Theorem 4, pp. 15–16 — conic characterization of feasible utilities

Let $n\ge2$ and let $v_i(\theta)=c_i\theta+d_i$ be normalized linear valuations on $[0,1]$, with $c_i/2+d_i=1$ and $2\ge d_1\ge\cdots\ge d_n\ge0$. A utility vector belongs to the feasible utility set exactly when it is nonnegative and there exist real $z_i,w_i,s_i,t_i$ satisfying all the linear bounds and parabolic constraints of Theorem 4:

The complete auxiliary system includes $u_1\le z_1$, $u_i\le z_i+w_{i-1}$ for interior buyers, $u_n\le1+w_{n-1}$, $z_i\in[0,1]$, $w_i\in[-1,0]$, and $z_i+w_i\ge0$. This is a finite convex representation of every feasible utility vector, including vectors below the Pareto frontier.

Formalization Note The printed indices require $n\ge2$. Adjacent intercepts may tie. The paper's further bound on $s_i,t_i$ is a separate companion statement because it fails at a tie.