{"title": "Bayesian Action-Graph Games", "book": "Advances in Neural Information Processing Systems", "page_first": 991, "page_last": 999, "abstract": "Games of incomplete information, or Bayesian games, are an important game-theoretic model and have many applications in economics. We propose Bayesian action-graph games (BAGGs), a novel graphical representation for Bayesian games. BAGGs can represent arbitrary Bayesian games, and furthermore can compactly express Bayesian games exhibiting commonly encountered types of structure including symmetry, action- and type-specific utility independence, and probabilistic independence of type distributions. We provide an algorithm for computing expected utility in BAGGs, and discuss conditions under which the algorithm runs in polynomial time.  Bayes-Nash equilibria of BAGGs can be computed by adapting existing algorithms for complete-information normal form games and leveraging our expected utility algorithm. We show both theoretically and empirically that our approaches improve significantly on the state of the art.", "full_text": "Bayesian Action-Graph Games\n\nAlbert Xin Jiang\n\nDepartment of Computer Science\nUniversity of British Columbia\n\njiang@cs.ubc.ca\n\nKevin Leyton-Brown\n\nDepartment of Computer Science\nUniversity of British Columbia\n\nkevinlb@cs.ubc.ca\n\nAbstract\n\nGames of incomplete information, or Bayesian games, are an important game-\ntheoretic model and have many applications in economics. We propose Bayesian\naction-graph games (BAGGs), a novel graphical representation for Bayesian games.\nBAGGs can represent arbitrary Bayesian games, and furthermore can compactly\nexpress Bayesian games exhibiting commonly encountered types of structure in-\ncluding symmetry, action- and type-speci\ufb01c utility independence, and probabilistic\nindependence of type distributions. We provide an algorithm for computing ex-\npected utility in BAGGs, and discuss conditions under which the algorithm runs in\npolynomial time. Bayes-Nash equilibria of BAGGs can be computed by adapting\nexisting algorithms for complete-information normal form games and leveraging\nour expected utility algorithm. We show both theoretically and empirically that our\napproaches improve signi\ufb01cantly on the state of the art.\n\n1\n\nIntroduction\n\nIn the last decade, there has been much research at the interface of computer science and game\ntheory (see e.g. [19, 22]). One fundamental class of computational problems in game theory is\nthe computation of solution concepts of a \ufb01nite game. Much of current research on computation\nof solution concepts has focused on complete-information games, in which the game being played\nis common knowledge among the players. However, in many multi-agent situations, players are\nuncertain about the game being played. Harsanyi [10] proposed games of incomplete information (or\nBayesian games) as a mathematical model of such interactions. Bayesian games have found many\napplications in economics, including most notably auction theory and mechanism design.\nOur interest is in computing with Bayesian games, and particularly in identifying sample Bayes-Nash\nequilibrium. There are two key obstacles to performing such computations ef\ufb01ciently. The \ufb01rst\nis representational: the straightforward tabular representation of Bayesian game utility functions\n(the Bayesian Normal Form) requires space exponential in the number of players. For large games,\nit becomes infeasible to store the game in memory, and performing even computations that are\npolynomial time in the input size are impractical. An analogous obstacle arises in the context of\ncomplete-information games: there the standard representation (normal form) also requires space\nexponential in the number of players. The second obstacle is the lack of existing algorithms for\nidentifying sample Bayes-Nash equilibrium for arbitrary Bayesian games. Harsanyi [10] showed\nthat a Bayesian game can be interpreted as an equivalent complete-information game via \u201cinduced\nnormal form\u201d or \u201cagent form\u201d interpretations. Thus one approach is to interpret a Bayesian game\nas a complete-information game, enabling the use of existing Nash-equilibrium-\ufb01nding algorithms\n(e.g. [24, 9]). However, generating the normal form representations under both of these complete-\ninformation interpretations causes a further exponential blowup in representation size.\nMost games of interest have highly-structured payoff functions, and thus it is possible to overcome\nthe \ufb01rst obstacle by representing them compactly. This has been done for complete information\ngames through (e.g.) the graphical games [16] and Action-Graph Games (AGGs) [1] representations.