Graphical Multiagent Models Quang Duong Computer Science and Engineering Chair: Michael P. Wellman 1.

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Graphical Multiagent Models

Quang DuongComputer Science and Engineering

Chair: Michael P. Wellman

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Example: Election In The City Of AA

Political discussion

Vote

May, political analyst

• Phone surveys• Demographic information• Party registration• …

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Modeling Objectives

Construct a model that takes into account people (agent) interactions (graph edges) in:

– Representing joint probability of all vote outcomes*– Computing marginal and conditional probabilities

Vote Republican or

Democrat?

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Modeling Objectives (cont.)

Generate predictions:– Individual actions, dynamic behavior induced by individual

decisions– Detailed or aggregate

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More Applications Of Modeling Multiagent Behavior

Financial Institutions

Computer Network/ Internet

Social Network

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Challenges: Uncertaintyfrom the system modeler’s perspective

1a. Agent choiceVote for personal favorite or conform with

others?

1b. CorrelationWill the historic district of AA unanimously pick

one candidate to support?

1c. InterdependenceMay does not know all friendship relations in AA

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Challenges: Complexity

2a. Representation and inferenceNumber of all action configurations (all vote

outcomes) is exponential in the number of agents (people).

2b. Historical informationPeople may change their minds about whom to

vote for after discussions.

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Existing Approaches That This Work Builds On

Game-theory Approach:• Assume game structure/perfect rationality

Statistical Modeling Approach:• Aggregate statistical measures/ make simplifying assumptions

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Approach Outline

Graphical Multiagent Models (GMMs) are probabilistic graphical models designed to

• Facilitate expressions of different knowledge sources about agent reasoning

• Capture correlated behaviors

while

• Exploiting dependence structure

uncertainty

complexity

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Roadmap

(Ch. 3) GMM(static)

(Ch. 4) History-

Dependent GMM

(Ch. 6) Application: Information

Diffusion

(Ch. 2) Background

(Ch. 5) Learning

Dependence Graph Structure

(Ch. 2) Background

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Multiagent Systems

• n agents {1,…,i,…,n}

• Agent i chooses action ai, joint action (action configuration) of the system: a = (a1,…, an)

• In dynamic settings: – time period t, time horizon T.– history Ht of history horizon h, Ht = (at-h,…,at-1)

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Game Theory

Each player (agent i) chooses a strategy (action ai).

Strategy profile (joint action a) of all players.

Payoff function: ui(ai,a-i)

Player i‘s regret εi(a): maximum gain if player i chooses strategy ai

’, instead of strategy ai, given than everyone else fixes their strategies.

a* is a Nash equilibrium (NE) if for every player i, regret εi(a) = 0.

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Graphical Representations of Multiagent Systems

1. Graphical Game Models [Kearns et al. ‘01]

An agent’s payoff depends on strategy chosen by itself and its neighbors Ji

Payoff/utility: ui(ai,aJi)

Similar approaches: Multiagent influence diagrams (MAIDs) [Koller & Milch

’03]Networks of Influence Diagrams [Gal & Pfeffer ’08]Action-graph games [Jiang et al ‘11].

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Graphical Representations (cont.)

2. Probabilistic graphical models

Markov random field (static) [Kindermann & Laurie ’80, KinKoller & Friedman ‘09]

Dynamic Bayesian Networks [Kanazawa & Dean ’89, Ghahramani ’98]

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Probabilistic Graphical Models

This Work

demonstrate and examine

the benefits of applying probabilistic graphical models to the problem of modeling multiagent behavior

in scenarios with different sets of assumptions and information available to the system modeler.

Building on

Game Models

incorporating

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Roadmap

(Ch. 3) GMM(static)

(Ch. 4) History-

Dependent GMM

(Ch. 5) Learning

Dependence Graph Structure

(Ch. 6) Application: Information

Diffusion

(Ch. 2) Background

1. Overview2. Examples3. Knowledge Combination4. Empirical Study

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Graphical Multiagent Models (GMMs)

[Duong, Wellman & Singh ‘08]

• Nodes: agents. Edges: dependencies among agent actions

• Dependence neighborhood Ni

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2 7

5

4

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GMMs

Pr(a) ∝ Πi πi(aNi)

Joint probabilitydistribution ofsystem’s actions

potential of neighborhood’s joint actions

Factor joint probability distribution into neighborhood potentials.

