Coalition Games:A Lesson in Multiagent System

Based on Jose Vidal’s bookFundamentals of Multiagent Systems

Henry Hexmoor

SIUC

2

5

3

4

1

6

A

BC

Coalition game _ characteristic from game

Agents

vector of utilities one for each agent

payoffs for teaming

V(s) – characteristic function / Value function

s – set of agents• v(S) R is defined for every S that is a subset of A.

|}|,...,2,1{ AA

},...,{ ||1 Auuu

...,...},{)( yxsV

Transferable Utility

• Players can exchange utilities in a team

• is feasible if there exists a set of coalitions

T = • Where

Are there a disjoint set of coalitions that add up to

T = Coalition structure

S V(s)

(1) i

(1 2) ii

(1 3) iii

(2 3) iv

( 1 2 3 ) v

uuss ,...,1

Ai

iTsTs

usuAS )(&

u

Feasibility property

• Nothing is lost by merging coalitions

is not feasible

is feasible

)(AVuAi

i

S V(S)

(1) 2

(2) 2

(3) 4

(1 3) 7

( 2 3 ) 8

( 1 2 3 ) 9

}5,5,5{u

}3,4,2{u

Super Additive property

• Nothing is lost by merging coalitions

)(AVuAi

i

Stability

• Feasibility does not imply stability. Defections are possible.

• is stable if x subset of agents gets paid more, as a whole, than they get paid in

.

uu

The Core

• An Outcome is in the core if 1. outcome > coalition payoff

2. It is stable

u

)(sVuSi

iAs

Core: Example 1

is in the core

is not in the core

is not in the core

S V(S)

(1) 1

(2) 2

(3) 2

(1 2) 4

( 1 3) 3

(2 3 ) 4

(1 2 3) 6

}2,2,2{u

}3,2,2{u

}2,2,1{u

The Core: Example 2: An empty core

S V(S)

(1) 0

(2) 0

(3) 0

(1 2) 10

( 1 3) 10

(2 3 ) 10

(1 2 3) 10

Core: Example 3

S V(S)

() 0

(1) 1

(2) 3

(1 2) 6

The Shapley Value (Fairness)

• Given an ordering of the agents in I, we denote the set of agents that appear before i in

• The Shapley value is defined as the marginal contribution of an agent to its set of predecessors, averaged on all permutations

Shapley value Example

S V(S)

() 0

(1) 1

(2) 3

(1 2) 6

F({1, 2}, 1) = ½ · (v(1) − v() + v(21) − v(2))=1/2· (1 − 0 + 6 − 3) = 2

F({1, 2}, 2) = ½ · (v(12) − v(1) + v(2) − v())=1/2· (6-1+3 -0) = 4

Relaxing the Core…

• The core is often empty…• Minimizing the total temptation felt by the agents called the nucleolus.

• A coalition S is more tempting the higher its value is over what the agents gets in . This is known as the excess.

• A coalition’s excess e(S) is v(S) - Σi in Su(i)u

References

1. Shapley (1953,1967,1971)

2. Aumann & Dreze (1974)