Concepts

Cooperative games

The characteristic function form of a cooperative game $G(v, N)$ consists of the following components:

1. A set $N = \{1, ..., n\}$ of players.

2. A characteristic function $v : 2^N \rightarrow \mathbb{R}$. Here $2^N$ is the set of all coalitions of $N$, where each coalition $C \subseteq N$ is a subset of players. In essence, the function $v$ assigns a value to each coalition.

An outcome

An outcome of a cooperative game $G(v, N)$ with transferable utility consists of the following:

1. A coalition partition $\rho = \{C_1, ..., C_k\}$ of $N$.

2. A payoff allocation $\psi = \{\psi_1, ..., \psi_n\} \in \mathbb{R}^n$.

From a high-level, $v(C)$ is shared with players in $C$ (i.e., transferable utility). Further, the payoff allocation must be feasible for each coalition $C \in \rho$. That is:

Cores

A core of a cooperative game with transferable utility is the set of all payoff allocations $\psi = \{\psi_1, ..., \psi_n\} \in \mathbb{R}^n$ (defined above) where:

1. The coalition partition is the grand coalition (i.e., $\rho = \{N\}$).

2. Collectively rational: $\sum_{i \in N} \psi_i = v(N)$.

3. Coalitionally rational: $\sum_{i \in C} \psi_i \geq v(C), \; \forall C \subseteq N$.

Intuitively, if a feasible allocation is in the core, then for each coalition $C \subset N$, there does not exist an alternative allocation such that under the feasibility constraint, at least one players in $C$ receives a higher payoff while all others are not worse off (i.e., an allocation in the core is pareto-dominant). As a result, at least one players in $C$ would not have the incentive to form the coalition.