A Polynomial-Time Algorithm for Action-Graph Games

Transcription

1 A Polynomal-Tme Algorthm for Acton-Graph Games Albert Xn Jang Kevn Leyton-Brown Department of Computer Scence Unversty of Brtsh Columba Abstract Acton-Graph Games (AGGs) (Bhat & Leyton-Brown 2004) are a fully expressve game representaton whch can compactly express strct and context-specfc ndependence and anonymty structure n players utlty functons. We present an effcent algorthm for computng expected payoffs under mxed strategy profles. Ths algorthm runs n tme polynomal n the sze of the AGG representaton (whch s tself polynomal n the number of players when the n-degree of the acton graph s bounded). We also present an extenson to the AGG representaton whch allows us to compactly represent a wder varety of structured utlty functons. 1 Introducton Game-theoretc models have recently been very nfluental n the computer scence communty. In partcular, smultaneous-acton games have receved consderable study, whch s reasonable as these games are n a sense the most fundamental. In order to analyze these models, t s often necessary to compute game-theoretc quanttes rangng from expected utlty to Nash equlbra. Most of the game theoretc lterature presumes that smultaneous-acton games wll be represented n normal form. Ths s problematc because qute often games of nterest have a large number of players and a large set of acton choces. In the normal form representaton, we store the game s payoff functon as a matrx wth one entry for each player s payoff under each combnaton of all players actons. As a result, the sze of the representaton grows exponentally wth the number of players. Even f we had enough space to store such games, most of the computatons we d le to perform on these exponental-szed obects tae exponental tme. Fortunately, most large games of any practcal nterest have hghly structured payoff functons, and thus t s possble to represent them compactly. (Intutvely, ths s why humans are able to reason about these games n the frst place: we understand the payoffs n terms of smple relatonshps rather than n terms of enormous loo-up tables.) One nfluental class of representatons explot strct ndependences between players utlty functons; ths class nclude graphcal games (Kearns, Lttman, & Sngh 2001), Copyrght c 2006, Amercan Assocaton for Artfcal Intellgence ( All rghts reserved. 1 We would le to acnowledge the contrbutons of Navn A.R. Bhat, who s one of the authors of the paper whch ths wor extends. mult-agent nfluence dagrams (Koller & Mlch 2001), and game nets (LaMura 2000). A second approach to compactly representng games focuses on context-specfc ndependences n agents utlty functons that s, games n whch agents abltes to affect each other depend on the actons they choose. Snce the context-specfc ndependences consdered here are condtoned on actons and not agents, t s often natural to also explot anonymty n utlty functons, where each agent s utltes depend on the dstrbuton of agents over the set of actons, but not on the denttes of the agents. Examples nclude congeston games (Rosenthal 1973) and local effect games (LEGs) (Leyton-Brown & Tennenholtz 2003). Both of these representatons mae assumptons about utlty functons, and as a result cannot represent arbtrary games. Bhat & Leyton-Brown (2004) ntroduced acton graph games (AGGs). Smlar to LEGs, AGGs use graphs to represent the context-specfc ndependences of agents utlty functons, but unle LEGs, AGGs can represent arbtrary games. Bhat & Leyton-Brown proposed an algorthm for computng expected payoffs usng the AGG representaton. For AGGs wth bounded n-degree, ther algorthm s exponentally faster than normal-form-based algorthms, yet stll exponental n the number of players. In ths paper we mae several sgnfcant mprovements to results n (Bhat & Leyton-Brown 2004). Frst, we present an mproved algorthm for computng expected payoffs. Our new algorthm s able to better explot anonymty structure n utlty functons. For AGGs wth bounded n-degree, our algorthm s polynomal n the