\nIn this paper we propose Bayesian Action-Graph Games (BAGGs), a compact representation for\n\n1\n\n\fBayesian games. BAGGs can represent arbitrary Bayesian games, and furthermore can compactly\nexpress Bayesian games with commonly encountered types of structure. The type pro\ufb01le distribution\nis represented as a Bayesian network, which can exploit conditional independence structure among\nthe types. BAGGs represent utility functions in a way similar to the AGG representation, and like\nAGGs, are able to exploit anonymity and action-speci\ufb01c utility independencies. Furthermore, BAGGs\ncan compactly express Bayesian games exhibiting type-speci\ufb01c independence: each player\u2019s utility\nfunction can have different kinds of structure depending on her instantiated type. We provide an\nalgorithm for computing expected utility in BAGGs, a key step in many algorithms for game-theoretic\nsolution concepts. Our approach interprets expected utility computation as a probabilistic inference\nproblem on an induced Bayesian Network. In particular, our algorithm runs in polynomial time\nfor the important case of independent type distributions. To compute Bayes-Nash equilibria for\nBAGGs, we consider the agent form interpretation of the BAGG. Although a naive normal form\nrepresentation would require an exponential blowup, BAGGs can act as a compact representation\nof the agent form. Computational tasks on the agent form can be done ef\ufb01ciently by leveraging our\nexpected utility algorithm for BAGGs. We have implemented our approach by adapting two Nash\nequilibrium algorithms, the simplicial subdivision algorithm [24] and Govindan and Wilson\u2019s global\nNewton method [9]. We show empirically that our approach outperforms the existing approaches of\nsolving for Nash on the induced normal form or on the normal form representation of the agent form.\nWe now discuss some related literature. There has been some research on heuristic methods for\n\ufb01nding Bayes-Nash equilibria for certain classes of auction games using iterated best response (see\ne.g. [21, 25]). Such methods are not guaranteed to converge to a solution. Howson and Rosenthal\n[12] applied the agent form transformation to 2-player Bayesian games, resulting in a complete-\ninformation polymatrix game. Our approach can be seen as a generalization of their method to\ngeneral Bayesian games. Singh et al. [23] proposed a incomplete information version of the graphical\ngame representation, and presented ef\ufb01cient algorithms for computing approximate Bayes-Nash\nequilibria in the case of tree games. Gottlob et al. [7] considered a similar extension of the graphical\ngame representation and analyzed the problem of \ufb01nding a pure-strategy Bayes-Nash equilibrium.\nLike graphical games, such representations are limited in that they can only exploit strict utility\nindependencies. Oliehoek et al. [20] proposed a heuristic search algorithm for common-payoff\nBayesian games, which has applications to cooperative multi-agent problems. Bayesian games can\nbe interpreted as dynamic games with a initial move by Nature; thus, also related is the literature\non representations for dynamic games, including multi-agent in\ufb02uence diagrams (MAIDs) [17]\nand temporal action-graph games (TAGGs) [14]. Compared to these representations for dynamic\ngames, BAGGs focus explicitly on structure common to Bayesian games; in particular, only BAGGs\ncan ef\ufb01ciently express type-speci\ufb01c utility structure. Also, by representing utility functions and\ntype distributions as separate components, BAGGs can be more versatile (e.g., a future direction\nis to answer computational questions that do not depend on the type distribution, such as ex-post\nequilibria). Furthermore, BAGGs can be solved by adapting Nash-equilibrium algorithms such as\nGovindan and Wilson\u2019s global Newton method [9] for static games; this is generally more practical\nthan their related Nash equilibrium algorithm [8] that directly works on dynamic games: while both\napproach avoids the exponential blowup of transforming to the induced normal form, the algorithm\nfor dynamic games has to solve an additional quadratic program at each step.\n\n2 Preliminaries\n2.1 Complete-information Games\n\nby ai \u2208 Ai one of i\u2019s actions. An action pro\ufb01le a = (a1, . . . , an) \u2208 (cid:81)\nagents\u2019 actions. Agent i\u2019s utility function is ui :(cid:81)\n\nWe assume readers are familiar with the basic concepts of complete-information games and here we\nonly establish essential notation. A complete-information game is a tuple (N,{Ai}i\u2208N ,{ui}i\u2208N )\nwhere N = {1, . . . , n} is the set of agents; for each agent i, Ai is the set of i\u2019s actions. We denote\ni\u2208N Ai is a tuple of the\nj\u2208N Aj \u2192 R. A mixed strategy \u03c3i for player i\nis a probability distribution over Ai. A mixed strategy pro\ufb01le \u03c3 is a tuple of the n players\u2019 mixed\nstrategies. We denote by ui(\u03c3) the expected utility of player i under the mixed strategy pro\ufb01le \u03c3. We\nadopt the following notational convention: for any n-tuple X we denote by X\u2212i the elements of X\ncorresponding to players other than i.\nA game representation is a data structure that stores all information needed to specify a game. A\nnormal form representation of a game uses a matrix to represent each utility function ui. The size of\n\nthis representation is n(cid:81)\n\nj\u2208N |Aj|, which grows exponentially in the number of players.\n\n2\n\n\f2.2 Bayesian Games\nWe now de\ufb01ne Bayesian games and discuss common types of structure.\nDe\ufb01nition 1. A Bayesian game is a tuple (N,{Ai}i\u2208N , \u0398, P,{ui}i\u2208N ) where N = {1, . . . , n} is\ni Ai is the set of action pro\ufb01les;\ni \u0398i is the set of type pro\ufb01les, where \u0398i is player i\u2019s set of types; P : \u0398 \u2192 R is the type\n\nthe set of players; each Ai is player i\u2019s action set, and A = (cid:81)\n\u0398 =(cid:81)\n\ndistribution and ui : A \u00d7 \u0398 \u2192 R is the utility function for player i.