the Tractability Frontier Y Zick G Chalkiadakis and E Elkind submitted to AAMAS 2012 Motivation Agents have limited integer resources The benefit of interaction may be freely divided ID: 424689
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Slide1
Overlapping Coalition Formation: Charting the Tractability Frontier
Y.
Zick
, G.
Chalkiadakis
and E.
Elkind
(submitted to AAMAS 2012)Slide2
MotivationAgents have limited integer resources
The
benefit of interaction
may be freely divided
Form Bilateral Trade
Contracts
:
coalitions
Slide3
QuestionsWhat is the optimal coalition structure
?
How should profits be divided?Slide4
Problem ComplexityAgents are
nodes
The problem can be modeled as a
graph
There is an
edge
between agents if they can profit from collaborating.
Goal: optimal allocationSlide5
v1(
x
) = 5
I
5(
x)
v1,2
(x,y
) = log
(x + y
+ 2)
v
2
(
x) = 0 w
1 = 8
w2
= 3Slide6
v1,2(
x
,
y
) = log
(x + y
+ 2)
v
2
(
x) = 0
w1
= 8w2
= 3
v
1
(x
) = 5I5
(x)
v1(5) = 5
v
1,2
(1,1) = 2
v
1,2
(1,1) = 2
v
1,2
(1,1) = 2Slide7
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 Slide8
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 Slide9
Computational complexity even when weights are at most 3, complex interactions cause NP-hardness (the X3C problem).
Optimal Coalition Structure Slide10
We assume that:Weights are polynomially boundedInteractions are simple.
Optimal Coalition Structure Slide11
Suppose that the interaction graph is a tree
Optimal Coalition Structure Slide12
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 Slide13
We set:ui(
x
i
)
– 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
Optimal Coalition Structure Slide14
1
8
7
6
4
5
3
2
9
OPT=max{
u
1
(
x
1
) +
§
u
1
,
j
(
x
1j
,
y
j
) +
T
j
(wj
- yj
)}
T
3
(x3)= max{u
3(y3)+
§u3,j
(y3j,zj
) + Tj
(wj
- zj
)} Slide15
Stability
Optimal
resource allocation
Which profit divisions ensure
group stability
?Slide16
17,15
10,5
1,5
4,3
10,13
5
5,7
16,5
7
1,1
10,9
4,5
13,12
(
CS
,
x
)
CS
x
Outcome
Is
(
CS
,
x
)
in the
core
?Slide17
Deviation“Coalitional game theory [...] considers a game of
n
players as a set of possible
2
n – 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.Slide18
17,15
10,5
1,5
4,3
10,13
5
5,7
16,5
7
1,1
10,9
4,5
13,12
20
15Slide19
StabilityArbitration functions:
agents may receive all or some of the payoff from unbroken/changed agreements.
Behavior can be very general. Slide20
Arbitration FunctionsOthers can react to deviation either
locally
or
globally
.Conservative – give nothing
Refined – give all from unhurt coalitionsOptimistic
– deviators absorb the marginal damage of deviation; get the difference. Slide21
17,15
10,5
1,5
4,3
10,13
5
5,7
16,5
7
1,1
10,9
4,5
13,12
8,15
Global
Local
8,10Slide22
StabilityTheorem:
if there is an efficient algorithm to compute the most one can get from
global
arbitration functions, then
P = NP. Slide23
1
7
6
5
4
3
2
1
2
3
4
5
6
7
1
2
3
4
5
6
7
0
5
1
0
1
0
1
0
1
0
1
0
1
0
1
0
"
"
"Slide24
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) timeSlide25
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? Slide26
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. Slide27
StabilityGiven an outcome
(
CS
,
x), the
excess of a set S
is the difference between the payoff to S
under (CS,
x), denoted
pS(CS
, x)
and A
*(S,CS,
x)
e
(S,CS,
x) = A
*(S,CS
, x) - pS
(CS, x
) Slide28
StabilityWe set:
E
i
(
x)
– the maximal excess of a set containing i, assuming i
invests x in working with that set.Slide29
E
1
(
x
) = max{
u
1
(
a
1
) +
§
i
2
{2,3,4}
(
u
1,
i
(
b
1,i
, y
i) + Ei(
wi – y
i))} – p1
1
2
3
4
56
7
8
9Slide30Slide31
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 Slide32
RecapOptimization/Stability:
Hard in general due to
Weights
Complex interactionSlide33
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.Slide34
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. Slide35
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.Slide36
Poly-time, but…Complexity is still high:
Order of
O
(n
kW5(k+1)
) for computing optimal allocation in a graph with treewidth
kCan probably do better if valuations are known. Slide37
Future WorkDeterministic, Exact:
randomized/ approximation algorithms?
Restricted classes of games:
convex,
subadditive…Slide38
Thank you!Questions?