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Graphical Models for Online Solutions to Interactive

POMDPs

Prashant Doshi Yifeng Zeng Qiongyu ChenUniversity of Georgia Aalborg University National Univ.

USA Denmark of Singapore

International Conference on Autonomous Agents and Multiagent Systems(AAMAS 2007)

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Decision-Making in Multiagent Settings

State (S)

Act to optimize preferences given beliefs

Actions (Ai)

Agent i

Observations (Oi)

Actions (Aj)

Observations (Oj)

Agent j

Belief over state and model of j

Belief over state and model of i

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Finitely Nested I-POMDP (Gmytrasiewicz&Doshi, 05) A finitely nested I-POMDP of agent i with a strategy

level l : Interactive states:

Beliefs about physical environments: Beliefs about other agents in terms of their preferences,

capabilities, and beliefs: Type: A Joint actions Possible observations Ti Transition function: S×A×S [0,1]

Oi Observation function: S×A× [0,1]

Ri Reward function: S×A

},,,,,;{:1, jjjjjjjjlj OCRAbM

S1,, ljli MSIS

iiiijili RAAAISPOMDPI li ,,,),(,,,

i

1, ljM

i

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Belief Update

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Forget It!

Different approach Use the language of Influence Diagrams (IDs) to

represent the problem more transparently Belief update

Use standard ID algorithms to solve it Solution

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Challenges

Representation of nested models for other agents Influence diagram is a single agent oriented

language

Update beliefs on models of other agents New models of other agents Over time agents revise beliefs over the models of

others as they receive observations

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Related Work

Multiagent Influence Diagrams (MAIDs) (Koller&Milch,2001) Uses IDs to represent incomplete information games Compute Nash equilibrium solutions efficiently by exploiting

conditional independence

Network of Influence Diagrams (NIDs) (Gal&Pfeffer,2003) Allows uncertainty over the game Allows multiple models of an individual agent Solution involves collapsing models into a MAID or ID

Both model static single play games Do not consider agent interactions over time (sequential decisio

n-making)

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Introduce Model Node and Policy Link

A generic level l Interactive-ID (I-ID) for agent i situated with one other agent j Model Node: Mj,l-1

Models of agent j at level l-1

Policy link: dashed line Distribution over the other

agent’s actions given its models

Beliefs on Mj,l-1

P(Mj,l-1|s) Update?

AiRi

Oi

S Aj

M j,l-1

Level l I-ID

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Details of the Model Node

Members of the model node Different chance nodes are

solutions of models mj,l-1

Mod[Mj] represents the different models of agent j

CPT of the chance node Aj is a multiplexer Assumes the distribution of

each of the action nodes (Aj

1, Aj2) depending on the valu

e of Mod[Mj]

Mod[Mj]

Aj1

Aj2

M j,l-1

S

m j,l-11

m j,l-12

Aj

m j,l-11 , m j,l-12 could be I-IDs or IDs

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Whole I-ID

AiRi

Oi

S Aj

Mod[Mj]

Aj1 Aj

2

m j,l-11 m j,l-12m j,l-11 , m j,l-12 could be I-IDs or IDs

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Interactive Dynamic Influence Diagrams (I-DIDs)

Ait+1

Ri

Oit+1

St+1 Ajt+1

M j,l-1t+1

Ait

Ri

Oit

St

Ajt

M j,l-1t

Model Update Link

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m j,l-1t,2

Semantics of Model Update Link

Mod[Mjt]

Aj1

M j,l-1t

st

m j,l-1t,1

Ajt

Aj2

Oj1

Oj2

Oj

Mod[Mjt+1]

Aj1

M j,l-1t+1

st+1

m j,l-1t+1,1

m j,l-1t+1,2

Ajt+1

Aj2

Aj3

Aj4

m j,l-1t+1,3

m j,l-1t+1,4

These models differ in their initial beliefs, each of which is the result of j updating its beliefs due to its actions and possible observations

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Notes

Updated set of models at time step (t+1) will have at most models :number of models at time step t :largest space of actions :largest space of observations