number of players. We then extend the AGG representaton by ntroducng functon nodes. Ths feature allows us to compactly represent a wder range of structured utlty functons. We also descrbe computatonal experments whch confrm our theoretcal predctons of compactness and computatonal speedup. Acton Graph Games An acton-graph game (AGG) s a tuple N, S, ν, u. Let N = {1,..., n} denote the set of agents. Denote by S = N S the set of acton profles, where s the Cartesan product and S s agent s set of actons. We denote by s S one of agent s actons, and s S an acton profle. Agents may have actons n common. Let S N S denote the set of dstnct acton choces n the game. Let denote the set of confguratons of agents over actons. A confguraton D s an ordered tuple of S ntegers (D(s), D(s ),...), wth one nteger for each acton n S. For each s S, D(s) specfes the number of agents that chose acton s S. Let D : S be the func-

2 ton that maps from an acton profle s to the correspondng confguraton D. These shared actons express the game s anonymty structure: agent s utlty depends only on her acton s and the confguraton D(s). Let G be the acton graph: a drected graph havng one node for each acton s S. The neghbor relaton s gven by ν : S 2 S. If s ν(s) there s an edge from s to s. Let D (s) denote a confguraton over ν(s),.e. D (s) s a tuple of ν(s) ntegers, one for each acton n ν(s). Intutvely, agents are only counted n D (s) f they tae an acton whch s an element of ν(s). (s) s the set of confguratons over ν(s) gven that some player has played s. 2 Smlarly we defne D (s) : S (s) whch maps from an acton profle to the correspondng confguraton over ν(s). The acton graph expresses context-specfc ndependences of utltes of the game: N, f chose acton s S, then s utlty depends only on the numbers of agents who chose actons connected to s, whch s the confguraton D (s) (s). In other words, the confguraton of actons not n ν(s ) does not affect s utlty. We represent the agents utltes usng a tuple of S functons u (u s, u s,...), one for each acton s S. Each u s s a functon u s : (s) R. So f agent chose acton s, and the confguraton over ν(s) s D (s), then agent s utlty s u s (D (s) ). Observe that all agents have the same utlty functon,.e. condtoned on choosng the same acton s, the utlty each agent receves does not depend on the dentty of the agent. For notatonal convenence, we defne u(s, D (s) ) u s (D (s) ) and u (s) u(s, D (s) (s)). Bhat & Leyton-Brown (2004) provded several examples of AGGs, showng that AGGs can represent arbtrary games, graphcal games and games exhbtng context-specfc ndependence wthout any strct ndependence. Due to space lmts we do not reproduce these examples here. Sze of an AGG Representaton We have clamed that acton graph games provde a way of representng games compactly. But what exactly s the sze of an AGG representaton? And how does ths sze grow as the number of agents n grows? Let I = max s ν(s),.e. the maxmum n-degree of the acton graph. The sze of an AGG representaton s domnated by the sze of ts utlty functons. 3 For each acton s, we need to specfy a utlty value for each dstnct confguraton D (s) (s). The set of confguratons (s) can be derved from the acton graph, and can be sorted n lexcographcal order. So we do not need to explctly specfy (s) ; we can ust specfy a lst of (s) utlty values that correspond to the (ordered) set of confguratons. 4 (s), the number of dstnct confgu- 2 If acton s s n multple players acton sets (say players, ), and these acton sets do not completely overlap, then t s possble that the set of confguratons gven that played s (denoted (s,) ) s dfferent from the set of confguratons gven that played s. (s) s the unon of these sets of confguratons. 3 The acton graph can be represented as neghbor lsts, wth space complexty O( S I). 