Coalition Games: A Lesson in Multiagent System

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Coalition Games: A Lesson in Multiagent System Based on Jose Vidals book Fundamentals of Multiagent Systems Henry Hexmoor SIUC Coalition game characteristic from game Agents vector of utilities one for each agent payoffs for teaming V(s)

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Coalition Games:A Lesson in Multiagent SystemBased on Jose Vidal’s bookFundamentals of Multiagent Systems Henry Hexmoor
SIUC

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.

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

Feasibility property Nothing is lost by merging coalitions

is not feasible

is feasible

Super Additive property Nothing is lost by merging coalitions

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
.

The Core An Outcome is in the core if
outcome > coalition payoff

It is stable

Core: Example 1 is in the core
is not in the core
is not in the core

The Core: Example 2: An empty core

Core: Example 3

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 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)

References Shapley (1953,1967,1971)
Aumann & Dreze (1974)

Coalition Games: A Lesson in Multiagent System

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