Markov Models for Multi-Agent Coordination

Maayan Roth

Multi-Robot Reading Group

April 13, 2005

Multi-Agent Coordination

Teams:– Agent work together to achieve a common goal– No individual motivations

Objective:– Generate policies (individually or globally) to yield

best team performance

Observability [Pynadath and Tambe, 2002]

“Observability” - degree to which agents can, either individually or as a team, identify current world state

– Individual observability MMDP [Boutilier, 1996]

– Collective observability O1 + O2 + … + On uniquely identify the state e.g. [Xuan, Lesser, Zilberstein, 2001]

– Collective partial observability DEC-POMDP [Bernstein et al., 2000] POIPSG [Peskin, Kim, Meuleau, Kaelbling, 2000]

– Non-observability

Communication [Pynadath and Tambe, 2002]

“Communication” - explicit message-passing from one agent to another

– Free Communication No cost to send messages Transforms MMDP to MDP, DEC-POMDP to POMDP

– General Communication Communication is available but has cost or is limited

– No Communication No explicit message-passing

Taxonomy of Cooperative MAS

No Communication

General Communication

Free Communication

Full Observability

Collective Observability

Partial Observability

Unobservability

Ope

n-lo

op c

ontr

ol?MMDP

MDPMDP POMDP

DEC-MDP DEC-POMDP

Complexity

Individually Observable

Collectively Observable

Collectively Partially Observable

No Comm. P-complete NEXP-complete NEXP-complete

General Comm. P-complete NEXP-complete NEXP-complete

Free Comm. P-complete P-complete PSPACE-complete

Multi-Agent MDP (MMDP) [Boutilier, 1996]

Also called IPSG (identical payoff stochastic games)

M = <S, {Ai}im, T, R>– S is set of possible world states– {Ai}im is set of joint actions, <a1, …, am> where ai Ai

– T defines transition probabilities over joint actions– R is team reward function

State is fully observable by each agent P-complete

Coordination Problems

Multiple equilibria – two optimal joint policies

s1

s2

s3

s4

s5

s6

+10

-10

+5

<a,*>

<b,*><*,*>

<*,*>

<*,*>

<*,*>

<a,a>; <b,b>

<a,b>; <b,a>

Randomization with Learning

Expand MMDP to include FSM that selects actions based on whether agents are coordinated or uncoordinated

Build the expanded MMDP by adding a coordination mechanism at every state where a coordination problem is discovered

U

A

Brandom(a,b)

b

a

<*,*>

<*,*>

<a,a>

<b,b>

<a,b>; <b,a>

Multi-Agent POMDP

DEC-POMDP [Bernstein et al., 2000], MTDP [Pynadath et al., 2002], POIPSG [Peshkin et al., 2000]

M = <S, {Ai}im, T, {i}im, O, R>– S is set of possible world states– {Ai}im is set of joint actions, <a1, …, am> where ai Ai

– T defines transition probabilities over joint actions– {i}im is set of joint observations, <1, …, m> where i i

– O defines observation probabilities over joint actions and joint observations

– R is team reward function

Complexity [Bernstein, Zilberstein, Immerman, 2000]

For all m 2, DEC-POMDP with m agents is NEXP-complete

– “For a given DEC-POMDP with finite time horizon T, and an integer K, is there a policy for which yields a total reward K?”

– Agents must reason about all possible actions and observations of their teammates

For all m 3, DEC-MDP with m agents is NEXP-complete

Dynamic Programming[Hansen, Bernstein, Zilberstein, 2004]

Finds optimal solution Partially observable stochastic game (POSG)

framework– Iterate between DP step and pruning step to remove

dominated strategies

Experimental results show speed-up in some domains

– Still NEXP-complete, so hyper-exponential in worst case

No communication

Transition Independence [Becker, Zilberstein, Lesser, Goldman, 2003]

DEC-MDP – collective observability Transition independence:

– Local state transitions Each agent observes local state Individual actions only affect local state transitions

– Team connected through joint reward

Coverage set algorithm – finds optimal policy quickly in experimental domains

No communication

Efficient Policy Computation [Nair, Pynadath, Yokoo, Tambe, Marsella, 2003]

Joint Equilibrium-Based Search for Policies– initialize random policies for all agents– for each agent i:

fix the policies for all other agents find the optimal policy for agent i

– repeat until no policies change Finds locally optimal policies with potentially

exponential speedups over finding the global optimum

No communication

Communication [Pynadath and Tambe, 2002]

COM-MTDP framework for analyzing communication– Enhance model with {i}im where i is the set of possible

messages agent i can send– Reward function includes cost ( 0) of communication

If communication is free:– DEC-POMDP reducible to single-agent POMDP (PSPACE-

complete)– Optimal communication policy is to communicate at every

time step Under general communication (no restrictions on

cost), DEC-POMDP still NEXP-complete i may contain every possible observation history

