Transferable-Utility Coalitional Game
A transferable-utility coalitional game consists of a finite player set $N$ and a characteristic function $$ v : 2^N \to \mathbb{R} $$ with $v(\varnothing)=0$. For each coalition $S \subseteq N$, the number $v(S)$ is the total payoff that members of $S$ can secure by coordinating among themselves.
For a payoff vector $x : N \to \mathbb{R}$, the payoff assigned to a coalition is $$ x(S) = \sum_{i \in S} x_i. $$ This derived coalition payoff is the quantity compared with $v(S)$ in imputation, core, and balancedness conditions.
Lean Status
The Lean structure CoalitionalGame N U is intentionally parameterized by
the utility type U; real-valued assumptions are introduced only for
theorems such as Bondareva-Shapley and Shapley-value results. The declaration
CoalitionalGame.coalitionPayoff supplies the finite sum
$\sum_{i \in S} x_i$.
References
- [MSZ Ch.16, Def 16.1] Maschler, Solan, Zamir, Game Theory.