(Markov random field for graphical games [Daskalakis & Papadimitriou ’06])

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Example GMMs

• Markov Random Field for computing pure strategy Nash equilibrium

• Markov Random Field for computing correlated equilibrium

• Information diffusion GMMs [Ch. 6]

• Regret GMMs [Ch. 3]

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Examples: Regret potential

Assume a graphical gameRegret ε(aNi)

πi(aNi) = exp(-λ εi(aNi))

Illustration:

Assume: prefers Republican to Democrat (fixing others’ choices)

Near zero λ: picks randomlyLarger λ: more likely to pick Republican

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Flexibility: Knowledge Combination• Assume known graph structures, given GMMs G1 and G2 that

represent 2 different knowledge sources

Final GMMfinalG

GMM 2GMM 1 Knowledge Combination

Regret GMMreG

Heuristic Rule-based

GMMhG

1. Direct update2. Opinion pool3. Mixing data

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Empirical Study

• Combining knowledge sources in one GMM improves predictions

• Combined models fail to improve on input models when input does not capture any underlying behavior

ratio > 1: combined model performs better than input model

Mixing data GMM vs. regret GMM

Mixing data GMM vs. heuristic GMM

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Summary Of Contributions (Ch. 3)

(I.A) GMMs accommodate expressions of different knowledge sources

(I.B) This flexibility allows the combination of models for improved predictions

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Roadmap

(Ch. 3) GMM(static)

(Ch. 4) History-

Dependent GMM

(Ch. 6) Application: Information

Diffusion

(Ch. 2) Background

1. Consensus Dynamics2. Description3. Joint vs. individual behavior4. Empirical study

(Ch. 5) Learning

Dependence Graph Structure

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Example: Consensus Dynamics

[Kearns et al. ’09] abstracted version of the AA mayor election example

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3

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Agent Blue consensus

Red consensus

neither

1 1.0 0.5 0

2 0.5 1.0 0

Observation graphAgent 1’s perspective

Examine the ability to make collective decisions with limited communication and observation

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Network structure here plays a large role in determining the outcomes

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Modeling Multiagent Behavior In Consensus Dynamics Scenario

time

Time series action data + observation graph

1. Predict detailed actions 2. Predict aggregate measures

or

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History-Dependent Graphical Multiagent Models (hGMMs)

[Duong, Wellman, Singh & Vorobeychik ’10]

We condition actions on abstracted history Ht

Note: dependence graphs can be different from observation graphs.

1 1 1

t-1 t t+1

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hGMMs

(Undirected) within-time edges: dependencies between agent actions in the same time period, and define dependence neighborhood Ni for each agent i.

A GMM at every time t

1 1 1

t-1 t t+1

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hGMMs

(Directed) across-time edges: dependencies of agent i’s action on some abstraction of prior actions by agents in i’s conditioning set Γi

Example: frequency function.

1 1 1

t-1 t+1

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hGMMs

Pr(at | H) ∝ Πi πi(atNi | Ht

Γi)

Joint probabilitydistribution ofsystem’s actions at time t

potential of neighborhood’s joint actions at t

history of the conditioning set

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Challenge: Dependence

• Conditional independence

• Dependence induced by history abstraction/summarization (*)

1

2

1

2

1

2

t-2 t-1 t

1

2

1

2

1

2

t-2 t-1 t

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Individual vs. Joint Behavior Models

Given complete history, autonomous agents’ behaviors are conditionally independent Individual behavior models:

πi(ati | Ht

Γi,complete)

Joint behavior models allow specifying any action dependence within one’s within-time neighborhood, given some (abstracted) history

πi(atNi | Ht

Γi,abstracted)

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Empirical Study: Summary

Evaluation: compares joint behavior and individual behavior models by likelihood of testing data (time-series votes)

* Observation graph defines both dependence neighborhoods N and conditioning sets Γ

1. Joint behavior outperform individual behavior models for shorter history lengths, which induce more action dependence.

2. Approximation does not deteriorate performance

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Summary Of Contributions (Ch. 4)

(II.A) hGMMs support inference about system dynamics

(II.B) hGMMs allow the specification of action dependence emerging from history abstraction

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Roadmap

(Ch. 3) GMM(static)

(Ch. 4) History-

Dependent GMM

(Ch. 6) Application: Information

Diffusion

(Ch. 2) Background

1. Learning Graphical Game Models

(Ch. 5) Learning

Dependence Graph structure

2. Learning hGMMs

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Learning History-Dependent Graphical Multiagent Models

ObjectiveGiven action data + observation graph, build a model that predicts:

– Detailed actions in next period– Aggregate measures of actions in the more distant future

Challenge: Learn dependence graph– (Within-time) Dependence graph ≠ observation graph – Complexity of the dependence graph

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Consensus Dynamics Joint Behavior Model

Extended Joint Behavior hGMM (eJCM)

πi(aNi | HtΓi) = ri(aNi) f(ai , Ht

Γi)γ Ι(ai , Ht

i)β

1. ri(aNi) = reward for action ai, discounted by the number of dissenting neighbors in Ni

2. frequency of ai chosen previously by agents in the conditioning set Γi 3. inertia proportional to how long i has maintained its most recent action

1 2 3

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Consensus Dynamics Individual Behavior Models

1. Extended Individual Behavior hGMM (eICM): similar to eJCM but assumes that Ni contains i only

πi(ai | HtΓi) = Pr(ai | Ht

Γi) ∝ ri(ai) f(ai , HtΓi)

γ Ι(ai , Hti)β

2. Proportional Response Model (PRM): only incorporates the most recent time period [Kearns et al., ‘09]:Pr(ai | Ht

Γi) ∝ ri(ai) f(ai , HtΓi)

