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 Agents ID: 1027386
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1. Coalition Games:A Lesson in Multiagent SystemBased on Jose Vidal’s bookFundamentals of Multiagent SystemsHenry HexmoorSIUC253416ABC
2. 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.
3. 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 structureSV(s)(1)i(1 2)ii(1 3)iii(2 3)iv ( 1 2 3 )v
4. Feasibility property Nothing is lost by merging coalitions is not feasible is feasible SV(S)(1)2(2)2(3)4(1 3)7( 2 3 )8( 1 2 3 )9
5. Super Additive property Nothing is lost by merging coalitions
6. 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.
7. The CoreAn Outcome is in the core if outcome > coalition payoff It is stable
8. Core: Example 1 is in the core is not in the core is not in the core SV(S)(1)1(2)2(3)2(1 2)4( 1 3)3(2 3 )4(1 2 3)6
9. 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)10
10. Core: Example 3 SV(S)()0(1)1(2)3(1 2)6
11. 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 permutations
12. Shapley value Example SV(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) = 4
13. 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)
14. ReferencesShapley (1953,1967,1971)Aumann & Dreze (1974)