\nAs in the complete-information case, we denote by ai an element of Ai, and a = (a1, . . . , an) an\naction pro\ufb01le. Furthermore we denote by \u03b8i an element of \u0398i, and by \u03b8 a type pro\ufb01le. The game\nis played as follows. A type pro\ufb01le \u03b8 = (\u03b81, . . . , \u03b8n) \u2208 \u0398 is drawn according to the distribution P .\nEach player i observes her type \u03b8i and, based on this observation, chooses from her set of actions Ai.\nEach player i\u2019s utility is then given by ui(a, \u03b8), where a is the resulting action pro\ufb01le.\nPlayer i can deterministically choose a pure strategy si, in which given each \u03b8i \u2208 \u0398i she deterministi-\ncally chooses an action si(\u03b8i). Player i can also randomize and play a mixed strategy \u03c3i, in which her\nprobability of choosing ai given \u03b8i is \u03c3i(ai|\u03b8i). That is, given a type \u03b8i \u2208 \u0398i, she plays according to\ndistribution \u03c3i(\u00b7|\u03b8i) over her set of actions Ai. A mixed strategy pro\ufb01le \u03c3 = (\u03c31, . . . , \u03c3n) is a tuple\nof the players\u2019 mixed strategies.\nThe expected utility of i given \u03b8i under a mixed strategy pro\ufb01le \u03c3 is the expected value of i\u2019s utility\nunder the resulting joint distribution of a and \u03b8, conditioned on i receiving type \u03b8i:\n\n(cid:88)\n\n(cid:88)\n\n(cid:89)\n\nui(\u03c3|\u03b8i) =\n\nP (\u03b8\u2212i|\u03b8i)\n\nui(a, \u03b8)\n\n\u03c3j(aj|\u03b8j).\n\n(1)\n\n\u03b8\u2212i\n\na\n\nj\n\nA mixed strategy pro\ufb01le \u03c3 is a Bayes-Nash equilibrium if for all i, for all \u03b8i, for all ai \u2208 Ai,\nui(\u03c3|\u03b8i) \u2265 ui(\u03c3\u03b8i\u2192ai|\u03b8i), where \u03c3\u03b8i\u2192ai is the mixed strategy pro\ufb01le that is identical to \u03c3 except\nthat i plays ai with probability 1 given \u03b8i.\nIn specifying a Bayesian game, the space bottlenecks are the type distribution and the utility functions.\nWithout additional structure, we cannot do better than representing each utility function ui : A\u00d7\u0398 \u2192\nR as a table and the type distribution as a table as well. We call this representation the Bayesian\n\nnormal form. The size of this representation is n \u00d7(cid:81)n\ni=1(|\u0398i| \u00d7 |Ai|) +(cid:81)n\ni.e. the type-pro\ufb01le distribution P (\u03b8) is a product distribution: P (\u03b8) =(cid:81)\ndistribution P can be represented compactly using(cid:80)\n\nWe say a Bayesian game has independent type distributions if players\u2019 types are drawn independently,\ni P (\u03b8i). In this case the\nGiven a permutation of players \u03c0 : N \u2192 N and an action pro\ufb01le a = (a1, . . . , an), let a\u03c0 =\n(a\u03c0(1), . . . , a\u03c0(n)). Similarly let \u03b8\u03c0 = (\u03b8\u03c0(1), . . . , \u03b8\u03c0(n)). We say the type distribution P is symmetric\nif |\u0398i| = |\u0398j| for all i, j \u2208 N, and if for all permutations \u03c0 : N \u2192 N, P (\u03b8) = P (\u03b8\u03c0). We say a\nBayesian game has symmetric utility functions if |Ai| = |Aj| and |\u0398i| = |\u0398j| for all i, j \u2208 N, and if\nfor all permutations \u03c0 : N \u2192 N, we have ui(a, \u03b8) = u\u03c0(i)(a\u03c0, \u03b8\u03c0) for all i \u2208 N. A Bayesian game\nis symmetric if its type distribution and utility functions are symmetric. The utility functions of such\n\ni |\u0398i| numbers.\n\ni=1 |\u0398i|.\n\na game range over at most |\u0398i||Ai|(cid:0)n\u22122+|\u0398i||Ai|\n\n|\u0398i||Ai|\u22121\n\n(cid:1) unique utility values.\n\nA Bayesian game exhibits conditional utility independence if each player i\u2019s utility depends on the\naction pro\ufb01le a and her own type \u03b8i, but does not depend on the other players\u2019 types. Then the utility\nfunction of each player i ranges over at most |A||\u0398i| unique utility values.\n\n2.2.1 Complete-information interpretations\n\nHarsanyi [10] showed that any Bayesian game can be interpreted as a complete-information game,\nsuch that Bayes-Nash equilibria of the Bayesian game correspond to Nash equilibria of the complete-\ninformation game. There are two complete-information interpretations of Bayesian games.\nA Bayesian game can be converted to its induced normal form, which is a complete-information game\nwith the same set of n players, in which each player\u2019s set of actions is her set of pure strategies in the\nBayesian game. Each player\u2019s utility under an action pro\ufb01le is de\ufb01ned to be equal to the player\u2019s\nexpected utility under the corresponding pure strategy pro\ufb01le in the Bayesian game.