Optimization and
Stability in Games with
Restricted Interactions
Reshef Meir, Yair Zick and Jeffrey S. Rosenschein
CoopMAS 2012
Lecture content
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Coalitional (TU) games
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Restricted cooperation
The Cost of Stability
Main result: a bound on the CoS
Discussion
TU Games - Notations
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Agents: N = (1,…,n)
Coalition: S µ N
Characteristic function: v: 2I → R
A TU game is simple, if every coalition
either wins or loses, i.e. v: 2I → {0,1}
A TU game is monotone, if the value of a
coalition can only increase by adding more
agents to it
TU Games – notations (2)
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A TU game is superadditive (SA), if there
is positive synergy. That is,
v(S [ T) ≥ v(S) + v(T) for disjoint S,T.
• No need to consider coalition structures
• Results can be generalized
• Every game has a superadditive cover
Weighted Voting Games
(WVG)
A class of simple TU games
 Each agent has a weight wi2 R
 A game has a quota q 2 R
 G = [w1,w2,…,wn;q]
 A coalition S wins if Σi2S wi ¸q
Payoffs
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Agents may freely distribute profits.
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An imputation is a vector x = (x1,…,xn)
such that Σi2Nxi= v(N)
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Individual rationality: each agent gets
at least what she can make on her own:
xi ≥ v({i})
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The payoff of a coalition x(S) is the sum
of payments to its agents.
The Core
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The core is the set of all stable
imputations: for all S µ N we have
Σi2S xi ¸ v(S)
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May be empty in many games:
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No stable imputations
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Example: G = [2,2,3;4]
Computational questions:
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Is the core empty?
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Is the vector x in the core?
Restricted cooperation
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Some coalitions may be impossible or
unlikely due to practical reasons
an underlying communication network
(Myerson’77).
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agents are nodes.
A coalition can form
only if its agents are
connected.
6
1
3
2
5
4
9
11
10
7
8
12
Restricted cooperation example
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The coalition {2,9,10,12} is allowed
The coalition {3,6,7,8} is not allowed
1
3
2
5
6
4
9
11
10
7
8
12
Restricted cooperation
increases stability
Theorem [Demange’04] : If the underlying
communication network H is a tree, then the
core is non-empty.
Moreover, a core imputation
can be computed efficiently.
1
3
2
5
6
4
9
11
10
7
8
12
What if the core is empty?
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A solution: subsidies
Sometimes an external party is interested
in the stability of a specific outcome
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Willing to spend money to increase stability
External Payments
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Originally, we divided v(N) between the agents.
We increase the value of v(N), creating a
“superimputation”:
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Division of the incremented value v’(N)= α∙v(N)
Create a new game G(α)
v(N)
The Cost Of Stability (CoS)
(Bachrach et al., SAGT’09)
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Observation: With a big enough payment,
every game can be stabilized
α≤n
 The Cost of Stability (CoS) is the minimal
subsidy α that stabilizes the grand coalition
i.e. allows a non-empty core in G(α)
 Can also stabilize coalition structures
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Back to our example
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G = [2,2,3;4] (core is empty)
By distributing a total payoff of 1½ (rather
than 1), the core of G(1½) is non-empty.
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Thus CoS(G) ≤ 1½
Is this bound tight?
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x = (½, ½, ½) is a stable super-imputation.
A lower payment cannot stabilize the game
Thus CoS(G) = 1½
Conceptual Issues
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How do properties of the game affect the
CoS?
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Superadditivity, restricted cooperation, convexity…
Can we stabilize other outcomes?
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A particular coalition, coalition structures…
Computational Issues
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How hard is computing the optimal
coalitional structure?
How hard is computing the CoS?
How hard is checking whether a specific
super-imputation is stable?
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The answer depends on game representation
We assume oracle access to v(S)
Bounds on the CoS
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In the general case can be as high as n
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If G is superadditive, CoS(G)≤√n
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For example, the WVG [1,1,1,…,1; 1]
Easier to achieve cooperation
If G is superadditive and symmetric,
CoS(G) ≤ 2
Bachrach et al., SAGT’09
Meir et al., SAGT ‘10
CoS with
restricted cooperation
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Recall that by [Demange’04] : if H is a tree,
then the core is non-empty (i.e. CoS = 0).
Theorem: If H contains a single cycle,
then CoS(G) ≤ 2, and this is tight
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Sparse graphs  lower subsidies?
Sparse graphs  easier computation?
Meir et al., IJCAI ‘11
Graphs and tree-width
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Combinatorial measures to the
“cyclicity” of a graph:
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1
4
9
3
11
5
10
6
Degree
Path-width
Tree-width
…
Many NP-hard combinatorial
problems become easy when
the tree-width is bounded.
2
8
7
1,2,3
2,4
2,5,9
5,9,10
5,6,8
6,7,8
5,8,10
9,11
Bounding the CoS
Conjecture [MRM’11]: Let d be the
maximal degree in H, then CoS(G) ≤ d
There are games on a 3-dimensional grid
(d = 6) with unbounded CoS
Conjecture (fixed): Let k be the treewidth of H, then CoS(G) ≤ k
Main result
Theorem: Let G be a superadditive
game, then CoS(G) ≤ (TW(H) + 1) ∙ log(n)
Also, a stable payoff vector can be found efficiently
Proof
a b x y z …
a d l m
a b ef
a b c d
b c k
c d i j
Proof
a b x y z …
a d l m
a b ef
a b c d
b c k
(k+1)v(N)
c d i j
Proof
a b x y z …
a d l m
a b ef
a b c d
b c k
(k+1)v(N)
c d i j
Proof
a b x y z …
a d l m
a b ef
f xz
a b c d
b c k
c d i j
ij…
(k+1)v(N) + (k+1)(v(S1) + v(S2) + …) ≤ (k+1)v(N) + (k+1)v(N) + …
Proof
a b x y z …
a d l m
a b ef
f xz
a b c d
b c k
c d i j
We pay at most (k+1)v(N) at each iteration
ij…
Proof
a b x y z …
a d l m
a b ef
f xz
a b c d
b c k
c d i j
We repeat at most log(|T |) ≤ log(n) times
ij…
Discussion
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The CoS depends on the tree-width of the
underlying graph
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New results…
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Bounded tree-width does not facilitate
computations (e.g. Greco et al.’11)
For more information:
http://www.huji.ac.il/~reshef24