computational problems associated with a solution concept why we need the compact representation of...
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Compact Representations of Coalitional Games
Jianing Yu
Computational problems associated with a solution concept
Why we need the compact representation of coalitional games
Compactly-represented coalitional games Weighted graph game Marginal contribution nets
Conclusion
Outline
Introduction WG Game MC-nets Conclusion
Computational problems
There are several problems associated with a solution concept.
Is the solution concept nonempty?Given a payoff vector, does it belong to the solution concept?……
Study the computational complexity of the problems associated with each solution concept.Criteria for judging whether a solution concept is appropriate
FairnessStability…….The computational complexity of the problems associated with it should not be too great
Introduction WG Game MC-nets Conclusion
Why we need the compact representation?
(1) (2) (3) (4)
(1,2) (1,3) (1,4) (2,3) (2,4) (3,4)
(1,2,3) (1,2,4) (1,3,4) (2,3,4)
(1,2,3,4)
v v v v
v v v v v v
v v v v
v
Example: Given a coalitional game (N=4, v), if we want to compute the Shapley value, we need
How do we represent the input when computing a solution concept?Straightforward representation by enumeration requires exponential spaceComplexity is measured in term of the input sizeWe need compact representation so that the input size is a polynomial in the number of agentsIn general, the more succinct a representation is, the harder it is to compute
Introduction WG Game MC-nets Conclusion
Weighted Graph Game
Definition: Let denote an undirected weighted graph, where is the set of vertices and is the set of edge weights; denote the weight of the edge between the vertices and as . This graph defines a weighted graph game (WGG), where the coalitional weighted graph game is constructed game as follows:
d
( , )V W VV VW R
i j ijw
N V
,( ) iji j Sv S w
Example:
1
34
212w
24w14w
23w13w
12 13 14 23 24( )v N w w w w w
(1) 0v
13 14({1,3,4})v w w
Introduction WG Game MC-nets Conclusion
Example - Revenue Sharing game
Question: How to divide the total revenues among the cities?
1
34
23
9
6 45
Nodes: citiesEdges: toll highwaysWeights: highway’s toll revenues
( ) 3 5 6 4 9 27v N ({1,2,4}) 3 6 9 18v
Fun game:Who can answer the following question the most quickly?Question:What is the Shapley value of city one in this game?
Introduction WG Game MC-nets Conclusion
Shapley value
Theorem: The Shapley value of the coalitional game induced by a weighted graph game is
1( , )
2i ijj i
N v w
( , )V W
( , )N v
1
34
212w
24w14w
23w13w1 12 13 14
1( , ) ( )
2N v w w w
Example:
Complexity: The Shapley value can be computed in time2( )O n
Introduction WG Game MC-nets Conclusion
Shapley value
Theorem:1
( , )2i ijj i
N v w
Proof:
1
3 4
212w
24w14w
23w13w
1
3
13w
4
14w
1
212w1
24w
2
4 3
223w
13G 14G 12G 24G 23GStep1: Symmetry axiom & Dummy axiom 1 13 3 13 13
1( ) ( )
2G G w 1 14 4 14 14
1( ) ( )
2G G w 1 12 2 12 12
1( ) ( )
2G G w
Step2: Additivity axiom
1 1 13 1 14 1 12 1 24 1 23
13 14 12
( ) ( ) ( ) ( ) ( ) ( )
1( )2
G G G G G G
w w w
Introduction WG Game MC-nets Conclusion
WGG with nonnegative weights is convex
Proposition: If all the weights are nonnegative then the game is convex.
1
3 4
212w
24w14w
23w13w
Convex: For all ,S T N ( ) ( ) ( ) ( )v S T v S v T v S T
Proof: ( ) ( ) ( ) ( )v S v T v S T v S T
1
34
14w13w
1
4
212w
24w
1
4
14w
1
34
14w13w
3
223w
24w
212w1
4
14w
{1,3,4} {1,2,4} {1,2,3,4} {1,4}S v S T S T
The core is nonempty if all the weights are nonnegative
Introduction WG Game MC-nets Conclusion
Is a given payoff vector in the core?
Proposition: If all the weights are nonnegative then membership of a payoff vector in the core can be tested in polynomial time.
How to test?Construct a flow network and calculate the maximum flow.
Introduction WG Game MC-nets Conclusion
How to test?Construct a flow network and calculate the maximum flow.
