Structured Models for Multi-Agent Interactions Daphne Koller Stanford University Joint work with Brian Milch, U.C. Berkeley

Scaling Up Question: –Modeling and solving small games is already hard –How can we scale up to larger ones? Answer: –Real-world situations have a lot of structure –Otherwise people wouldn’t be able to handle them Goal: construct –languages based on structured representations, allowing compact models of complex situations –algorithms that exploit this structure to support effective reasoning

Representations of Games Normal form –basic units: strategies –game representation loses all structure –matrix size exponentially larger than game tree Extensive form –basic units: events –game structure explicitly encodes time, information –game tree size can still be very large strategies of player II strategies of player I

Representation & Inference solution time (sec) size of tree Normal form Sequence form Minimax linear program for two-player zero-sum games Applied to abstract 2-player poker [Koller + Pfeffer] [Romanovskii, 1962; Koller, Megiddo & von Stengel, 1994]

MAID Representation MAID form –basic units: variables & dependencies between them –game structure explicitly encodes time, information, independence –can be exponentially smaller than game tree –game structure supports new forms of decomposition & backward inductions solving can be exponentially more efficient than extensive form Sales-A B Sales Strategy Cost Commission Resource Allocation Sales-B A Sales Strategy Commission Revenue

Outline  Probabilistic Reasoning: Bayesian networks [Pearl, Jensen, …]  Influence Diagrams  Strategic Relevance  Exploiting Structure for Solving Games

Probability Distributions Probabilistic model (e.g., a la Savage): –set of possible states in which the world can be; –probability distribution over this space. State: assignment of values to variables –diseases, symptoms, predisposing factors, … Problem: –n variables  2 n states (or more); –representing the joint distribution is infeasible.

P(A | B,E) a function Val(B,E)   (Val(A)) Bayesian Network nodes = random variables edges = direct probabilistic influence Network structure encodes conditional independencies: Phone-Call is independent of Burglary given Alarm PhoneCall Newscast EarthquakeBurglary b e b eb e e b EBP(A | B, E) Alarm

BN Semantics: Probability Model Compact & natural representation: –nodes have  k parents  2 k n vs. 2 n parameters –parameters natural and easy to elicit. qualitative BN structure + local probability models full joint distribution over domain = C A N EB

BN Semantics: Independencies The graph structure of the BN implies a set of conditional independence assumptions –satisfied by every distribution over this graph C A N EB Burglary and Call independent given Alarm Newscast and Alarm independent given Earthquake Burglary and Earthquake independent C A N EB C A N EB C A N EB

BN Semantics: Dependencies BN structure also specifies potential dependencies –those that might hold for some distribution over graph C A N EB Burglary and Earthquake dependent given Alarm C A N EB

Active paths C A D B Probabilistic influence “flows” along “active” paths “d-separation” if there is no active path B, C can be dependent B, C are independent given A B, C can be dependent given A,D A D Simple linear-time algorithm for testing conditional independence using only graphical structure: Sound: d-separation  independence for all P Complete: no d-separation  dependence for almost all P

CPCS  states

Bayesian Networks Explicit representation of domain structure Cognitively intuitive compact models of complex domains Same model allows relevant probabilities to be computed in any evidence state Algorithms that exploit structure for effective inference even in very large models

Outline  Probabilistic Reasoning: Bayesian networks  Influence Diagrams [Howard, Shachter, Jensen, …]  Strategic Relevance  Exploiting Structure for Solving Games

Example: The Tree Killer Alice wants a patio, but the benefit outweighs the cost only if she gets an ocean view Bob’s tree blocks her view Alice chooses whether to poison the tree Tree may become sick Bob chooses whether to call a tree doctor –Alice can see whether tree doctor comes Alice chooses whether to build her patio Tree may die when winter comes

Standard Representation: Game Tree Poison Tree? Tree Sick? Call Tree Doctor? Build Patio? Tree Dead? 5 levels; 2 5 = 32 terminal nodes

Multi-Agent Influence Diagrams (MAIDs) TreeSick Tree Doctor View Cost Spike Tree Build Patio TreeDead Tree Influence diagram representation easily extended to multiple agents “Tree killer” example

MAIDs  Trees Same idea as for single-agent Ids Information is different for different agents TreeSick Tree Doctor Spike Tree Build Patio

Decision Nodes Incoming edges are information edges –variables whose values the agent knows when deciding –agent’s strategy can depend on values of parents Each parent instantiation –u  Val(Parents(D)) is an information set Perfect recall: if D 1 precedes D 2 –at D 2 agent remembers: his decision at D 1 everything he knew at D 1 –formally: {D 1,Parents(D 1 )}  Parents(D 2 ) –usually perfect recall edges are implicit, not drawn TreeSick Tree Doctor Spike Tree Build Patio

