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Graphical Multiagent Models

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Example: Election In The City Of AA

May, political analyst

Political discussion

- Phone surveys
- Demographic information
- Party registration
- …

Vote

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?

Modeling Objectives (cont.)

Generate predictions:

- Individual actions, dynamic behavior induced by individual decisions
- Detailed or aggregate

More Applications Of Modeling Multiagent Behavior

Computer Network/

Internet

Financial Institutions

Social Network

Challenges: Uncertainty

from the system modeler’s perspective

1a. Agent choice

Vote for personal favorite or conform with others?

1b. Correlation

Will the historic district of AA unanimously pick one candidate to support?

1c. Interdependence

May does not know all friendship relations in AA

Challenges: Complexity

2a. Representation and inference

Number of all action configurations (all vote outcomes) is exponential in the number of agents (people).

2b. Historical information

People may change their minds about whom to vote for after discussions.

Existing Approaches That This Work Builds On

Game-theory Approach:

- Assume game structure/perfect rationality
Statistical Modeling Approach:

- Aggregate statistical measures/ make simplifying assumptions

Approach Outline

Graphical Multiagent Models (GMMs) areprobabilistic graphical modelsdesigned to

- Facilitate expressions of different knowledge sources about agent reasoning
- Capture correlated behaviors
while

- Exploiting dependence structure

uncertainty

complexity

Roadmap

(Ch. 3)

GMM

(static)

(Ch. 4) History-Dependent GMM

(Ch. 2) Background

(Ch. 2) Background

(Ch. 5)

Learning Dependence Graph Structure

(Ch. 6) Application: Information Diffusion

Multiagent Systems

- nagents {1,…,i,…,n}
- Agent ichooses actionai, joint action (action configuration)of the system: a = (a1,…,an)
- In dynamic settings:
- time period t, time horizon T.
- historyHt of history horizon h, Ht= (at-h,…,at-1)

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 regretεi(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.

Graphical Representations of Multiagent Systems

- 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:

Multiagentinfluence diagrams (MAIDs) [Koller & Milch’03]

Networks of Influence Diagrams [Gal & Pfeffer’08]

Action-graph games [Jiang et al ‘11].

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]

This Work

Building on

incorporating

Probabilistic Graphical Models

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.

Game Models

Roadmap

(Ch. 3)

GMM

(static)

(Ch. 4) History-Dependent GMM

(Ch. 2) Background

1. Overview

2. Examples

3. Knowledge Combination

4. Empirical Study

(Ch. 5)

Learning Dependence Graph Structure

(Ch. 6) Application: Information Diffusion

Graphical Multiagent Models (GMMs)

2

7

4

[Duong, Wellman & Singh ‘08]

- Nodes: agents. Edges: dependencies among agent actions
- Dependence neighborhood Ni

6

11

3

5

GMMs

Factor joint probability distribution into neighborhood potentials.

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

Joint probability

distribution of

system’s actions

potential of neighborhood’s joint actions

Pr(a) ∝Πi πi(aNi)

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]

Examples: Regret potential

Assume a graphical game

Regret ε(aNi)

πi(aNi) = exp(-λεi(aNi))

Illustration:

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

Near zero λ: picks randomly

Larger λ: more likely to pick Republican

Flexibility: Knowledge Combination

- Assume known graph structures, given GMMs G1 and G2 that represent 2 different knowledge sources

Heuristic Rule-based

GMM

hG

GMM 2

GMM 1

Regret GMM

reG

Knowledge Combination

Direct update

Opinion pool

Mixing data

Final GMM

finalG

Empirical Study

ratio > 1: combined model performs better than input model

Mixing data GMM vs. regret GMM

Mixing data GMM vs. heuristic GMM

- Combining knowledge sources in one GMM improves predictions
- Combined models fail to improve on input models when input does not capture any underlying behavior

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

Roadmap

(Ch. 3)

GMM

(static)

(Ch. 4) History-Dependent GMM

(Ch. 2) Background

1. Consensus Dynamics

2. Description

3. Joint vs. individual behavior

4. Empirical study

(Ch. 5)

Learning Dependence Graph Structure

(Ch. 6) Application: Information Diffusion

Example: Consensus Dynamics

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

2

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

5

3

1

Observation graph

Agent 1’s perspective

6

4

Network structure here plays a large role in determining the outcomes

time

Modeling MultiagentBehavior In Consensus Dynamics Scenario

Time series action data + observation graph

2. Predict aggregate measures

1. Predict detailed actions

or

time

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

hGMMs

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

A GMM at every time t

1

1

1

t-1

t

t+1

hGMMs

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

Example: frequency function.

