Shapley Value Axioms

Game Theory · Axiom Academy

Unit 6 - Cooperative Game Theory The axiom: The sum of all payoffs equals the total value created by the grand coalition. What it means: All the value created should be distributed among the players - nothing is wasted, and nothing appears from nowhere. If the grand coalition creates value v(N), then the payoffs must sum exactly to v(N). The axiom: If two players contribute equally to every coalition, they should receive equal payoffs. What it means: Players i and j are symmetric if v(S ∪ i ) = v(S ∪ j ) for every coalition S not containing i or j. If they're symmetric, then they must receive the same payoff: φᵢ = φⱼ. The axiom: If a player contributes nothing to every coalition, they should receive zero payoff. What it means: Player i is a null player if v(S ∪ i ) = v(S) for every coalition S. Adding them to any group doesn't change that group's value. Such a player should receive φᵢ = 0. The axiom: If we combine two separate games, payoffs should add up accordingly. What it means: Suppose we have two separate cooperative games v and w on the same player set N. If we play both games simultaneously (creating a combined game where (v + w)(S) = v(S) + w(S)), then each player's payoff should be the sum of their payoffs from the individual games: φᵢ(v + w) = φᵢ(v) + φᵢ(w). 5. Uniqueness and Why Axioms Matter This is a remarkable result! It tells us that if we agree on these four basic principles of fairness, we are forced to use the Shapley value - there's no other option.

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