Overlapping Coalition Formation: Charting the Tractability Frontier

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Overlapping Coalition Formation: Charting the Tractability Frontier

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Overlapping Coalition Formation: Charting the Tractability Frontier

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Overlapping Coalition Formation: Charting the Tractability Frontier

Y. Zick, G. Chalkiadakis and E. Elkind

(AAMAS 2012)

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]:

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]).

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.

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)

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

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.

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.

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.

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

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 resource allocation

- Which profit divisions ensure group stability?

(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

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

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

Theorem:

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

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.

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.

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?