4 Ths s the most compact way of representng the utlty functons, but does not provde easy random access of the utltes. ratons over ν(s), n general does not have a closed-form expresson. Instead, we consder the operaton of extendng all agents acton sets va : S S. Now the number of confguratons over ν(s) s an upper bound on (s). The bound s the number of (ordered) combnatoral compostons of n 1 (snce one player has already chosen s) nto ν(s) + 1 nonnegatve ntegers, whch s (n 1+ ν(s) )! (n 1)! ν(s)!. Then the total space requred for the utltes s bounded from above by S (n 1+I)! If I s bounded by a constant (n 1)!I!. as n grows, the representaton sze grows le O( S n I ),.e. polynomally wth respect to n. For each AGG, there exsts a unque nduced normal form representaton wth the same set of players and S actons for each ; ts utlty functon s a matrx that specfes each player s payoff for each possble acton profle s S. Ths mples a space complexty of n n =1 S. When S S for all, ths becomes n S n, whch grows exponentally wth respect to n. The number of payoff values stored n an AGG representaton s always less than or equal to the number of payoff values n the nduced normal form representaton. For each entry n the nduced normal form whch represents s utlty under acton profle s, there exsts a unque acton profle s n the AGG wth the correspondng acton for each player. Ths s nduces a unque confguraton D(s) over the AGG s acton nodes. By constructon of the AGG utlty functons, D(s) together wth s determnes a unque utlty u s (D (s) (s)) n the AGG. Furthermore, there are no entres n the AGG utlty functons that do not correspond to any acton profle (s, s ) n the normal form. Ths means that there exsts a many-to-one mappng from entres of normal form to utltes n the AGG. Of course, the AGG representaton has the extra overhead of representng the acton graph, whch s bounded by S I. But asymptotcally, AGG s space complexty s never worse than the equvalent normal form. Computng wth AGGs One of the man motvatons of compactly representng games s to do effcent computaton on the games. We focus on the computatonal tas of computng expected payoffs under a mxed strategy profle. Besdes beng mportant n tself, ths tas s an essental component of many game-theoretc applcatons, e.g. computng best responses, Govndan and Wlson s contnuaton methods for fndng Nash equlbra (2003; 2004), the smplcal subdvson algorthm for fndng Nash equlbra (van der Laan, Talman, & van der Heyden 1987), and fndng correlated equlbra usng Papadmtrou s algorthm (2005). Let ϕ(x) denote the set of all probablty dstrbutons over a set X. Defne the set of mxed strateges for as Σ ϕ(s ), and the set of all mxed strategy profles as Σ N Σ. We denote an element of Σ by σ, an element of Σ by σ, and the probablty that plays acton s as σ (s). Therefore, when we want to do computaton usng AGG, we may convert each utlty functon u s to a data structure that effcently mplements a mappng from sequences of ntegers to (floatngpont) numbers, (e.g. tres, hash tables or Red-Blac trees), wth space complexty n the order of O(I (s) ).

3 Defne the expected utlty to agent for playng pure strategy s, gven that all other agents play the mxed strategy profle σ, as V s (σ ) s S u (s, s ) Pr(s σ ). (1) where Pr(s σ ) = σ (s ) s the probablty of s under the mxed strategy σ. Equaton (1) s a sum over the set S of acton profles of players other than. The number of terms s S, whch grows exponentally n n. Thus (1) s an exponental tme algorthm for computng Vs (σ ). If we were usng the normal form representaton, there really would be S dfferent outcomes to consder, each wth potentally dstnct payoff values, so evaluaton Equaton (1) s the best we could do. Can we do better usng the AGG representaton? Snce AGGs are fully expressve, representng a game wthout any structure as an AGG would not gve us any computatonal savngs compared to the normal form. Instead, we are nterested n structured games that have a compact AGG representaton. In ths secton we present an algorthm that gven any, s and σ, computes the expected payoff Vs (σ ) n tme polynomal wth respect to the sze of the AGG representaton. In other words, our algorthm s effcent f the AGG s compact, and requres tme exponental n n f t s not. In