Transferable Utility

EC487_Advanced_Microeconomics

Transferable Utility is a special case in the Cooperative Game Theory.

See some basic setting from Cooperative Game Theory:

The function $V(\cdot )$ assigns to each coalition a set of possible utility vectors:

$$V:\mathcal{P}(\{1,\dots ,N\})⟹{\mathbb{R}}^{|S|}$$

If $(u^j{)}_{j\in S}\in V(S)$, it means the members in coalition $S$ can use some matching system to reach to this utility profile.

here we add some definitions:

A coalitional game with transferable payoffs consists of:

• a finite set $N$ (the set of players)
• a function $v$ that associates with every nonempty subset $S$ of $N$ (a coalition) a real number $v(S)$ (the worth of coalition).

$$V(S)=\left\{(u^j{)}_{j\in S}:\sum _{j\in S}{u}^j=v(S)\right\}$$

Remark: Here, $v(S)$ is a scalar, it represents the worth of the coalition. But $V(S)$ could still be a set, it represents all the utility profiles that sum up to $v(S)$.

For example, if $S=\{1,2\}$, and $v(S)=10$, then

$$V(S)=\{(u_1,u_2):u_1+u_2=10\}$$