New distribution over the updated models uses original distribution over the models probability of the other agent performing the action, and receiving the observation that led to the updated model

|||||| 1, jjtlj AM

|| 1,tljM

|| jA

|| j

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Ait+1

Ri

Oit+1

St+1Oit

Ait

Ri

St

m j,l-1t,1

m j,l-1t,2

Aj1 Oj

1

Aj2 Oj

2

Aj1

Aj2

Aj3

Aj4

m j,l-1t+1,1

m j,l-1t+1,2

m j,l-1t+1,3

m j,l-1t+1,4

Ajt+1

Mod[Mjt]

Ajt

Oj

Mod[Mjt+1]

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Example Applications: Emergence of Social Behaviors Followership and Leadership in the persistent

multiagent tiger problem

Altruism and Reciprocity in the public good problem with punishment

Strategies in a simple version of two-player Poker

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Followership and Leadership in Multiagent Persistent Tiger Experimental Setup:

Agent j has a better hearing capability (95% accurate) compared to i’s (65% accuracy)

Agent i does not have initial information about the tiger’s location

Agent i considers two models of agent j which differ in j’s level 0 initial beliefs Agent j likely thinks that the tiger is behind the left door Agent j likely thinks that the tiger is behind the right door

Solve the corresponding level 1 I-DID expanded over three time steps and get the normative behavioral policy of agent i

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Level 1 I-ID in the Tiger Problem

Expand over threetime steps

Mapping decision nodes to chance nodes

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Policy Tree 1: Agent i has hearing accuracy of 65%

LL

LL LL

OROR LLLL LL LL OLOL

GL,* GR,*

GL,CRGL,S/CL

GR,*GL,*

GR,S/CR

GR,CL

Conditional Followership

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Policy Tree 2: Agent i loses hearing ability (accuracy is 0.5)

LL

LL

OROR OLOLLL

*,*

*,CR *,S *,CL

Unconditional (Blind) Followership

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Example 2: Altruism and Reciprocity in the Public Good Problem Public Good Game

Two agents initially endowed with XT amount of resources Each agent may choose

contribute (C) a fixed amount of the resources to a public pot not contribute ie. defect (D)

Agents’ actions and pot are not observable, but agents receive an observation symbolizing the state of the public pot plenty (PY) meager (MR)

Value of resources in the public pot is discounted by ci (<1) for each agent i, where ci is the marginal private return

In order to encourage contributions, the contributing agents punish free riders P but incur a small cost cp for administering the punishment

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Agent Types

Altruistic and Non-altruistic types Altruistic agent has a high marginal private return (ci

is close to 1) and does not punish others who defect Optimal Behavior

One action remaining: both types of agents choose to contribute to avoid being punished

Two actions to go: altruistic type chooses to contribute, while the other defects Why?

Three steps to go: the altruistic agent contributes to avoid punishment and the non-altruistic type defects

Greater than three steps: altruistic agent continues to contribute to the public pot depending on how close its marginal return is to 1, the non-altruistic type prescribes defection

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Level 1 I-ID in the Public Good Game

Expand over threetime steps

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Policy Tree 1: Altruism in PG

If agent i (altruistic type) believes with a probability 1 that j is altruistic, i chooses to contribute for each of the three steps.

This behavior persists when i is unaware of whether j is altruistic, and when i assigns a high probability to j being the non-altruistic type

CC

CC

CC

*

*

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Policy Tree 2: Reciprocal Agents Reciprocal Type

The reciprocal type’s marginal private return is less and obtains a greater payoff when its action is similar to that of the other

Experimental Setup Consider the case when the

reciprocal agent i is unsure of whether j is altruistic and believes that the public pot is likely to be half full

Optimal Behavior From this prior belief, i chooses to

defect On receiving an observation of

plenty, i decides to contribute, while an observation of meager makes it defect

With one action to go, i believes that j contributes, will choose to contribute too to avoid punishment regardless of its observations

DD

CC

CC

DD

CC

*

PY

*

MR

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Conclusion and Future Work

I-DIDs: A general ID-based formalism for sequential decision-making in multiagent settings Online counterparts of I-POMDPs

Solving I-DIDs approximately for computational efficiency (see AAAI ’07 paper on model clustering)

Apply I-DIDs to other application domains

Visit our poster on I-DIDs today for more information

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Thank You!