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

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Example: Election In The City Of AA 2 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? 3

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Modeling Objectives (cont.) Generate predictions: – Individual actions, dynamic behavior induced by individual decisions – Detailed or aggregate 4

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More Applications Of Modeling Multiagent Behavior 5 Financial Institutions Computer Network/ Internet Social Network

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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 6

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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. 7

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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 8

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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 9 uncertainty complexity

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Roadmap (Ch. 3) GMM (static) (Ch. 4) History- Dependent GMM (Ch. 6) Application: Information Diffusion 10 (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 a i, joint action (action configuration) of the system: a = (a 1,…, a n ) In dynamic settings: – time period t, time horizon T. – history H t of history horizon h, H t = (a t-h,…,a t-1 ) 11

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Game Theory Each player (agent i) chooses a strategy (action a i ). Strategy profile (joint action a) of all players. Payoff function: u i (a i,a -i ) Player i‘s regret ε i (a): maximum gain if player i chooses strategy a i ’, instead of strategy a i, given than everyone else fixes their strategies. a * is a Nash equilibrium (NE) if for every player i, regret ε i (a) = 0. 12

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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 J i Payoff/utility: u i (a i,a J i ) Similar approaches: Multiagent influence diagrams (MAIDs) [Koller & Milch ’03] Networks of Influence Diagrams [Gal & Pfeffer ’08] Action-graph games [Jiang et al ‘11]. 13

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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] 14

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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. 15 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 16 (Ch. 2) Background 1. Overview 2. Examples 3. Knowledge Combination 4. Empirical Study

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Graphical Multiagent Models (GMMs) [Duong, Wellman & Singh ‘08] Nodes: agents. Edges: dependencies among agent actions Dependence neighborhood N i 17 1 3 6 2 7 5 4

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GMMs Pr(a) ∝ Π i π i (a N i ) Joint probability distribution of system’s actions Joint probability distribution of system’s actions potential of neighborhood’s joint actions 18 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] 19

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Examples: Regret potential Assume a graphical game Regret ε(a N i ) π i (a N i ) = exp(-λ ε i (a N i )) Illustration: 20 Assume: prefers Republican to Democrat (fixing others’ choices) Near zero λ: picks randomly Larger λ: more likely to pick Republican

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Flexibility: Knowledge Combination Assume known graph structures, given GMMs G 1 and G 2 that represent 2 different knowledge sources Final GMM finalG GMM 2 GMM 1 Knowledge Combination Regret GMM reG Heuristic Rule-based GMMhG 21 1.Direct update 2.Opinion pool 3.Mixing data

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Example Domain: Technology Upgrade Node: company Edge: partnership Action: upgrade or retain existing technology Ground truth: Reinforcement learning process (derived from both reG and hG) [Krebs ’02] 22

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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 23 Mixing data GMM vs. regret GMM Mixing data GMM vs. heuristic GMM

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Empirical Results Combining knowledge sources in one GMM improves predictions ratio > 1: combined model performs better than input model 24

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Empirical Results (cont.) Combined models fail to improve on input models when input does not capture any underlying behavior 25 Input D & test D’: similar behavior Input E & test E’: similar behavior

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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 26

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Roadmap (Ch. 3) GMM (static) (Ch. 4) History- Dependent GMM (Ch. 6) Application: Information Diffusion 27 (Ch. 2) Background 1. Consensus Dynamics 2. Description 3. Joint vs. individual behavior 4. 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 2 3 4 5 1 6 AgentBlue consensus Red consensus neither 11.0 0.50 2 1.00 Observation graph Agent 1’s perspective 28 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 29 time

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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 H t Note: dependence graphs can be different from observation graphs. 1 1 1 t-1tt+1 31

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hGMMs (Undirected) within-time edges: dependencies between agent actions in the same time period, and define dependence neighborhood N i for each agent i. A GMM at every time t 1 1 1 t-1tt+1 32

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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 Γ i Example: frequency function. 1 1 1 t-1t+1 33