partcular, recall that for classes of AGGs whose n-degrees are bounded by a constant, ther szes are polynomal n n. As a result our algorthm wll be polynomal n n for such games. Frst we consder how to tae advantage of the contextspecfc ndependence structure of the AGG,.e. the fact that s payoff when playng s only depends on the confguratons n the neghborhood of. Ths allows us to proect the other players strateges nto smaller acton spaces that are relevant gven s. Intutvely we construct a graph from the pont of vew of an agent who too a partcular acton, expressng hs ndfference between actons that do not affect hs chosen acton. Ths can be thought of as nducng a context-specfc graphcal game. Formally, for every acton s S defne a reduced graph G (s) by ncludng only the nodes ν(s) and a new node denoted. The only edges ncluded n G (s) are the drected edges from each of the nodes ν(s) to the node s. Player s acton s s proected to a node s (s) n the reduced graph G (s) by the followng map- { png: s (s) s s ν(s) s ν(s). In other words, actons that are not n ν(s) (and therefore do not affect the payoffs of agents playng s) are proected to. The resultng proected acton set S (s) has cardnalty at most mn( S, ν(s) + 1). We defne the set of mxed strateges on the proected acton set S (s) by Σ (s) ϕ(s (s) the orgnal acton set S s proected to σ (s) followng mappng: { σ (s) (s (s) ) ). A mxed strategy σ on σ (s ) s S \ν(s) σ (s ) s (s) Σ (s) s ν(s) = by the. (2) So gven s and σ, we can compute σ (s ) n O(n S ) tme n the worst case. Now we can operate entrely on the proected space, and wrte the expected payoff as Vs (σ ) = u(s, D (s) (s, s )) Pr(s (s) σ(s) ) s (s ) S(s ) where Pr(s (s ) σ(s ) ) = σ(s ) (s (s ) ). The summaton s over S (s ), whch n the worst case has ( ν(s ) + 1) (n 1) terms. So for AGGs wth strct or context-specfc ndependence structure, computng Vs (σ ) ths way s much faster than dong the summaton n (1) drectly. However, the tme complexty of ths approach s stll exponental n n. Next we want to tae advantage of the anonymty structure of the AGG. Recall from our dscusson of representaton sze that the number of dstnct confguratons s usually smaller than the number of dstnct pure acton profles. So deally, we want to compute the expected payoff Vs (σ ) as a sum over the possble confguratons, weghted by ther probabltes: V s (σ ) = D (s ) (s,) u (s, D (s) )P r(d (s) σ (s) ) (3) where σ (s ) (s, σ (s ) ) and Pr(D (s ) σ (s ) ) = s:d (s ) (s)=d (s ) =1 N σ (s ) (4) whch s the probablty of D (s) gven the mxed strategy profle σ (s). Equaton (3) s a summaton of sze (s,), the number of confguratons gven that played s, whch s polynomal n n f I s bounded. The dffcult tas s to compute Pr(D (s) σ (s) ) for all D (s) (s,),.e. the probablty dstrbuton over (s,) nduced by σ (s). We observe that the sum n Equaton (4) s over the set of all acton profles correspondng to the confguraton D (s). The sze of ths set s exponental n the number of players. Therefore drectly computng the probablty dstrbuton usng Equaton (4) would tae exponental tme n n. Indeed ths s the approach proposed n (Bhat & Leyton-Brown 2004). Can we do better? We observe that the players mxed strateges are ndependent,.e. σ s a product probablty dstrbuton σ(s) = σ (s ). Also, each player affects the confguraton D ndependently. Ths structure allows us to use dynamc programmng (DP) to effcently compute the probablty dstrbuton Pr(D (s) σ (s) ). The ntuton behnd our algorthm s to apply one agent s mxed strategy at a tme. Let σ (s) 1... denote the proected strategy profle of agents {1,..., }. Denote by (s ) the set of confguratons nduced by actons of agents {1,..., }. Smlarly denote D (s) (s). Denote by P the probablty dstrbuton on (s) nduced by σ (s) 1..., and by P [D] the probablty of confguraton D. At teraton of the algorthm, we compute P from P 1 and σ (s ). After teraton n, the algorthm stops and returns P n. The pseudocode of our DP algorthm s shown as Algorthm 1. Due to space lmts we omt the proof of correctness of our algorthm.

