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). 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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Overlapping coalition formation charting the tractability frontier

Overlapping Coalition Formation: Charting the Tractability Frontier

Y. Zick, G. Chalkiadakis and E. Elkind

(AAMAS 2012)


Framework

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]:


Framework1

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

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

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)


Problem model

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


Optimal coalition structure

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.


Optimal coalition structure1

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.


Overlapping coalition formation charting the tractability frontier

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 structure2

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


Optimal coalition structure3

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).


Stability

Stability

Optimal resource allocation

  • Which profit divisions ensure group stability?


Overlapping coalition formation charting the tractability frontier

(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


Stability1

Stability

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


Overlapping coalition formation charting the tractability frontier

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


Stability2

Stability

Theorem:

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


Stability3

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

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

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.


Overlapping coalition formation charting the tractability frontier

Thank you!

Questions?


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