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

Questions

What 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.

Optimal Coalition Structure

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

Optimal Coalition Structure

We assume that:

•Weights are polynomially bounded• Interactions are simple.

Optimal Coalition Structure

Suppose that the interaction graph is a tree

Optimal Coalition Structure

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.

Optimal Coalition Structure

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

Optimal Coalition Structure

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)}

Stability

Optimal resource allocation

Which profit divisions ensure group stability?

17,15

10,5

1,5

4,310,13

5

5,7

16,5

71,1

10,9

4,5

13,12

(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.

17,15

10,5

1,5

4,310,13

5

5,7

16,5

71,1

10,9

4,5

13,12

20

15

Stability

Arbitration 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.

17,15

10,5

1,5

4,310,13

5

5,7

16,5

71,1

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.

1

7

6

5

4

3

2

1 2 3 4 5 6 7

1 2 3 4 5 6 7

05

10

10

10

10

10

10

10

"""

Stability

Theorem: 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

Stability

Denote 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?

Stability

Theorem: 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.

Stability

Corollary: 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

Recap

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

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.

More Results

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

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

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.

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 Work

Deterministic, Exact: randomized/ approximation algorithms?

Restricted classes of games: convex, subadditive…

Thank you!

Questions?