4 Algorthm 1 Computng the nduced probablty dstrbuton Pr(D (s) σ (s) ). Algorthm ComputeP Input: s, σ (s ) Output: P n, whch s the dstrbuton P r(d (s) σ (s) ) represented as a tre. D (s ) 0 = (0,..., 0) P 0 [D (s ) 0 ] = 1.0 // Intalzaton: (s ) 0 = {D (s ) 0 } for = 1 to n do Intalze P to be an empty tre for all D (s ) 1 from P 1 do for all s (s ) S (s ) such that σ (s ) (s (s ) ) > 0 do D (s ) = D (s ) 1 f s (s ) then D (s ) ) += 1 // Apply acton s (s ) end for end for end for return P n (s (s ) end f f P [D (s ) P [D (s ) ] = 0.0 ] does not exst yet then end f P [D (s ) ] += P 1 [D (s ) 1 ] σ(s ) (s (s ) ) Each D (s) s represented as a sequence of ntegers, so P s a mappng from sequences of ntegers to real numbers. We need a data structure to manpulate such probablty dstrbutons over confguratons (sequences of ntegers) whch permts quc looup, nserton and enumeraton. An effcent data structure for ths purpose s a tre (Fredn 1962). Tres are commonly used n text processng to store strngs of characters, e.g. as dctonares for spell checers. Here we use tres to store strngs of ntegers rather than characters. Both looup and nserton complexty s lnear n ν(s ). To acheve effcent enumeraton of all elements of a tre, we store the elements n a lst, n the order of ther nsertons. Our algorthm for computng Vs (σ ) conssts of frst computng the proected strateges usng (2), then followng Algorthm 1, and fnally dong the weghted sum gven n (3). The overall complexty s O(n S + n ν(s ) 2 (s,) (σ ) ), where (s,) (σ ) denotes the set of confguratons over ν(s ) that have postve probablty of occurrng under the mxed strategy (s, σ ). Due to space lmts we omt the dervaton of ths complexty result. Snce (s,) (σ ) (s,) (s), and (s) s the number of payoff values stored n payoff functon u s, ths means that expected payoffs can be computed n polynomal tme wth respect to the sze of the AGG. Furthermore, our algorthm s able to explot strateges wth small supports whch lead to a small (s,) (σ ). Snce (s) s bounded by (n 1+ ν(s) )! (n 1)! ν(s )!, ths mples that f the n-degree of the graph s bounded by a constant, then the complexty of computng expected payoffs s O(n S + n I+1 ). Theorem 1. Gven an AGG representaton of a game, s expected payoff Vs (σ ) can be computed n tme polynomal n the sze of the representaton. If I, the n-degree of the acton graph, s bounded by a constant, V s (σ ) can be computed n tme polynomal n n. AGG wth Functon Nodes There are games wth certan nds of context-specfc ndependence structures that AGGs are not able to explot. Example 1. In the Coffee Shop Game there are n players; each player s plannng to open a new coffee shop n a downtown area, but has to decde on the locaton. The downtown area s represented by a r c grd. Each player can choose to open the shop at any of the B rc blocs, or decde not to enter the maret. Condtoned on player choosng some locaton s, her utlty depends on the number of players that chose the same bloc, the number of players that chose any of the surroundng blocs, and the number of players that chose any other locaton. The normal form representaton of ths game has sze n S n = n(b + 1) n. Let us now represent the game as an AGG. We observe that f agent chooses an acton s correspondng to one of the B locatons, then her payoff s affected by the confguraton over all B locatons. Hence, ν(s) would consst of B acton nodes correspondng to the B locatons. The acton graph has n-degree I = B. Snce the acton sets completely overlap, the representaton sze s O( S (s) ) = O(B (n 1+B)! (n 1)!B! ). If we hold B constant, ths becomes O(Bn B ), whch s exponentally more compact than the normal form representaton. If we nstead hold n constant, the sze of the representaton s O(B n ), whch s only slghtly better than the normal form. Intutvely, the AGG representaton s only able to explot the anonymty structure n ths game. However, ths