 0. Since all the activit y happ ens th en in the ﬁr st comp onent of th e game, whic h is 1-sum, an av eraging argument implies that b oth p la y ers’ pa y oﬀ is ab out 1 / 2 at equilibr iu m. O bserv e further that, for a giv en mixed str ateg y of the ro w pla y er, the str ategy of the column play er that guaran tees h er the h ighest pa y oﬀ is th e sub set cont aining the ℓ/ 2 elemen ts of S to wh ic h the r o w pla ye r assigns the smallest probabilit y m ass. Hence, if the pr obabilit y distr ib ution of the row play er were far fr om un iform, then, contrary to what w e argued, the corresp onding p a y oﬀ for the column play er w ould b e larger than 1 / 2—this is established via a delicate geometric argument for ℓ 1 distances of probabilit y measures. See Lemma 3 in App end ix A.2. Supp ose no w that there is an oblivious PT AS for the Nash equilibrium, that is, a distrib ution D o ve r pairs of mixed strategi es su c h that, for an y game G S , the p robabilit y that an ǫ -appro ximate 7 Nash equilibr ium is sampled is in ve rse p olynomial in n . Let us consider the pr obabilit y d istr ibution D R induced b y D on th e mixed strategies of the row pla y er and d en ote by B S the ℓ 1 ball of radius 8 ǫ around u S . Lemma 2 implies that D R should b e assigning probability mass at least inv erse p olynomial in n to eac h b all B S , S ⊆ [ n ], | S | = ℓ . This is imp ossib le, since by the f ollo wing claim there is a su p er-p olynomial n umber of disjoint suc h balls. Th e pro of of the claim is via a counti ng argumen t. See App en dix A.2 . Claim 3. Ther e is a family of Ω  n ( . 8 − 34 ǫ ) log 2 n  disjoint such b al ls. The p ro of of Theorem 4 implies in particular that an y oblivious PT AS for general t w o-pla y er games requires exp ected runnin g time of at least Ω  n ( . 8 − 34 ǫ ) log 2 n  . Compare this b ound to the n O (log n/ǫ 2 ) upp er b ound ob tained b y Lipton, Mark akis and Meht a [18]. 4 A Lo w er B ound for Anonymous G ames Recall the deﬁnition of anonymous games from Section 1.1. In [13] we give a PT AS for anon ymous games with tw o strategies, ru nning in time n O ((1 / ǫ ) 2 ) , and in [7 ] a more eﬃcient one, with ru n ning time n O (1) · (1 /ǫ ) O ((1 / ǫ ) 2 ) . (In [14] we also giv e a muc h more soph isticat ed PT AS for anonymous games with more than t w o strategies). All these PT AS’s ha ve 1 ǫ in the exp onen t of the r unning time, and th ey w ork ev en if there is a ﬁxed num b er t of t yp es (in whic h case t m ultiplies the exp onen t). F urthermore, it turns out that all of these algorithms are oblivious , in a s p ecialize d sense app ropriate for anonymous games d eﬁned next. An oblivious ǫ -appr oximation algorithm for anonymous games with n players is deﬁned in terms of a d istribution, ind exed b y n , on unor der e d n -tuples of mixe d str ate gies . 3 The algorithm sam- ples from this d istribution, and for eac h { p 1 , . . . , p n } samp led, it determines wh ether there is an assignmen t of these probabilities to the n p la y ers such that the resulting strategy proﬁle (with eac h pla y er playing strategy 1 w ith the assigned probabilit y) is an ǫ -app ro ximate Nash equilibrium; this latter test can b e carried out by max-ﬂo w tec hniques, see, e.g., [13]. Th e exp ected runnin g time of this app ro ximation algorithm is then the inv erse of its p robabilit y of success. W e show the f ollo wing result, implying that an y oblivious ǫ -appro ximation algorithm for anon y- mous games w hose exp ected r unning time is p olynomial in th e num b er of play ers must hav e ex- p ected runn in g time exp onen tial in ( 1 ǫ ) 1 / 3 . Hence, our PT AS from [7] is essentiall y optimal among oblivious PT AS’s. Theorem 5. F or any c onstant c ≥ 0 , ǫ < 1 , no oblivious ǫ -appr oximation algorithm for anony mous games with 2 str ate gi es and 3 player typ es has pr o b ability of suc c e ss lar ger than n − c · 2 − Ω(1 /ǫ 1 / 3 ) . W e only ske tc h the p r oof next and p ostp one th e d etai ls for App end ix B. W e ﬁrst sho w the follo wing (see Theorem 10 in Ap p endix B.1): giv en an y ordered n -tuple ( p 1 , . . . , p n ) of probabilities, w e can construct a norm alized anonymous game with n p lay ers of type A, and t w o more pla ye rs of their o wn t yp e, suc h th at in any ǫ -Nash equilibr ium of the game the i -th pla ye r of t yp e A pla ys strategy 1 with probability v ery close (dep end ing on ǫ and n ) to p i . T o obtain this game, we need to understand ho w to exploit the diﬀerence in the pa y oﬀ f unctions of the pla ye rs of t yp e A to enforce diﬀeren t b eha viors at equ ilibrium, despite th e fact that in all other asp ects of the game the pla ye rs of group A are ind istinguishable. 3 F or ordered sets of mixed strategies, th e lo w er b ound we are ab out to show b ecomes trivial and un in teresting. 8 The construction is based on the follo wing idea: F or all i , let u s denote by µ − i := P j 6 = i p j the tar get exp ected num b er of t yp e-A pla ye rs d iﬀeren t than i who pla y strategy 1; and let u s giv e th is pa y oﬀ to pla ye r i if she pla ys strategy 0, regardless of wh at the other p la y ers are doing. If i c ho oses 1, we giv e her pay oﬀ t where t is the num b er of pla ye rs diﬀerent th an i who p la y 1. By setting the pa y oﬀs in this wa y we ensu r e that ( p 1 , . . . , p n ) is in fact an equilibrium , since for every play er the pa y oﬀ she gets from strategy 0 matc hes the exp ected pay oﬀ sh e gets f rom strategy 1. How ev er, enforcing that ( p 1 , . . . , p n ) is also the unique equilibriu m is a more challe nging task; and to do this w e need to include t w o other pla y ers of their o wn type: we u se these p la y ers to ens u re that the sum of the mixed stategies of the p la y ers of t yp e A matc hes P p i at equilibrium, so that a p la y er i d eviating from h er prescrib ed strategy p i is pu shed bac k to w ards p i . W e sho w h o w this can b e done in App endix B.1. W e also provi de guarantees for the ǫ -Nash equilibria of th e resulting game. The constru ctio n outlined ab o v e enables us to deﬁne a family of 2 Ω(1 /ǫ 1 / 3 ) anon ymous games with th e prop ert y that no t w o games in the family share an ǫ -Nash equilibriu m , ev en as an unordered set of mixed strategies (Claims 7 and 8). T hen, b y an av eraging argumen t, we can deduce that for an y oblivious algorithm th er e is a game in the ensem ble for whic h the probabilit y of success is at most 2 − Ω(1 /ǫ 1 / 3 ) . It is imp ortan t for our construction to w ork that the anon ymous game deﬁned for a giv en collectio n of p robabilit y v alues ( p 1 , p 2 , . . . , p n ) do es not deviate to o muc h from the pr escrib ed set of mixed strategies p 1 , p 2 , . . . , p n ev en in an ǫ -Nash equilibrium. T he b ou n d of 2 Ω(1 /ǫ 1 / 3 ) emerges from a quantiﬁca tion of this deviation as a fu nction of ǫ . The p ro of of Th eorem 5 is giv en in App endix B.2. Remark 2. We c an show an e quivalent of The or em 5 for oblivious ǫ - appr oximation algorithms for anonymous games with 2 -player typ es and 3 str ate gi e s p er player. The details ar e omitte d. 5 A Quasi-p olynomial PT AS W e circumv en t the lo w er b ound of the previous sectio n by pro viding a PT AS for anon ymous games with run ning time p olynomial in the num b er of pla ye rs n times a factor of ( 1 ǫ ) O (log 2 1 ǫ ) . The PT AS is, of course, non-oblivious, and in fact in the follo wing interesting w a y: Instead of enumerating a ﬁxed set of un ordered collections of pr obabilit y v alues, we enumerate a ﬁxed set of log(1 /ǫ ) -tuples, r ep r esenting the ﬁrst log (1 /ǫ ) moments of these pr ob ability values. W e can think of the these momen ts as mor e suc cinct aggr e gates of mixed strategy proﬁles th an the unord er ed collections of strategies consid ered in [7, 13], since seve ral of these collectio ns m ay share the same moments. T o put the idea into con text, let us recall the follo wing th eorem. Theorem 6 ([7 , 8]) . F or every ǫ > 0 , ther e exists some inte ger k = O (1 /ǫ ) such that for any n -player 2 -str ate gy anonymous game with p ayoﬀs in [ − 1 , 1] ther e is an ǫ -Nash e quilibrium such that 1. either at most k 3 players r ando mize, and their mixe d str ate gie s ar e inte ger multiples of 1 /k 2 ; 2. or al l players who r an domize use the same mixe d str ate gy, and this str ate gy is an inte ger multiple of 1 k n . This stru ctural result can b e tur ned in to an oblivious PT AS using max-ﬂo w arguments (see App endix D.1 for