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hGMMs Pr(a t | H) ∝ Π i π i (a t N i | H t Γ i ) Joint probability distribution of system’s actions at time t Joint probability distribution of system’s actions at time t potential of neighborhood’s joint actions at t history of the conditioning set 34

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Challenge: Dependence Conditional independence Dependence induced by history abstraction/summarization (*) 35 1 2 1 2 1 2 t-2t-1t 1 2 1 2 1 2 t-2t-1t

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Individual vs. Joint Behavior Models Given complete history, autonomous agents’ behaviors are conditionally independent Individual behavior models: π i (a t i | H t Γ i,complete ) Joint behavior models allow specifying any action dependence within one’s within-time neighborhood, given some (abstracted) history π i (a t N i | H t Γ i,abstracted ) 36

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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. 1.Approximation does not deteriorate performance 37

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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 38

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Roadmap (Ch. 3) GMM (static) (Ch. 4) History- Dependent GMM (Ch. 6) Application: Information Diffusion 39 (Ch. 2) Background 1. Learning Graphical Game Models (Ch. 5) Learning Dependence Graph structure 2. Learning hGMMs

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Learning Graphical Game Models [Duong, Vorobeychik, Singh & Wellman ‘09] Payoff dependence (graphical games) ≠ Probabilistic dependence (GMMs) Example: u 1 (a 1,a 3,a 2, a 5,a 6 ) Do not directly observe the underlying graphical structures, but strategy profiles and their corresponding payoffs 40 1 3 6 2 7 5 4

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Contributions (Ch. 5.1) (III.A) The learning problem’s definition, theoretical characteristics and evaluation metrics are formally introduced and formulated. (III.B) An evaluation of structure learning algorithms reveals that a greedy approach often offers best time-performance tradeoff. 41

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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 42

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Consensus Dynamics Joint Behavior Model Extended Joint Behavior hGMM (eJCM) π i (a N i | H t Γ i ) = r i (a N i ) f(a i, H t Γ i ) γ Ι(a i, H t i ) β 1.r i (a N i ) = reward for action a i, discounted by the number of dissenting neighbors in N i 1.frequency of a i chosen previously by agents in the conditioning set Γ i 2.inertia proportional to how long i has maintained its most recent action 43 123

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Consensus Dynamics Individual Behavior Models 1. Extended Individual Behavior hGMM (eICM): similar to eJCM but assumes that N i contains i only π i (a i | H t Γ i ) = Pr(a i | H t Γ i ) ∝ r i (a i ) f(a i, H t Γ i ) γ Ι(a i, H t i ) β 2. Proportional Response Model (PRM): only incorporates the most recent time period [Kearns et al., ‘09]: Pr(a i | H t Γ i ) ∝ r i (a i ) f(a i, H t Γ i ) 3. Sticky Proportional Response Model (sPRM) 44

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Learning hGMMS 45 Input: observation graph Search space: 1.Model parameters γ, β 2.Within-time edges Output: hGMM Objective: likelihood of data Constraint: 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 46

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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 47

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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 48

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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. 49

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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 50

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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 51

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Modeling Multiagent Systems: Step By Step 52 Given as input Learn from data Intuition, background information Approximation Dependence graph structure Potential function GMM hGMM Observation graph structure

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Roadmap (Ch. 3) GMM (static) (Ch. 4) History- Dependent GMM (Ch. 6) Application: Information Diffusion 53 (Ch. 2) Background (Ch. 5) Learning Dependence Graph structure 1. Definition 2. Joint behavior modeling 3. Learning missing edges 4. 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 54 True network G* Observed Network 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 55 [Duong, Wellman & Singh ‘11] 1.Network G 2.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 56

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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) 57

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Summary of Contributions (Ch. 6) (II.C) Joint behavior hGMM, can capture state dependence caused by missing edges 58

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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 59

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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. 60

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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] 61

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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 62

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THANK YOU! 63

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