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
(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.
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)}
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?
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