details). A t its heart the pro of of th e theorem r elies on the follo wing intuitiv e fact ab out sums of ind icat or random v ariables: If two su ms of indep endent indic at ors have close me a ns and v arianc es, then their total variatio n distanc e should b e smal l. The w a y this fact b ecomes 9 relev ant to anon ymous games is that, if th ere are 2 str ateg ies p er pla y er, then the mixed str ateg y of a play er can b e describ ed by an indicator random v a riable; and as it tur ns out, if we replace one set of indicators b y another, the c hange in p a y oﬀ that every play er will exp erience is b oun ded b y the total v ariation distance b et w een the sum of the indicators b efore and after the c hange. Nev ertheless, the b ound obtained b y approximat ing the ﬁ rst t w o moments of the sum of the Nash equilibrium strategies is weak, and the space of un ordered sets of pr obabilit y v alues th at we need to search o ve r b ecomes exp onential in 1 /ǫ . T o obtain an exp onen tial pru ning of the search space, we turn to higher moments of th e Nash equilibrium. W e s ho w the follo wing theorem, whic h pr ovides a rather strong quant iﬁcation of ho w the total v ariati on distance b etw een t w o su ms of in dicato rs d ep ends on the num b er of their ﬁr st momen ts that are equal. Theorem 7. L et P := ( p i ) n i =1 ∈ (0 , 1 / 2] n and Q := ( q i ) n i =1 ∈ (0 , 1 / 2] n b e two c ol le ctions of pr ob a- bility values in (0 , 1 / 2 ] . L et also X := ( X i ) n i =1 and Y := ( Y i ) n i =1 b e two c ol le ctions of indep endent indic at ors with E [ X i ] = p i and E [ Y i ] = q i , for al l i ∈ [ n ] . If for some d ∈ [ n ] the fol lowing c onditio n is satisﬁe d : ( C d ) : n X i =1 p ℓ i = n X i =1 q ℓ i , for al l ℓ = 1 , . . . , d, then           X i X i ; X i Y i           ≤ 20( d + 1) 1 / 4 2 − ( d +1) / 2 . (2) Condition ( C d ) considers the p o wer su ms of the exp ectations of the ind icato r r andom v ariables. Using the theory of symmetric p olynomials we can show that ( C d ) is equiv alen t to the follo w- ing condition on the momen ts of th e sum s of the in dicators (for the pro of see Prop osition 1 in App endix C): ( V d ) : E   n X i =1 X i ! ℓ   = E   n X i =1 Y i ! ℓ   , for all ℓ ∈ [ d ] . Theorem 7 p ro vides then th e follo wing s trong appro ximation guaran tee for su ms of indicator random v aria bles, that w e think should b e imp ortant in other settings. Ou r result is related to the classica l momen t method in probability theory [15], but to our kn o wledge it is no ve l and signiﬁcantly stronger than kn o wn resu lts: If two sums of indep endent indic ato rs with exp e ctations b ounde d by 1/2 have e qual ﬁrst d moments, then their total variation distanc e is 2 − Ω( d ) . It is imp ortan t to note that our b ound in (2) d oes not rely on summing up a large num b er of indicators n . This is quite critical since the previous tec hniques b r eak down for small n ’s—for large n ’s Berry-Ess´ een t yp e b ounds are suﬃcient to obtain strong guarantee s (this is th e heart of the probabilistic results used in [13, 7]). The pro of of Theorem 7 (and its complemen t for probabilit y v alues in [1 / 2 , 1)) is giv en in App endix C . It pro ceeds by expressin g the distribution of the sum of n indicators, with exp ecta tions p 1 , . . . , p n , as a we igh ted sum of the binomial distribu tion B ( n, p ) (with p = ¯ p = P p i /n ) and its n d er iv ativ es with r esp ect to p , at the v alue p = ¯ p (these deriv ativ es corresp ond to ﬁnite signed measures). It turns out that the co eﬃcien ts of the ﬁ rst d terms of this expans ion are sy m metric p olynomials with resp ect to the probabilit y v alues p 1 , . . . , p n , of degree at most d ; hence, from the theory of symmetric p olynomials, eac h of these co eﬃcien ts can b e written as a fu nction of the p o w er-sum symmetric p olynomials P i p ℓ i , ℓ = 1 , . . . , d (see, e.g., [22]). So, if t wo su m s of in dicators 10 satisfy Condition ( C d ), the ﬁr st terms cancel, and the total v a riation distance of the t wo sum s dep ends only on the other terms of th e expansion (those corresp ond ing to higher deriv at ive s of the binomial d istr ibution). The p ro of is concluded by sh o wing that the joint con tribution of these terms is inv erse p olynomial in 2 Ω( d ) . (The details are giv en in App end ix C). Our algorithm, sh o wn b elo w, exploits th e strong appr o ximatio n guaran tee of Theorem 7 to impro ve up on the algorithm of [7] in the case where only O (1 /ǫ 3 ) play ers mix at the ǫ -Nash equi- librium (this corresp onds to case 1 in Theorem 6). Th e complement ary case (case 2 in Theorem 6 ) can b e tr eated easily in time p olynomial in n and 1 /ǫ by exhaustiv e searc h and max-ﬂow arguments (see App endix D.1). Algorithm Moment S earch Input: An anon ymous game G , the desired approxima tion ǫ . Output: An ǫ -Nash equilibrium of G in whic h all probabilities are integ er m ultiples of 1 k 2 , where k = ⌈ c ǫ ⌉ and c is u niv ersal (indep end en t of n ) constan t, determined by Theorem 6. F or tec hnical reasons th at will b e clear shortly , w e c ho ose a v alue for k th at is by a factor of 2 larger than the v al ue required b y Th eorem 6; this is the v alue k that guaran tees an ǫ /2-Nash equilibrium in m ultiples of 1 /k 2 . Finally , we assume th at we ha v e already p erformed the s earc h corresp ond ing to Case 2 of Theorem 6 for this v alue of k and we hav e not found an ǫ/ 2-Nash equilibriu m. So there m ust exist an ǫ/ 2-Nash equilibrium in w hic h at most k 3 pla y ers rand omize in inte ger multiples of 1 /k 2 . 1. Guess in tegers t 0 , t 1 , t s , t b ≤ n , t s + t b ≤ k 3 , where t 0 pla y ers will pla y pure strategy 0, t 1 will pla y pure strategy 1, t s will mix with probabilit y ≤ 1 2 , and t b = n − t 0 − t 1 − t s will mix with pr obabilit y > 1 2 . (Note that w e ha v e to handle lo w and h igh probabilities separately , b ecause Theorem 7 only applies to indicators with exp ectations in (0 , 1 / 2]; we handle ind icators with exp ectations in (1 / 2 , 1) by taking their complement and employing Theorem 7 —see Corollary 1.) 2. F or d = ⌈ 3 log 2 (320 / ǫ ) ⌉ , guess µ 1 , µ 2 , . . . , µ d , µ ′ 1 , µ ′ 2 , . . . , µ ′ d , wh ere, for all ℓ ∈ [ d ]: µ ℓ ∈ ( j  1 k 2  ℓ : t s ≤ j ≤ t s  k 2 2  ℓ ) , and µ ′ ℓ ∈ ( j  1 k 2  ℓ : t b  k 2 2 + 1  ℓ ≤ j ≤ t b ( k 2 − 1) ℓ ) . F or all ℓ , µ ℓ represent s the ℓ -p o w er sum of the mixed strategies of the p lay ers who mix and c ho ose strategies from the set { 1 /k 2 , . . . , 1 / 2 } . Similarly , µ ′ ℓ represent s the ℓ -p o wer sum of the m ixed strategies of the p la y ers who mix and c ho ose strategies f r om the set { 1 / 2 + 1 /k 2 , . . . , ( k 2 − 1) /k 2 } . Remark: Whether there actually exist probabilit y v alues π 1 , . . . , π t s ∈ { 1 /k 2 , . . . , 1 / 2 } and θ 1 , . . . , θ t b ∈ { 1 / 2 + 1 /k 2 , . . . , ( k 2 − 1) / k 2 } such that µ ℓ = P t s i =1 π ℓ i and µ ′ ℓ = P t b i =1 θ ℓ i , for all ℓ = 1 , 2 , . . . , d , will b e determined later. 3. F or eac h pla y er i = 1 , . . . , n , ﬁ nd a su bset S i ⊆  0 , 1 k 2 , . . . , k 2 − 1 k 2 , 1  of p ermitted strategies for th at p lay er in an ǫ 2 -Nash equilibriu m, conditioned on the guesses in the previous steps. By this, w e mean determinin g the answer to the follo wing: “Give n our 11 guesses for the aggrega tes t 0 , t 1 , t s , t b , µ ℓ , µ ′ ℓ , f or all ℓ ∈ [ d ], what m ultiples of 1 /k 2 could pla y er i b e pla ying in an ǫ/ 2-Nash equilibrium ?” Our test exploits the anon ymit y of the game and uses T heorem 7 to ac hiev e the follo wing: • if a multi ple of 1 /k 2 can b e assigned to play er i and complemente d b y c hoices of m ultiples for the other pla y ers, so that the agg regate conditions are satisﬁed and pla y er i is at 3 ǫ/ 4- b est resp onse (that is, sh e exp eriences at most 3 ǫ/ 4 regret), then this multiple of 1 /k 2 is included in the set S i ; • if, giv en a multiple of 1 /k 2 to play er i , there exists no assignment of m ultiples to the other play ers so that the aggregate conditions are satisﬁed and play er i is at 3 ǫ/ 4-b est resp onse, th e multiple is rejected from set S i . Observe that the v alue of 3 ǫ/ 4 for the regret used in th e classiﬁer is inten tionally c hosen midwa y b et w een ǫ/ 2 and ǫ . The reason for this v alue is that, if we only matc h the ﬁrst d momen ts of a mixed strategy pr oﬁle, our estimati on of the real regret in that strategy proﬁle is distorted by an additiv e err or of ǫ / 4 (coming f rom (2) and the choic e of d ). Hence, with a threshold at 3 ǫ/ 4 w e mak e sure that: a. we are not going to “miss” th e ǫ/ 2-Nash equilibrium (that we kn o w exists in multiples of 1 /k 2 b y virtue of our choic e of a larger k ), and b. any strategy pr oﬁle that is consisten t with the aggregate conditions and the sets S i found in this step is going to ha v e regret at most 3 ǫ/ 4 + ǫ/ 4 = ǫ . The fairly in v olv ed details of our test are giv en in App end ix D.2, and the wa y its analysis ties in with th e search for an ǫ -Nash equilibrium is giv en in the pro ofs of Claims 9 and 10 of App endix D.3. 