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Algorithmic aspects of the core in cooperative games over

graphs

Vangelis MarkakisVangelis Markakis

Athens University of Economics and Athens University of Economics and BusinessBusiness

Dept. of InformaticsDept. of Informatics

Joint work with:Joint work with:

Georgios Chalkiadakis, (University of Georgios Chalkiadakis, (University of Southampton) Southampton)

Nicholas R. Jennings (University of Southampton)Nicholas R. Jennings (University of Southampton)

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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?

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Outline

Cooperative games with restricted cooperation

TU games on lines, trees, and rings A

lgorithmic and NP-hardness results

Extensions to Partition Function Games

Conclusions / Future work

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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)

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The standard model of TU games

Let Π = {C1,…,Ck} be a partition of N

Ι(Π) = imputations of Π:

Core:

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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

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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:

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TU games defined on graph structures

●

●

● ●

● ● ● ● ●

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1

3

54 6 7 8 9

10 11● ● -{1,2,3,5,8}F(G), {2,3,5,8} F(G)

- ({2,4,5,6,7}, {1,3,8,9,10,11}) P(G)

-({1,2,4}, {5,6,7}, {1,3,8,9,10,11}) P(G)

Example:

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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 ?

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The state of the art

Lines Superadd. trees General trees Superadd Rings

General Rings

NON-EMPTINESS

O(1) O(1) O(1) P ?

FIND P P NP-hard P ?

MEMBER

SHIPP ? (conjecture:

co-NP- complete)

co-NP-complete

P P

Note: For many classes of general TU-games without restrictions, the problems are known to be NP-hard or co-NP-hard

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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.

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Algorithmic results

(Worst case) running time analysis of the algorithmTheorem: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

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Hardness results

Theorem:

For general games on trees, CORE-FIND

is NP-hard and CORE-MEMBERSHIP is co-NP-

complete, 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

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Hardness results–Proof Sketch

Given n numbers a1,…,an, we construct a tree with n leaves

●

●

● ● ●

0

21 n-1 n

Finding an element in the core corresponds to finding the “right” subset of the leaves that will join the root

…

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Outline

Cooperative games with restricted cooperation

TU games on lines, trees, and rings A

lgorithmic and NP-hardness results

Extensions to Partition Function Games

Conclusions / Future work

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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?

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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!