\nAlternatively, a Bayesian game can be transformed to its agent form, where each type of each player\nin the Bayesian game is turned into one player in a complete-information game. Formally, given a\n\n3\n\n\fgame ( \u02dcN ,{ \u02dcAj,\u03b8j}(j,\u03b8j )\u2208 \u02dcN ,{\u02dcuj,\u03b8j}(j,\u03b8j )\u2208 \u02dcN ), where \u02dcN consists of(cid:80)\naction set of j in the Bayesian game. The set of action pro\ufb01les is then \u02dcA =(cid:81)\n\nBayesian game (N,{Ai}i\u2208N , \u0398, P,{ui}i\u2208N ), we de\ufb01ne its agent form as the complete-information\nj\u2208N |\u0398j| players, one for every\ntype of every player of the Bayesian game. We index the players by the tuple (j, \u03b8j) where j \u2208 N\nand \u03b8j \u2208 \u0398j. For each player (j, \u03b8j) \u2208 \u02dcN of the agent form game, her action set \u02dcA(j,\u03b8j ) is Aj, the\nA(j,\u03b8j ). The utility\nfunction of player (j, \u03b8j) is \u02dcuj,\u03b8j : \u02dcA \u2192 R. For all \u02dca \u2208 \u02dcA, \u02dcuj,\u03b8j (\u02dca) is equal to the expected utility of\nplayer j of the Bayesian game given type \u03b8j, under the pure strategy pro\ufb01le s\u02dca, where for all i and all\n\u03b8i, s\u02dca\ni (\u03b8i) = \u02dca(i,\u03b8i). Observe that there is a one-to-one correspondence between action pro\ufb01les in\nthe agent form and pure strategies of the Bayesian game. A similar correspondence exists for mixed\nstrategy pro\ufb01les: each mixed strategy pro\ufb01le \u03c3 of the Bayesian game corresponds to a mixed strategy\n\u02dc\u03c3 of the agent form, with \u02dc\u03c3(i,\u03b8i)(ai) = \u03c3i(ai|\u03b8i) for all i, \u03b8i, ai. It is straightforward to verify that\n\u02dcui,\u03b8i(\u02dc\u03c3) = ui(\u03c3|\u03b8i) for all i, \u03b8i. This implies a correspondence between Bayes Nash equilibria of a\nBayesian game and Nash equilibria of its agent form.\nProposition 2. \u03c3 is a Bayes-Nash equilibrium of a Bayesian game if and only if \u02dc\u03c3 is a Nash\nequilibrium of its agent form.\n\nj,\u03b8j\n\n3 Bayesian Action-Graph Games\n\nIn this section we introduce Bayesian Action-Graph Games (BAGGs), a compact representation of\nBayesian games. First consider representing the type distributions. Speci\ufb01cally, the type distribution\nP is speci\ufb01ed by a Bayesian network (BN) containing at least n random variables corresponding to\nthe n players\u2019 types \u03b81, . . . , \u03b8n. For example, when the types are independently distributed, then P\ncan be speci\ufb01ed by the simple BN with n variables \u03b81, . . . , \u03b8n and no edges.\nNow consider representing the utility functions. Our approach is to adapt concepts from the AGG\nrepresentation [1, 13] to the Bayesian game setting. At a high level, a BAGG is a Bayesian game on\nan action graph, a directed graph on a set of action nodes A. To play the game, each player i, given\nher type \u03b8i, simultaneously chooses an action node from her type-action set Ai,\u03b8i \u2286 A. Each action\nnode thus corresponds to an action choice that is available to one or more of the players. Once the\nplayers have made their choices, an action count is tallied for each action node \u03b1 \u2208 A, which is the\nnumber of agents that have chosen \u03b1. A player\u2019s utility depends only on the action node she chose\nand the action counts on the neighbors of the chosen node.\nWe now turn to a formal description of BAGG\u2019s utility function representation. Central to our model\nis the action graph. An action graph G = (A, E) is a directed graph where A is the set of action\nnodes, and E is a set of directed edges, with self edges allowed. We say \u03b1(cid:48) is a neighbor of \u03b1 if there\nis an edge from \u03b1(cid:48) to \u03b1, i.e., if (\u03b1(cid:48), \u03b1) \u2208 E. Let the neighborhood of \u03b1, denoted \u03bd(\u03b1), be the set of\nneighbors of \u03b1.\nFor each player i and each instantiation of her type \u03b8i \u2208 \u0398i, her type-action set Ai,\u03b8i \u2286 A is the set\nof possible action choices of i given \u03b8i. These subsets are unrestricted: different type-action sets\nmay (partially or completely) overlap. De\ufb01ne player i\u2019s total action set to be A\u222a\nAi,\u03b8i.\nthe set of action pro\ufb01les, and by a \u2208 A an action pro\ufb01le. Observe that\nthe action pro\ufb01le a provides suf\ufb01cient information about the type pro\ufb01le to be able to determine the\noutcome of the game; there is no need to additionally encode the realized type distribution. We note\nthat for different types \u03b8i, \u03b8(cid:48)\nmay have different sizes; i.e., i may have different\nnumbers of available action choices depending on her realized type.\nA con\ufb01guration c is a vector of |A| non-negative integers, specifying for each action node the\nnumbers of players choosing that action. Let c(\u03b1) be the element of c corresponding to the action\n\u03b1. Let C : A (cid:55)\u2192 C be the function that maps from an action pro\ufb01le a to the corresponding\ncon\ufb01guration c. Formally, if c = C(a) then c(\u03b1) = |{i \u2208 N : ai = \u03b1}| for all \u03b1 \u2208 A. De\ufb01ne\nC = {c : \u2203a \u2208 A such that c = C(a)}. In other words, C is the set of all possible con\ufb01gurations.\nWe can also de\ufb01ne a con\ufb01guration over a subset of nodes. In particular, we will be interested in\ncon\ufb01gurations over a node\u2019s neighborhood. Given a con\ufb01guration c \u2208 C and a node \u03b1 \u2208 A, let\nthe con\ufb01guration over the neighborhood of \u03b1, denoted c(\u03b1), be the restriction of c to \u03bd(\u03b1), i.e.,\nc(\u03b1) = (c(\u03b1(cid:48)))\u03b1(cid:48)\u2208\u03bd(\u03b1). Similarly, let C (\u03b1) denote the set of con\ufb01gurations over \u03bd(\u03b1) in which at\nleast one player plays \u03b1. Let C(\u03b1) : A (cid:55)\u2192 C (\u03b1) be the function which maps from an action pro\ufb01le to\nthe corresponding con\ufb01guration over \u03bd(\u03b1).