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Using Dialogue Games to Form Coalitions with Self-Interested Agents

Using Dialogue Games to Form Coalitions with Self-Interested Agents. Luke Riley Department of Computer Science University of Liverpool L.J.Riley@Liverpool.ac.uk. Supervisors: Katie Atkinson & Terry Payne. Talk Overview . 1. Coalition Formation in Cooperative Game Theory.

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Using Dialogue Games to Form Coalitions with Self-Interested Agents

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  1. Using Dialogue Games to Form Coalitions with Self-Interested Agents Luke Riley Department of Computer Science University of Liverpool L.J.Riley@Liverpool.ac.uk Supervisors: Katie Atkinson & Terry Payne

  2. Talk Overview 1. Coalition Formation in Cooperative Game Theory. 2. Coalition Formation in Argumentation. 3. The Issues and Problems Between these Two Approaches. 4. My Research. 2

  3. 1. Coalition Formation in Cooperative Game Theory (CGT) 3

  4. Background • N-person cooperative games (coalition games) were first proposed in detail by von Neumann & Morgenstern in 1944[1]: Where... Agent set: Characteristic Function: 4 [1] J. von Neumann and O. Morgenstern. The Theory of Games and Economic Behavior. Princeton University Press, 1944.

  5. Solving a Coalition Game • In its most traditional style the CGT outcome of a coalition game is: Where... CS = a set of coalitions (the coalition structure) x = a vector of each individual agent's payoff in the game. e.g. When Ag = {a,b,c} a possible CS could be {{a},{b,c}} and a possible payoff vector could be x(1,2,3) where: 5

  6. Finding a Stable Outcome – The Core • A Coalition structure is core-stable if no subset of agents can benefit from defecting to another coalition. • The core [2] is the set: e.g. [3] Example 1: Given a coalition game where Ag = {a,b}, v({a}) = v({b}) = 5 and v({a,b}) = 20 the proposed core outcome is <{a,b}, x(10,10) > 6 [2] D. Gillies. Some theorems on n-person games. PhD thesis, Princeton University, 1953. [3] Wooldridge, M. An Introduction to MultiAgent Systems Second Edition. John Wiley & Sons, 2009

  7. Finding a Stable Outcome – The Core • A Coalition structure is core-stable if no subset of agents can benefit from defecting to another coalition. • The core [2] is the set: e.g. [3] Example 2: Given a coalition game where Ag = {a,b}, v({a}) = v({b}) = 5 and v({a,b}) = 20 the proposed core outcome is <{a,b}, x(15,5) > • Yet core payoffs can sometimes be unfair 7 [2] D. Gillies. Some theorems on n-person games. PhD thesis, Princeton University, 1953. [3] Wooldridge, M. An Introduction to MultiAgent Systems Second Edition. John Wiley & Sons, 2009

  8. Epsilon-Core • Also the core can sometimes be empty e.g. [4] Example 3: Given a coalition game where Ag = {a,b,c}, forall subsets C if |C| = 2 agents then v(C) = 1 else v(C) = 0 • Solution [5] → • The epsilon value can be seen as the cost of deviating. e.g. [4] Example 4: Given the coalition game of example 3, the payoff vector x(1/3,1/3,1/3) is 1/3-core stable. 8 [4] G. Chalkiadakis, E. Elkind, and M. Wooldridge. Computational Aspects of Cooperative Game Theory. Morgan & Claypool Publishers, 2011. • [5] Shapley, Lloyd S. and Shubik, M. Quasi-cores in a monetary economy with non-convex preferences , Econometrica (The Econometric Society) 34(4): 805–827, 1966.

  9. 2. Coalition Formation in Argumentation 9

  10. Argumentation Background • Argumentation Frameworks are a means to represent and reason with different possibly conflicting data. • AFs use graphs of nodes and arcs: = Preferred extension • A set of arguments S are acceptable if for every argument a1 that attacks an element a2 in S then there exists another a2 in S that defeats a1. • The preferred extension is the maximal acceptable set 10 .

  11. Dung's Initial Work • Dung showed that Argumentation Frameworks were natural ways to represent n-person games, for example theorem 6 of [6]: x(3,4,8) The AF represents 3 possible payoff vectors of the coalition game: v({a}) = v({b}) = v({c}) = 3 v({a,c}) = 8 v({b,c}) = 12 or v(C) = 0 x(3,3,5) x(3,3,3) 11 [6] P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77:321–357, 1995.