game s payoff functon does have context-specfc structure. Observe that u s depends only on three quanttes: the number of players that chose the same bloc, the surroundng blocs, and other locatons. In other words, u s can be wrtten as a functon g of only 3 ntegers: u s (D (s) ) = g(d(s), s S D(s ), s S D(s )) where S s the set of actons that surrounds s and S the set of actons correspondng to the other locatons. Because the AGG representaton s not able to explot ths context-specfc nformaton, utlty values are duplcated n the representaton. We can fnd smlar examples where u s could be wrtten as a functon of a small number of ntermedate parameters. One example s a party game where u s depends only on whether s ν(s) D(s ) s even or odd. Thus u s would have ust two dstnct values, but the AGG representaton would have to specfy a value for every confguraton D (s). Ths nd of structure can be exploted wthn the AGG framewor by ntroducng functon nodes to the acton graph G. Now G s vertces consst of both the set of acton nodes S and the set of functon nodes P. We requre that no functon node p P can be n any player s acton set,.e. S P = {}. Each node n G can have acton nodes and/or functon nodes as neghbors. For each p P, we ntroduce a functon f p : (p) N, where D (p) (p) denotes confguratons over p s neghbors. The confguratons D are extended over the entre set of nodes, by defnng D(p) f p (D (p) ). Intutvely, D(p) are the ntermedate parameters that players utltes depend on.

5 Fgure 1: A 5 6 Coffee Shop Game: Left: the AGG representaton wthout functon nodes (loong at only the neghborhood of the a node s). Rght: after ntroducng two functon nodes, s now has only 3 ncomng edges. To ensure that the AGG s meanngful, the graph G restrcted to nodes n P s requred to be a drected acyclc graph (DAG). Furthermore t s requred that every p P has at least one neghbor (.e. ncomng edge). These condtons ensure that D(s) for all s and D(p) for all p are well-defned. To ensure that every p P s useful, we also requre that p has at least one out-gong edge. As before, for each acton node s we defne a utlty functon u s : (s) R. We call ths extended representaton (N, S, P, ν, {f p } p P, u) an Acton Graph Game wth Functon Nodes (AGGFN). Representaton Sze Gven an AGGFN, we can construct an equvalent AGG wth the same players N and actons S and equvalent utlty functons, but represented wthout any functon nodes. We put an edge from s to s n the AGG f ether there s an edge from s to s n the AGGFN, or there s a path from s to s through a chan of functon nodes. The number of utltes stored n an AGGFN s no greater than the number of utltes n the equvalent AGG wthout functon nodes. We can show ths by followng smlar arguments as before, establshng a many-to-one mappng from utltes n the AGG representaton to utltes n the AGGFN. On the other hand, AGGFNs have to represent the functons f p, whch can ether be mplemented usng elementary operatons, or represented as mappngs smlar to u s. We want to add functon nodes only when they represent meanngful ntermedate parameters and hence reduce the number of ncomng edges on acton nodes. Consder our coffee shop example. For each acton node s correspondng to a locaton, we ntroduce functon nodes p s and p s. Let ν(p s) consst of actons surroundng s, and ν(p s ) consst of actons for the other locatons. Then we modfy ν(s) so that t has 3 nodes: ν(s) = {s, p s, p s }, as shown n Fgure 1. For all functon nodes p P, we defne f p (D (p) ) = m ν(p) D(m). Now each D(s) s a confguraton over only 3 nodes. Snce f p s a summaton operator, (s) s the number of compostons of n 1 nto 4 nonnegatve ntegers, (n+2)! (n 1)!3! = n(n + 1)(n + 2)/6 = O(n3 ). We must therefore store O(Bn 3 ) utlty values. Computng wth AGGFNs Our expected-payoff algorthm cannot be drectly appled to AGGFNs wth arbtrary f p. Frst of all, proecton of strateges does not wor