4. Find an assignment of mixed strategies v 1 ∈ S 1 , . . . , v n ∈ S n to pla ye rs, su c h that: • t 0 pla y ers are assigned v alue 0 and t 1 pla y ers are assigned v alue 1; • t s pla y ers are assigned a v alue in (0 , 1 / 2] and P i : v i ∈ (0 , 1 / 2] v ℓ i = µ ℓ , for all ℓ ∈ [ d ]; • t b pla y ers are assigned a v alue in (1 / 2 , 1) and P i : v i ∈ (1 / 2 , 1) v ℓ i = µ ′ ℓ , for all ℓ ∈ [ d ]. Solving this assignment problem is n on-trivial, but it can b e done by dynamic programming in time O ( n 3 ) ·  1 ǫ  O (log 2 (1 /ǫ )) , b ecause the sets S i are sub sets of { 0 , 1 /k 2 , . . . , 1 } . The algorithm is giv en in th e pro of of Claim 11 in App endix D.3. 5. If an assignment is found , then th e vec tor ( v 1 , . . . , v n ) constitutes an ǫ -Nash equilibr ium. Theorem 8. Momen t S ear ch is a PT AS for n -player 2 -str ate gy anonymous games with running time U · p oly( n ) · (1 /ǫ ) O (log 2 (1 /ǫ )) , wher e U is the numb er of bits r e quir e d to r epr esent a p ayo ﬀ value of the game. The algorithm gener alizes to a c onstan t numb er of player typ es with the numb er of typ es multiplying the exp o nent of the running time. Sk etc h: Correctness follo ws from this observ a tion: The resu lts in [7] and the c hoice of k guarante e that an ǫ 2 -appro ximate Nash equilibrium in discretized probabilit y v alues exists; therefore, S tep 3 will ﬁn d non-empty S i ’s for all play ers (for some guesses in S teps 1 and 2, sin ce in particular the ǫ/ 2-Nash equilibr iu m w ill sur v ive th e tests of Step 3—b y Theorem 7 an d the c hoice of d , at m ost ǫ/ 4 accuracy is lost if the correct v a lues for the momen ts are guessed); and th us Step 4 will ﬁn d an ǫ -appro ximate Nash equilibr iu m (another ǫ/ 4 m igh t b e lost in this step). The f u ll pr oof and the runn in g time analysis are pro vided in App end ix D.3. 12 6 A Sparse ǫ -co v er for Sums of Indicators A b ypr o du ct of our pro of is sho wing the existence of a sparse (and eﬃcien tly computable) ǫ -co v er of the set of sums of in d ep enden t indicators, und er the total v aria tion distance. T o state our co v er theorem, let S := {{ X i } i | X 1 , . . . , X n are indep endent in dicators } . W e show the follo wing. Theorem 9 (Co v er for sum s of in d icato rs) . F or al l ǫ > 0 , ther e exists a set S ǫ ⊆ S such that (i) |S ǫ | ≤ n 3 · O (1 /ǫ ) + n ·  1 ǫ  O (log 2 1 /ǫ ) ; (ii) F or e very { X i } i ∈ S ther e exists some { Y i } i ∈ S ǫ such that d TV ( P i X i , P i Y i ) ≤ ǫ ; and (iii) the set S ǫ c an b e c onstructe d in time O  n 3 · O (1 /ǫ ) + n ·  1 ǫ  O (log 2 1 /ǫ )  . Mor e over, if { Y i } i ∈ S ǫ , then the c ol le ction { Y i } i has one of the fol lowing forms, wher e k = k ( ǫ ) = O (1 /ǫ ) is a p ositive i nte ger: • (Sp ar se F o rm) Ther e is a value ℓ ≤ k 3 = O (1 /ǫ 3 ) such that for al l i ≤ ℓ we have E [ Y i ] ∈ n 1 k 2 , 2 k 2 , . . . , k 2 − 1 k 2 o , and for al l i > ℓ we have E [ Y i ] ∈ { 0 , 1 } . • ( k -he avy Binomial F orm) Ther e is a value ℓ ∈ { 0 , 1 , . . . , n } and a value q ∈  1 k n , 2 k n , . . . , k n − 1 k n  such that for al l i ≤ ℓ we have E [ Y i ] = q ; for al l i > ℓ we have E [ Y i ] ∈ { 0 , 1 } ; and ℓ, q satisfy the b ounds ℓq ≥ k 2 − 1 k and ℓq (1 − q ) ≥ k 2 − k − 1 − 3 k . Pro of of Theorem 9: Da sk alakis [8] establishes the same th eorem, except that the s ize of the co v er he pro duces, as well as the time needed to pro du ce it, are n 3 · O (1 /ǫ ) + n ·  1 ǫ  O (1 /ǫ 2 ) . Indeed, this b ound is obtained by enumerating ov er all p ossible collections { Y i } i in s p arse form and all p ossible collections in k -hea vy Binomial F orm, for k = O (1 /ǫ ) sp eciﬁed by the theorem. I ndeed, the tota l num b er of collectio ns in k -hea vy Binomial form is at most ( n + 1) 2 nk = n 3 · O (1 /ǫ ), since there are at most n + 1 c hoices for the v alue of ℓ , at most k n c hoices for the v alue of q , and at most n + 1 c hoices for the num b er of v ariables ind exed b y i > ℓ that ha v e exp ectation equal to 1 (the p recise su bset of these that hav e exp ectation 1 is not imp ortan t, since this d oes n ot aﬀect the distr ib ution of P i Y i ). On the other hand , the num b er of collectio ns { Y i } i in sp arse form is at most ( k 3 + 1) · k 3 k 2 · ( n + 1) = n ·  1 ǫ  O (1 /ǫ 2 ) , since there are k 3 + 1 c hoices for ℓ , k 3 k 2 c hoices for the exp ectatio ns of v ariables Y 1 , . . . , Y ℓ up to p ermuta tions of the indices of th ese v ariables (namely w e need to c ho ose ho w many of these ℓ v ariables hav e exp ectation 1 /k 2 , ho w many h a v e exp ectation 2 /k 2 , etc.), and at most n + 1 c hoices for the num b er of v ariables indexed b y i > ℓ that ha ve exp ectatio n equal to 1. T o imp ro v e on the size of the co v er w e sho w that we can r emo v e from the aforement ioned co v er a large fraction of collec tions in sparse form. In particular, w e shall only k eep n ·  1 ǫ  O (log 2 1 /ǫ ) collect ions in sparse form, making us e of Theorem 7. Ind eed, consider a collection Y = { Y i } i in sparse f orm and let L Y = { i | E [ Y i ] ∈ (0 , 1 / 2] } ⊆ [ n ], R Y = { i | E [ Y i ] ∈ (1 / 2 , 1) } ⊆ [ n ]. Theorem 7 implies that, if we compare { Y i } i with another collecti on { Z i } i satisfying the follo wing: X i ∈L E [ Y i ] t = X i ∈L E [ Z i ] t , for all t = 1 , . . . , d ; (3) X i ∈R E [ Y i ] t = X i ∈R E [ Z i ] t , for all t = 1 , . . . , d ; (4) E [ Y i ] = E [ Z i ] , for all i ∈ [ n ] \ ( L ∪ R ) , (5 ) then d TV ( P i Y i , P i Z i ) ≤ 2 · 20( d + 1) 1 / 4 2 − ( d +1) / 2 . In p articular, f or some d ( ǫ ) = O (log 1 /ǫ ), this b ound b ecomes at most ǫ . 13 F or a collection Y = { Y i } i , w e deﬁn e the moment pr oﬁle m Y of the collection to b e a (2 d ( ǫ ) + 1)- dimensional v ector m Y =   X i ∈L Y E [ Y i ] , X i ∈L Y E [ Y i ] 2 , . . . , X i ∈L Y E [ Y i ] d ( ǫ ) ; X i ∈R Y E [ Y i ] , . . . , X i ∈R Y E [ Y i ] d ( ǫ ) ; |{ i | E [ Y i ] = 1 }|   . By the previous discussion, for t wo collections Y = { Y i } i and Z = { Z i } i , if m Y = m Z then d TV ( P i Y i , P i Z i ) ≤ ǫ . No w give n the ǫ -co v er pro du ced in [8] we p erform the follo wing sp arsiﬁcation op eration: for ev ery p ossible moment ve ctor that can arise from a collectio n { Y i } i in sp arse form, we only k eep in our co ve r one collection w ith such momen t vecto r. Th e co v er resu lting fr om the sparsiﬁcation op eration is a 2 ǫ -co ve r, since th e sparsiﬁcation loses us an additiv e ǫ in total v ariatio n distance, as argued ab ov e. W e now compu te the size of the n ew co v er. Th e total num b er of m omen t vec tors arising from sparse-form collections of indicators is at most k O ( d ( ǫ ) 2 ) · ( n + 1). Indeed, consider a collect ion Y in sparse form. T here are at most k 3 + 1 choic es for |L Y | , at most k 3 + 1 choic es for |R Y | , and at most ( n + 1) choic es for |{ i | E [ Y i ] = 1 }| . W e claim next that the total num b er of p ossible vec tors   X i ∈L Y E [ Y i ] , X i ∈L Y E [ Y i ] 2 , . . . , X i ∈L Y E [ Y i ] d ( ǫ )   is at most k O ( d ( ǫ ) 2 ) . In d eed, for all t = 1 , . . . , d ( ǫ ), P i ∈L Y E [ Y i ] t ≤ |L Y | and it m ust b e a m ultiple of 1 /k 2 t . So the total n umb er of p ossible v alues for P i ∈L Y E [ Y i ] t is at most ( k 2 t |L Y | + 1) ≤ ( k 2 t k 3 + 1). It’s easy to see then that the num b er of p ossible moment vec tors   X i ∈L Y E [ Y i ] , X i ∈L Y E [ Y i ] 2 , . . . , X i ∈L Y E [ Y i ] d ( ǫ )   is at m ost d ( ǫ ) Y t =1 ( k 2 t k 3 + 1) ≤ k O ( d ( ǫ ) 2 ) . The s ame upp er b ound applies to the total num b er of p ossib le momen t v ectors   X i ∈R Y E [ Y i ] , X i ∈R Y E [ Y i ] 2 , . . . , X i ∈R Y E [ Y i ] d ( ǫ )   . It follo w s then that the tota l num b er of sparse-form coll ections of indicators that we h a v e k ept in our co v er after the s parsiﬁcation op eration is at most k O ( d ( ǫ ) 2 ) · ( n + 1) = n ·  1 ǫ  O (log 2 1 /ǫ ) . The n umber of collections in heavy Binomial f orm that we ha v e in our co v er is the same as b efore and hence it is at m ost n 3 · O (1 /ǫ ). So the size of the sp arsiﬁed co v er is at m ost n 3 · O (1 /ǫ ) + n ·  1 ǫ  O (log 2 1 /ǫ ) . T o ﬁnish the p ro of it remains to argue that we d on ’t actually n eed to pro du ce the co v er of [8] and sub sequen tly sparsify it to obtain our co v er, but we can pro du ce it directly in time n 3 · O (1 /ǫ ) + n ·  1 ǫ  O (log 2 1 /ǫ ) . W e claim that given a momen t vec tor m we can compute a collectio n Y = { Y i } i suc h that m Y = m , if su c h a collection exists, in time  1 ǫ  O (log 2 1 /ǫ ) . Th is follo ws f rom Claim 11 in 14 App endix D.3. 