\n\nWe denote by A =(cid:81)\n\ni \u2208 \u0398i, Ai,\u03b8i and Ai,\u03b8(cid:48)\n\ni A\u222a\n\ni\n\ni =(cid:83)\n\n\u03b8i\u2208\u0398i\n\ni\n\n4\n\n\fG,{u\u03b1}\u03b1\u2208A) where N is the set of agents; \u0398 = (cid:81)\n\nDe\ufb01nition 3. A Bayesian action-graph game (BAGG) is a tuple (N, \u0398, P, {Ai,\u03b8i}i\u2208N,\u03b8i\u2208\u0398i ,\ni \u0398i is the set of type pro\ufb01les; P is the type\ndistribution, represented as a Bayesian network; Ai,\u03b8i \u2286 A is the type-action set of i given \u03b8i;\nG = (A, E) is the action graph; and for each \u03b1 \u2208 A, the utility function is u\u03b1 : C (\u03b1) \u2192 R.\n\nIntuitively, this representation captures two types of structure in utility functions: \ufb01rstly, shared\nactions capture the game\u2019s anonymity structure: if two action choices from different type-action sets\nshare an action node \u03b1, it means that these two actions are interchangeable as far as the other players\u2019\nutilities are concerned. In other words, their utilities may depend on the number of players that chose\nthe action node \u03b1, but not the identities of those players. Secondly, the (lack of) edges between\nnodes in the action graph expresses action- and type-speci\ufb01c independencies of utilities of the game:\ndepending on player i\u2019s chosen action node (which also encodes information about her type), her\nutility depends on con\ufb01gurations over different sets of nodes.\nLemma 4. An arbitrary Bayesian game given in Bayesian normal form can be encoded as a BAGG\nstoring the same number of utility values.\n\nProof. Provided in the supplementary material.\n\nBayesian games with symmetric utility functions exhibit anonymity structure, which can be expressed\nin BAGGs by sharing action nodes. Speci\ufb01cally, we label each \u0398i as {1, . . . , T}, so that each\nt \u2208 {1, . . . , T} corresponds to a class of equivalent types. Then for each t \u2208 {1, . . . , T}, we have\nAi,t = Aj,t for all i, j \u2208 N, i.e. type-action sets for equivalent types are identical.\n3.1 BAGGs with function nodes\nIn this section we extend the basic BAGG representation by introducing function nodes to the action\ngraph. The concept of function nodes was \ufb01rst introduced in the (complete-information) AGG setting\n[13]. Function nodes allow us to exploit a much wider variety of utility structures in BAGGs.\nIn this extended representation, the action graph G\u2019s vertices consist of both the set of action nodes A\nand the set of function nodes F. We require that no function node p \u2208 F can be in any player\u2019s action\nset. Each function node p \u2208 F is associated with a function f p : C (p) \u2192 R. We extend c by de\ufb01ning\nc(p) to be the result of applying f p to the con\ufb01guration over p\u2019s neighbors, f p(c(p)). Intuitively, c(p)\ncan be used to describe intermediate parameters that players\u2019 utilities depend on. To ensure that the\nBAGG is meaningful, the graph restricted to nodes in F is required to be a directed acyclic graph. As\nbefore, for each action node \u03b1 we de\ufb01ne a utility function u\u03b1 : C (\u03b1) \u2192 R.\nOf particular computational interest is the subclass of contribution-independent function nodes\n(also introduced by [13]). A function node p in a BAGG is contribution-independent if \u03bd(p) \u2286 A,\nthere exists a commutative and associative operator \u2217, and for each \u03b1 \u2208 \u03bd(p) an integer w\u03b1, such\nthat given an action pro\ufb01le a = (a1, . . . , an), c(p) = \u2217i\u2208N :ai\u2208\u03bd(p) wai. A BAGG is contribution-\nindependent if all its function nodes are contribution-independent. Intuitively, if function node p is\ncontribution-independent, each player\u2019s strategy affects c(p) independently.\nA very useful kind of contribution-independent function nodes are counting function nodes, which\nset \u2217 to the summation operator + and the weights to 1. Such a function node p simply counts the\nfor P is exponential only in the in-degree of the BN. The utility functions store(cid:80)\nnumber of players that chose any action in \u03bd(p).\nLet us consider the size of a BAGG representation. The representation size of the Bayesian network\n\u03b1 |C (\u03b1)| values. As\nin similar analysis for AGGs [15], estimations of this size generally depend on what types of function\nnodes are included. We state only the following (relatively straightforward) result since in this paper\nwe are mostly concerned with BAGGs with counting function nodes.\nTheorem 5. Consider BAGGs whose only function nodes, if any, are counting function nodes. If the\nin-degrees of the action nodes as well as the in-degrees of the Bayesian networks for P are bounded\ni |\u0398i| and\n\nby a constant, then the sizes of the BAGGs are bounded by a polynomial in n, |A|, |F|,(cid:80)\n\nthe sizes of domains of variables in the BN.\n\nThis theorem shows a nice property of counting function nodes: representation size does not grow\nexponentially in the in-degrees of these counting function nodes. The next example illustrates the\nusefulness of counting function nodes, including for expressing conditional utility independence.