Proposition 2, p. 17 — (1 − 1/e)σ⁽¹⁾ < 1 − e^{−σ⁽¹⁾} < 1 − e^{−α⁽¹⁾}, 1/(1 + ln(r⁽ᵐ⁾/r⁽¹⁾)) < σ⁽¹⁾, and 1 − e^{−α} < 1 − e^{−α⁽¹⁾}

Let $m \ge 2$ and let $0 < r^{(1)} < \dots < r^{(m)}$ be a price set. Let $\alpha^{(1)}, \dots, \alpha^{(m)}$ be its booking limits (positive, summing to $1$, satisfying (7)) and $\sigma^{(1)}, \dots, \sigma^{(m)}$ its Ball–Queyranne limits (positive, summing to $1$, satisfying (8)). Then

and, for the real number $\alpha$ solving $1 - e^{-\alpha} = (1 - \alpha)/\ln(r^{(m)}/r^{(1)})$,

The proposition places the paper's tight ratio $F(\mathcal P) = 1 - e^{-\alpha^{(1)}}$ among known ratios: $\sigma^{(1)}$ is the tight ratio for a single item with price set $\mathcal P$ and $1 - 1/e$ the tight ratio for many single-price items, and (11) shows that $F(\mathcal P)$ strictly exceeds their naive combination $(1 - 1/e)\sigma^{(1)}$. Inequalities (12) and (13) say that restricting the price continuum $[r^{(1)}, r^{(m)}]$ to the discrete set $\mathcal P$ strictly raises the tight ratio, for one item and for many items respectively.

Formalization Note The booking limits enter as arbitrary families satisfying (7) resp. (8) with positivity and sum $1$; by Proposition 1 these are the paper's $\alpha^{(j)}$, $\sigma^{(j)}$. The constant $1 - 1/e$ is written $1 - e^{-1}$. Clause (13) is stated for every real root $a$ of $1 - e^{-a} = (1-a)/\ln(r^{(m)}/r^{(1)})$; since this equation has exactly one real root for $m \ge 2$ (milestone (34)), this is the paper's statement about "the unique solution", and it is not vacuous. The hypothesis $m \ge 2$ is the paper's: for $m = 1$ one has $\sigma^{(1)} = \alpha^{(1)} = 1$ and $\ln(r^{(m)}/r^{(1)}) = 0$.