drectly, because a player playng an acton s ν(s) could stll affect D (s) va functon nodes. Furthermore, our DP algorthm for computng the probabltes does not wor because for an arbtrary functon node p ν(s), each player would not be guaranteed to affect D(p) ndependently. Therefore n the worst case we need to convert the AGGFN to an AGG wthout functon nodes n order to apply our algorthm. Ths means that we are not always able to translate the extra compactness of AGGFNs over AGGs nto more effcent computaton. Defnton 1. An AGGFN s contrbuton-ndependent (CI) f For all p P, ν(p) S,.e. the neghbors of functon nodes are acton nodes. There exsts a commutatve and assocatve operator, and for each node s S an nteger w s, such that gven an acton profle s, for all p P, D(p) = N:s ν(p) w s. Note that ths defnton entals that D(p) can be wrtten as a functon of D (p) by collectng terms: D(p) f p (D (p) ) = s ν(p) ( D(s) =1 w s). The coffee shop game s an example of a contrbutonndependent AGGFN, wth the summaton operator servng as, and w s = 1 for all s. For the party game mentoned earler, s nstead addton mod 2. If we are modelng an aucton, and want D(p) to represent the amount of the wnnng bd, we would let w s be the bd amount correspondng to acton s, and be the max operator. For contrbuton-ndependent AGGFNs, t s the case that for all functon nodes p, each player s strategy affects D(p) ndependently. Ths fact allows us to adapt our algorthm to effcently compute the expected payoff Vs (σ ). For smplcty we present the algorthm for the case where we have one operator for all p P, but our approach can be drectly appled to games wth dfferent operators assocated wth dfferent functon nodes, and lewse wth a dfferent set of w s for each operator. We defne the contrbuton of acton s to node m S P, denoted C s (m), as 1 f m = s, 0 f m S \ {s}, and m ν(m)( C s(m ) =1 w s ) f m P. Then t s easy to verfy that gven an acton profle s, D(s) = n =1 C s (s) for all s S and D(p) = n =1 C s (p) for all p P. Gven that player played s, we defne the proected contrbuton of acton s, denoted C (s ) s, as the tuple (C s (m)) m ν(s ). Note that dfferent actons may have dentcal proected contrbutons. Player s mxed strategy σ nduces a probablty dstrbuton over s proected contrbutons, Pr(C (s) σ ) = s :C (s σ ) s =C (s ) (s ). Now we can operate entrely usng the probabltes on proected contrbutons nstead of the mxed strategy probabltes. Ths s analogous to the proecton of σ to σ (s ) n our algorthm for AGGs wthout functon nodes. Algorthm 1 for computng the dstrbuton P r(d (s) σ) can be straghtforwardly adopted to wor wth contrbutonndependent AGGFNs: whenever we apply player s contrbuton C (s) s to D (s) 1, the resultng confguraton D(s) s computed componentwse as follows: D (s ) (m) =

6 C (s) s 1 C (s ) s (m) + D (s ) 1 (m) f m S, and D(s ) (m) = (m) D (s) (m) f m P. Followng smlar complexty analyss, f an AGGFN s CI, expected payoffs can be computed n polynomal tme wth respect to the representaton sze. Appled to the coffee shop example, snce (s) = O(n 3 ), our algorthm taes O(n S + n 4 ) tme, whch grows lnearly n S. Experments We mplemented the AGG representaton and our algorthm for computng expected payoffs n C++. We ran several experments to compare the performance of our mplementaton aganst the (heavly optmzed) GameTracer mplementaton (Blum, Shelton, & Koller 2002) whch performs the same computaton for a normal form representaton. We used the Coffee Shop game (wth randomly-chosen payoff values) as a benchmar. We vared both the number of players and the number of actons. Frst, we compared the AGGFNs representaton sze to that of the normal form. The results confrmed our theoretcal predctons that the AGGFN representaton grows polynomally wth n whle the normal form representaton grows exponentally wth n. (The graph s omtted because of space constrants.) Second, we tested the