\n\n5\n\n\fExample 6 (Coffee Shop game). Consider a symmetric Bayesian game involving n players; each\nplayer plans to open a new coffee shop in a downtown area, but has to decide on the location. The\ndowntown area is represented by a r \u00d7 k grid. Each player can choose to open a shop located within\nany of the B \u2261 rk blocks or decide not to enter the market. Each player has T types, representing\nher private information about her cost of opening a coffee shop. Players\u2019 types are independently\ndistributed. Conditioned on player i choosing some location, her utility depends on: (a) her own\ntype; (b) the number of players that chose the same block; (c) the number of players that chose any of\nthe surrounding blocks; and (d) the number of players that chose any other location.\n\nThe Bayesian normal form representation of this game has size n[T (B + 1)]n. The game can be\nexpressed as a BAGG as follows. Since the game is symmetric, we label the types as {1, . . . , T}. A\ncontains one action O corresponding to not entering and T B other action nodes, with each location\ncorresponding to a set of T action nodes, each representing the choice of that location by a player\nwith a different type. For each t \u2208 {1, . . . , T}, the type-action sets Ai,t = Aj,t for all i, j \u2208 N and\neach consists of the action O and B actions corresponding to locations for type t. For each location\n(x, y) we create three function nodes: pxy representing the number of players choosing this location,\np(cid:48)\nxy representing the number of players choosing any surrounding blocks, and p(cid:48)(cid:48)\nxy representing the\nnumber of players choosing any other block. Each of these function nodes is a counting function\nnode, whose neighbors are action nodes corresponding to the appropriate locations (for all types).\nEach action node for location (x, y) has three neighbors, pxy, p(cid:48)\nxy. Since the BAGG action\ngraph has maximum in-degree 3, by Theorem 5 the representation size is polynomial in n, B and T .\n\nxy, and p(cid:48)(cid:48)\n\n4 Computing a Bayes-Nash Equilibrium\n\nIn this section we consider the problem of \ufb01nding a sample Bayes-Nash equilibrium given a BAGG.\nOur overall approach is to interpret the Bayesian game as a complete-information game, and then to\napply existing algorithms for \ufb01nding Nash equilibria of complete-information games. We consider\ntwo state-of-the-art Nash equilibrium algorithms, van der Laan et al\u2019s simplicial subdivision [24]\nand Govindan and Wilson\u2019s global Newton method [9]. Both run in exponential time in the worst\ncase, and indeed recent complexity theoretic results [3, 6, 4] imply that a polynomial-time algorithm\nfor Nash equilibrium is unlikely to exist.1 Nevertheless, we show that we can achieve exponential\nspeedups in these algorithms by exploiting the structure of BAGGs.\nRecall from Section 2.2.1 that a Bayesian game can be transformed into its induced normal form or\nits agent form. In the induced normal form, each player i has |Ai||\u0398i| actions (corresponding to her\npure strategies of the Bayesian game). Solving such a game would be infeasible for large |\u0398i|; just to\nrepresent an Nash equilibrium requires space exponential in |\u0398i|.\nA more promising approach is to consider the agent form. Note that we can straightforwardly adapt\nthe agent-form transformation described in Section 2.2.1 to the setting of BAGGs: now the action set\nof player (i, \u03b8i) of the agent form corresponds to the type-action set Ai,\u03b8i of the BAGG. The resulting\ni\u2208N |\u0398i| players and |Ai,\u03b8i| actions for each player (i, \u03b8i); a\n|Ai,\u03b8i| numbers. However, the normal form\n|Ai,\u03b8i|, which grows exponentially in n\nand |\u0398i|. Applying the Nash equilibrium algorithms to this normal form would be infeasible in terms\nof time and space. Fortunately, we do not have to explicitly represent the agent form as a normal\nform game. Instead, we treat a BAGG as a compact representation of its agent form, and carry out\nany required computation on the agent form by operating on the BAGG. A key computational task\nrequired by both Nash equilibrium algorithms in their inner loops is the computation of expected\nutility of the agent form. Recall from Section 2.2.1 that for all (i, \u03b8i) the expected utility \u02dcui,\u03b8i(\u02dc\u03c3) of\nthe agent form is equal to the expected utility ui(\u03c3|\u03b8i) of the Bayesian game. Thus in the remainder\nof this section we focus on the problem of computing expected utility in BAGGs.\n\ncomplete-information game has(cid:80)\nNash equilibrium can be represented using just(cid:80)\nrepresentation of the agent form has size(cid:80)\n\ni,\u03b8i\n\n(cid:80)\nj\u2208N |\u0398j|(cid:81)\n\n\u03b8i\n\ni\n\n4.1 Computing Expected Utility in BAGGs\nRecall that \u03c3\u03b8i\u2192ai is the mixed strategy pro\ufb01le that is identical to \u03c3 except that i plays ai given \u03b8i.\nThe main quantity we are interested in is ui(\u03c3\u03b8i\u2192ai|\u03b8i), player i\u2019s expected utility given \u03b8i under\n1There has been some research on ef\ufb01cient Nash-equilibrium-\ufb01nding algorithms for subclasses of games,\nsuch as Daskalakis and Papadimitriou\u2019s [5] PTAS for anonymous games with \ufb01xed numbers of actions. One\nfuture direction would be to adapt these algorithms to subclasses of Bayesian games.