Strategies Strategy  at D : –A pure (deterministic) strategy specifies an action at D for every information set u –A behavior strategy specifies a distribution over actions for every u Strategy  specifies distribution P  (D | Parents(D)) –turns a decision node into a chance node –information parents play exactly the same role as parents of chance node

MAID Semantics MAID M defines a set of possible strategy profiles M plus any strategy profile  defines a BN M [  ] –Each decision node D becomes a chance node, with  [ D ] as its CPD M [  ] defines a probability distribution, from which we can derive an expected utility for each agent: Thus, a MAID defines a mapping from strategy profiles to expected utility vectors

Readability P1 HandP2 Hand Bet Flop Cards Bet Card 4 Bet U

Compactness Suitability 1W Building 1E Building 1W Suitability 1E Suitability 2W Building 2E Building 2W Suitability 2E Suitability 3W Building 3E Building 3W Suitability 3E Util 1E Util 2W Util 2E Util 1W Util 3W Util 3E “Road” example

Compactness Assume all variables have three values Each decision node observes three variables Number of information sets per agent: 3 3 = 27 Size of MAID: –n chance nodes of “size” 3 –n decision nodes of “size” 27·3 Size of game tree: –2n splits, each over three values Size of normal (matrix) form: –n players, each with 3 27 pure strategies  54n  3 2n  (3 27 ) n

Outline  Probabilistic Reasoning: Bayesian networks  Influence Diagrams  Strategic Relevance  Exploiting Structure for Solving Games

Optimality and Equilibrium Let E be a subset of D a, and let  be a partial strategy over E Is  the best partial strategy for agent a to adopt? –Depends on decision rules for other decision nodes  is optimal for a strategy profile  if for all partial strategies  ’ over E : A strategy profile  is a Nash equilibrium if for every agent a,  D a is optimal for 

MAIDs and Games A MAID is equivalent to a game tree: it defines a mapping from strategy profiles to payoff vectors Finding equilibria in the MAID is equivalent to finding equilibria in the game tree One way to find equilibrium in MAID: –construct the game tree –solve the game Incurs exponential blowup in representation size Question: can we find equilibria in a MAID directly?

Local Optimization Consider finding a decision rule for a single decision node D that is optimal for  For each instantiation pa of Pa(D), must find P* that maximizes: Some decision rules in  may not affect this maximization problem

Strategic Relevance Intuitively, D relies on D’ if we need to know the decision rule for D’ in order to determine the optimal decision rule for D. We define a relevance graph, with: –a node for each decision –an edge from D to D’ if D relies on D’ D D’

Examples I: Information D D’ D U don’t care U D D’ D perfect enough U D D’ D perfect info U D D’ D simultaneous move U

Examples II: Simple Card Game Bet 2 relies on Bet 1 even though Bet 2 observes Bet 1 –Bet 2 can depend on Deal –Deal influences U –Need probability model of Bet 2 to derive posterior on Deal and compute expectation over U Bet 1 Bet 2 Decision D can require D’ even if D’ is observed at D ! Bet 1 Bet 2 Deal U

Examples III: Decoupled Utilities Bet 2 relies on Bet 1 even without influence on utility –Bet 2 can depend on Deal –Deal influences U –Need probability model of Bet 2 to derive posterior on Deal and compute expectation over U Bet 1 Bet 2 Bet 1 Bet 2 Deal UU

Examples IV: Tree Killer Poison Tree Build Patio Tree Doctor TreeSick Tree Doctor View Cost Poison Tree Build Patio TreeDead Tree

s-Reachability D’ is s-reachable from D if there is some among the descendants of D, such that if a new parent were added to D, there would be an active path from to U given D and Pa(D). D U D U given exists CPD of D’ influences P(U | D,Pa(D)) D relies on D’ ( D’ relevant to D ) D’

s-Reachability Theorem: s-reachability is sound & complete for strategic relevance Nodes that D relies on are the nodes that are s-reachable from D. Sound: no s-reachability  strategic irrelevance  P,U Complete: s-reachability  relevance for some P,U Theorem: Can build the relevance graph in quadratic time using only structure of MAID

Outline  Probabilistic Reasoning: Bayesian networks  Influence Diagrams  Strategic Relevance  Exploiting Structure for Solving Games

