Computing Correlated Equilibria in Multi-Player Games
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1 Computng Correlated Equlbra n Mult-Player Games Chrstos H. Papadmtrou Presented by Zhanxang Huang December 7th,
2 The Author Dr. Chrstos H. Papadmtrou CS professor at UC Berkley (taught at Harvard, MIT, Athens Polytechnc, Stanford, and UCSD) Books: Elements of the theory of computaton (Prentce- Hall 1982, wth Harry Lews, second edton September 1997 ) Computatonal Complexty (Addson Wesley, 1994) 2
3 Outlne Motvaton & Related Work Correlated Equlbra (CE) Revew Explct Descrptons n Symmetrc Games Applcatons n General Games Polynomal Algorthm for Fndng a CE Dscusson Concluson 3
4 Motvaton Computatonal Complexty of Equlbra Intractablty of an equlbrum concept would make the game model mplausble. Polynomal tme algorthms are promsng. Mult-player Games Most algorthmc research n game theory amed at 2- player case so far. Expanson to nclude mult-player games s requred by models for markets, auctons, and networks. 4
5 Related Research Fcttous Play Converge to the Nash equlbrum n zero sum cases but not general cases. Adaptve Procedure The prob. a player changes ts strategy s proportonal to the player s regret for havng played the present strategy Converge to correlated equlbra but n exponental tme. 5
6 Three Relevant Papers [23] C. H. Papadmtrou, T. Roughgarden Computng equlbra n multplayer games, 2005 SODA. [11] S. Hart and D. Schmedler Exstence of Correlated Equlbra, Mathematcs of Operatons Research 14, 1, 18-25, Ths one. 6
7 Man Results Develop a general framework based on the lnear programmng for obtanng polynomaltme algorthms for optmzng over correlated equlbra, and present complexty results mplyng that such algorthms are not possble n certan other such games. Prove the exstence of polynomal-tme algorthms and gve an algorthm for computng correlated equlbra n a broad classes of succnctly representable multplayer games. 7
8 Correlated Equlbra (CE) Revew Game Let G= ({ S },{ u}), = 1.. n, be an n-player game, where S 's are fnte strategy sets and 1 2 n Correlated Equlbra u ( s) s the payoff functon for player. S=S S S s the strategy profle space. Let p be a probablty dstrbuton on S. Dstrbuton q s a correlated equlbrum f for each player and each par ( l, l') of strateges n S satsfes (1), where s' s obtaned from s by reassgnng 's strategy to be l'. 8
9 Example: Chcken Game (a) (b) (c) (d) (e) 9
10 Explct Descrpton of CE (1) Am for a descrpton equvalent to q. Consequently, every lnear functon can be effcently optmzed over the set of CE and n partcular, one can be found. Frst, look at symmetrc games, especally S =2 case. 10
11 Explct Descrpton of CE (2) Let G = ( S = {1, 2}, u, u ) be an n-player 2-strategy symmetrc game. 1 n Am for the probablty dstrbuton on S, q( s). qs ( ) p( j) (Basc Varables) & p( j) (Auxlary Varables) S ( j) S, n whch exactly j players, ncludng player, choose strategy 1. S( j) S, n whch exactly j players choose strategy 1. p( j): the aggregate probablty assgned to S ( j). p( j): the total probablty of S( j). 11
12 Explct Descrpton of CE (3) The constrants for CE are as follows, where u ( j, l) denotes the payoff to player n a strategy profle n whch player chooses strategy l and a total of j players choose strategy 1. (2)-(6) are called the basc lnear system of G. 12
13 Explct Descrpton of CE (4) q p( j) & p( j) Snce constrants (2)-(3) are effectvely aggregated versons of the CE constrants (1), every correlated equlbrum of an n-player, 2-strategy symmetrc game G nduces a soluton to G's basc lnear systme va the ntended aggregatons of probablty. p( j) & p( j) q The most nterestng part!!! We say p extends to S f there s a functon q: S R+ wth qs () = p() j & qs () = p( j). ( ) s S ( j) s S j Theorem 3.1 Let G be a 2-strategy symmetrc game. Then every soluton to G's basc lnear system can be extended to a correlated equlbrum of G. 13
14 Explct Descrpton of CE (5) Theorem 3.1 can be proved by usng Lemma 3.3 & 3.4 shown below. Snce the lnear program can be solved n polynomal tme n worst case by usng the ellpsod algorthm and the converson from p ( j ) & p( j) to q can be done also n polynomal tme, the CE can be found and the every lnear optmzaton over CE can be done n polynomal tme. 14