\n\n6\n\n\fui(\u03c3|\u03b8i) =(cid:80)\n\nui(\u03c3\u03b8i\u2192ai|\u03b8i)\u03c3i(ai|\u03b8i).\n\nai\n\nthe strategy pro\ufb01le \u03c3\u03b8i\u2192ai. Note that the expected utility ui(\u03c3|\u03b8i) can then be computed as the sum\n\nj\n\nj , and given its parent \u03b8j, its CPD chooses an action from A\u222a\n\nOne approach is to directly apply Equation (1), which has (|\u0398\u2212i| \u00d7 |A|) terms in the summation.\nFor games represented in Bayesian normal form, this algorithm runs in time polynomial in the\nrepresentation size. Since BAGGs can be exponentially more compact than their equivalent Bayesian\nnormal form representations, this algorithm runs in exponential time for BAGGs.\nIn this section we present a more ef\ufb01cient algorithm that exploits BAGG structure. We \ufb01rst formulate\nthe expected utility problem as a Bayesian network inference problem. Given a BAGG and a mixed\nstrategy pro\ufb01le \u03c3\u03b8i\u2192ai, we construct the induced Bayesian network (IBN) as follows.\nWe start with the BN representing the type distribution P , which includes (at least) the random\nvariables \u03b81, . . . , \u03b8n. The conditional probability distributions (CPDs) for the network are unchanged.\nWe add the following random variables: one strategy variable Dj for each player j; one action\ncount variable for each action node \u03b1 \u2208 A, representing its action count, denoted c(\u03b1); one function\nvariable for each function node p \u2208 F, representing its con\ufb01guration value, denoted c(p); and one\nutility variable U \u03b1 for each action node \u03b1. We then add the following edges: an edge from \u03b8j to Dj\nfor each player j; for each player j and each \u03b1 \u2208 A\u222a\nj , an edge from Dj to c(\u03b1); for each function\nvariable c(p), all incoming edges corresponding to those in the action graph G; and for each \u03b1 \u2208 A,\nfor each action or function node m \u2208 \u03bd(\u03b1) in G, an edge from c(m) to U \u03b1 in the IBN.\nThe CPDs of the newly added random variables are de\ufb01ned as follows. Each strategy variable\nDj has domain A\u222a\nj according to the\n. In other words, if j (cid:54)= i then Pr(Dj = aj|\u03b8j) is equal to \u03c3j(aj|\u03b8j) for all\nmixed strategy \u03c3\u03b8i\u2192ai\nj \\ Aj,\u03b8j ; and if j = i we have Pr(Dj = ai|\u03b8j) = 1. For each\naj \u2208 Aj,\u03b8j and 0 for all aj \u2208 A\u222a\naction node \u03b1, the parents of its action-count variable c(\u03b1) are strategy variables that have \u03b1 in their\ndomains. The CPD is a deterministic function that returns the number of its parents that take value \u03b1;\ni.e., it calculates the action count of \u03b1. For each function variable c(p), its CPD is the deterministic\nfunction f p. The CPD for each utility variable U \u03b1 is a deterministic function speci\ufb01ed by u\u03b1.\nIt is straightforward to verify that the IBN is a directed acyclic graph (DAG) and thus represents a\nvalid joint distribution. Furthermore, the expected utility ui(\u03c3ti\u2192ai|\u03b8i) is exactly the expected value\nof the variable U ai conditioned on the instantiated type \u03b8i.\nLemma 7. For all i \u2208 N, all \u03b8i \u2208 \u0398i and all ai \u2208 Ai,\u03b8i, we have ui(\u03c3\u03b8i\u2192ai|\u03b8i) = E[U ai|\u03b8i].\nStandard BN inference methods could be used to compute E[U ai|\u03b8i]. However, such standard\nalgorithms do not take advantage of structure that is inherent in BAGGs. In particular, recall that\nin the induced network, each action count variable c(\u03b1)\u2019s parents are all strategy variables that\nhave \u03b1 in their domains, implying large in-degrees for action count variables. Applying (e.g.) the\nclique-tree algorithm would yield large clique sizes, which is problematic because running time scales\nexponentially in the largest clique size of the clique tree. However, the CPDs of these action count\nvariables are structured counting functions. Such structure is an instance of causal independence in\nBNs [11]. It also corresponds to anonymity structure for complete-information game representations\nlike symmetric games and AGGs [13]. We can exploit this structure to speed up computation of\nexpected utility in BAGGs. Our approach is a specialization of Heckerman and Breese\u2019s method\n[11] for exploiting causal independence in BNs, which transforms the original BN by creating new\nnodes that represent intermediate results, and re-wiring some of the arcs, resulting in an equivalent\nBN with small in-degree. Given an action count variable c(\u03b1) with parents (say) {D1 . . . Dn}, for\neach i \u2208 {1 . . . n \u2212 1} we create a node M\u03b1,i, representing the count induced by D1 . . . Di. Then,\ninstead of having D1 . . . Dn as parents of c(\u03b1), its parents become Dn and M\u03b1,n\u22121, and each M\u03b1,i\u2019s\nparents are Di and M\u03b1,i\u22121. The resulting graph has in-degree at most 2 for c(\u03b1) and the M\u03b1,i\u2019s. The\nCPDs of function variables corresponding to contribution-independent function nodes also exhibit\ncausal independence, and thus we can use a similar transformation to reduce their in-degree to 2. We\ncall the resulting Bayesian network the transformed Bayesian network (TBN) of the BAGG.