Proposition 1, p. 10 — the impulse selects the introspective equilibrium under In/Out demand

Suppose a market's inverse demand curve is In/Out, with thresholds $p_{\min}=P(Q_L)$ and $p_{\max}=P(Q_H)$. Start the level-$k$ recursion $I_k=D(p,\cdot)^{\circ k}(I_0)$ from an impulse $I_0\in[0,1]$.

For $p_{\min}\le p\le p_{\max}$, the out, middle, and in branch equilibria all exist, and the introspective equilibrium selected by the impulse is

For $p>p_{\max}$ or $p<p_{\min}$ there is exactly one feasible equilibrium quantity $q$ satisfying $D(p,q)=q$, and the level-$k$ sequence converges to $q$ for every allowed impulse.

This is the paper's selection rule for continuation-game equilibrium quantities.

Formalization Note At a threshold price two branch quantities coincide. The existence clause prevents empty branch predicates from making the selection rule vacuous. Outside the threshold interval the equilibrium is defined by $D(p,q)=q$, so the statement covers saturated equilibria where $P(q)=p$ has no solution. The market's strict demand signs at inverse-demand points disclose the reading of “decreasing” used in footnote 12; weak monotonicity in quantity replaces the paper's stronger phrase, which fails in its saturated consumer model.