Game Theory
Cooperative Games and the Shapley Value
Cooperative Game Model
In 2019, Apple, Google, Amazon, and Meta negotiated a patent revenue split worth $4.7 billion across four corporations. Without a formal allocation rule, talks lasted three years. Cooperative game theory gives an axiomatically justified answer in seconds.
If v({1,2}) = 50 and v({1}) + v({2}) = 60, the game is:
Subadditivity means coalition is not beneficial: joint payoff is less than the sum of individual payoffs.
The Shapley Value
Lloyd Shapley proved in 1953 that exactly one allocation rule satisfies four reasonable axioms -- efficiency, symmetry, null player, and additivity. That rule is the Shapley value. Today it underpins SHAP (SHapley Additive exPlanations), the de facto standard for ML interpretability at Google, Microsoft, and Amazon.
Which Shapley axiom guarantees that the sum of all phi_i equals v(N)?
Efficiency is the key axiom: the grand coalition payoff is split without remainder among all participants.