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Overlapping Coalition Formation: Charting the Tractability Frontier

Overlapping Coalition Formation: Charting the Tractability Frontier. Y. Zick , G. Chalkiadakis and E. Elkind (submitted to AAMAS 2012). Motivation. Agents have limited integer resources. Form Bilateral Trade Contracts : coalitions. The benefit of interaction may be freely divided.

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Overlapping Coalition Formation: Charting the Tractability Frontier

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  1. Overlapping Coalition Formation: Charting the Tractability Frontier Y. Zick, G. Chalkiadakis and E. Elkind (submitted to AAMAS 2012)

  2. Motivation Agents have limited integer resources Form Bilateral TradeContracts: coalitions • The benefit of interaction may be freely divided

  3. Questions What is the optimal coalition structure? How should profits be divided?

  4. Problem Complexity The problem can be modeled as a graph Agents are nodes • There is an edge between agents if they can profit from collaborating. • Goal: optimal allocation

  5. w2 = 3 v1,2(x,y) = log(x + y + 2) v2(x) = 0 w1 = 8 v1(x) = 5I5(x)

  6. w2 = 3 v1,2(x,y) = log(x + y + 2) v2(x) = 0 w1 = 8 v1,2(1,1) = 2 v1,2(1,1) = 2 v1,2(1,1) = 2 v1(x) = 5I5(x) v1(5) = 5

  7. Optimal Coalition Structure Computational complexity computing an optimal allocation is NP-hard even for a single agent (the KNAPSACKproblem). One agent with large weight – find the optimal set of tasks to complete.

  8. Optimal Coalition Structure Theorem: computing an optimal allocation is in P for constant # of agents and poly size weights. Proof: can be done by dynamic programming.

  9. Optimal Coalition Structure Computational complexity even when weights are at most 3, complex interactions cause NP-hardness (the X3C problem).

  10. Optimal Coalition Structure We assume that: • Weights are polynomially bounded • Interactions are simple.

  11. Optimal Coalition Structure Suppose that the interaction graph is a tree

  12. Optimal Coalition Structure Theorem: if the maximal weight is W and there are n nodes, an optimal allocation can be computed in time linear in n and polynomial in W.

  13. Optimal Coalition Structure We set: ui(xi) – the most an agent can make working alone ui,j(xi,xj) – the most two agents can make by working together Ti(xi) – the most the subtree rooted at i can make

  14. OPT=max{u1(x1) +§u1,j(x1j,yj) + Tj(wj-yj)} T3(x3)= max{u3(y3)+§u3,j(y3j,zj) + Tj(wj-zj)} 1 8 5 2 9 3 7 4 6

  15. Stability Optimal resource allocation • Which profit divisions ensure group stability?

  16. (CS,x) CS x 17,15 5 Outcome 10,5 Is (CS,x) in the core? 1,5 10,13 4,3 13,12 4,5 5,7 7 1,1 16,5 10,9

  17. Deviation “Coalitional game theory [...] considers a game of n players as a set of possible 2n – 1 coalitions, each of which, call it S, can achieve a particular value v(S) […] against worst case behavior of players in N\S” C.H. Papadimitriou, STOC 2001 • Players assume they are “on their own” if they deviate.

  18. 15 17,15 5 10,5 20 1,5 10,13 4,3 13,12 4,5 5,7 7 1,1 16,5 10,9

  19. Stability Arbitration functions: agents may receive all or some of the payoff from unbroken/changed agreements. Behavior can be very general.

  20. Arbitration Functions Others can react to deviation either locally or globally. Conservative – give nothing Refined – give all from unhurt coalitions Optimistic– deviators absorb the marginal damage of deviation; get the difference.

  21. Global Local 17,15 8,15 5 10,5 1,5 8,10 10,13 4,3 13,12 4,5 5,7 7 1,1 16,5 10,9

  22. Stability Theorem: if there is an efficient algorithm to compute the most one can get from global arbitration functions, then P = NP.

  23. 4 1 6 2 3 5 7 1 3 2 4 7 5 6 0 5 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 3 2 4 7 5 6 " " "

  24. Stability Theorem: if the arbitration function is local, and the interaction graph is a tree, computing the most a set can get from deviating is possible in poly(n,W) time

  25. Stability Denote the most that a set S can get by deviating by A*(S,CS, x) Having divided payoffs, can we verify that no set wants to deviate?

  26. Stability Theorem: if the arbitration function is local, and the interaction graph is a tree, then one can verify if an outcome is A -stable in poly(n,W) time.

  27. Stability Corollary: Given a coalition structure CS, we can find x such that (CS, x) is A -stable in poly(n,W) time. Proof: ellipsoid method to solve an LP

  28. Recap Optimization/Stability: Hard in general due to • Weights • Complex interaction

  29. More Results Bounded hyper-treewidth: Our results can be extended to graphs with bounded hyper-treewidth. If the graph is “tree-like” we can still obtain efficient algorithms.

  30. More Results Stable conservative core: We can find a stable outcome against worst case behavior. Each agent receives the minimum needed to make his subtree stable.

  31. Summary Computational Issues: A major obstacle in OCF games. But: if interactions are (somewhat) local, both for values and arbitration functions, we can obtain poly-time algorithms.

  32. Poly-time, but… Complexity is still high: Order of O(nkW5(k+1)) for computing optimal allocation in a graph with treewidthk Can probably do better if valuations are known.

  33. Future Work Deterministic, Exact: randomized/ approximation algorithms? Restricted classes of games: convex, subadditive…

  34. Thank you! Questions?

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