COMM-JESP [Nair, Roth, Yokoo, Tambe, 2004]

Add SYNC action to domain– If one agent chooses SYNC,

all other agents SYNC– At SYNC, send entire

observation history since last SYNC

SYNC brings agents to synchronized belief over world states

Policies indexed by root synchronized belief and observation history since last SYNC

t=0(SL () 0.5)(SR () 0.5)

(SL (HR) 0.1275)(SL (HL) 0.7225)(SR (HR) 0.1275)(SR (HL) 0.0225)

(SL (HR) 0.0225)(SL (HL) 0.1275)(SR (HR) 0.7225)(SR (HL) 0.1275)

a = {Listen, Listen}

= HL = HR

a = SYNC

t = 2(SL () 0.5)(SR () 0.5)

t = 2(SL () 0.97)(SR () 0.03)

“At-most K” heuristic – there must be a SYNC within at most K timesteps

Dec-Comm [Roth, Simmons, Veloso, 2005]

At plan-time, assume communication is free– generate joint policy using single-agent POMDP-

solver

Reason over possible joint beliefs to execute policy

Use communication to integrate local observations into team belief

Tiger Domain: (States, Actions)

Two-agent tiger problem [Nair et al., 2003]:

S: {SL, SR}

Tiger is either behind left door or behind right door

Individual Actions:

ai {OpenL, OpenR, Listen}

Robot can open left door, open right door, or listen

Tiger Domain: (Observations)

Individual Observations:

I {HL, HR}

Robot can hear tiger behind left door or hear tiger behind right door

Observations are noisy and independent.

Tiger Domain:(Reward)

Coordination problem – agents must act together for maximum reward

Maximum reward (+20) when both agents open door with treasure

Minimum reward (-100) when only one agent opens door with tiger

Listen has small cost (-1 per agent)

Both agents opening door with tiger leads to medium negative reward (-50)

Joint Beliefs

Joint belief (bt) – distribution over world states Why compute possible joint beliefs?

– action coordination– transition and observation functions depend on joint action– agent can’t accurately estimate belief if joint action is

unknown

To ensure action coordination, agents can only reason over information known by all teammates

Possible Joint Beliefs

b: P(SL) = 0.5

p: p(b) = 1.0

= {}

L0

a = <Listen, Listen>

HL

HL

b: P(SL) = 0.8

p: p(b) = 0.29

= {HL,HL}

b: P(SL) = 0.5

p: p(b) = 0.21

= {HL,HR}

b: P(SL) = 0.5

p: p(b) = 0.21

= {HR,HL}

b: P(SL) = 0.2

p: p(b) = 0.29

= {HR,HR}

L1

How should agents select actions over joint beliefs?

),|( 111 ttttt abPpp

Q-POMDP Heuristic

Select joint action over possible joint beliefs Q-MDP (Littman et. al., 1995)

– approximate solution to large POMDP using underlying MDP

Q-POMDP– approximate solution to DEC-POMDP using

underlying single-agent POMDP

)()(maxarg)( sVsbbQSs

aa

MDP

)),(()(maxarg)( aLbQLpLQ ti

LL

ti

a

tPOMDP

tti

Q-POMDP Heuristic

)),(()(maxarg)( aLbQLpLQ ti

LL

ti

a

tPOMDP

tti

b: P(SL) = 0.5

p: p(b) = 0.21

= {HR,HL}

b: P(SL) = 0.8

p: p(b) = 0.29

= {HL,HL}

b: P(SL) = 0.5

p: p(b) = 0.21

= {HL,HR}

b: P(SL) = 0.5

p: p(b) = 1.0

= {}

b: P(SL) = 0.2

p: p(b) = 0.29

= {HR,HR}

Choose joint action by computing expected reward over all leaves

Agents will independently select same joint action…

but action choice is very conservative (always <Listen,Listen>)

DEC-COMM: Use communication to add local observations to joint belief

Dec-Comm Example

b: P(SL) = 0.5

p: p(b) = 0.21

= {HR,HL}

b: P(SL) = 0.8

p: p(b) = 0.29

= {HL,HL}

b: P(SL) = 0.5

p: p(b) = 0.21

= {HL,HR}

b: P(SL) = 0.5

p: p(b) = 1.0

= {}

b: P(SL) = 0.2

p: p(b) = 0.29

= {HR,HR}

HL

HL

L1

aNC = Q-POMDP(L1) = <Listen,Listen>

L* = circled nodes

aC = Q-POMDP(L*) = <Listen,Listen>

Don’t communicate

Dec-Comm Example (cont’d)