4 Hence, our algorithm pr od ucing the co v er enumerates o v er all p ossible moment v ectors and for eac h moment v ector inv ok es Claim 11 to ﬁnd a consisten t sparse collectio n of indicators, if suc h collection exists, adding that collectio n into th e co v er. Then it en umer ates o v er collect ions of ind icato rs in hea vy Binomial form and adds them to the co ve r. The ov erall ru nning time is as pr omised.  7 Discussion and Op en P roblems The m ystery of PT AS for Nash equilibria deep ens. There are simple algorithms for in teresting sp ecial cases well within r eac h, and in fact w e ha v e seen that the existence of a PT AS is not incompatible w ith PP AD-complete ness. But oblivious algorithms cannot tak e us all the w a y to the co v eted PT AS for the general case. In the imp ortan t sp ecial case of anonymous games, the approac h of [13, 14, 7 ] — by design in v olving oblivious algorithms — hits a b ric k wall of ( 1 ǫ ) 1 ǫ α , but then a more elab orate p robabilistic result ab out m oments and Bernoulli sums breaks th at barrier. Pseudop olynomial b ounds , familiar from [18], sho w up in appro ximation algorithms for anonymous games as well. Man y op en problems remain, of cour se: • Is th ere a PT AS for Nash equilibria in general games? A PT AS for b im atrix games that exploits the lin ear pr ogramming-like nature of the problem w ould not b e u nthink able. • Find a truly p r actic al, and hop efu lly ev o cativ e of strategica lly in teracting crowds, PT AS for anon ymous games with t wo strategies. • Pr o v e that ﬁnd ing an exact Nash equilibrium in an anonymous game with a ﬁnite n umb er of strategies is PP AD-complete . • Find a PT AS for 2-strategy grap h ical games — the other imp ortan t class of m ulti-pla y er games. • Alternative ly , it is not unt hin k able that th e graphical games sp ecial case ab o v e is PP AD- complete to app ro ximate suﬃcien tly close. References [1] I. Alth¨ ofer. O n sparse approximat ions to ran d omized strategie s and co nv ex com binations. Line ar Algebr a and Applic ations, 199:339–35 5, 1994. [2] M. Blonski. Anonymous Games with Binary Actions. Games and Ec onomic Behavior , 28(2): 171–1 80, 1999. [3] M. Blonski. The women of Cairo: Equ ilibr ia in large anon ymous games. Journal of Mathe- matic al Ec onomics, 41(3): 253–264, 2005. [4] H. Bosse, J . Byrk a and E. Mark akis. New Algorithms for Approxima te Nash E quilibria in Bimatrix Games. WINE , 2007. 4 A n aiv e application of Claim 11 results in running time n 3 ·  1 ǫ  O (log 2 1 /ǫ ) . Ho wev er, w e can p roceed as foll ows: w e can guess |L Y | and |R Y | (at most O ( k 6 ) guesses) and inv oke Claim 11 with m 0 = m 1 = 0, m = 1, m s = |L Y | and m b = |R Y | just to ﬁnd { Y i } i ∈L Y ∪R Y . T o obtain th e sough t after Y = { Y i } i w e then add m 2 d ( ǫ )+1 indicators with exp ectation 1 and make the remaining indicators 0. 15 [5] X. Chen and X. Deng. Settling the Comp lexit y of Two-Pla y er Nash Equilibrium. FOCS , 2006. [6] X. Chen, X. Deng and S.-H. T eng. Sp arse Games Are Hard. WINE , 2006. [7] C. Dask alakis. An Eﬃcien t PT AS for Two -Strategy Anonymous Games. WINE , 2008. [8] C. Dask alakis. An Eﬃcien t PT AS for Two -Strategy Anonymous Games. ArXiv R ep ort , 2008. [9] C. Dask a lakis, P . W. Goldb erg, and C. H. P apadimitriou. Th e C omp lexit y of C omputing a Nash Equ ilibrium. STOC , 2006. [10] C . Dask alakis, A. Meh ta, and C. H. Pa padimitriou. A Note on Appr o ximate Nash Equilibr ia. WINE , 2006. [11] C . Dask a lakis, A. Meh ta, and C . H. Papadimitriou. P rogress in Appro ximate Nash Equilibria. EC , 2007. [12] C . Dask alakis and C. H. Pa padimitriou. T hree-Pla y er Games Are Hard. Ele ctr onic Col lo qu ium in Computational Complexity , TR 05-139, 2005. [13] C . Dask alakis and C . H. P apadimitriou. Comp uting Equ ilibria in Anonymo us Games. F OCS , 2007. [14] C . Dask alakis and C. H. P apadimitriou. Di scretized Multinomial Distrib u tions and Nash Equilibria in An onymous Games. F OCS , 2008. [15] R . Durr ett. Pr ob ability: The o ry and Examples. Second Edition, Duxbu ry P ress, 1996. [16] P . W. Goldb erg and C. H. P apadimitriou. Redu cibilit y Among Equilibrium Problems. STOC , 2006. [17] E . Kalai. Pa rtially-Sp eciﬁed Large Games. WIN E , 2005. [18] R . L ip ton, E. Mark akis, and A. Meh ta. Pla ying Large Games Using Simp le S trategi es. EC , 2003. [19] I . Milc h taic h. Congestion Games with Pla y er-Sp eciﬁc P a y oﬀ F unctions. Games and Ec onomic Behavior , 13:111 –124. [20] B. Ro os. Binomial App ro ximation to the P oisson Binomial Distribution: The K r a wtc houk Expansion. The ory of Pr ob ability and its A pplic ations , 45(2):258–2 72, 2000. [21] H. Tsaknakis and P . G. Spirakis. An Optimization Ap p roac h for Appro ximate Nash Equilibria. WINE , 2007. [22] V. M. Zolotarev. R an d om S ymmetric P olynomials. J ournal of M athema tic al Scienc es , 38(5): 2262–227 2, 1987. 16 A Bimatrix Games A.1 PT AS for Small P robabilit y Games Pro of of Lemma 1: Let X and Y b e ind ep enden t random v ariables suc h that X = e i , with prob ab ility x i , for all i ∈ [ n ] , Y = e j , with p robabilit y y j , for all j ∈ [ n ] . Let then X 1 , X 2 , . . . , X t b e t copies of v ariable X and Y 1 , Y 2 , . . . , Y t b e t copies of v ariable Y , where the v ariables X 1 , . . . , X t , Y 1 , . . . , Y t are m utually in dep enden t. Setting X = 1 t P t k =1 X k and Y = 1 t P t k =1 Y k , as in the statemen t of the theorem, w e will argue that, with high pr obabilit y , ( X , Y ) is an ǫ -Nash equilibr ium of G . Let U i = e T i R Y and V i = X T C e i b e the pa y oﬀ of the row and column play er resp ectiv ely for pla ying strategy i ∈ [ n ]. Fixing i ∈ [ n ], we hav e th at U i = 1 t t X k =1 e T i RY k , E [ U i ] = 1 t t X k =1 E [ e T i RY k ] = 1 t t X k =1 E [ e T i RY ] = E [ e T i RY ] = X j ∈ [ n ] y j e T i Re j = e T i Ry . An application of McDiarmid’s inequalit y on the function f (Λ 1 , . . . , Λ t ) = 1 t P t k =1 Λ k , where Λ k := e T i RY k , k ∈ [ t ], are ind ep enden t random v ariables, giv es Pr[ |U i − E [ U i ] | ≥ ǫ/ 2] ≤ 2 e − tǫ 2 8 ≤ 2 n 2 . Applying a un ion b ound, it follo ws that with p r obabilit y at least 1 − 4 n the follo wing prop erties are satisﬁed by the pair of strategies ( X , Y ): | e T i R Y − e T i Ry | ≤ ǫ/ 2 , f or all i ∈ [ n ]; (6) |X T C e j − x T C e j | ≤ ǫ/ 2 , for all j ∈ [ n ] . (7) The ab ov e imply that ( X , Y ) is an ǫ -Nash equilibr ium. Indeed, for all i, i ′ ∈ [ n ], we ha v e that e T i R Y > e T i ′ R Y + ǫ ⇒ e T i Ry ≥ e T i R Y − ǫ/ 2 > e T i ′ R Y + ǫ/ 2 > e T i ′ Ry ⇒ x i ′ = 0 ⇒ X i ′ = 0 , where the last implication follo ws from the f act that X is formed b y taking a d istribution ov er samples fr om x . Similarly , it can b e argued that, for all j, j ′ ∈ [ n ], X T C e j > X T C e j ′ + ǫ ⇒ Y j ′ = 0 .  17 Pro of of Claim 1: W e are only going to lo w er b ound the cardinalit y of the set A . A similar argumen t app lies for B . Recall that ( x, y ) is a Nash equ ilibrium in which x i ≤ 1 δn , for all i ∈ [ n ], and similarly for y . Let u s now deﬁne the set A ′ as follo ws A ′ := { A A ∈ A and x i > 0 , ∀ i ∈ A } ⊆ A , that is, A ′ is the s u bset of A con taining those multisets that could arise by taking t samples from x . W e are going to use the stru cture of the p robabilit y d istribution x and Lemma 1 to argue that A ′ is large. F or this, let us consider the random exp erimen t of taking t in dep enden t samples fr om x and forming the corresp onding multiset A . Let also X b e th e u niform distribu tion ov er A . Lemma 1 asserts that with prob ab ility at least 1 − 4 n the distr ib ution X will satisfy assertion 3 of Lemma 1. Ho w ev er, this do es not directly imp ly a lo w er b ound on the cardinalit y of A ′ —it could b e th at all the p robabilit y mass is concen trated on a s ingle elemen t of A ′ . Ho wev er, th e probab ility that a m ultiset arises by sampling x is at most  1 δ n  t , since the p robabilit y m ass that x assigns to eve ry i ∈ [ n ] is at most 1 δn . Th erefore, the total num b er of distinct go o d multise ts in A ′ is at least 1 − 4 n  1 δn  t . Therefore, |A ′ | ≥ (1 − 4 n )( δ n ) t and |A | ≥ |A ′ | ≥ (1 − 4 n )( δ n ) t .  