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Understanding Coalition Games: Key Concepts from Multiagent Systems

This text explores the fundamental concepts of coalition games based on the insights from Jose Vidal's work on Multiagent Systems. It covers critical elements such as characteristic functions, transferable utility, and coalition structure feasibility. We delve into core stability, the properties of super-additivity, and the significance of the Shapley value in determining fair resource distribution among agents. Additionally, we discuss concepts like the nucleolus and excess in coalition formation, providing a comprehensive understanding for researchers and practitioners in the field.

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Understanding Coalition Games: Key Concepts from Multiagent Systems

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  1. A 1 3 C 2 B 4 5 6 Coalition Games:A Lesson in Multiagent SystemBased on Jose Vidal’s bookFundamentals of Multiagent Systems Henry Hexmoor SIUC

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

  3. 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

  4. Feasibility property • Nothing is lost by merging coalitions is not feasible is feasible

  5. Super Additive property • Nothing is lost by merging coalitions

  6. 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 .

  7. The Core • An 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

  9. The Core: Example 2: An empty core

  10. Core: Example 3

  11. 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

  12. Shapley value Example 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

  13. Relaxing the Core… • The core is often empty… • Minimizing the total temptation felt by the agentscalled the nucleolus. • Acoalition 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. References Shapley (1953,1967,1971) Aumann& Dreze (1974)

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