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

Framework OCF Games [Chalkiadakis et. al, 2010] : 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.

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] ).

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.

2-OCF Games Agents may form coalitions of at most size 2: If agent i contributes x and agent j contributes y, the value of interaction is v i, j ( x, y ) If an agent i invests x in working alone, he makes v i ( x )

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

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

Theorem: computing an optimal allocation for a constant # of agents can be done in poly(W+ 1 ) time, where W is the maximal weight of any agent. 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. Optimal Coalition Structure

Suppose that the interaction graph is a tree Optimal Coalition Structure w1w1 w2w2 w3w3 w4w4 w6w6 w5w5 w8w8 w7w7 w9w9 v1(x)v1(x) v2v2 v3v3 v4v4 v6v6 v8v8 v9v9 v7v7 v5v5 v 1,5 ( x, y ) v 1,3 ( x, y ) v 1,2 ( x, y ) v 3,6 v 3,4 v 5,7 v 5,9 v 5,8

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). Optimal Coalition Structure

Stability Optimal resource allocation Which profit divisions ensure group stability?

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

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

17,15 10,5 1,5 4,3 10,13 5 5,7 16,5 7 1,1 10,9 4,5 13,12 8,15 GlobalLocal 8,10

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

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.

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.

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.

Thank you! Questions?