Pro of o f Claim 2: F r om Claim 1, it follo ws that, if a random multiset A ′ is sampled, the probabilit y that it is go od is Pr  A ′ ∈ A  ≥  1 − 4 n  ( δ n ) t n t =  1 − 4 n  δ t = Ω  δ · n − 16 log(1 /δ ) /ǫ 2  , Similarly , if a rand om m ultiset B ′ is sampled, the pr obabilit y that it is go od is Pr  B ′ ∈ B  = Ω  δ · n − 16 log(1 /δ ) /ǫ 2  . By indep endence, it follo ws that Pr  A ′ ∈ A and B ′ ∈ B  = Ω  δ 2 · n − 32 log(1 /δ ) /ǫ 2  . But, if A ′ ∈ A , B ′ ∈ B , then the un if orm distribu tion X ′ o v er A ′ and th e un if orm distribu tion Y ′ o v er B ′ comprise an ǫ -Nash equilibrium (see Pro of of Lemm a 1 f or a justiﬁcation). Hence, Pr  ( X ′ , Y ′ ) is an ǫ -Nash equilibriu m  = Ω  δ 2 · n − 32 log(1 /δ ) /ǫ 2  .  18         1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1         Figure 1: R S for the case n = 6, ℓ = 4, S = { 1 , 2 , 3 , 4 } . A.2 The Lo we r Bound for Bimatrix Games In Figure 1, w e illustrate the game matrix R S in our construction in the Pro of of Theorem 4 (Section 3), for the case n = 6, ℓ = 4, S = { 1 , 2 , 3 , 4 } . E ac h column corr esp onds to a subset of S of size 2. W e add then t wo extra rows to mak e the matrix square. Pro of of Lemma 2: The ﬁrst assertion is easy to ju stify . Ind eed, no m atte r what y is, the p a y oﬀ u i of the row pla y er for pla ying strategy i , i ∈ [ n ], satisﬁes: • u i = − 1, for all i / ∈ S ; • u i ≥ 0, for all i ∈ S ; Hence, ﬁxing any i ′ ∈ S , we ha v e, for all i / ∈ S , that u i ′ > u i + ǫ , whic h by the deﬁnition of an ǫ -Nash equilibrium implies that x i = 0. T o w ards justifying the second assertion, let u b e the utilit y of the r o w play er at th e ǫ -Nash equilibrium ( x, y ). Sin ce, by the ﬁr st assertion, x i = 0 for all i / ∈ S , it follo ws th at the pa yo ﬀ of the column pla y er is 1 − u , since the game restricted to the ro ws of the s et S is 1-sum. Since ev ery column of R S restricted to the r o ws of the set S h as exactly half of the ent ries equal to 1 and the other half equ al to 0, it follo ws that X i ∈ S u i = X j ∈ [ n ] ℓ 2 y j = ℓ 2 . It follo ws that there exists some i ∗ ∈ S su c h that u i ∗ ≥ 1 / 2. By the deﬁnition of ǫ -Nash equilibrium, it f ollo ws then that u ≥ u i ∗ − ǫ ≥ 1 / 2 − ǫ, (8) otherwise the r ow pla y er wo uld b e including in h is supp ort strategie s whic h are more than ǫ wo rse than i ∗ . Let us no w consider the p a y oﬀ of the column pla y er for c ho osing v arious strategies j ∈ [ n ]. Without loss of generalit y let u s assu m e that S = { 1 , 2 , . . . , ℓ } and that x 1 ≥ x 2 ≥ . . . ≥ x ℓ . Let j ∗ b e suc h that S j ∗ = { ℓ/ 2 + 1 , ℓ/ 2 + 2 , . . . , ℓ } . Then, by the deﬁ n ition of C S , the pa yo ﬀ of the column pla y er for choosing strategy j ∗ is v j ∗ = ℓ/ 2 X i =1 x i . Since ( x, y ) is an ǫ -Nash equilibr ium, it follo ws that v j ∗ should b e within ǫ from the pay oﬀ of the column p la y er. Hence, v j ∗ ≤ 1 − u + ǫ. 19 Com bining the ab o v e with (8), it follo ws that v j ∗ − 1 2 ≤ 2 ǫ ⇒ ℓ/ 2 X i =1  x i − 1 ℓ  ≤ 2 ǫ. The right hand side of the ab o v e, implies ℓ 1 ( x, u S ) ≤ 8 ǫ, b y an application of Lemma 3 b elo w, w ith a i = x i − 1 ℓ , for all i ∈ [ ℓ ], k = 2 ǫ . Lemma 3. L et { a i } ℓ i =1 b e r e al numb ers satisfying the fol lowing pr op erties for some k ∈ R + : 1. a 1 ≥ a 2 ≥ . . . ≥ a ℓ ; 2. P ℓ i =1 a i = 0 ; 3. P ℓ/ 2 i =1 a i ≤ k . Then ℓ X i =1 | a i | ≤ 4 k . (9) Pro of of Lemma 3 : W e d istinguish t w o cases: (i) a ℓ/ 2 ≥ 0 and (ii) a ℓ/ 2 < 0. I n th e ﬁrst case, we ha v e f rom Conditions 1 and 3 that ℓ/ 2 X i =1 | a i | ≤ k . Using Condition 1 s ome more w e get X i> ℓ 2 : a i ≥ 0 | a i | ≤ X i> ℓ 2 : a i ≥ 0 a ℓ/ 2 ≤ ℓ 2 a ℓ/ 2 ≤ ℓ/ 2 X i =1 | a i | ≤ k . Com bining the ab o v e we get X i : a i ≥ 0 | a i | ≤ 2 k . (10) No w w e employ Condition 2, to get X i : a i < 0 | a i | = X i : a i < 0 ( − a i ) = X i : a i ≥ 0 a i = X i : a i ≥ 0 | a i | ≤ 2 k . (11) W e com bine (10), (11) to d educe (9). Case (ii) is treated by rep eating the ab o v e argumen t with a i ← ( − a ℓ − i +1 ), f or all i ∈ [ ℓ ] .   Pro of of Claim 3: Consid er the set V := { u S | S ⊆ [ n ] , | S | = ℓ } ; note that | V | = Ω( n . 8 log 2 n ). F or ev ery u S ∈ V c onsider the set N ( u S ) of all other u S ′ ’s that are w ithin ℓ 1 distance 17 ǫ from u S ; it is easy to see that | N ( u S ) | = O ( n 34 ǫ log 2 n ). Therefore, we can select a su bset V ′ ⊆ V of at least | V | max S | N ( u S ) | = Ω  n ( . 8 − 34 ǫ ) log 2 n  elemen ts of V so that ev ery pair of elemen ts is at ℓ 1 distance at least 17 ǫ apart; it follo ws that for eve ry pair of element s in V ′ their ℓ 1 balls of r adius 8 ǫ are disjoint. The p ro of is complete.  20 B The Lo w er B ound for Oblivious PT AS’s for Anon ymous Games B.1 Constructing Anon ymous Games with P rescrib ed Equilibria Theorem 10. F or any c o l le ction P := ( p i ) i ∈ [ k ] , wher e p i ∈ [3 δ k , 1] , for some δ > 0 and k ∈ N , ther e exists an anonymous game G P with k + 2 players, 2 str ate gi e s, 0 and 1 , p ayoﬀs in [ − 1 , 1] , and thr e e player typ es A,B,C, i n which k pla yers, 1 , . . . , k , b elong to typ e A, 1 player b elongs to typ e B, and 1 player b elongs to typ e C, and suc h that in every δ ′ -Nash e quilibrium, wher e δ ′ < δ , the fol lowing is satisﬁe d: F or every i , player i ’s mixe d str ate gy b elongs to the set [ p i − 7 k 2 δ , p i + 7 k 2 δ ] ; mor e over, at le ast one of the players b elonging to typ e s B and C play str ate gy 1 with pr ob ability 0 . Pr o of. Let us call B the single pla y er of t yp e B and C the single play er of t yp e C . Let u s also use the notation: µ = P i ∈ [ k ] p i , and µ − i = P j ∈ [ k ] \{ i } p j , for all i . No w, let u s assign th e follo wing pa y oﬀs to the pla ye rs B and C : • u B 1 = 1 k · ( t A − µ ), w here t A is the num b er of play ers of typ e A who pla y strategy 1; • u B 0 = 2 δ ; • u C 1 = 1 k · ( µ − t A ), w here t A is the num b er of pla y ers of typ e A who pla y strategy 1; • u C 0 = 2 δ ; The p a y oﬀ functions of th e play ers of t yp e A are deﬁned as f ollo ws. F or all i ∈ [ k ]: • u i 0 = 1 k ( µ − i · X B plays 0 · X C pla ys 0 − δk · X C plays 1 ), where X B plays 0 , X C plays 0 and X C plays 1 are the indicators of the even ts ‘B plays 0’, ‘C pla ys 0’ and ‘C pla ys 1’ resp ectiv ely . • u i 1 = 1 k ( t A, − i · X B pla ys 0 · X C pla ys 0 − δ k · X B plays 1 ), where t A, − i is the n umb er of p la y ers of t yp e A who are diﬀerent than i and pla y 1, and X B plays 0 , X C pla ys 0 and X B pla ys 1 are the indicators of the even ts ‘B pla ys 0’, ‘C p la ys 0 ’ and ‘B pla ys 1’ resp ectiv ely . Note th at the r ange of all pa y oﬀ fun ctions of the game th us d eﬁned is [ − 1 , 1]. W e n o w claim the follo wing: Claim 4. In every δ ′ -Nash e quilibrium, wher e δ ′ < δ , it must b e that X i ∈ [ k ] q i = µ ± 3 δk , wher e q 1 , . . . , q k ar e the mixe d str ate gies of the players 1 , . . . , k . Pro of of C laim 4: L et µ ′ = P i ∈ [ k ] q i . Supp ose for a cont radiction that in a δ ′ -Nash equilibriu m µ ′ > µ + 3 δk ; then 1 k ( µ ′ − µ ) > 3 δ . Note ho w eve r that E [ u B 1 ] = 1 k ( µ ′ − µ ) and E [ u C 1 ] = − 1 k ( µ ′ − µ ). Hence, the ab o v e implies E [ u B 1 ] > E [ u B 0 ] + δ, (12) E [ u C 1 ] < E [ u C 0 ] − δ. (13) 21 Since δ ′ < δ , it must b e then that Pr[B plays 1] = 1 and Pr[C pla ys 1 ] = 0. It follo ws then that for all i ∈ [ k ]: E [ u i 0 ] = 0 , E [ u i 1 ] = − δ. Hence, in a δ ′ -Nash equilibriu m w ith δ ′ < δ , it must b e that Pr[ i p la ys 1 ] = q i = 0, for all i ∈ [ k ]. This is a con tradiction since we assumed that µ ′ = P i ∈ [ k ] q i > µ + 3 δ k , and µ is non-n egativ e. Via similar argumen ts w e show that the assumption µ ′ < µ − 3 δ k also leads to a contradictio n. Hence, in every δ ′ -Nash equilibrium with δ ′ < δ , it must b e th at µ ′ = µ ± 3 δk .  