1

1

1

t-1

t+1

hGMMs

potential of neighborhood’s joint actions at t

Joint probability

distribution of

system’s actions at time t

Pr(at | H) ∝Πi πi(atNi | HtΓi)

history of the conditioning set

Challenge: Dependence

- Conditional independence
- Dependence induced by history abstraction/summarization (*)

2

2

2

2

2

2

1

1

1

1

1

1

t-2

t-1

t

t-2

t-1

t

Individual vs. Joint Behavior Models

Given completehistory, autonomous agents’ behaviors are conditionally independent

Individual behavior models:

πi(ati | HtΓi,complete)

Joint behavior modelsallow specifying any action dependence within one’s within-timeneighborhood, given some (abstracted) history

πi(atNi | HtΓi,abstracted)

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

- Joint behavior outperform individual behavior models for shorter history lengths, which induce more action dependence.
- Approximation does not deteriorate performance

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

Roadmap

(Ch. 3)

GMM

(static)

(Ch. 4) History-Dependent GMM

(Ch. 2) Background

(Ch. 5)

Learning Dependence Graph structure

1. Learning Graphical Game Models

2. Learning hGMMs

(Ch. 6) Application: Information Diffusion

Learning History-Dependent Graphical Multiagent Models

Objective

Given 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

Consensus Dynamics Joint Behavior Model

Extended JointBehavior hGMM(eJCM)

πi(aNi | HtΓi) = ri(aNi)f(ai, HtΓi)γΙ(ai, Hti)β

- ri(aNi) = reward for action ai, discounted by the number of dissenting neighbors in Ni
- frequency of ai chosen previously by agents in the conditioning set Γi
- inertia proportional to how long i has maintained its most recent action

1

2

3

Consensus Dynamics Individual Behavior Models

1. Extended IndividualBehaviorhGMM(eICM): similar to eJCMbut assumes that Nicontains ionly

πi(ai | HtΓi) = Pr(ai | HtΓi) ∝ ri(ai)f(ai, HtΓi)γΙ(ai, Hti)β

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

Pr(ai | HtΓi) ∝ ri(ai)f(ai, HtΓi)

3. Sticky Proportional Response Model (sPRM)

Learning hGMMS

- Input:
- <action observations (time series)>
- observation graph

- Search space:
- Model parameters
- γ, β
- 2.Within-time edges

Output:

hGMM

Objective: likelihood of data

Constraint: max node degree

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

Empirical Study: Learning from human-subject data

Use asynchronous human-subject data

Vary 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

Predicting Dynamic Behavior

eJCMs and eICMsoutperform the existing PRMs/sPRMs

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

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.

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

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 bothdependence graph structure and model parameters

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

Modeling Multiagent Systems: Step By Step

Given as input

Dependence graph structure

Observation graph structure

Learn from data

GMM

hGMM

Potential function

Approximation

Intuition, background information

Roadmap

(Ch. 3)

GMM

(static)

(Ch. 4) History-Dependent GMM

(Ch. 2) Background

1. Definition

2. Joint behavior modeling

3. Learning missing edges

4. Experiments

(Ch. 5)

Learning Dependence Graph structure

(Ch. 6) Application: Information Diffusion

Networks with Unobserved Links

True

network G*

- Links facilitate how information diffuses from one node to another
- Real-world nodes have links unobserved by third parties

Observed

Network G

Problem

[Duong, Wellman & Singh ‘11]

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

Objective: model information diffusions on this network

- Network G
- Diffusion traces (on G*)

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

Approach 2: Potential Learning

Construct an hGMMon G without

recovering missing links

- hGMMs allow capturing state correlationsbetween 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)

Summary of Contributions (Ch. 6)

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

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

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.

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]

Acknowledgments

- Advisor: Professor Michael P. Wellman
- Committee members: Prof. Satinder Singh Baveja, Prof. Edmund H. Durfee, and Asst. Prof. Long Nguyen
- Research collaborators: YevgeniyVorobeychik (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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