\nIt is straightforward to verify that the representation size of the TBN is polynomial in the size of the\nBAGG. We can then use standard inference algorithms to compute E[U \u03b1|\u03b8i] on the TBN. For classes\nof BNs with bounded treewidths, this can be computed in polynomial time. Since the graph structure\n(and thus the treewidth) of the TBN does not depend on the strategy pro\ufb01le and only depends on the\nBAGG, we have the following result.\n\n7\n\n\fFigure 1: GW, varying\nplayers.\n\nFigure 2: GW, varying\nlocations.\n\nFigure 3: GW, varying\ntypes.\n\nFigure 4:\nsubdivision.\n\nsimplicial\n\ntime polynomial in n, |A|, |F| and |(cid:80)\n\ni \u0398i|.\n\nTheorem 8. For BAGGs whose TBNs have bounded treewidths, expected utility can be computed in\n\nBayesian games with independent type distributions are an important class of games and have many\napplications, such as independent-private-value auctions. When contribution-independent BAGGs\nhave independent type distributions, expected utility can be ef\ufb01ciently computed.\nTheorem 9. For contribution-independent BAGGs with independent type distributions, expected\nutility can be computed in time polynomial in the size of the BAGG.\nProof. Provided in the supplementary material.\n\nNote that this result is stronger than that of Theorem 8, which only guarantees ef\ufb01cient computation\nwhen TBNs have constant treewidth.\n\n5 Experiments\n\nWe have implemented our approach for computing a Bayes-Nash equilibrium given a BAGG by\napplying Nash equilibrium algorithms on the agent form of the BAGG. We adapted two algorithms,\nGAMBIT\u2019s [18] implementation of simplicial subdivision and GameTracer\u2019s [2] implementation of\nGovindan and Wilson\u2019s global Newton method, by replacing calls to expected utility computations\nof the complete-information game with corresponding expected utility computations of the BAGG.\nWe ran experiments that tested the performance of our approach (denoted by BAGG-AF) against\ntwo approaches that compute a Bayes-Nash equilibrium for arbitrary Bayesian games. The \ufb01rst\n(denoted INF) computes a Nash equilibrium on the induced normal form; the second (denoted NF-\nAF) computes a Nash equilibrium on the normal form representation of the agent form. Both were\nimplemented using the original, normal-form-based implementations of simplicial subdivision and\nglobal Newton method. We thus studied six concrete algorithms, two for each game representation.\nWe tested these algorithms on instances of the Coffee Shop Bayesian game described in Example 6.\nWe created games of different sizes by varying the number of players, the number of types per player\nand the number of locations. For each size we generated 10 game instances with random integer\npayoffs, and measured the running (CPU) times. Each run was cut off after 10 hours if it had not yet\n\ufb01nished. All our experiments were performed using a computer cluster consisting of 55 machines\nwith dual Intel Xeon 3.2GHz CPUs, 2MB cache and 2GB RAM, running Suse Linux 11.1.\nWe \ufb01rst tested the three approaches based on the Govindan-Wilson (GW) algorithm. Figure 1 shows\nrunning time results for Coffee Shop games with n players, 2 types per player on a 2 \u00d7 3 grid, with\nn varying from 3 to 7. Figure 2 shows running time results for Coffee Shop games with 3 players,\n2 types per player on a 2 \u00d7 x grid, with x varying from 3 to 10. Figure 3 shows results for Coffee\nShop games with 3 players, T types per player on a 1 \u00d7 3 grid, with T varying from 2 to 8. The data\npoints represent the median running time of 10 game instances, with the error bars indicating the\nmaximum and minimum running times. All results show that our BAGG-based approach (BAGG-AF)\nsigni\ufb01cantly outperformed the two normal-form-based approaches (INF and NF-AF). Furthermore,\nas we increased the dimensions of the games the normal-form based approaches quickly ran out of\nmemory (hence the missing data points), whereas BAGG-NF did not.\nWe also did some preliminary experiments on BAGG-AF and NF-AF running the simplicial subdivi-\nsion algorithm. Figure 4 shows running time results for Coffee Shop games with n players, 2 types\nper player on a 1 \u00d7 3 grid, with n varying from 3 to 6. Again, BAGG-AF signi\ufb01cantly outperformed\nNF-AF, and NF-AF ran out of memory for game instances with more than 4 players.\n\n8\n\n110100100010000100000in secondsBAGG-AFNF-AFINF0.111010010001000010000034567CPU time in secondsnumber of playersBAGG-AFNF-AFINF10000100000100010000100000nds10100100010000100000econds110100100010000100000in seconds0.1110100100010000100000time in seconds0.111010010001000010000068101214161820CPU time in secondsnumberoflocations0.111010010001000010000068101214161820CPU time in secondsnumber of locations10000100100010000ds10100100010000econds01110100100010000in seconds0.010.1110100100010000time in seconds0.010.11101001000100002345678CPU time in secondstypes perplayer0.010.11101001000100002345678CPU time in secondstypes perplayer10100100010000e in secondsBAGG-AFNF-AF110100100010000234567CPU time in secondsnumber of playersBAGG-AFNF-AF\fReferences\n[1] N. Bhat and K. Leyton-Brown. Computing Nash equilibria of action-graph games. 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