W e next show that in ev ery δ ′ -Nash equilibrium w ith δ ′ < δ , at least one of the p lay ers B and C will not include strategy 1 in her supp ort. Claim 5. In every δ ′ -Nash e quilibrium with δ ′ < δ , Pr[ B plays 1] = 0 or Pr[ C plays 1] = 0 . Pro of of C laim 5: Let q 1 , . . . , q k b e the mixed strategies of pla ye rs 1 , . . . , k at some δ ′ -Nash equilibrium of th e game with δ ′ < δ . Let us consider the quan tit y M = 1 k ( µ ′ − µ ), where µ ′ = P i ∈ [ k ] q i . W e distinguish the follo wing cases: • M ≤ δ : In this case, E [ u B 1 ] = 1 k ( µ ′ − µ ) ≤ δ ≤ 2 δ − δ = E [ u B 0 ] − δ . Since δ ′ < δ , Pr [B plays 1] = 0. • M ≥ δ : In this case, E [ u C 1 ] = − 1 k ( µ ′ − µ ) ≤ − δ ≤ 2 δ − δ = E [ u C 0 ] − δ . Since δ ′ < δ , Pr[C pla ys 1] = 0.  Finally , we establish th e follo wing. Claim 6. In every δ ′ -Nash e quilibrium with δ ′ < δ , it must b e that f or every player i ∈ [ k ] : µ ′ − i := X j ∈ [ k ] \{ i } q j = µ − i ± 4 δ k 2 , wher e q 1 , . . . , q k ar e the mixe d str ate gies of the players 1 , . . . , k . Pro of of C laim 6: Let us ﬁ x any δ ′ -Nash equilibrium. F rom Claim 5 it follo ws that either pla y er B or C p la ys strategy 1 w ith p r obabilit y 0. Without loss of generalit y , we will assum e th at Pr[C pla ys 1] = 0 (the argumen t f or the case Pr[B pla ys 1 ] = 0 is identic al to the one th at follo ws). Let u s now ﬁx a pla y er i ∈ [ k ]. W e show ﬁ rst that u n der the assump tion Pr[C plays 1] = 0, Pr[C pla ys 0] = 1, it must b e th at µ − i ≤ µ ′ − i + δ k . (14) 22 Assume for a contradict ion that µ − i > µ ′ − i + δ k . It follo ws then that µ − i (1 − Pr[B pla ys 1 ]) ≥ µ ′ − i (1 − Pr[B p la ys 1]) + δ k (1 − Pr[B pla ys 1]) ⇒ µ − i Pr[B pla ys 0] ≥ µ ′ − i Pr[B pla ys 0] + δ k (1 − Pr[B pla ys 1]) ⇒ µ − i Pr[B pla ys 0] Pr[C pla ys 0] ≥ µ ′ − i Pr[B p lays 0] Pr[C p la ys 0 ] − δ k Pr[B p la ys 1] + δk ⇒ E [ u i 0 ] ≥ E [ u i 1 ] + δ. But we assum ed that we ﬁ xed a δ ′ -Nash equilibriu m with δ ′ < δ ; hence the last equation implies that q i = 0. Bu t this leads quickl y to a con tradiction s in ce, if q i = 0, then us in g Claim 4 w e ha v e µ ′ − i = µ ′ ≥ µ − 3 δk ≥ µ − p i = µ − i , where we also used that p i ≥ 3 δ k . Th e ab o v e inequalit y contradicts our assu mption th at µ − i > µ ′ − i + δ k . Hence, (14) must b e satisﬁed. Using that µ ′ ≤ µ + 3 δ k , whic h is implied by Claim 4) w e get q i ≤ p i + 4 δ k . F rom th e ab ov e discuss ion it follo ws that q j ≤ p j + 4 δ k , for all j . (15) No w ﬁx i ∈ [ k ] again. S umming (15 ) o v er all j 6 = i , we get that µ ′ − i ≤ µ − i + 4 δ k 2 . (16) Com bining (14 ) and (16) w e get µ ′ − i = µ − i ± 4 δ k 2 .  T o conclude the pr oof of Theorem 10 , we com bine Claims 4 and 6 , as follo ws. F or every pla y er i ∈ [ k ], we hav e from Claims 4 and 6 that in eve ry δ ′ -Nash equilibrium w ith δ ′ < δ , µ ′ − i = µ − i ± 4 δ k 2 and µ ′ = µ ± 3 δk . By com bining these equations w e get q i = p i ± 7 δk 2 . B.2 The Lo we r Bound Giv en Theorem 10, w e can establish our lo w er b ound. Pro of of Theorem 5 : L et us ﬁ x any oblivious ǫ -appr o ximatio n algorithm for anon ymous games with 2-strategies and 3-pla y er types. The algorithm comes together with a distribution ov er un- ordered s ets of m ixed strategies—parametrized by the num b er of play ers n —wh ic h we denote by D n . 23 W e will consider the p erformance of the algorithm on the family of games sp eciﬁed in the statemen t of Theorem 10 for the f ollo wing s etting of parameters: k = ⌊ (1 /ǫ ) 1 / 3 ⌋ , δ = 1 . 01 ǫ, P ∈ T k ǫ where T ǫ := n j · 15 ǫ 1 / 3 j = 1 , . . . , t ǫ o , t ǫ =  1 15 ǫ − 1 / 3  . F or tec hnical reasons, let u s deﬁn e the follo wing notion of distance b et we en P , Q ∈ T k ǫ . d ( P , Q ) := t ǫ X j =1    v P j − v Q j    . where v P = ( v P 1 , v P 2 , . . . , v P t ǫ ) is a v ector storing the fr equ encies of v arious elemen ts of the set T ǫ in the collection P , i.e. v P j := |{ i i ∈ [ k ] , p i = j · 1 5 ǫ 1 / 3 }| . T o ﬁnd the distance b et w een tw o collections P , Q we compute the ℓ 1 distance of their frequen cy vec tors. Notic e in particular that this distance m ust b e an eve n num b er. W e also need th e follo wing deﬁn ition. Deﬁnition 2. We say that two anonymous games G and G ′ shar e an ǫ -N ash e quilibrium in un- or der e d form if ther e exists an ǫ - Nash e quilibrium σ G of game G and an ǫ -Nash e quilibrium σ G ′ of game G ′ such that σ G and σ G ′ ar e e qual as unor d er e d sets of mixe d str ate gies. W e show ﬁ rst the follo wing ab out the shareabilit y of ǫ -Nash equilibr ia among the games G P , P ∈ T k ǫ . Claim 7. If, for P , Q ∈ T k ǫ , d ( P , Q ) > 0 , then ther e i s no ǫ - N ash e quilibrium that is shar e d b etwe en the games G P and G Q in unor der e d form. Pro of of Claim 7: F or all j , let u s deﬁne the 7 . 07 k 2 ǫ ball aroun d probability j · 15 ǫ 1 / 3 in the natural wa y: B j := [ j · 15 ǫ 1 / 3 − 7 . 07 k 2 ǫ, j · 15 ǫ 1 / 3 + 7 . 07 k 2 ǫ ] . Observe that for all j ≥ 2: ( j + 1) · 15 ǫ 1 / 3 − j · 15 ǫ 1 / 3 = 15 ǫ 1 / 3 > 2 · 7 . 07 k 2 ǫ. Hence, for all j, j ′ : B j ∩ B ′ j = ∅ . No w, let us consider an y p air of ǫ -Nash equilibria σ G P , σ G Q of the games G P and G Q and let u s consider the ve ctors v σ G P = ( v σ G P 1 , . . . , v σ G P t ǫ ) and v σ G Q = ( v σ G Q 1 , . . . , v σ G Q t ǫ ) whose j -th comp onents are deﬁned as f ollo ws: v σ G P j =   n umb er of pla yers w ho are assigned a mixed s trateg y fr om the set B j in σ G P   , v σ G Q j =   n umb er of p la y ers who are assigned a mixed s trateg y from the set B j in σ G Q   . It is not hard to see that T h eorem 10 and our assum ption d ( P , Q ) > 0 imply that k v σ G P − v σ G Q k 1 > 0 , h ence σ G P and σ G Q cannot b e p ermutations of eac h other. T h is concludes the p roof.  Next, we sho w that there exists a large family of games s u c h that no tw o mem b ers of the family share an ǫ -Nash equilibr ium. 24 Claim 8. Ther e exists a subset T ⊆ T k ǫ such that: 1. for every P , Q ∈ T : d ( P , Q ) > 0 ; 2. | T | ≥ 2 Ω  ( 1 ǫ ) 1 / 3  ; Pro of of claim 8: The total num b er of d istinct multi-set s of cardinalit y k with elemen ts from T ǫ is  t ǫ + k − 1 k  . Hence, it is easy to create a sub set T ⊆ T k ǫ suc h that: • f or eve ry P , Q ∈ T : d ( P , Q ) > 0; • | T | =  t ǫ + k − 1 k  . Clearly , the set T satisﬁes Prop erty 1 in the statemen t. F or th e cardinalit y b ound we hav e: | T | ≥  t ǫ + k − 1 k  ≥  t ǫ + k − 1 k  k ≥  1 + 1 15 − 2 k  k ≥ 2 Ω  ( 1 ǫ ) 1 / 3  .  No w let us consider the p erform ance of the distr ibution D k on the f amily of anon ymous games {G P } P ∈ T , where T is the set deﬁn ed in Claim 8. By Claims 7 and 8, no tw o games in the family share an ǫ -Nash equilibrium in u n ordered form. Hence, no matter wh at D k is, there will b e some game in our family for which the pr obabilit y that D k samples an ǫ -Nash equilibrium of that game is at m ost 1 / | T | ≤ 2 − Ω  ( 1 ǫ ) 1 / 3  . This concludes the p ro of of T h eorem 5.  C The Binomial Appro ximation T heorem Prop osition 1. Condition ( C d ) i n the statement of The or em 7 is e quivalent to the fol lowing c on- dition: ( V d ) : E   n X i =1 X i ! ℓ   = E   n X i =1 Y i ! ℓ   , for al l ℓ ∈ [ d ] . Pro of of Prop osition 1: ( V d ) ⇒ ( C d ): It is not h ard to see that E h ( P n i =1 X i ) ℓ i can b e w ritten as a weig hte d sum of the elementary symmetric p olynomia ls ψ 1 ( P ), ψ 2 ( P ),..., ψ ℓ ( P ) with p ositiv e co eﬃcien ts, where, for all ℓ , ψ ℓ ( P ) is d eﬁ ned as ψ ℓ ( P ) := X S ⊆ [ n ] | S | = ℓ Y i ∈ S p i . 25 ( V d ) then imp lies b y indu ction ψ ℓ ( P ) = ψ ℓ ( Q ) , for all ℓ = 1 , . . . , d. (17) No w, for all ℓ , d eﬁne π ℓ ( P ) as the p o we r s um symmetric p olynomial of d egree ℓ π ℓ ( P ) := n X i =1 p ℓ i . No w ﬁx any ℓ ≤ d . Observ e that π ℓ ( P ) is a symmetric p olynomial of degree ℓ on the v ariables p 1 , . . . , p n . It follo ws (see, e.g., [22]) th at π ℓ ( P ) can b e expressed as a fun ctio n of the elementa ry symmetric p olynomials ψ 1 ( P ) , . . . , ψ ℓ ( P ). Sin ce, by (17 ), ψ j ( P ) = ψ j ( Q ), for all j ≤ ℓ , we get that π ℓ ( P ) = π ℓ ( Q ). Since this h olds for an y ℓ ≤ d , ( C d ) is satisﬁed. The im p licat ion ( C d ) ⇒ ( V d ) is established in a similar fashion. ( C d ) implies π ℓ ( P ) = π ℓ ( Q ) , for all ℓ = 1 , . . . , d. F or an y ℓ ≤ d , E h ( P n i =1 X i ) ℓ i is a symmetric p olynomial of degree ℓ on the v ariables p 1 , . . . , p n . It follo ws (see, e.g., [2 2]) that E h ( P n i =1 X i ) ℓ i can be expressed as a function of the p ow er-sum symmet- ric p olynomials π 1 ( P ) , . . . , π ℓ ( P ). And since π j ( P ) = π j ( Q ), for all j ≤ ℓ , w e get E h ( P n i =1 X i ) ℓ i = E h ( P n i =1 Y i ) ℓ i . Since this holds for an y ℓ ≤ d , ( V d ) is satisﬁed.  