Some subgroups of a population can win an election. A winning coalition gets 1, losers get 0. (Or, a variant, if $S$ wins the coalition gets $|S|$.)
Individual $i$ in coalition $S$ gets utility $u_i(m,g)$ from money $m$ and local public good $g$. She can choose to contribute $c_i\leq\omega_i$ to the production of public good. Public goods comes from the contributions, $g=f(\sum_{i\in S}c_i)$. The set of all feasible allocations $((c_i)_{i\in S},g)$ determines the (vector-valued) characteristic functions for coalition $S$. If utilities are quasi-linear in money, the characteristic function is the maximum achievable total utility.
Individual $i$ has endowment $\omega_i$. A coalition $S$ can achieve any allocation $a$, such that $\sum_{j\in S}a_j\leq\sum_{j\in S}\omega_j$.