1. 1 Graphical Models for Online Solutions to Interactive POMDPs Prashant Doshi Yifeng Zeng Qiongyu Chen University of Georgia Aalborg University National Univ. USA Denmark of Singapore International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2007)

2. 2 Decision-Making in Multiagent Settings State (S) Act to optimize preferences given beliefs Actions (A i ) Agent i Observations (O i ) Actions (A j ) Observations (O j ) Agent j Belief over state and model of j Belief over state and model of i

3. 3 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  T i Transition function: S×A×S  [0,1]  O i Observation function: S×A×  [0,1]  R i Reward function: S×A 

4. 4 Belief Update

5. 5 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

6. 6 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

7. 7 RelatedWork 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 decision- making)

8. 8 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: M j,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 M j,l-1 P(M j,l-1 |s) Update? AiAi RiRi OiOi S AjAj M j,l-1 Level l I-ID

9. 9 Details of the Model Node Members of the model node  Different chance nodes are solutions of models m j,l-1  Mod[M j ] represents the different models of agent j CPT of the chance node A j is a multiplexer  Assumes the distribution of each of the action nodes (A j 1, A j 2 ) depending on the value of Mod[M j ] Mod[M j ] Aj1Aj1 Aj2Aj2 M j,l-1 S m j,l-1 1 m j,l-1 2 AjAj m j,l-1 1, m j,l-1 2 could be I-IDs or IDs

10. 10 Whole I-ID AiAi RiRi OiOi SAjAj Mod[M j ] Aj1Aj1 Aj2Aj2 m j,l-1 1 m j,l-1 2 m j,l-1 1, m j,l-1 2 could be I-IDs or IDs

11. 11 Interactive Dynamic Influence Diagrams (I-DIDs) A i t+1 RiRi O i t+1 S t+1 A j t+1 M j,l-1 t+1 AitAit RiRi OitOit StSt AjtAjt M j,l-1 t Model Update Link

12. 12 m j,l-1 t,2 Semantics of Model Update Link Mod[M j t ] Aj1Aj1 M j,l-1 t stst m j,l-1 t,1 AjtAjt Aj2Aj2 Oj1Oj1 Oj2Oj2 OjOj Mod[M j t+1 ] Aj1Aj1 M j,l-1 t+1 s t+1 m j,l-1 t+1,1 m j,l-1 t+1,2 A j t+1 Aj2Aj2 Aj3Aj3 Aj4Aj4 m j,l-1 t+1,3 m j,l-1 t+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

13. 13 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

14. 14 A i t+1 RiRi O i t+1 S t+1 OitOit AitAit RiRi StSt m j,l-1 t,1 m j,l-1 t,2 Aj1Aj1 Oj1Oj1 Aj2Aj2 Oj2Oj2 Aj1Aj1 Aj2Aj2 Aj3Aj3 Aj4Aj4 m j,l-1 t+1,1 m j,l-1 t+1,2 m j,l-1 t+1,3 m j,l-1 t+1,4 A j t+1 Mod[M j t ] AjtAjt OjOj Mod[M j t+1 ]

15. 15 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

16. 16 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

17. 17 Level 1 I-ID in the Tiger Problem Expand over three time steps Mapping decision nodes to chance nodes

18. 18 Policy Tree 1: Agent i has hearing accuracy of 65% L L L L L L OR L L L L L L L L OL GL,*GR,* GL,CR GL,S/CL GR,* GL,* GR,S/CR GR,CL Conditional Followership

19. 19 Policy Tree 2: Agent i loses hearing ability (accuracy is 0.5) L L L L OR OL L L *,* *,CR *,S*,CL Unconditional (Blind) Followership

20. 20 Example 2: Altruism and Reciprocity in the Public Good Problem Public Good Game  Two agents initially endowed with X T 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 c i (<1) for each agent i, where c i is the marginal private return  In order to encourage contributions, the contributing agents punish free riders P but incur a small cost c p for administering the punishment

21. 21 Agent Types Altruistic and Non-altruistic types  Altruistic agent has a high marginal private return (c i 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

22. 22 Level 1 I-ID in the Public Good Game Expand over three time steps

23. 23 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 C C C C C C * *

24. 24 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 D D C C C C D D C C * PY * MR

25. 25 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

26. 26 Thank You!