Between Collaboration and Competition: An Initial Formalization using Distributed POMDPs Praveen Paruchuri, Milind Tambe University of Southern California Spiros Kapetanakis University of York, UK Sarit Kraus Bar-Ilan University, Israel University of Maryland, College Park July 2003 University of Southern California 1
Motivation Many domains present where agents act in team but need to maintain some self interest. Electric Elves – Agents take decisions for users but act as a team like arranging a meeting etc. SDR – Software for Distributed Robotics where 100+ robots must locate and protect objects. Ø Robots must ensure their survival like refilling batteries University of Southern California 2
The Problem Framework for teams of agents maintaining private goals for stochastic, complex and dynamic environments. Agents need to maximize joint objectives and yet honor private preferences. Ø Private versus Team Interest – Might be conflicting Build framework based on Distributed POMDPs for policy generation Analyze complexity of policy generation University of Southern California 3
Previous work Distributed POMDPs like COM-MTDP Ø Have single joint reward Ø Optimal policy maximizes joint value (Ex 1) – Solution not stable Stochastic Games Ø Have individual rewards. Ø Policy finds equilibrium solution. Stability, key concept (Ex 2) – Solution not favorable to both individually and as team Ex 1: Ex 2: 4(6, -2) 2(1, 1) 5, 5(10) -1, 6(5) 0(0, 0) 3(-2, 5) 6, -1(5) 0, 0(0) University of Southern California 4
Motivation : Simple examples One shot game without stochastic elements Ex 1: Two people need to meet, one prefers 4 pm, other 5 pm When should they meet ? ? Need to compromise some extent, but not totally. No meeting is bad for both. Agree on mutually acceptable solution. Ex 2: Team of robots work on task Limited battery Last n% battery for re-fuelling itself. Otherwise die. Need to achieve team goal while they don’t die. University of Southern California 5
MTDP: A Distributed POMDP Model An MTDP is a tuple <S, A(α), P, Ω(α), O(α), B(α), R> where, Ø S is a set of world states. Ø A(α) is a set of allowed team actions. A(α) = π ( A(i) ) , A(i) is a set of domain level actions for each agent i. Ø P is a probability distribution that governs the effect of domain level actions. ( P( s, a, s 1) = Pr ( s 1/s, a) ) Ø Ω(α) is the joint set of observations. Ø B(α) is the combination of all the agent’s set of possible belief states. Ø R is the common reward for the team. R: S * A(α) R University of Southern California 6
E-MTDP: Formally Defined An E-MTDP is a tuple <S, A(α), P, Ω(α), O(α), B(α), R> where S, A(α), P, Ω(α), O(α), B(α) are as defined in MTDP. R = < R 1, R 2, ……. . , Rn, Rα > where, Ø R 1, R 2, . . , Rn are rewards of agents 1, 2, . . , n Ø Rα is the joint reward for the n agents where Rα = γ*R 1 + δ*R 2 +……… Both individual and joint rewards can be expressed. University of Southern California 7
E-MTDP Policy maps belief states to actions - Π : Bi Ai Centralized Policy generator. Policy π is such that: V 1(π) > T 1 , V 2(π) > T 2 For π’ <> π, where V 1(π’) > T 1 and V 2(π’)>T 2, V(π) > V(π’) where, T 1 and T 2 are thresholds for agents 1 and 2. V 1 is value from policy for agent 1 and V 2 for agent 2. V is overall value of policy without splitting. University of Southern California 8
Novelties of E-MTDP Maintains individual rewards for each agent and a joint reward for the team. Solution concept is novel because optimal policy both Ø Maximizes joint reward and Ø Ensures certain minimum expected value for individual team members. University of Southern California 9
Experimental Validation Goal: Show utility of EMTDP A real system called Electric(E)-Elves based on MDPs. Ø Based on maximizing single joint reward. Expressed as EMTDP and helped improve performance. E-Elves- A published real world multi agent system Used at USC/ISI for 6 months. Agents called proxies - Reschedule meetings, Decide to present talks on behalf of user, Order meals, Track user location etc. University of Southern California 10
Electric Elves Focus on task of rescheduling meetings. Used single agent MDP to model an agent Actions like delaying/canceling meeting, asking user etc. Asking user for his input is critical. Time constraints might prevent agent asking user for input. Policy generator uses the notion of team reward for deciding actions. No notion of individual reward. University of Southern California 11
![Perceived Problem and Improvement Original formulation had R(α) and R(user) terms[1]. However R(α) +](https://present5.com/presentation/9ba349875cb1c46542a1d0fc78d42db6/image-12.jpg "Perceived Problem and Improvement Original formulation had R(α) and R(user) terms[1]. However R(α) +")
Perceived Problem and Improvement Original formulation had R(α) and R(user) terms[1]. However R(α) + R(user) is maximized in policy generation. As R(α) increased with R(user) constant, agent stopped asking user. As R(α) increases, cost(Uncertainty in getting response from user) > δ ( Increase in quality of decision due to user’s feedback ). Hence, decision taken without asking. User might want to have different decision. User can set his importance to meeting using R(user) Ø If user important, agent needs to make a correct decision regarding user. Ø User’s opinion becomes important affecting # of asks. University of Southern California 12
Original Elves Result x-axis: Value of meeting without the user. y-axis: # of times the agent asks the user. Number of asks decrease as R(alpha) increases. Agents sometime cancel important meeting without asking user ( Very high cost )[1]. University of Southern California 13
E-MTDP based E-Elves Solving using E-MTDP – Let there be two agents Ø Priv 1 = R(user), agent 1’s private reward Ø Priv 2 = R(alpha), agent 2’s private reward Set priv 1 >= Threshold. # of asks now dependent on Threshold. User importance(priv 1) set high. Agent asks the user for his input before deciding unlike earlier. Setting threshold is important to obtain the required behavior. University of Southern California 14
E-MTDP result From graph above, giving flexibility to the user to set his threshold can result in agent asking him more times. ØUser opinion taken into consideration. “Flexibility” is the key word. Users like control over their agents. University of Southern California 15
Conclusions A framework for teams of self-interested agents. E-MTDP presented as a solution concept. E-MTDP applied to E-Elves Improvement in performance of system measured in terms of number of asks. Fine-tuning of agents, according to user needs, now possible. University of Southern California 16
Future Work Fine tune the existing E-MTDP framework. Need to analyze complexity of E-MTDP policies. Analyze stability of the E-MTDP solutions. References 1. Towards Adjustable Autonomy for the Real World Paul Scerri, David V. Pynadath and Milind Tambe, JAIR-02 THANK YOU Any Questions ? ? University of Southern California 17
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Stability of solution Designed a multistage game for E-MTDP policy to be stable. University of Southern California 19
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