Algorithmic aspects of the core
in cooperative games over graphs
Vangelis Markakis
Athens University of Economics and Business
Dept. of Informatics
Joint work with:
Georgios Chalkiadakis, (University of Southampton)
Nicholas R. Jennings (University of Southampton)
Our focus

Cooperative games with restrictions on the set of
allowable coalitions


Restrictions defined via a graph structure
Computational Complexity issues

Can we compute a core element or decide if an element
belongs to the core with efficient algorithms?
Outline

Cooperative games with restricted cooperation

TU games on lines, trees, and rings
Algorithmic and NP-hardness results

Extensions to Partition Function Games

Conclusions / Future work
3
The standard model of
TU games

Set of players N = {1,…,n}

A TU (Transferable Utility) game is a pair

Value of coalition S:

Usually superadditivity is also assumed (we will
not insist on this here)
4
The standard model of
TU games

Let Π = {C1,…,Ck} be a partition of N

Ι(Π) = imputations of Π:

Core:
5
TU games defined on
graph structures

In some settings not all coalitions are allowed to form
[Aumann-Dreze ’74, Myerson ’77]





Physical limitations on communication,
Legal banishments,
Players with similar expertise need not participate in the same
coalition
…
Cooperation structures suggested so far:


Hierarchies (directed trees), general graphs (directed, undirected),
antimatroids [van den Brink ’08]
Applications: sensor networks, telecommunication networks,
various multi-agent and multi-robot systems
6
TU games defined on
graph structures
Let G be an undirected graph
 Feasible coalitions:
F(G) = connected coalitions, i.e., all sets S such that the
subgraph induced by S is connected
 Feasible partitions:
P(G) = partitions into feasible coalitions
 New version of core:

7
TU games defined on
graph structures
1
●
Example:
3
2
●
●
4
● 5● 6● 7●
●
●
8
-{1,2,3,5,8}F(G), {2,3,5,8} F(G)
9
●
10
- ({2,4,5,6,7}, {1,3,8,9,10,11}) P(G)
●11
-({1,2,4}, {5,6,7}, {1,3,8,9,10,11}) P(G)
8
Algorithmic issues
Computational problems:
CORE-NONEMPTINESS: Given a game on a graph
G, is the core of the game non-empty?
CORE-FIND: Given a game, find an element in the
core or output that the core is empty.
CORE-MEMBERSHIP: Given (Π,x), does it belong
to the core?
Polynomial time algorithms / impossibility results ?
9
The state of the art
Lines Superadd. trees General trees Superadd General
Rings
Rings
NONO(1)
EMPTINESS
O(1)
O(1)
P
?
FIND
P
P
NP-hard
P
?
MEMBER
SHIP
P
? (conjecture:
co-NPcomplete)
co-NPcomplete
P
P
Note: For many classes of general TU-games without restrictions,
the problems are known to be NP-hard or co-NP-hard
10
Algorithmic results
An algorithm for CORE-FIND on trees [Demange ’04]


Pick a vertex r as the root (think of the tree as oriented
from the root downwards to the leaves)
Step 1: We compute a guarantee level gi for every node i.


For leaves, gi is the reservation value.
For an internal node i,
where max taken over subtrees starting at i

Step 2: Start from root and go downwards. Pick the
coalition T (subtree) where the root achieves its guarantee
level. Continue downwards in same manner for remaining
nodes, not included in T.
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Algorithmic results
Note: For superadditive games on a line, player i
simply receives its marginal contribution of being
added to coalition {1,…,i-1}
Theorem [Demange ’04]: The outcome of the
algorithm belongs to the core.
12
Algorithmic results
(Worst case) running time analysis of the algorithm
Theorem:
i.
For superadditive games on a line the
complexity of the algorithm is O(n)
ii.
For non-superadditive games on a line, the
complexity is O(n2)
iii.
For superadditive games on a tree, the
complexity is O(n)
iv.
For non-superadditive games on a tree the
algorithm requires exponential time
13
Hardness results
Theorem:
For general games on trees, CORE-FIND
is NP-hard and CORE-MEMBERSHIP is co-NPcomplete, even when the tree is a star.
Proof: By reductions from the PARTITION problem:
Given n numbers a1,…,an is there a subset S such that
14
Hardness results–Proof Sketch
Given n numbers a1,…,an, we construct a tree with n
leaves
0
●
1 ● 2●
…
● n●
n-1
Finding an element in the core corresponds to finding the “right”
subset of the leaves that will join the root
15
Outline

Cooperative games with restricted cooperation

TU games on lines, trees, and rings
Algorithmic and NP-hardness results

Extensions to Partition Function Games

Conclusions / Future work
16
Partition Function Games




Externalities in cooperative games [Thrall, Lucas ’63]
The value of a coalition may depend on the partitioning of
the other players
V(S, Π): value of S in partition Π
How should a deviating coalition reason about the
behavior of the non-deviators?
17
Partition Function Games

Pessimistic approach: assume the rest of the
players will partition themselves so as to hurt you
the most

Optimistic approach: assume the rest of the
players will form a partition, most profitable for
you.

Other approaches have also been considered, each
resulting in a different notion of core [Koczy ’07]
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PFGs on graphs
Theorem: For PFGs on trees,
(i)
The pessimistic core is always non-empty.
(ii) There exist games where the optimistic
core is empty.
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Extensions and future work





Resolve open questions regarding the core
Polynomial time approximation algorithms
for the core?
Other cooperative solution concepts?
Other classes of graphs or cooperation
structures?
Bayesian Partition Function games
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Thank you!