Structured Models forMulti-Agent Interactions

Daphne KollerStanford 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

stra

tegi

es o

f pl

ayer

I

Representation & Inferenceso

lutio

n tim

e (s

ec)

size of tree

0

5

10

15

20

25

30

0 5000 10000 15000 20000

Normal form

Sequence form

Minimax linear program for two-player zero-sum gamesApplied 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 SalesStrategy

Cost

Commission

ResourceAllocation

Sales-B

A SalesStrategy

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 2n states (or more);– representing the joint distribution is infeasible.

P(A | B,E)a function

Val(B,E) (Val(A))

Bayesian Network

nodes = random variablesedges = direct probabilistic influence

Network structure encodes conditional independencies: Phone-Call is independent of Burglary given Alarm

PhoneCall Newscast

EarthquakeBurglary0.8 0.2

b

e

b

0.6 0.4

0.01 0.99

0.2 0.8

eb

e

e

b

EB P(A | B, E)

Alarm

BN Semantics: Probability Model

• Compact & natural representation:– nodes have k parents 2kn vs. 2n parameters– parameters natural and easy to elicit.

qualitativeBN structure

+local

probabilitymodels

full jointdistribution

over domain=

e)|nP(a)|P(ce),b|P(aP(e))bP()nc,a,e,,bP(

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 AlarmNewscast 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 independentgiven A

B, C can be dependentgiven A,D

A

D

Simple linear-time algorithm for testing conditionalSimple linear-time algorithm for testing conditionalindependence using only graphical structure:independence using only graphical structure:

• Sound: d-separation independence for all P

• Complete: no d-separation dependence for almost all P

CPCS

21000 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; 25 = 32 terminal nodes

Multi-Agent Influence Diagrams (MAIDs)

TreeSick

TreeDoctor

View

Cost

SpikeTree

BuildPatio

TreeDead

Tree

Influence diagram representation easily extended to multiple agents

“Tree killer”example

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 D1 precedes D2

– at D2 agent remembers:• his decision at D1

• everything he knew at D1

– formally: {D1,Parents(D1)} Parents(D2)– usually perfect recall edges are implicit, not drawn

TreeSick

TreeDoctor

SpikeTree

BuildPatio

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

aU UValu

Ma uuUPEUU )(

][ )()(

ReadabilityP1 Hand P2 Hand

Bet

Flop Cards

BetBet

BetBet

Card 4

Bet

Bet

Bet

U

Bet

CompactnessSuitability

1W

Building1E

Building1W

Suitability 1E

Suitability 2W

Building2E

Building2W

Suitability 2E

Suitability 3W

Building3E

Building3W

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: 33 = 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 327 pure strategies

54n

32n

(327)n

Outline

Probabilistic Reasoning: Bayesian networks

Influence Diagrams

Strategic Relevance

Exploiting Structure for Solving Games

Optimality and Equilibrium

• Let E be a subset of Da, 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, Da

is optimal for

)',(),( EE aa EUEU

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

)( )(

][* ),|()|(

DVald U UValuM

a

upaduUPpadPU

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

D’

U

don’tcare

U

D

D’

D

D’

perfectenough

U

D

D’

D

D’

perfectinfo

U

D

D’

D

D’

simultaneousmove

U

Examples II: Simple Card Game

Bet2 relies on Bet1 even though Bet2 observes Bet1

– Bet2 can depend on Deal

– Deal influences U

– Need probability model of Bet2 to derive posterior on Deal and compute expectation over U

Bet1

Bet2

Decision D can require D’ even if D’ is observed at D !

Bet1

Bet2

Deal

U

Examples III: Decoupled Utilities

Bet2 relies on Bet1 even without influence on utility

– Bet2 can depend on Deal

– Deal influences U

– Need probability model of Bet2 to derive posterior on Deal and compute expectation over U

Bet1

Bet2

Bet1

Bet2

Deal

U U

Examples IV: Tree Killer

PoisonTree

BuildPatio

TreeDoctor

TreeSick

TreeDoctor

View

Cost

PoisonTree

BuildPatio

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

existsCPD of D’ influences P(U | D,Pa(D))

D relies on D’(D’ relevant to D)

D’

'̂D

'̂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

D’

U

U

D

D’

D

D’

U

Generalized Backward InductionIdea:Idea: Solve decisions by order of relevance graph 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

D’

U

UD

D’

D

D’

U

Finding Equilibria: Acyclic Relevance Graphs

• Choose any strategy profile for D1,…,Dn-1

• Derive decision rule for Dn that is optimal for • Node Dn does not rely on preceding ones

is optimal for any other strategy profile as well!

D1 D2 DnDn-1…D1 D2 Dn-1Dn-1

• We can now set as CPD for Dn

• And continue by optimizing Dn-1

Dn

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 D1,…,Dn of relevance graph:

• Begin with arbitrary fully mixed strategy profile • For i = n down to 1:

– Find decision rule for Di that is optimal for • Decision rules at previous decisions fixed earlier

• Decision rules at subsequent decisions irrelevant

– Let (Di) =

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 GraphsQuestion:Question: What if the relevance graph is cyclic? 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 InductionGiven topological sort C1,…,Cm of SCCs in relevance

graph:• Begin with arbitrary fully mixed strategy profile • For i = m down to 1:

– Construct reduced MAID M[-Ci]

• Strategies for previous SCCs selected before• Strategies for subsequent SCCs irrelevant

– Create game tree for M[-Ci]

– Use game solver to find equilibrium strategy profile for Ci in this reduced game

– Let (Ci) = Theorem: If find equilibrium for each SCC, the result

is equilibrium for whole game

“Road” Relevance Graph

1W 1E

2W 2E

3W 3E

Note: Reduced games over SCCs are not subgames!

Experiment: “Road” Example

Reminder, for n=4: Tree size: 6561 nodes Matrix size: 4.71027

Running Time of Backward Induction Algorithm

0

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600

0 10 20 30 40

Number of Plots of Land

Tim

e (s

eco

nd

s)

For n=40: Tree size: 1.47 1038 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 decisioncompletely 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 agentsby allowing real-world games to be easily representedand 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?

http://robotics.stanford.edu/~koller

koller@cs.stanford.edu