Transcript zick2aamas2012.pptx

Overlapping Coalition
Formation: Charting
the Tractability
Frontier
Y. Zick, G. Chalkiadakis and
E. Elkind
(submitted to AAMAS 2012)
Motivation
Agents have limited
integer resources
Form Bilateral
Trade Contracts:
coalitions
The benefit of interaction
may be freely divided
Questions
What is the optimal
coalition structure?
How should profits be
divided?
Problem Complexity
The problem can be
modeled as a graph
Agents are nodes
There is an edge
between agents if
they can profit from
collaborating.
Goal: optimal allocation
v1,2(x,y) = log(x + y + 2)
w2 = 3
v2(x) = 0
w1 = 8
v1(x) = 5I5(x)
v1,2(x,y) = log(x + y + 2)
w2 = 3
v2(x) = 0
w1 = 8
vv11(x)
(5) == 5I
5 5(x)
v1,2(1,1) = 2
v1,2(1,1) = 2
v1,2(1,1) = 2
Optimal Coalition Structure
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 NPhardness (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 alone
ui,j(xi, xj) – the most two
agents can make by
working together
Ti(xi) – the most the subtree
rooted at i can make
OPT=max{u1(x1) + §u1,j(x1j,yj) + Tj(wj - yj)}
T3(x3)= max{u3(y3)+§u3,j(y3j,zj) + Tj(wj - zj)}
1
8
5
2
4
3
9
6
7
Stability
Optimal resource
allocation
Which profit
divisions ensure
group stability?
17,15
CS xx)
(CS,
5
Outcome
10,5
Is (CS, x) in
the core?
1,5
10,13
4,3
13,12
4,5
5,7
7
16,5
1,1
10,9
Deviation
“Coalitional game theory [...] considers a
n
game of n players as a set of possible 2 – 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
15
5
10,5
20
1,5
10,13
4,3
13,12
4,5
5,7
7
16,5
1,1
10,9
Stability
Arbitration functions:
agents may receive all or
some of the payoff from
unbroken/changed
agreements.
Behavior can be very
general.
Arbitration Functions
Others can react to deviation
either locally or globally.
Conservative – give nothing
Refined – give all from unhurt
coalitions
Optimistic – deviators absorb
the marginal damage of
deviation; get the difference.
17,15
8,15
5
Global
Local
10,5
1,5
8,10
10,13
4,3
13,12
4,5
5,7
7
16,5
1,1
10,9
Stability
Theorem: if there is an
efficient algorithm to
compute the most one
can get from global
arbitration functions,
then P = NP.
4
1
6
2
3
5
7
1 2 3 4 5 6 7
0
5
1
0
1
0
1
0
1
0
1 1
0 0
1
0
1 2 3 4 5 6 7
"
" "
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:
k
5(k+1)
Order of O(n W
) 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?