Cooperative Game

A game where binding agreements among coalitions are allowed; analysis focuses on joint payoffs.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A TU (transferable-utility) game is a pair $(N,v)$ where $N$ is the set of players and $v:2^N\to\mathbb{R}$ with $v(\emptyset)=0$ ; $v(S)$ is the worth of coalition $S$ .

Solutions include the core, the Shapley value, the nucleolus, the bargaining set, and the kernel.

Intuition

Players can form coalitions and freely redistribute the resulting surplus.

The question shifts from "what will I do" to "how should the joint value be divided".

Worked example

Bankruptcy/airport cost-sharing games have rich core and Shapley-value solutions.

A 3-player majority game with $v(S)=1$ iff $|S|\ge 2$ has empty core but a clear Shapley value $(1/3,1/3,1/3)$ .

The math

Key properties: superadditivity $v(S\cup T)\ge v(S)+v(T)$ for disjoint $S,T$ ; convexity $v(S\cup T)+v(S\cap T)\ge v(S)+v(T)$ . Convex games always have a nonempty core containing the Shapley value.

Bondareva-Shapley theorem characterizes nonempty core via balanced collections of coalitions.

Where it is used

Cost allocation in utilities, revenue sharing in alliances, voting power, and collaborative ML.

Formalizes fairness notions like symmetry, efficiency, and additivity.

More in Game theory

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.