Computational problems associated with a solution concept Why we need the compact representation of coalitional games Compactly-represented coalitional

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


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


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


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?


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


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


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


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.


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.



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



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


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


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.


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!


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.


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



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


Documents

Computational problems associated with a solution concept Why we need the compact representation of coalitional games Compactly-represented coalitional