3. Sticky Proportional Response Model (sPRM)

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Learning hGMMS

Input:• <action observations (time series)>

• observation graph

Search space:1.Model parameters γ, β

2.Within-time edges

Output:

hGMM

Objective: likelihood of dataConstraint: max node degree

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Greedy Learning

Initialize the graph with no edges

Repeat:Add edges that generate the biggest increase (>0) in the training data’s likelihood

Until no edge can be added without violating the maximum node degree constraint

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Empirical Study: Learning from human-subject data

Use asynchronous human-subject dataVary the following environment parameters: • Discretization intervals, delta (0.5 and 1.5 seconds)• History lengths, h• Graph structures/payoff functions: coER_2, coPA_2, & power22

(strongly connected minority)

Goal: evaluate eJCM, eICM, PRM, and sPRM using 2 metrics• Negative likelihood of agents’ actions• Convergence rates/outcomes

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Predicting Dynamic Behavior

eJCMs and eICMs outperform the existing PRMs/sPRMs

eJCMs predict actions in the next time period noticeably more accurately than PRMs and sPRMs, and (statistically significantly) more accurate than eICMs

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Predicting Consensus Outcomes

eJCMs have comparable prediction performance with other models in 2 settings: coER_2 and coPA_2.In power22, eJCM predict consensus probability and colors much more accurately.

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Graph Analysis

In learned graphs, intra edges >> inter edges.

In power22, a large majority of edges are intra red identify the presence of a strongly connected red minority

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Summary Of Contributions (Ch. 5.2)

(II.B) [revisit] This study highlights the importance of joint behavior modeling

(III.C) It is feasible to learn both dependence graph structure and model parameters

(III.D) Learned dependence graphs can be substantially different from observation graphs

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Modeling Multiagent Systems: Step By Step

Given as input

Learn from data

Intuition, background information

Approximation

Dependence graph structure

Potential function

GMMhGMM

Observation graph structure

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Roadmap

(Ch. 3) GMM(static)

(Ch. 4) History-

Dependent GMM

(Ch. 6) Application: Information

Diffusion

(Ch. 2) Background

(Ch. 5) Learning

Dependence Graph structure

1. Definition2. Joint behavior modeling 3. Learning missing edges4. Experiments

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Networks with Unobserved Links

• Links facilitate how information diffuses from one node to another

• Real-world nodes have links unobserved by third parties

Truenetwork G*

ObservedNetwork G

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Problem

Given: a network (with missing links) and snapshots of the network states over time.

Objective: model information diffusions on this network

[Duong, Wellman & Singh ‘11]

1. Network G2. Diffusion traces (on G*)

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Approach 1: Structure Learning

Recover missing edges

• Learn network G’• Learn parameters of an individual behavior model built on G’• Learning algorithms: NetInf [Gomez-Rodriguez et al. ’10] and

MaxInf

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Approach 2: Potential Learning

Construct an hGMM on G without recovering missing links

• hGMMs allow capturing state correlations between neighbors who appear disconnected in the input network

• Theoretical evidence [6.3.2]• Empirical illustrations: hGMMs outperform individual behavior

models on learned graph– random graph with sufficient training data– preferential attachment graph (varying amounts of data)

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Summary of Contributions (Ch. 6)

(II.C) Joint behavior hGMM, can capture state dependence caused by missing edges

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Conclusions

1. The machinery of probabilistic graphical models helps to improve modeling in multiagent systems by: • allowing the representation and combination of different

knowledge sources of agent reasoning • relaxing assumptions about action dependence (which may be a

result of history abstraction or missing edges)

2. One can learn from action data both: (i) model parameters, and (ii) dependence graph structure, which can be different from interaction/observation graph structure

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Conclusions (cont.)

3. The GMM framework contributes to the integration of:

• strategic behavior modeling techniques from AI and economics• probabilistic models from statistics that can efficiently extract

behavior patterns from massive amount of data

for the goal of understanding fast-changing and complex multiagent systems.

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Summary

• Graphical multiagent models: flexibility to represent different knowledge sources and combine them [UAI ’08]

• History-dependent GMM: capture dependence in dynamic settings [AAMAS ’10, AAMAS ’12]

• Learning graphical game models [AAAI ’09]• Learning hGMM dependence graph, distinguishing

observation/interactions graphs and probabilistic dependence graphs [AAMAS ‘12]

• Modeling information diffusion in networks with unobserved links [SocialCom ‘11]

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Acknowledgments• Advisor: Professor Michael P. Wellman• Committee members: Prof. Satinder Singh Baveja, Prof. Edmund H.

Durfee, and Asst. Prof. Long Nguyen• Research collaborators: Yevgeniy Vorobeychik (Sandia Labs), Michael

Kearns (U Penn), Gregory Frazier (Apogee Research), David Pennock and others (Yahoo/Microsoft Research)

• Undergraduate advisor: David Parkes.

• Family• Friends• CSE staff

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THANK YOU!