Pro of of Theorem 7: Let X = P i X i . The follo wing theorem du e to Ro os [20], sp eciﬁes an expansion of the distribution fu nction of X as a sum of a ﬁ nite num b er of signed m easur es: the binomial distribution B n,p ( m ) (for an arbitrary c hoice of p ) and its ﬁ rst n d eriv ative s with resp ect to the p arameter p , at the chosen v alue of p . More precisely , Theorem 11 ([20 ]) . L et P := ( p i ) n i =1 b e an arbitr ar y set of pr ob ability values in [0 , 1] and X := ( X i ) n i =1 a c ol le ction of indep endent indic ators with E [ X i ] = p i , for al l i ∈ [ n ] ; also let X := P i X i . Then, for al l m ∈ { 0 , . . . , n } and any p ∈ (0 , 1) , P r [ X = m ] = n X ℓ =0 α ℓ ( P , p ) · δ ℓ B n,p ( m ) , (18) wher e in the ab ove α 0 ( P , p ) := 1 , α ℓ ( P , p ) := X 1 ≤ k (1) <... k 3 , guess an integ er multiple i/k n of 1 /k n and, solving a max-ﬂo w instance, chec k if there is an ǫ -Nash equ ilibrium in which t p la y ers p la y i/kn , t 0 pla y ers pla y 0, and t 1 pla y ers play 1. Figure 2: The oblivious PT AS of [Dask ala kis, 2008] Clearly , there are O ( n 2 ) p ossible c hoices for Step 2 of the algorithm. Moreo v er, the search of Step 3b can b e completed in time (see [7]) U · p oly( n ) · (1 /ǫ ) log 2 (1 /ǫ ) , whic h is p olynomial in 1 /ǫ . On the other hand , Step 3a in vo lv es searc hing ov er all p artitions of t balls in to k 2 − 2 b ins. T he resulting ru nning time for this step (see details in [7]) is U · p oly( n ) · (1 /ǫ ) O (1 /ǫ 2 ) , whic h is exp onenti al in 1 /ǫ . D.2 Moment Search : Missing Det ails W e describ e in detail the third step of Moment S earch . 3. F or eac h pla y er i = 1 , . . . , n , ﬁ nd a su bset S i ⊆  0 , 1 k 2 , . . . , k 2 − 1 k 2 , 1  of p ermitted mixed strategie s for that pla y er in an ǫ -Nash equilibr ium, “conditioning” on the total num b er of play ers p la ying 0 b eing t 0 , th e total num b er of pla yers pla ying 1 b eing t 1 , an d the probabilities of th e p la y ers who mix resulting in th e p o wer-sums µ 1 , . . . , µ d and µ ′ 1 , . . . , µ ′ d . The wa y w e compu te the set S i is as follo ws: (a) T o determine wh ether 0 ∈ S i : i. Find any set of mixed strategies q 1 , . . . , q t s ⊆ { 1 k 2 , 2 k 2 , . . . , 1 2 } suc h that P t s i =1 q ℓ i = µ ℓ , for all ℓ = 1 , . . . , d . Find any set of mixed strategies r 1 , . . . , r t b ⊆ { 1 2 + 1 k 2 , 1 2 + 2 k 2 , . . . , 1 − 1 k 2 } such that P t b i =1 r ℓ i = µ ′ ℓ , for all ℓ = 1 , . . . , d . If suc h v alues do not exist F a il . Remark: An eﬃcien t algorithm to solv e this optimization problem is describ ed in the pro of of Claim 11. 29 ii. Deﬁn e th e random v ariable Y = ( t 0 − 1) · 0 + t s X i =1 S i + t b X i =1 B i + t 1 · 1 , where the v ariables S 1 , . . . , S t s , B 1 , . . . , B t b are m utually ind ep enden t with exp ecta- tions E [ S i ] = q i , for all i = 1 , . . . , t s , and E [ L i ] = r i , for all i = 1 , . . . , t b . iii. Comp ute the exp ected p ay oﬀ U i 0 = E [ u i 0 ( Y )] and U i 1 = E [ u i 1 ( Y )] of pla ye r i for pla y- ing 0 and 1 resp ectiv ely , if the num b er of the other pla y ers pla ying 1 is distr ibuted iden tically to Y . iv. if U i 0 ≥ U i 1 − 3 ǫ/ 4, then includ e 0 to th e set S i , otherwise do n ot. (b) T o determine whether 1 ∈ S i , follo w the same strategy except n ow Y is deﬁned as follo ws Y = t 0 · 0 + t s X i =1 S i + t b X i =1 B i + ( t 1 − 1) · 1 , to accoun t for the f act that we are testing f or the cand idate s trateg y 1 for pla y er i . Also, the test th at determines whether 1 ∈ S i is no w U i 1 ≥ U i 0 − 3 ǫ/ 4. (c) F or all j ∈ { 1 , . . . , k 2 / 2 } , to determine whether j /k 2 ∈ S i do the f ollo wing sligh tly up dated test: i. Find an y set of mixed strategi es q 1 , . . . , q t s − 1 ⊆ { 1 k 2 , 2 k 2 , . . . , 1 / 2 } su c h that P t s − 1 i =1 q ℓ i = µ ℓ − ( j /k 2 ) ℓ , for all ℓ = 1 , . . . , d . Find any set of mixed strategies r 1 , . . . , r t b ⊆ { 1 2 + 1 k 2 , 1 2 + 2 k 2 , . . . , 1 − 1 k 2 } suc h that P t b i =1 r ℓ i = µ ′ ℓ , for all ℓ = 1 , . . . , d . I f such v alues do not exist F ail . ii. Deﬁn e th e random v ariable Y = t 0 · 0 + t s − 1 X i =1 S i + t b X i =1 B i + t 1 · 1 , where the v ariables S 1 , . . . , S t s − 1 , B 1 , . . . , B t b are m utually ind ep enden t with E [ S i ] = q i , for all i = 1 , . . . , t s − 1, and E [ L i ] = r i , for all i = 1 , . . . , t b . iii. Comp ute the exp ected p ay oﬀ U i 0 = E [ u i 0 ( Y )] and U i 1 = E [ u i 1 ( Y )] of pla ye r i for pla y- ing 0 and 1 resp ectiv ely , if the num b er of the other pla y ers pla ying 1 is distr ibuted iden tically to Y . iv. if U i 0 ∈ [ U i 1 − 3 ǫ/ 4 , U i 1 + 3 ǫ/ 4], then include j /k 2 to the s et S i , otherwise do n ot. (d) F or all j ∈ { ( k 2 + 2) / 2 , . . . , k 2 − 1 } , to d etermine wh ether j /k 2 ∈ S i do the app ropriate mo diﬁcations to the m etho d describ ed in Step 3c . D.3 The Analysis of Moment Se ar ch Correctness The correctness of Momen t S ear ch follo ws from the follo wing t wo claims. Claim 9. If ther e exists an ǫ /2-Nash e quilibrium in which t ≤ k 3 players mix, and their mixe d str ate gies ar e inte ger multiples of 1 /k 2 , then Mome nt S earch wil l not f ail, i.e. it wil l output a set of mixe d str a te gies ( v 1 , . . . , v n ) . 30 Claim 10. If Moment Search outputs a set of mixe d str ate gies ( v 1 , . . . , v n ) , then these str ate gies c onst itute an ǫ -Nash e quilibrium. Pro of of Claim 9: Let ( p 1 , . . . , p n ) b e an ǫ/ 2-Nash equilibrium in wh ic h t 0 pla y ers pla y 0, t 1 pla y ers play 1, and t ≤ k 3 pla y ers mix, and their mixed str ateg ies are intege r multiples of 1 /k 2 . It suﬃces to show that there exist gu esses for t 0 , t 1 , t s , t b , µ 1 , . . . , µ d , µ ′ 1 , . . . , µ ′ d , suc h that p 1 ∈ S 1 , p 2 ∈ S 2 , . . . , p n ∈ S n . Indeed, let I 0 := { i | p i = 0 } , I s := { i | p i ∈ (0 , 1 / 2] } , I b := { i | p i ∈ (1 / 2 , 1) } , I 1 := { i | p i = 1 } , and let u s choose the follo wing v alues for our guesses t 0 := |I 0 | , t s = |I s | , t b = |I b | , t 1 := |I 1 | and, f or all ℓ ∈ [ d ], µ ℓ = X i ∈I s p ℓ i , µ ′ ℓ = X i ∈I b p ℓ i . W e will sh o w that for the guesses that we deﬁ n ed ab o v e p i ∈ S i , for all i . W e distinguish the follo wing cases: i ∈ I 0 , i ∈ I s , i ∈ I b , i ∈ I 1 . The p ro of for all the cases pro ceeds in the same fashion. W e will only argue ab out the case i ∈ I s ; in particular, we will sho w that in Step 3c of Moment S earch th e test su cceeds for j /k 2 = p i . A t the equilibriu m p oin t ( p 1 , . . . , p n ), the num b er of the other pla y ers wh o c ho ose strategy 1, from th e p ersp ectiv e of pla ye r i , is distribu ted identi cally to the rand om v ariable: Z := X j ∈I s \{ i } X j + X j ∈I b X j + t 1 · 1 , where E [ X j ] = p j for all j . Since ( p 1 , . . . , p n ) is an ǫ / 2- Nash equilibr ium it must b e the case that |E [ u i 0 ( Z )] − E [ u i 1 ( Z )] | ≤ ǫ/ 2 . (23) W e w ill argue that, if in the ab o v e equation, w e replace Z b y Y , w here Y is the random v ariable deﬁned in Step 3(c)ii of Mom ent Sea r ch , the inequalit y still holds with slightly up dated u p p er b ound: |E [ u i 0 ( Y )] − E [ u i 1 ( Y )] | ≤ 3 ǫ/ 4 . (24) If (24) is established, the p ro of is completed since S tep 3(c)iv will include j /k 2 in to the set S i . Let S 1 , . . . , S t s − 1 , B 1 , . . . , B t b b e the random v ariables with exp ectatio ns q 1 , . . . , q t s − 1 , r 1 , . . . , r t b deﬁned in S tep 3(c)ii of Momen t S earch . Observe that, f or all ℓ = 1 , . . . , d , t s − 1 X j =1 q ℓ j = µ ℓ − ( j /k 2 ) ℓ = X j ∈I s \{ i } p ℓ j , since p i = j /k 2 . Hence, by Theorem 7,       t s − 1 X j =1 S j − X j ∈I s \{ i } X j       ≤ 20( d + 1) 1 / 4 2 − ( d +1) / 2 ≤ ǫ / 16 . (25) 31 Via similar argum en ts and Corollary 1, we get       t b X j =1 B j − X j ∈I b X j       ≤ ǫ / 16 . (26) (25) an d (26) imp ly via the couplin g lemma k Y ; Z k ≤ ǫ 8 . (27) It is n ot hard to s ee that |E [ u i 0 ( Y )] − E [ u i 0 ( Z )] | ≤ k Y ; Z k ≤ ǫ 8 , where w e used (27 ). Similarly , |E [ u i 1 ( Y )] − E [ u i 1 ( Z )] | ≤ k Y ; Z k ≤ ǫ 8 . Com bining the ab o v e with (23) we get (24). This conclud es the p r oof.  