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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 (AAMAS 2012). Framework. Each p layer has a divisible resource (in our model, an integer weight) .

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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 (AAMAS 2012)

  2. Framework Each player has a divisible resource (in our model, an integer weight). A coalition is formed by agents contributing some of their weight to a certain collaborative task. OCF Games [Chalkiadakis et. al, 2010]:

  3. Framework Goal: find an optimal coalition structure; divide coalitional payoffs in a stable manner. • One Key Issue: the stability of a payoff division highly depends on the behavior of non-deviators (arbitration functions [Zick and Elkind, 2011]).

  4. NP-hardness Finding optimal coalition structures/stable payoff allocations is known to be NP-hard: The objective of our work is to identify conditions that make optimization and stability tractable.

  5. 2-OCF Games Agents may form coalitions of at most size 2: • If agent icontributesxand agent jcontributesy, the value of interaction isvi,j(x,y) • If an agent iinvests xin working alone, he makes vi(x)

  6. Problem Model The problem can be modeled as a graph Agents are weighted nodes • Node value: vi(x) • Edge value: vi,j(x,y) • Goal #1: optimal allocation

  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 for a constant # of agents can be done in poly(W+1) time, where Wis the maximal weight of any agent.

  9. Optimal Coalition Structure Computational complexity: even when weights are at most 3, complex interactions cause NP-hardness (the X3C problem). We assume that: • Weights are small • Interactions are simple.

  10. Optimal Coalition Structure Suppose that the interaction graph is a tree w8 w1 v8 v5,8 v1,2(x,y) v1,5(x,y) v1(x) w2 w5 v5,9 v1,3(x,y) w9 v2 v5 v9 w3 v5,7 v3,4 v3,6 v3 w7 w6 w4 v7 v6 v4

  11. Optimal Coalition Structure Theorem: if the interaction graph is a tree, an optimal allocation can be computed in time linear in the # of agents and polynomial in (W+1).

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

  13. (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

  14. Stability Arbitration functions: Given a set’s deviation from an outcome, how much will it get from surviving agreements with non-deviators?

  15. 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

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

  17. Stability Theorem: if the arbitration function is local, and the interaction graph is a tree, then one can verify if an outcome is stable in poly(n,W+1) time.

  18. 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.

  19. 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.

  20. Thank you! Questions?

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