performance of our dynamc programmng algorthm aganst GameTracer s normal form based algorthm for computng expected payoffs, on Coffee Shop games of dfferent szes. For each game nstance, we generated 1000 random strategy profles wth full support, and measured the CPU (user) tme spent computng the expected payoffs under these strategy profles. We fxed the sze of blocs at 5 5 and vared the number of players. Fgure 2 shows plots of the results. For very small games the normal form based algorthm s faster due to ts smaller booeepng overhead; as the number of players grows larger, our AGGFN-based algorthm s runnng tme grows polynomally, whle the normal form based algorthm scales exponentally. For more than fve players, we were not able to store the normal form representaton n memory. Next, we fxed the number of players at 4 and number of columns at 5, and vared the number of rows. Our algorthm s runnng tme grew roughly lnearly n the number of rows, whle the normal form based algorthm grew le a hgher-order polynomal. Ths was consstent wth our theoretcal predcton that our algorthm tae O(n S + n 4 ) tme for ths class of games whle normal-form based algorthms tae O( S n 1 ) tme. Last, we consdered strategy profles havng partal support (though space prevents showng the fgure). Whle ensurng that each player s support ncluded at least one acton, we generated strategy profles wth each acton ncluded n the support wth probablty 0.4. GameTracer too about 60% of ts full-support runnng tmes to compute expected payoffs n ths doman, whle our algorthm requred about 20% of ts full-support runnng tmes. Conclusons We presented a polynomal-tme algorthm for computng expected payoffs n acton-graph games. For AGGs wth CPU tme n seconds number of players AGG NF CPU tme n seconds AGG NF number of rows Fgure 2: Runnng tmes for payoff computaton n the Coffee Shop Game. Left: 5 5 grd wth 3 to 16 players. Rght: 4-player r 5 grd wth r varyng from 3 to 10. bounded n-degree, our algorthm acheves an exponental speed-up compared to normal-form based algorthms and Bhat & Leyton-Brown s algorthm (2004). We also extended the AGG representaton by ntroducng functon nodes, whch allows us to compactly represent a wder range of structured utlty functons. We showed that f an AG- GFN s contrbuton-ndependent, expected payoffs can be computed n polynomal tme. In the full verson of ths paper we wll also dscuss speedng up the computaton of Nash and correlated equlbra. We have combned our expected-payoff algorthm wth GameTracer s mplementaton of Govndan & Wlson s algorthm (2003) for computng Nash equlbra, and acheved exponental speedup compared to the normal form. Also, as a drect corollary of our Theorem 1 and Papadmtrou s result (2005), correlated equlbra can be computed n tme polynomal n the sze of the AGG. References Bhat, N., and Leyton-Brown, K Computng Nash equlbra of acton-graph games. In UAI. Blum, B.; Shelton, C.; and Koller, D Gametracer. Fredn, E Tre memory. Comm. ACM 3: Govndan, S., and Wlson, R A global Newton method to compute Nash equlbra. Journal of Economc Theory 110: Govndan, S., and Wlson, R Computng Nash equlbra by terated polymatrx approxmaton. Journal of Economc Dynamcs and Control 28: Kearns, M.; Lttman, M.; and Sngh, S Graphcal models for game theory. In UAI. Koller, D., and Mlch, B Mult-agent nfluence dagrams for representng and solvng games. In IJCAI. LaMura, P Game networs. In UAI. Leyton-Brown, K., and Tennenholtz, M Local-effect games. In IJCAI. Papadmtrou, C Computng correlated equlbra n multplayer games. In STOC. Avalable at chrstos/papers/cor.ps. Rosenthal, R A class of games possessng pure-strategy Nash equlbra. Int. J. Game Theory 2: van der Laan, G.; Talman, A.; and van der Heyden, L Smplcal varable dmenson algorthms for solvng the nonlnear complementarty problem on a product of unt smplces usng a general labellng. Mathematcs of OR 12(3):