b: P(SL) = 0.85

p: p(b) = 0.29

= {HL,HL}

b: P(SL) = 0.5

p: p(b) = 1.0

= {}

L1b: P(SL) = 0.5

p: p(b) = 0.21

= {HL,HR}

b: P(SL) = 0.5

p: p(b) = 0.21

= {HR,HL}

b: P(SL) = 0.15

p: p(b) = 0.29

= {HR,HR}

…b: P(SL) = 0.97

p: p(b) = 0.12

: {<HL,HL>,

<HL,HL>}

b: P(SL) = 0.85

p: p(b) = 0.06

: {<HL,HL>,

<HL,HR>}

b: P(SL) = 0.85

p: p(b) = 0.06

: {<HL,HL>,

<HR,HL>}

b: P(SL) = 0.5

p: p(b) = 0.04

: {<HL,HL>,

<HR,HR>}

b: P(SL) = 0.85

p: p(b) = 0.06

: {<HL,HR>,

<HL,HL>}

b: P(SL) = 0.5

p: p(b) = 0.04

: {<HL,HR>,

<HL,HR>}

b: P(SL) = 0.5

p: p(b) = 0.04

: {<HL,HR>,

<HR,HL>}

b: P(SL) = 0.16

p: p(b) = 0.06

: {<HL,HR>,

<HR,HR>}

L2

a = <Listen, Listen>

<HL,HL>

<HL,HL>

aNC = Q-POMDP(L2) = <Listen, Listen>

L* = circled nodes

V(aC) - V(aNC) > ε

Agent 1 communicatesaC = Q-POMDP(L*) = <OpenR,OpenR>

Dec-Comm Example (cont’d)

b: P(SL) = 0.85

p: p(b) = 0.29

= {HL,HL}

b: P(SL) = 0.5

p: p(b) = 1.0

= {}

L1b: P(SL) = 0.5

p: p(b) = 0.21

= {HL,HR}

b: P(SL) = 0.5

p: p(b) = 0.21

= {HR,HL}

b: P(SL) = 0.15

p: p(b) = 0.29

= {HR,HR}

…b: P(SL) = 0.97

p: p(b) = 0.12

: {<HL,HL>,

<HL,HL>}

b: P(SL) = 0.85

p: p(b) = 0.06

: {<HL,HL>,

<HL,HR>}

b: P(SL) = 0.85

p: p(b) = 0.06

: {<HL,HL>,

<HR,HL>}

b: P(SL) = 0.5

p: p(b) = 0.04

: {<HL,HL>,

<HR,HR>}

b: P(SL) = 0.85

p: p(b) = 0.06

: {<HL,HR>,

<HL,HL>}

b: P(SL) = 0.5

p: p(b) = 0.04

: {<HL,HR>,

<HL,HR>}

b: P(SL) = 0.5

p: p(b) = 0.04

: {<HL,HR>,

<HR,HL>}

b: P(SL) = 0.16

p: p(b) = 0.06

: {<HL,HR>,

<HR,HR>}

L2

a = <Listen, Listen>

<HL,HL>

<HL,HL>

Agent 1 communicates <HL,HL>

Agent 2 communicates <HL,HL>

Q-POMDP(L2) = <OpenR, OpenR>

Agents open right door!

References

Becker, R., Zilberstein, S., Lesser, V., Goldman, C. V. Transition-independent decentralized Markov decision processes. In International Joint Conference on Autonomous Agents and Multi-Agent Systems, 2003.

Bernstein, D., Zilberstein, S., Immerman, N. The complexity of decentralized control of Markov decision processes. In Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence, 2000.

Boutilier, C. Sequential optimality and coordination in multiagent systems. In Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999.

Hansen, E. A., Bernstein, D. S., Zilberstein, S. Dynamic programming for partially observable stochastic games. In National Conference on Artificial Intelligence, 2004.

Nair, R., Roth, M., Yokoo, M., Tambe, M. Communication for improving policy computation in distributed POMDPs. To appear in Proceedings of the International Conference on Autonomous Agents and Multiagent Systems, 2004.

Nair, R., Pynadath, D., Yokoo, M., Tambe, M., Marsella, S. Taming decentralized POMDPs: Towards efficient policy computation for multiagent settings. In Proceedings of International Joint Conference on Artificial Intelligence, 2003.

Peshkin, L., Kim, K.-E., Meuleau, N., Kaelbling, L. P. Learning to cooperate via policy search. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, 2000.

Pynadath, D. and Tambe, M. The communicative multiagent team decision problem: Analyzing teamwork theories and models. In Journal of Artificial Intelligence Research, 2002.

Roth, M., Simmons, R., Veloso, M. Reasoning about joint beliefs for execution-time communication decisions. To appear in International Joint Conference on Autonomous Agents and Multi-Agent Systems, 2005.

Xuan, P., Lesser, V., Zilberstein, S. Communication decisions in multi-agent cooperation: Model and experiments. In Proceedings of the International Joint Conference on Artificial Intelligence, 2001.