The Core

Allocations no coalition can profitably abandon.

The idea

The core is the solution concept for coalitional games: the set of divisions of the total payoff that no group of players can improve on by breaking away.

A coalitional game with transferable payoff is a pair $(N, v)$: a set $N$ of $n$ players, and a function $v$ assigning each coalition $S \subseteq N$ a worth $v(S)$, the total payoff the members of $S$ can guarantee themselves with no help from anyone outside. Suppose all the players cooperate and produce $v(N)$. An allocation $x = (x_{1}, \ldots, x_{n})$ is feasible when it distributes exactly what was produced, $\sum_{i} x_{i} = v(N)$. Feasibility does not settle the division: a coalition receiving less than its own worth can secede and do better.

The core is the set of feasible allocations satisfying

$\sum_{i \in S} x_{i} \ge v(S) \quad \text{for every coalition } S \subseteq N.$

Every coalition then receives at least what it could produce by itself, so seceding gains it nothing. The deviating unit is a group, not a single player.

The core can be empty: when the coalitions' worths add up to more than $v(N)$, no feasible allocation meets every constraint. When it is nonempty, it is usually a region rather than a single point, because the definition selects the stable divisions, not a unique fair one. The figure shows the three-player picture: the feasible allocations form a triangle, each pairwise constraint $\sum_{i \in S} x_{i} \ge v(S)$ cuts a strip off one corner, and the core is the region that survives every cut.

Ways to work on it

• Walkthrough. Feasibility, coalition constraints, and testing whether an allocation is in the core.
• Practice. Check one coalition's core constraint for a proposed allocation.
• Hardest. Locate the core of a three-player game and find the binding coalition.

Not sure where to start? Take the ten-question placement test.