§5.3, pp. 20–21 — for k ≤ 1/2 free information raises the manufacturer's profit from Π^{NX_j}_M to Π^{SX_j}_M

Let $1/3<k\le1/2$, $0\le\gamma_C<1$, $t>0$, $\sigma^2>0$, and let $\Pi^{X_iX_j}_{M_i,C}$ be manufacturer $i$'s equilibrium ex-ante profit under Cournot competition. Then for each $X_j\in\{S,N\}$,

When the retailer shares for free, the manufacturer strictly gains, so sharing is a strictly dominant choice for each manufacturer in the stage-one game when $k\le1/2$.

Formalization Note The page states the case $k\le1/2$, and the item keeps that hypothesis. The proof of Proposition 7 (p. 31) prints "$\Pi^{NS}_{M_i,C}>\Pi^{NN}_{M_i,C}$" at this point; those two profits are equal, and the inequality needed there is this one.