Characteristic Function

Game Theory · Axiom Academy

LESSON Characteristic Function Understanding how coalitions create value in cooperative games 1. The Characteristic Function v The characteristic function maps each possible coalition (subset of players) to a real number representing the value that coalition can guarantee. 2 N denotes the power set (all possible subsets) of N v(S) is the value coalition S can guarantee for itself v(∅) = 0 by convention (the empty coalition has no value) 2. Calculating Coalition Values For each coalition S, v(S) represents what the members of S can achieve by working together, regardless of how the remaining players behave. v( A ) = 0 (one player cannot win alone) v( A,B ) = 100 (two players can win) v( A,B,C ) = 100 (all three together still just win once) The characteristic function must specify v(S) for all 2 n possible coalitions, including the empty set and the grand coalition of all players. A characteristic function is superadditive if merging two disjoint coalitions never decreases their combined value. This is a natural assumption in cooperative games. Interpretation: There is no disadvantage to cooperation. Two groups working together can achieve at least as much as they could working separately. This property justifies why players might want to form coalitions in the first place. 4. Constructing a Characteristic Function To fully specify a cooperative game, we need to determine v(S) for every coalition. This involves analyzing what each group can guarantee through optimal play.

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