Intuition: Backward Induction D’ observes D Can optimize decision rule at D’ without knowing decision rule at D Having optimized D, can optimize D’ D doesn’t care about D’ Can optimize decision rule at D without knowing decision rule at D’ Having optimized D’, can optimize D D D’ D U U D D U

Generalized Backward Induction Idea: Solve decisions by order of relevance graph Generalized Backward Induction: Choose decision node D that relies on no other Find optimal strategy for D by maximizing its local expected utility Replace D by chance node D D’ D U U D D U

Finding Equilibria: Acyclic Relevance Graphs Choose any strategy profile  for D 1,…,D n-1 Derive decision rule  for D n that is optimal for  Node D n does not rely on preceding ones   is optimal for any other strategy profile as well! D1D1 D2D2 DnDn D n-1 … D1D1 D2D2 We can now set  as CPD for D n And continue by optimizing D n-1 DnDn

Generalized Backward Induction Theorem: If the relevance graph of a MAID is acyclic, it can be solved by generalized backward induction, and the result is a pure-strategy Nash equilibrium Given topological sort D 1,…,D n of relevance graph: Begin with arbitrary fully mixed strategy profile  For i = n down to 1: –Find decision rule  for D i that is optimal for  Decision rules at previous decisions fixed earlier Decision rules at subsequent decisions irrelevant –Let  (D i ) = 

When is the Relevance Graph Acyclic? Single-agent influence diagrams with perfect recall Multi-agent games with perfect information Some games with imperfect information –e.g., Tree Killer example But in many MAIDs the relevance graph has cycles…

Cyclic Relevance Graphs Question: What if the relevance graph is cyclic? Strongly connected component (SCC): –maximal subgraph s.t.  directed path between every pair of nodes The decisions in each SCC require each other –They must be optimized together Different SCCs can be solved separately

Generalized Backward Induction Given topological sort C 1,…,C m of SCCs in relevance graph: Begin with arbitrary fully mixed strategy profile  For i = m down to 1 : –Construct reduced MAID M [  - C i ] Strategies for previous SCCs selected before Strategies for subsequent SCCs irrelevant –Create game tree for M [  - C i ] –Use game solver to find equilibrium strategy profile  for C i in this reduced game –Let  ( C i ) =  Theorem: If find equilibrium for each SCC, the result is equilibrium for whole game

“Road” Relevance Graph 1W1E 2W2E 3W3E Note: Reduced games over SCCs are not subgames!

Experiment: “Road” Example Reminder, for n=4 : Tree size: 6561 nodes Matrix size: 4.7  For n=40 : Tree size: 1.47  nodes

Cutting Cycles Idea: enumerate possible values d for some decision D –Once we determine D, residual MAID has acyclic relevance graph –Solve residual MAID using generalized backward induction –Check whether combined strategy with d is an equilibrium Theorem: Can find all pure strategy equilibria in time linear in # of SCCs, exponential in max # of decisions required to cut all loops in component May need to instantiate several decision nodes to cut cycle Can deal with each SCC separately D

Irrelevant Information Sales-A B Sales Cost Commission Resource Sales-B A Sales Commission Revenue What if B can observe A ’s decision completely irrelevant to him We can automatically –analyze relevance based on graph structure –eliminate irrelevant information edges In associated tree, safe merging of information sets Leads to exponential decrease in # of decisions to optimize in influence diagram!

Related Work Suryadi and Gmytrasiewicz (1999) use multi-agent influence diagrams, but with recursive modeling Milch and Koller (2000) use the MAID representation described here, but have no algorithm for finding equilibria Nilsson and Lauritzen (2000) discuss limited memory influence diagrams (LIMIDs) and derive s-reachability, but do not apply it to multi-agent case La Mura (2000) proposes game networks, with an undirected notion of strategic dependence

Future Work Take advantage of structure within SCCs Represent asymmetric scenarios compactly Detect irrelevant observations

Computational Game Theory Expert analysis of: –“Prototypical” examples that highlight key issues –Abstracted problems for big organizations Autonomous agents interacting economically Decision support systems for consumers Complex problems: –many relevant variables –interacting decisions Simplified examples –small enough to be analyzed by hand Game theory: Past Game theory: Future Goals: Make game theory a broadly usable tool even for lay people a formal basis for interacting autonomous agents by allowing real-world games to be easily represented and solved.

Conclusions Multi-agent influence diagrams: –compact intuitive language for multi-agent interactions –basic units: variables rather than strategies or events MAIDs make explicit structure that is lost in game trees Can exploit structure to find equilibria efficiently –sometimes exponentially faster than existing algorithms Exciting question: What else does structure buy us?