  12. Amgoud's Further Research • Amgoud in [7] extended this research, where she highlighted: • Attacks can come from multiple attributes such as coalitions sharing an agent. • How to always find a solution to a coalition game. • Outlines how agents can collaboratively build AFs for coalition games. • How a dialogue game can be used to check if a certain coalition was in the best coalition structure. 12 [7] L. Amgoud. An argumentation-based model for reasoning about coalition structures. In ArgMAS, pages 217-228, 2005.

  13. 3. The Issues of Joining the Two Approaches 13

  14. Various Issues • CGT: Lacks flexible communication protocols to form stable coalition structures. • CGT: Generally does not take into account the computation and communication costs of finding stable coalition structures from a MAS perspective. • Arg: There is little research showing how payoff vectors are found and justified by MAS. • Arg: No research on how to stabilise coalitions games, using the epsilon-core • Arg: Only some limited direct mapping between the argumentation models and the CGT coalition game types (e.g. static, dynamic, skill games,...) 14

  15. My Current Research Question How can self-interested agents make use of argumentation within their communication to enable them to form a stable optimal coalition structure with an approximately fair payoff distribution? 15

  16. 4. The Proposed Method 16

  17. Dialogue Games & Argumentation Schemes • Dialogue Games can be used to build argumentation frameworks in real time, where agents can assert and retract arguments. • Argumentation schemes are patterns of reasoning that when instantiated provide presumptive justification for the particular conclusion of the scheme • e.g:... 17

  18. Approximately fair payoffs • Solution – restrict the payoffs allowed: • Agents have to propose an equal split of the payoff oreach agent should be given at least the same value it can get from a coalition of agents willing to defect (must assert the best coalition for itself) • Agents can object to a proposed payoff by finding a better one. • Once a core payoff is found, the dialogue stops • AFs can easily represent the core • ...But the core can be unfair 18

  19. Dialogue Games & Argumentation Schemes • I have devised a dialogue game [8] to find an optimal coalition structure with a restricted core payoff • Moves: e.g: 19 [8] L. Riley, K. Atkinson, and T. Payne. Coalition structure generation for self interested agents in a dialogue game. Technical Report ULCS-12-004, University of Liverpool, 2012.

  20. Core example Coalitions and associated payoff Move number Join phase The coalition structure of move A3 is {{a},{b},{c}}, the payoff vector is x(4,3,2) 20 = Preferred extension

  21. Core example Coalitions and associated payoff Move number Join phase Negotiation phase No cycles are created as later arguments have ‘’fairer’’ payoffs than earlier arguments The coalition structure of move A4 is {{a,c}, {b}}, the payoff vector is x(9,3,9) 21 = Preferred extension

  22. Core example Coalitions and associated payoff Join phase Move number Negotiation phase The coalition structure of move A5 is {{a,b}, {c}}, the payoff vector is x(10,4,2) 22 = Preferred extension

  23. Core example Coalitions and associated payoff Join phase Move number Negotiation phase The coalition structure of move A6 is {{a,c}, {b}}, the payoff vector is x(11,3,7) and is core stable 23 = Preferred extension

  24. Changes in agents payoff in a core stable game Core payoffs for agent j Core payoffs for agent j and k 24

  25. When no payoff changes are needed Core payoffs for agent j Core payoffs for agent j and k 25

  26. Recognising when the core is empty When there does not exist an agent in the coalition whose payoff is strictly increasing or decreasing → then the core is empty (given rules I have outlined) Core payoffs for agent j Core payoffs for agent j and k 26

  27. Empty core example Coalitions and associated payoff Move number No agent of coalition {b,c} has a strictly increasing or decreasing payoff in the arguments for that coalition 27

  28. Epsilon-Core Example Coalition Structure of move 11 is {{a},{b,c}}, the payoff vector is x(8,9.5,9.5) and is 3-core stable 28

  29. Next Steps…? • Extend the dialogue game to find epsilon-core stability and identify under what conditions the least core can be found. • Experiment with ideas further and find proofs. • Modify dialogue game so that other coalition games can be modeled. • Optimise process: Combine mechanism design approach of [9] with efficient distribution methods of [10]. [9] T. Sandholm, K. Larson, M. Andersson, O. Shehory and F. Tohmé, Coalition structure generation with worst case guarantees, Artificial Intelligence, Volume 111, Issues 1–2, July 1999, Pages 209-238. 29 [10] T. Rahwan. Algorithms for Coalition Formation in Multi-Agent Systems. PhD thesis, University of Southampton, 2007.

  30. Thanks For Listening 30 Questions?

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