Games A Lesson in Multiagent System Based on Jose Vidals book Fundamentals of Multiagent Systems Henry Hexmoor SIUC 2 5 3 4 1 6 A B C Coalition game characteristic from game ID: 187742
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Coalition Games:A Lesson in Multiagent SystemBased on Jose Vidal’s bookFundamentals of Multiagent Systems
Henry HexmoorSIUC
2
5
3
4
1
6
A
B
CSlide2
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 agentsv(S) R is defined for every S that is a subset of A.Slide3
Transferable UtilityPlayers 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
SV(s)(1)i(1 2)ii(1 3)iii(2 3)iv ( 1 2 3 )vSlide4
Feasibility property Nothing is lost by merging coalitions is not feasible is feasible
S
V(S)
(1)2(2)2(3)4(1 3)7( 2 3 )8( 1 2 3 )9Slide5
Super Additive property Nothing is lost by merging coalitions Slide6
StabilityFeasibility 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.Slide7
The CoreAn Outcome is in the core if outcome > coalition payoff It is stable Slide8
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)6Slide9
The Core: Example 2: An empty coreSV(S)
(1)0(2)0
(3)
0(1 2)10
( 1 3)10(2 3 )10(1 2 3)10Slide10
Core: Example 3
SV(S)()0
(1)
1(2)3
(1 2)6Slide11
The Shapley Value (Fairness)Given an ordering of the agents in I, we denote the set of agents that appear before i inThe Shapley value is defined as the marginal contribution of an agent to its set of predecessors, averaged on all permutationsSlide12
Shapley value Example S
V(S)()0
(1)
1(2)3(1 2)
6F({1, 2}, 1) = ½ · (v(1) − v() + v(21) − v(2))=1/2· (1 − 0 + 6 − 3) = 2F({1, 2}, 2) = ½ · (v(12) − v(1) + v(2) − v())=1/2· (6-1+3 -0) = 4Slide13
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
)Slide14
ReferencesShapley (1953,1967,1971)Aumann &
Dreze (1974)