Pro of of Claim 10 : Let I 0 := { i | v i = 0 } , I s := { i | v i ∈ (0 , 1 / 2] } , I b := { i | v i ∈ (1 / 2 , 1) } , I 1 := { i | v i = 1 } , t s = |I s | , t b = |I b | , Observe that the moment v alues that we re guessed in S tep 2 of Mom ent Search satisfy µ ℓ = X i ∈I s v ℓ i , µ ′ ℓ = X i ∈I b v ℓ i , f or all ℓ = 1 , . . . , d . W e will argue that ( v 1 , . . . , v n ) is an ǫ -Nash equilibriu m. T o do this we need to argue th at, for eac h p la y er i , v i is an ǫ -we ll sup p orted str ateg y against the strategies of her opp onen ts. W e distinguish the follo wing cases: i ∈ I 0 , i ∈ I s , i ∈ I b , i ∈ I 1 . The pro of f or all the cases pro ceeds in a similar f ashion. W e will only present the argument for th e case i ∈ I s . Let v i = j /k 2 for some j ∈ { 1 , . . . , k 2 2 } . F rom th e p ersp ectiv e of pla y er i , the num b er of other pla y ers wh o pla y 1 in the mixed strategy p roﬁle ( v 1 , . . . , v n ) is distribu ted iden tically to the random v aria ble Z := X j ∈ [ n ] \{ i } X j , where E [ X j ] = v j for all j . T o argue that v i is an ǫ -we ll sup p orted strategy against th e strategies of i ’s opp onents, w e need to sho w that |E [ u i 0 ( Z )] − E [ u i 1 ( Z )] | ≤ ǫ. (28) Let us now go bac k to the iteration of Step 3c in whic h the prob ab ility v a lue j /k 2 w as inserted in to the set S i . Let q 1 , . . . , q t s − 1 , r 1 , . . . , r t b b e the v alues th at w ere selected at Step 3(c)i of that iteration, and let Y = t s − 1 X j =1 S j + t b X j =1 B j + t 1 · 1 , 32 b e the random v aria ble d eﬁned in Step 3(c)ii , where the v ariables S 1 , . . . , S t s − 1 , B 1 , . . . , B t b are m utually in dep enden t with exp ectations E [ S i ] = q i , for all i = 1 , . . . , t s − 1, and E [ B i ] = r i , for all i = 1 , . . . , t b . Observ e that the q j ’s and r j ’s wh er e c hosen by Step 3(c)i so that the follo wing are satisﬁed t s − 1 X j =1 q ℓ j = µ ℓ − ( j /k 2 ) ℓ = µ ℓ − v ℓ i = X j ∈I s \{ i } v ℓ j , for all ℓ ∈ [ d ], (29) and t b X j =1 r ℓ j = µ ′ ℓ = X j ∈I b v ℓ j , for all ℓ = 1 , . . . , d . (30) Equation (29) implies via Th eorem 7 that       t s − 1 X j =1 S j − X j ∈I s \{ i } X j       ≤ 20( d + 1) 1 / 4 2 − ( d +1) / 2 ≤ ǫ / 16 . (31) Equation (30) and Corollary 1 imply       t b X j =1 B j − X j ∈I b X j       ≤ ǫ / 16 . (32) (31) an d (32) imp ly via the couplin g lemma k Y ; Z k ≤ ǫ 8 . (33) It is n ot hard to s ee that |E [ u i 0 ( Y )] − E [ u i 0 ( Z )] | ≤ k Y ; Z k ≤ ǫ 8 , (34) where w e used (33 ). Similarly , |E [ u i 1 ( Y )] − E [ u i 1 ( Z )] | ≤ k Y ; Z k ≤ ǫ 8 . (35) Moreo v er, n otice that the random v ariable Y satisﬁes the follo wing condition |E [ u i 0 ( Y )] − E [ u i 1 ( Y )] | ≤ 3 ǫ/ 4 , (36) since, in order for v i to b e included int o S i , the test in Step 3(c)iv of Moment Search must ha ve succeeded. Com binin g (34), (35) and (36) we get (28). This concludes the pro of.  Computational Complexity W e will argue that there exists an implementat ion of Mom ent Search whic h on input k = O (1 /ǫ ) run s in time U · p oly( n ) · (1 /ǫ ) O (log 2 (1 /ǫ )) , where U is the num b er of bits required to represent a pa y oﬀ v alue of the game. Observe ﬁr s t that the n umber of p ossible guesses for S tep 1 of Moment Search is at m ost n 2 O ((1 /ǫ ) 6 ). Observe further that the num b er of p ossible guesses for µ ℓ in Step 2 is at m ost 33 t  k 2 2  ℓ (where t ≤ k 3 is the num b er of pla ye rs who mix), so jointly the n umb er of p ossible guesses for µ 1 , . . . , µ d is at most d Y ℓ =1 t  k 2 2  ℓ = t d  k 2 2  d ( d +1) / 2 =  1 ǫ  O ( log 2 1 ǫ ) . The s ame asym p totic up p er b oun d applies to the total num b er of guesses for µ ′ 1 , . . . , µ ′ d . Giv en the ab o v e the total n umb er of guesses that Moment S earch has to do is n 2  1 ǫ  O ( log 2 1 ǫ ) . W e next argue that the r u nning time r equ ired to complete Steps 3, 4, and 5 is at most O ( n 3 ) · U ·  1 ǫ  O (log 2 (1 /ǫ )) . F or this we establish the follo wing; we giv e the pr oof in the en d of this section. Claim 11. Given a set of v alues µ 1 , . . . , µ d , µ ′ 1 , . . . , µ ′ d , wher e, for al l ℓ = 1 , . . . , d , µ ℓ , µ ′ ℓ ∈ ( 0 ,  1 k 2  ℓ , 2  1 k 2  ℓ , . . . , B ) , for some B ∈ N , discr ete sets T 1 , . . . , T m ⊆  0 , 1 k 2 , 2 k 2 , . . . , 1  , and four inte gers m 0 , m 1 ≤ m , m s , m b ≤ B , it is p ossible to solve the system of e quations: (Σ) : X p i ∈ (0 , 1 / 2] p ℓ i = µ ℓ , for al l ℓ = 1 , . . . , d, X p i ∈ (1 / 2 , 1) p ℓ i = µ ′ ℓ , for al l ℓ = 1 , . . . , d, |{ i | p i = 0 }| = m 0 |{ i | p i = 1 }| = m 1 |{ i | p i ∈ (0 , 1 / 2] }| = m s |{ i | p i ∈ (1 / 2 , 1) }| = m b with r esp e ct to the variables p 1 ∈ T 1 , . . . , p m ∈ T m , or to determine that no solution exists, in time O ( m 3 ) B O ( d ) k O ( d 2 ) . Applying Claim 11 with m ≤ t , B ≤ t (where t ≤ k 3 is the num b er of play ers wh o mix), m 0 = 0, m 1 = 0, shows that Steps 3(a)i , 3(c)i can b e completed in time O ( t 3 ) t O ( d ) k O ( d 2 ) =  1 ǫ  O (log 2 (1 /ǫ )) . 34 Another application of Claim 11 with m = n , B ≤ t , m 0 ≤ n , m 1 ≤ n sho ws that Step 4 of Moment S earch can b e completed in time O ( n 3 ) t O ( d ) k O ( d 2 ) = O ( n 3 ) ·  1 ǫ  O (log 2 (1 /ǫ )) . Finally , we argue that the computation of the exp ected utilities U i 0 and U i 1 required in Steps 3(a)iii, 3(c)iii of Mo ment S ear ch can b e d one eﬃcien tly us ing dynamic programming with O ( n 2 ) op era- tions on n umb er s with at most b ( n, k ) := ⌈ 1 + n log 2 ( k 2 ) + U ) ⌉ bits, w h ere U is the n um b er of bits required to sp ecify a pa yo ﬀ v alue of the game. 5 Therefore, the ov erall time required f or the execution of Moment Search is O ( n 3 ) · U ·  1 ǫ  O (log 2 (1 /ǫ )) . Pro of of C laim 11: W e u se dynamic programming. Let us consider th e follo wing tensor of dimension 2 d + 5: A ( i, z 0 , z 1 , z s , z b ; ν 1 , . . . , ν d ; ν ′ 1 , . . . , ν ′ d ) , where i ∈ [ m ], z 0 , z 1 ∈ { 0 , . . . , m } , z s , z b ∈ { 0 , . . . , B } and ν ℓ , ν ′ ℓ ∈ ( 0 ,  1 k 2  ℓ , 2  1 k 2  ℓ , . . . , B ) , for ℓ = 1 , . . . , d. T he total num b er of cells in A is m · ( m + 1) 2 · ( B + 1) 2 · d Y ℓ =1 ( B k 2 ℓ + 1) ! 2 ≤ O ( m 3 ) B O ( d ) k 2 d ( d +1) . Ev ery cell of A is assigned v alue 0 or 1, as follo ws: A ( i, z 0 , z 1 , z s , z b ; ν 1 , . . . , ν d , ν ′ 1 , . . . , ν ′ d ) = 1 ⇔             There exist p 1 ∈ T 1 , . . . , p i ∈ T i suc h that |{ j ≤ i | p j = 0 }| = z 0 , |{ j ≤ i | p j = 1 }| = z 1 , |{ j ≤ i | p j ∈ (0 , 1 / 2] }| = z s , |{ j ≤ i | p j ∈ (1 / 2 , 1) }| = z b , P j ≤ i : p j ∈ (0 , 1 / 2] p ℓ j = ν ℓ , for all ℓ = 1 , . . . , d , P j ≤ i : p j ∈ (1 / 2 , 1) p ℓ j = ν ′ ℓ , for all ℓ = 1 , . . . , d .             . 5 T o compute a b ound on the num b er of bits requ ired for the exp ected utility computations, note that every non- zero probabilit y v alue that is computed along the execution of the algorithm must b e an in teger multiple of ( 1 k 2 ) n − 1 , since the mixed strategies of all p la yers are from the set { 0 , 1 /k 2 , 2 /k 2 , . . . , 1 } . F urther note that the exp ected utilit y is a weigh ted sum of ( n − 1) p a yoﬀ val ues, with U bits required t o represent each val ue, and all weig hts b eing probabilities. 35 It is easy to complete A working in la y ers of increasing i . W e initialize all en tries to v alue 0. Then, the ﬁrst lay er A (1 , · , · ; · , . . . , · ) can b e completed easily as follo ws: A (1 , 1 , 0 , 0 , 0; 0 , 0 , . . . , 0; 0 , 0 , . . . , 0) = 1 ⇔ 0 ∈ T 1 A (1 , 0 , 1 , 0 , 0; 0 , 0 , . . . , 0; 0 , 0 . . . , 0) = 1 ⇔ 1 ∈ T 1 A (1 , 0 , 0 , 1 , 0; p, p 2 , . . . , p d ; 0 , . . . , 0) = 1 ⇔ p ∈ T 1 ∩ (0 , 1 / 2 ] A (1 , 0 , 0 , 0 , 1; 0 , . . . , 0; p, p 2 , . . . , p d ) = 1 ⇔ p ∈ T 1 ∩ (1 / 2 , 1) Inductive ly , to complete lay er i + 1, w e consider all the non-zero entries of la ye r i and for ev ery su c h non-zero entry and f or ev ery v i +1 ∈ T i +1 , we ﬁnd w h ic h en try of lay er i + 1 we w ould tran s ition to if w e c hose p i +1 = v i +1 . W e set that en try equal to 1 and w e also sa v e a p oin ter to this entry from the corresp onding en try of la y er i , lab eling that p ointer with the v alue v i +1 . The time we need to complete la y er i + 1 is b ounded b y |T i +1 | . ( m + 1) 2 B O ( d ) k 2 d ( d +1) ≤ O ( m 2 ) B O ( d ) k O ( d 2 ) . Therefore, th e ov erall time needed to complete A is O ( m 3 ) B O ( d ) k O ( d 2 ) . After completing tensor A , it is easy to c hec k if there exists a solution to (Σ). A solution exists if and only if A ( m, m 0 , m 1 , m s , m b ; µ 1 , . . . , µ d ; µ ′ 1 , . . . , µ ′ d ) = 1 , and can b e found by tracing bac k the p ointe rs from this cell of A . The o ve rall run ning time is dominated by the time needed to ﬁll in A .  36

On Oblivious PTASs for Nash Equilibrium

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