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)

MotivationAgents have limited

integer resources

The benefit of interaction may be freely divided

Form Bilateral Trade Contracts: coalitions

QuestionsWhat is the optimal coalition structure?

How should profits be divided?

Problem Complexity

Agents are nodes

The problem can be modeled as a graph

There is an edge between agents if they can profit from collaborating.

Goal: optimal allocation

v1(x) = 5I5(x)

v1,2(x,y) = log(x + y + 2)

v2(x) = 0 w1 = 8

w2 = 3

v1,2(x,y) = log(x + y + 2)

v2(x) = 0 w1 = 8

w2 = 3

v1(x) = 5I5(x)v1(5) = 5

v1,2(1,1) = 2v1,2(1,1) = 2v1,2(1,1) = 2

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 is in P for constant # of agents and poly size weights.

Proof: can be done by dynamic programming.


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


We assume that:•Weights are polynomially bounded• Interactions are simple.


Suppose that the interaction graph is a tree


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.


We set:ui(xi) – the most an agent can

make working aloneui,j(xi, xj) – the most two agents

can make by working togetherTi(xi) – the most the subtree

rooted at i can make


1 8

764

532 9

OPT=max{u1(x1) + §u1,j(x1j,yj) + Tj(wj - yj)}

T3(x3)= max{u3(y3)+§u3,j(y3j,zj) + Tj(wj - zj)}

StabilityOptimal resource allocation

Which profit divisions ensure group stability?

17,1510,5

1,5

4,310,13

5

5,7

16,5

7 1,1

10,9

4,513,12

(CS, x)CS xOutcome

Is (CS, x) in the core?

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.

17,1510,5

1,5

4,310,13

5

5,7

16,5

7 1,1

10,9

4,513,12

20

15

StabilityArbitration functions: agents may receive all or some of the payoff from unbroken/changed agreements.

Behavior can be very general.

Arbitration FunctionsOthers can react to deviation

either locally or globally.Conservative – give nothingRefined – give all from unhurt

coalitionsOptimistic – deviators absorb the

marginal damage of deviation; get the difference.

17,1510,5

1,5

4,310,13

5

5,7

16,5

7 1,1

10,9

4,513,12

8,15 GlobalLocal

8,10

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

1

7

6

5

4

32

1 2 3 4 5 6 7

1 2 3 4 5 6 7

05

10

10

10

10

10

10

10

"""

StabilityTheorem: 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

StabilityDenote 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?

StabilityTheorem: 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.

StabilityCorollary: 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

RecapOptimization/Stability: Hard in general due to•Weights• Complex interaction

More ResultsBounded 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.

More ResultsStable conservative core: We can find a stable outcome against worst case behavior.

Each agent receives the minimum needed to make his subtree stable.

SummaryComputational 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.

Poly-time, but…Complexity is still high: Order of O(nkW5(k+1)) for computing optimal allocation in a graph with treewidth k

Can probably do better if valuations are known.

Future WorkDeterministic, Exact: randomized/ approximation algorithms?

Restricted classes of games: convex, subadditive…

Thank you!

Questions?

Overlapping Coalition Formation: Charting the Tractability Frontier

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