15 Explct Descrpton of CE (6) Extend the defnton of a basc lnear system to k-strategy symmetrc games. There are varables p ( j, l) and p( j), where j s now an ordered partton of n nto k non-negatve ntegers, and l S = {1,2,, k}, wth p ( j, l) = p( j) for every player. l Theorem 3.6 Let G be a symmetrc game. Then p s a soluton to G's basc lnear system f and only f t can be extended to a correlated equlbrum of G. 15
16 Fndng CE of General Compact Games (1) 16
17 Fndng CE of General Compact Games (2) The proof of Theorem 4.3 shows that every lnear functon can be effcently optmzed over the set of correlated equlbra of such a game. Next, the applcatons are demonstrated to show the power of Theorem
18 Applcaton (1) Symmetrc Games A n-player, k-strategy symmetrc game admts a compact representaton j P = { P }, where represents a player and j represents an ordered partton of n 1 nto k parts correspondng to a dstrbuton of the other n 1 players among the k avalable strateges. The separaton problem for P: Gven y ( j, l), s there a s S wth j y ( j, l) < 0? (, jl,): s= ls, P Ths problem can be solved n polynomal tme usng mn-cost flow method. Corollary 4.4 A correlated equlbrum of a symmetrc game can be found n tme polynomal n ts natural compact representaton. 18
19 Applcaton (2) Graphcal Games j A graphcal game admts a compact representaton P= { P }, where represents a player and j represents an assgnment of strateges to the players that are neghbors of. The separaton problem for P: Gven y ( j, l), s there a s S wth y( j, l) < 0?, (, j,): l s = l, s P can be solved dynamc programmng. j Corollary 4.5 A correlated equlbrum of a graphcal game wth a tree topology can be found n tme polynomal n ts natural compact representaton. Proposton 4.6 The separaton problem for the natural compact representaton of a graphcal game s NP-complete, even n bpartte graphs. Proposton 4.7 Assumng P NP, there s no polynomal-tme algorthm for computng a correlated equlbrum of a compactly represented graphcal game that maxmzes the expected sum of player payoffs. 19
20 Applcaton (3) Congeston Games A congeston G = ( E, S,, S, n,, n, { c } ), where E s the ground set, S's are strategy set for player, n's are quanttes of player, 1 k 1 k e e E {c e} e E s the const functons defned on {1, 2,, n} Proposton 4.8 Assumng P NP, there s no polynomal-tme algorthm for computng a correlated equlbrum of a compactly represented congeston game that maxmzes the expected sum of player payoffs. 20
21 Pessmsm In a polymatrx game, t s NP-hard to compute the correlated equlbrum that optmzes the sum of all utltes n such games. In a graphcal game fndng the optmum correlated equlbrum n certan smple nontree graphcal games s NP-hard It s also true n hypergraphcal games, congeston games, local effect games, schedulng games and etc. 21
22 Fnd a CE n Every Game (1) Theorem Every game has a correlated equlbrum. Method: Utlze the dualty of lnear programs. Prmal: Dual: max x s T Ux 0 U y -1 x 0 y 0 Ux 0 means the CE constrants n (1): qsu ( ) ( s) qsu ( ) ( s') (1) ss : = l ss : = l (1) has soluton Prmal s unbounded Dual s nfeasble 22
23 Fnd a CE n Every Game (2) Polynamal Algorthm for Fndng a CE: 1) Apply ellpsod algorthm to Dual; T 2) At each step fnd the x that volated nequlty xu Y 1; 3) When the elllpsod algorthm termnates at the Lth step: T We have X=(x,, x ) such that each x volates xu Y 1 1 L T [ XU ] y 1, y 0 s nfeasble; α 0 s unbouned; The soluton α vector provdes the CE for the orgnal game, T Its dual [UX ] a convex combnaton of the x 's that satsfes Prmal; 23
24 Applcaton (4) Man Result 24
25 Applcaton (5) Games wth a poly-ce Scheme 1. Polymatrx games. 2. Graphcal games. 3. Hypergraphcal games. 4. Congeston games 5. Local effect games 6. Schedulng games 7. Faclty locaton games 8. Network desgn games 9. Symmetrc games 25
26 Open Questons Is there method other than ellpsod-based algorthm? Is there an approach to fndng CE to games of exponental type (e.g. congeston games)? Is there any game not found yet, on whch ths approach wll fal? 26
27 Thank You! 27
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