1
3 4
212w
24w14w
23w13w
12w
24w
14w
23w
13w
1
3
4
2
E12
E13
E14
E23
E24
S T
1x
2x
3x
4x
W is the weight. X={X1, x2, x3, x4} is the payoff vector.
Is a given payoff vector in the core?
Introduction WG Game MC-nets Conclusion
The value of the maximum flow is if and only if is in the coreIf part: If max-flow = , is in the core.Only if part: If max-flow < , is not in the core.
1
3 4
212w
24w14w
23w13w
12w
24w
14w
23w
13w
1
3
4
2
E12
E13
E14
E23
E24
S T
1x
2x
3x
4x
A max-flow problem can be solved in polynomial time.
( )v N x
( )v N x( )v N x
Is a given payoff vector in the core?
Introduction WG Game MC-nets Conclusion
Marginal contribution nets – a logical approach
Definition: An MC-net consists of a set of rules where the valuation function is given by
where evaluates to 1 if the Boolean formula evaluates to true for the truth assignment and 0 otherwise.
1 1{( , ),..., ( , )}k kp w p w
1( ) ( )
k Si ii
v S p e w
( )Sip e ip
Se
Example: {( ,5),( ,2),( ,4),( , 2)}a b b c b c
( ) 0 ( ) 0 ( ) 2 2 0
({ , }) 4 ({ , }) 5 2 2 5
({ , }) 4 2 6
v v a v b
v a c v a b
v b c
Introduction WG Game MC-nets Conclusion
Marginal contribution nets
Theorem: MC-nets can represent any game when negative literals are allowed in the patterns, or when the weights can be negative.
1 2 12 1 3 13 1 4 14
2 3 23 2 4 24
{( , ),( , ),( , ),
( , ),( , )}
v v w v v w v v w
v v w v v w
Proposition: MC-nets generalize Weighted Graph game representation.
1
34
212w
24w14w
23w13w
Introduction WG Game MC-nets Conclusion
Marginal contribution nets
Theorem: Given a TU game represented by an MC-net limited to conjunctive patterns, the Shapley value can be computed in time linear in the size of the input.
Proposition: Determining whether the core is empty or checking whether a payoff vector lies in the core are coNP-hard.
Introduction WG Game MC-nets Conclusion
Compact representation of coalitional games is necessary when the number of agents is large.representation for specific games (not complete):
weighted graph game.general representations, that may require less space in
some cases: Marginal contribution nets.
Conclusion
References Deng, Xiaotie, and Christos H. Papadimitriou. "On the
complexity of cooperative solution concepts." Mathematics of Operations Research 19.2 (1994): 257-266.
Ieong, Samuel, and Yoav Shoham. "Marginal contribution nets: a compact representation scheme for coalitional games." Proceedings of the 6th ACM conference on Electronic commerce. ACM, 2005.
Stéphane Airiau. “Cooperative Games Lecture 9: Representation and Complexitity issues”
Multiagent systems: Algorithmic, game-theoretic, and logical foundations, Y Shoham, K Leyton-Brown, Cambridge University Press, 2009
“Maximum flow problem”, Wikipedia,http://en.wikipedia.org/wiki/Maximum_flow_problem
Thank you!
Introduction WG Game MC-nets Conclusion
Background -- Flow network
A
S
E
D
B
C T
5/55/5
4/4
2/60/34/4
6/8
5/5
10/10
1/2
4/5
0/110: flow10: capacity
Each edge has a capacity and each edge receives a flow A maximum feasible flow through the flow network A cut divides the vertices of a graph into two disjoint subsets
The capacity of the cut is 5+6+10 = 21 A min-cut is a cut with the minimum capacity
Max-flow = Min-cut A max-flow problem can be solved in polynomial time.
Introduction WG Game MC-nets Conclusion
1
3 4
212w
24w14w
23w13wFor subgame S={1,2}
If part: If max-flow = , is in the core.( )v N x
12w
24w
14w
23w
13w
1
3
4
2
E12
E13
E14
E23
E24
S T
1x
2x
3x
4x
1 2 12x x w
Is a given payoff vector in the core?
Introduction WG Game MC-nets Conclusion
1
3 4
212w
24w14w
23w13wThe capacity of this cut is
Only if part: If max-flow < , is not in the core.( )v N x
12 3w
24 2w
14 2w
23 2w
13 2w
1
3
4
2
E12
E13
E14
E23
E24
S T
1 1x
2 1x
3 4x
4 5x
1 2 12
1 2 12
( ) ( ) 10 ( ) 11C cut x x v N w v N
x x w
Is a given payoff vector in the core?